Dalek splinter group

As recorded in the Dalek Combat Training Manual the Time Lords made reference to Dalek splinter groups which deviated from conventional Dalek mentality, purity and authority. (PROSE: Dalek Combat Training Manual)

Factions
The Time Lords believed the first instance of a Dalek splinter group (PROSE: Dalek Combat Training Manual) to be the Humanised Dalek, originally test Daleks who were implanted with the Human Factor as part of the Dalek Emperor's Operation Human Factor. Though this was intended by the Emperor to isolate the Dalek Factor and implant it in humans, the Second Doctor saw that many Humanised Daleks were created. These Daleks went on to question their orders, leading to a Dalek Civil War within the Dalek City on Skaro. (TV: The Evil of the Daleks) The rebel Daleks fled Skaro and settled on the planet Kyrol. (COMIC: Children of the Revolution)

Challenging the authority of the Supreme Dalek, Davros, the Daleks' creator, used an indoctrination serum hypodermic to condition a number of Daleks and Dalek duplicates to serve him. Though they were killed in battle against the Daleks in the Duplicate Incident, (TV: Resurrection of the Daleks) Davros himself escaped to Necros, where he created a new race of Daleks from the Resting Ones on Tranquil Repose. Having been alerted by Lancelot Takis, Daleks from Skaro came to Necros and fought through first of these Daleks whilst Orcini set off an explosion in Davros' bunker, killing many of Davros's prototypes. Davros was then apprehended by the Daleks, who intended to take Davros to Skaro for trial whilst the surviving Necros Daleks were to be reconditioned to obey the Supreme Dalek. (TV: Revelation of the Daleks)

Eventually, Davros succeeded in usurping the rule of the Dalek Emperor and "Imperial Daleks" on Skaro. (COMIC: Emperor of the Daleks!) As he took the position of Emperor Dalek, Davros's new army of Daleks became Imperial Daleks and engaged in a war with the now Renegade Daleks following the Supreme Dalek. As Davros had modified his Dalek mutants, they conflicted with the purity of the Renegade Daleks. (TV: Remembrance of the Daleks)

Devised by pacifist scientist Gonzo Orcini as a "refined, highly intellectual version" of the "old Daleks" created by Davros, the Orcini New Daleks initially followed his ideals but ultimately fell in with conventional Dalek mentality. (PROSE: War and Peace)

Created by Professor Martez on Red Rocket Rising, the Mutant Daleks welcomed the Daleks of Skaro to their planet, as they had been summoned by Martez to assist in their development. The Skaro Daleks, however, identified them as a "deviation from the purity of the Dalek bloodline" and so saw to their extermination. (AUDIO: Blood of the Daleks)

In anticipation of the Last Great Time War, the Dalek Emperor assembled four Daleks into the Cult of Skaro, who were tasked to use imagination, (PROSE: Birth of a Legend) to "think as the enemy thinks". Escaping the Time War, the Cult were embroiled in the Battle of Canary Wharf (TV: Doomsday) before retreating to Manhattan in 1930, where they conducted the Final Experiment to continue the Dalek race. After attempts at creating new mutant Dalek embryos failed, Dalek Sec, leader of the Cult of Skaro, controversially chose to combine with the human Diagoras, thus transforming into a Human-Dalek hybrid. (TV: Daleks in Manhattan) Having acquired over a thousand human corpses, the now Human-Dalek Sec, whose DNA was to be administered to mentally transform them into Dalek-humans, sought the Tenth Doctor's assistance to make them more human, admitting that he had come to believe that Davros was wrong to remove the Daleks' emotions, that emotions were responsible for humans surviving whilst the Daleks were now endangered. However, Sec was usurped by the rest of the Cult, with Dalek Caan launching an invasion of Manhattan. In the ensuing confrontation, Sec was exterminated whilst the Dalek-humans, having been imbued with Time Lord DNA when the Doctor caught the gamma strike for the Final Experiment, turned against the Cult, destroying Daleks Jast and Thay. Dalek Caan, the last survivor, responded by terminating the Dalek-human army before initiating an emergency temporal shift to escape, refusing the Doctor's offer to help him. (TV: Evolution of the Daleks)

Breaking through the time lock protecting the Time War timeline, Dalek Caan saved Davros from death before they returned together to the post-Time War universe, with Davros using his own DNA to create a New Dalek Empire. By passing through the time lock, Caan saw time and was regarded by observers as having gone "insane". (TV: The Stolen Earth) He had in fact come to believe that the Daleks must be destroyed and so maneuvered events so that the Doctor and Donna Noble would be together when the New Dalek Empire planned to use a reality bomb to wipe out all non-Dalek life across the multiverse. This led to the creation of the DoctorDonna, who outwitted the Daleks, and the Meta-Crisis Tenth Doctor who, once given access to the Dalekanium power feeds, was encouraged by Caan to destroy the New Dalek Empire, with Caan himself last being seen aboard the Crucible before it exploded. (TV: Journey's End, PROSE: Dalek Caan)

The last survivors of the New Dalek Empire (PROSE: Dalek: The Astounding Untold History of the Greatest Enemies of the Universe) came across a Progenitor device, which was capable of creating new Daleks. Knowing that they were "impure" and so the Progenitor would not respond to them, the Daleks goaded the Eleventh Doctor into identifying them as Daleks, providing a testimony for the Progenitor which created a New Dalek Paradigm. Acknowledging that they were "inferior", the old Daleks allowed themselves to be destroyed by the new Daleks. (TV: Victory of the Daleks)

Protected by a force field, the Dalek Asylum planet was home to many battle-scarred and insane Daleks that could not be controlled and so were confined by the Daleks, who hesitated to destroy them so as to preserve their "divine hatred". However, when the crash landing of the starliner Alaska created a breach in the force field through which the Asylum Daleks could potentially escape and threaten the New Dalek Paradigm, the Parliament of the Daleks summoned the Eleventh Doctor and sent him down to the planet and lower the defenses. This was done by Oswin Oswald, a human who had been converted into a Dalek by the Asylum Daleks. With the force field down, the Parliament's saucer saw to the destruction of the planet but not before Oswin removed the record of the Doctor from the Dalek pathweb. (TV: Asylum of the Daleks)

After being cloned and thus resurrected by Leo Rugazzi, the Reconnaissance Dalek created a New Dalek Army which invaded Earth. The Thirteenth Doctor was aware that, as a result of the cloning process, the new Daleks contained traces of human DNA and so would be deemed as "impure". Receiving the Reconnaissance Scout Signal from the Doctor, Death Squad Daleks intercepted and destroyed the new Daleks. Insisting that it served "the Dalek cause", the Reconnaissance Dalek claimed that its genetics could be modified and so purified only to be rejected and exterminated by the Death Squad. (TV: Revolution of the Daleks)

Individual cases
Zeg was an early Dalek scientist who serendipitously discovered the secret of metalert. Having been granted a casing that was invulnerable to all known Dalek weaponry, Zeg was effectively invincible and declared himself the new Emperor, hoping to unseat the Golden Emperor. Recognising that both Daleks had a claim to the title, as Zeg lacked provable intelligence but the Golden Emperor lacked provable strength, the Brain Machine ordered that a duel be fought between them to demonstrate who was truly superior. Through guile, the Golden Emperor defeated and killed Zeg, cementing his status as Emperor once more. (COMIC: Duel of the Daleks)

Having come into contact with Rose Tyler's human DNA, the Metaltron, which was at the time believed to be the last Dalek in existence, found that it was mutating and developing emotions. Deeming this an unacceptable outcome, the Dalek initiated self-destruct. (TV: Dalek)

During the Space Security Service's conflict with the New Dalek Paradigm, Professor Weston engineered a peaceful Dalek mutant, which came to be known to the Daleks as the Abomination. (COMIC: The Only Good Dalek)

After sustaining damage, Rusty was affected mentally by radiation which caused him to recognise the beauty of life and intending destroy the Daleks who threatened it. During the conflict with the Combined Galactic Resistance, a skeptical Twelfth Doctor investigated the possibility of a "good Dalek" before finding and inadvertently rectifying the cause, causing Rusty to revert to conventional Dalek mentality. By making telepathic contact, the Doctor convinced Rusty to turn against the Daleks but was dismayed to know that Rusty was now motivated by the Doctor's own hatred for the Daleks. (TV: Into the Dalek) Standing alone, Rusty continued his fight against the Daleks for billions of years and made a home on Villengard, where he destroyed many Daleks that came to exterminate him. (TV: Twice Upon a Time)

In a similar instance to Rusty, a Dalek that the Twelfth Doctor named Lumpy also deviated from the Daleks in their quest to stop Cybermen and Dalek alike from getting their hands on the Orb of Fates and thus keep them out of reach of the Starbane, a Gallifreyan warship. This turned out to be an intentional plan from the Daleks however, but the Doctor remained in control of him and blew up the Starbane with Lumpy still in it. (GAME: The Doctor and the Dalek)

Designated as a traitor, one bronze Dalek came to believe that the Daleks should be destroyed since while they were originally created to ensure the survival of the Kaleds, their mission had deviated and eroded Kaled identity. Upon hearing this, the Thirteenth Doctor remarked that Davros would be both impressed and horrified. Though offering to assist the Doctor in destroying the Daleks, this Dalek was itself intercepted and exterminated. (TV: The Power of the Doctor)

Behind the scenes

 * The Dalek Hierarchy feature of DWFC TLV 1 states that the Prime Minister of the Daleks led the restored Dalek Empire after a civil war between the New Dalek Paradigm and the "Children of Davros", which is glimpsed in The Eleventh Doctor Interactive Story.