Cults of the Nine Hells

As Lord of the Nine Hells, Asmodeus receives power through the cults of his subordinates as well as his own worshipers. Likewise, followers of the other Devil Lords all pay homage to Asmodeus. Thus, the Cults of the Nine Hells are best thought of as a sort of mini-pantheon with Asmodeus as the chief deity. They have their own complex web of alliances and rivalries, but in the end, they all answer to him. In ascending order, they are:

Zariel - A fallen angel who rules over Avernus. Her realm connects to the Abyss via the River Styx, and she is charged with leading legions of devils against the hordes of demons that constantly cross over to invade.
Dispater - The paranoid kingpin of Dis, who oversees the biggest weapons forges and foundries in all the planes. He deals in blackmail, trading power in exchange for secrets.
Mammon - The embodiment of greed. His realm, Minauros, hosts the Hells' main soul market in the Sinking City. It's a wretched plane, because Mammon refuses to spend more than the bare minimum of coin to keep things functional.
Fierna & Belial - The two rulers of Phlegethos, whose exact relationship is unclear. They are in a constant cycle of mutual backstabbing, yet somehow manage to not kill each other. None of the other devils can seem to take advantage of the situation, because whenever an actual threat arises, they drop their current schemes to gang up on the attackers.
Levistus - King of Stygia, a realm that he took by overthrowing Geryon. He is eternally imprisoned in an iceberg, and ironically, he specializes in preying on people who are in desperate, inescapable situations.
Glasya - The rebellious child of Asmodeus, ruler of Malbolge. She is the warden of the Nine Hells' prisons and torture chambers, a job that usually keeps her from causing too much trouble for her daddy. Her knowledge of the law makes her the patroness of people who exploit it.
Baalzebul - The devil who once tried to overthrow Asmodeus, now relegated to pushing papers in the endless archives of Maladomini. He is the devil in the details, a master at luring in people who desire some form of redemption with seemingly generous deals, only to trap them with clauses hidden in the fine print.
Mephistopheles - The ruler of Cania is a scholarly type who holds a great deal of lore unknown to most other beings. Secrets are the bait he uses to ensnare sages and wizards who are willing to pay anything for a breakthrough in their research.

Tenets of Faith

The tenets of Asmodeus are:

Assert dominance and power over others. Show your strength of will in the image of the Asmodeus.
Repay cruelty done unto you with further evil.
If others show you kindness, exploit it. As you ascend to power, do not pity or show mercy to those you climb over to get there. Compassion is for the weak.

Each of his subordinates has their own rules, but they all agree with Asmodeus' laws.

Worship

Worship of the Archdevils usually involves blood sacrifices, torture, ritual fights to the death, and other unpleasantness. The particulars depend on which devil they're worshiping.

Political Influence & Intrigue

Followers of the Nine are everywhere, in every part of society. They hide their allegiances and meet only in secret, but even the quietest little hamlet might be home to a cult. Through their mortal followers, the Nine (and devils of comparable power) exert influence in every government, guild, and organization of power to some degree.

Sects

Followers of Asmodeus himself are ambitious to the nth degree. He has little interest in dealing with them directly when he could be pulling the strings of more powerful entities, so his cultists do not expect to be heard; instead, they look to him as the ultimate embodiment of ambition.
The Cult of Mephistopheles is dedicated to gaining power through knowledge at all costs. Followers include power-hungry mages, scholars desperate to one-up a rival, and people who want to break the known boundaries of magic.
The Cult of Baalzebul is made up of those who have fallen from grace in some way. A rich merchant who lost their fortune on a bad deal, a knight who has besmirched their honor, a noble whose reputation has been slandered; followers of the Devil in the Details want nothing more than to take back what they believe is theirs.
The Cult of Glasya is all about exploiting loopholes in deals and laws. She is the patron of con artists, corrupt politicians, and anyone else who plays by the rules in the worst ways possible.
The Cult of Levistus is not so much an organization as it is a very loose association of people who have made deals with him. He appeals to the desperate and the wronged, promising them the means to survive and eventually take revenge on their enemies in exchange for their souls.
The Cults of Fierna and Belial appeal to those who want to make people into their adoring puppets. Some of their followers desire a particular person's love, while others just want to become famous.
The Cult of Mammon exists to accumulate wealth, just like its patron. Greed is the force that draws people into his trap, and some of them even gain the ability to infect others with their greed.
The Cult of Dispater is basically a series of interlocking conspiracies, with each group using blackmail and subterfuge to bend the wills of those in power. Each individual cult is running multiple plots at any given moment, all of which further their patron's influence on the world.
The Cult of Zariel preys on the desire for strength of arms. Most of her followers are people who were displaced by war and had the will but not the ability to fight back, or seasoned warriors and gladiators looking for an unfair advantage.

Bael is Mammon's chief enforcer. Rather than following him willingly, those in service to Bael are largely those he has defeated in battle; the vanquished have a choice between death and pledging their lives and souls to him. Others are shrewd individuals who can do the politicking for him at Mammon's court. Bael is more of a general than a politician, so he gladly welcomes mortals with the acumen to represent him. Some deluded followers believe him to be the most powerful of all devils.
One of the most powerful ex-archdevils is Gargauth, known to some as the Tenth Lord. He was once the treasurer of Asmodeus' court, but either left or was cast out from the Hells for reasons unknown. Few people know his name now, and even fewer are willing to even speak it for fear of invoking him, but a small number of foolishly ambitious mortals strike deals with Gargauth.
Geryon attracts brutes and murderers -- anyone who enjoys the taste of blood and the feeling of bones cracking under their hand as much as he does. He hopes to use their souls to build up enough resources to stage a comeback against Levistus.
Moloch, the former lord of Malbolge, is nearly powerless after having failed to take down Asmodeus. Most of his followers left, but a few holdouts still remain, and he is actively recruiting.
Titivulus is the trusted advisor of Dispater, and the biggest threat to his rule. He has a small but loyal following, mostly recruited through intermediaries so he can maintain some plausible deniability. They are largely politicians and other people who want to get ahead by any means necessary.