Demonic factions are groups of demons, warlocks or Darklighters that join together for one common goal; destroying magical entities, gaining control over the the Underworld, survival purposes, glory and power, etc.
The Source of All Evil traditionally, through the immense power inherent in being the ruler the Underworld, remains in control of the numerous tribes and groups of demons. However, the balance of power between the Source and the various factions of Evil is far more delicate than it may appear on the surface. If the factions like the Triad sense any weakness or opportunity to seize power for themselves, they will take it even if that means taking on the Source. This was the case for the demons Kurzon and Zankou both who lead mighty rebellions against the Source and while Kurzon failed in his quest for power, Zankou nearly succeeded in taking control of the Underworld with his army before being defeated and sealed into a mystic prison.
To preserve and gain further power, Demonic Factions will eliminate and annihilate those who dare challenge them. This factor is also one which must tempered by the ruling Source to ensure that the various armies of the Underworld don't begin their own civil wars. Yet even under strong leadership infighting amongst the factions is a constant threat.
Key Faction Leaders and Movements
By stealing the Ring of Inspiration, he tried to use the Passion and Creativity of the muses to grant his wish of becoming leader of the Underworld. Warlocks who questioned Devlin were vanquished, but his plan was not to be; he was found out and destroyed by the Charmed Ones and the Power of Three.
An upper-level demon who was banished from the Underworld by the Source for inciting a failed coup. Although he was exiled, he attempted to destroy the Source, ultimately being vanquished by the Charmed Ones.
Zankou was a legendary demon who once lead a great rebel against the Source whose power was so vast that the Source after putting down his army had to seal him away in an eternal prison. Years later he was freed by some demons who were seeking help against the Avatars, and after their defeat Zankou moved with the rest of his plan to take over the underworld and the ultimate power. Although, some factions of demons and warlocks banded to challenge Zankou thinking that releasing him from his prison was a mistake, while some others decide to help him in killing the Charmed Ones and take all the power for himself hoping to gain something in return. He was vanquished in the end when The Charmed Ones destroyed the Nexus when he took it in, destroying him too.
The demonic trio known as The Triad were a feared and influential force in the Underworld for an unclear but considerably long period of time. They often acted on their own personal motives particularly in defeating the Charmed Ones. It's not exactly known what the extent of their power was. However, it appears that even the Source recognized them as a force to be respected. The group didn't seem to be direct enemies of the Source, but it's thought that they eventually did aspire to one day replace him as the ruler of Evil. During the chaos following the defeat of the Source and the manipulative Seer they became more determined to take over by killing the Charmed Ones using the "Ultimate Power". They were initially vanquished by Cole Turner but eventually managed to return. Asmodeus was vanquished when he was orbed in front of a fireball thrown by Baliel, Baliel was blown up by Piper and Candor was vanquished by Christy Jenkins. They returned as spirits and were vanquished for good by the Charmed Ones with Electrokinesis from the Hollow in the original timeline and potions in the new timeline.
After The Charmed Ones had faked their deaths, a new generation of younger demons led by Paul Haas formed a new faction. Believing the sisters were dead, they saw it as their time to rise and take control over both the Underworld and the mortal world. They wanted to claim the Halliwell Manor as a symbolic seat of power.  He was later vanquished by the Charmed Ones with potions when he tried to draw them out of hiding.
A faction of Harpies, led by their Queen, appeared at the meeting organized by Cole Turner. The Queen was angered over an attack on her faction by the Grimlocks. The Queen was later reprimanded by Cole after sending her Harpies after The Charmed Ones.