Government Conspiracy. A secret project uniting a small number of members of a powerful but otherwise benevolent organization, usually but not necessarily The Government (hence the name). The project can hardly be older than the government itself. Their power comes from their Front Organization's resources and the trust that Normal People put into it.
Examples of works/series featuring conspiracies of more than one type:
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The less known thing about Bee Train's Girls with Guns trilogy is that in addition to playing into three distinct action movie genres (see Trivia.BeeTrain), they also each center on a different type of conspiracies:
Hellsing has the Millennium Organization, a group of Nazi scientists that was thought to be wiped out by Hellsing but some of whom escaped out of Europe into Brazil in the aftermath of WWII. Their purpose was to develop a means of engineering vampires and build an army of them. To what end? Why, world domination of course. A large portion of their funding came from the Vatican, although we're not quite sure why. In the years leading up to the main plot of the story, they periodically released groups of ghouls as a covert way of testing the defences of the Hellsing Organization and the world in general, as well as keeping them occupied until the moment came to unleash their undead army on the world.
In HHB-era Narnia fanfiction In The Fledgling Year, the princeís entourage gets caught up in a conspiracy by a group called the Finii, whose goal seems to be to systematically kill off the royal line of Archenland in order to put a distant cousin on the throne. (The fic is still a WIP, so itís not yet fully developed.)
One of the driving plots in Families is a conspiracy by corrupt noble Olive Branch to use his media resources to discredit Princess Celestia, playing off of the public's fear after the Changeling invasion to eventually gain enough support to overthrow her. As it turns out, he didn't really care about her position, he was just trying to get into a position to force her to give him immortality.
Jewel Of Darkness: One of the major components of the Myth Arc is the existence of the White Glove, a group with definite negative vibes that is so far reaching that its leader, the "Good Man", can walk in and out of the Justice League Watchtower whenever he wants. Jinx is their mole on the Titans, albeit one who is Becoming the Mask.
Scandal: It's not yet clear what the nature of this is, but a large part of Season Two's backstory seems to involve one including Olivia, First Lady Mellie Grant, Cyrus Beene, Hollis Doyle, and Justice Verna Thornton. The conspiracy resulted in throwing the election to get Grant elected, but he appears to be unaware of it, and the depths appear to go much further than that. Quinn's framing appears to have been related to this conspiracy, and it probably is what resulted in her former office being firebombed. Hollis seems to be the Token Evil Teammate; it is not clear how malicious or how implicated the other members are, but it has been stated that their actions would get them locked up for the rest of their lives. We know Olivia generally has good intentions, but it is possible that she is The Atoner for actions in her past.
The Masons in Dancing on the Edge are presumably involved with all sorts of conspiracies but they're efforts to frame Louie for Jessie's murder deserve particular notice as they are able to do things like fake passport stamps thanks to their government ties.
Orphan Black features the clones taking on two conspiracies:
A group called the Neolutionists created the clones, monitor them and run tests on them. Known members are Olivier, Astrid, Dr. Leekie, Delphine, and Paul (though he doesn't know what's going on at first.
A group of religious fanatics are hunting and killing the clones. This group includes Maggie Chen and Thomas and they used Helena as their assassin.
Eclipse Phase has Firewall, a "heroic" (well, anti-heroic) conspiracy that does not work for any government or criminal organization, and is devoted to the survival of transhumanity at all costs. Though there's also Project Ozma, SETI turned MIB.
And on a truly epic scale, Assassin's Creed III reveals that Juno has been manipulating humanity from before recorded history so that she could blackmail Desmond into bringing her back to rule over humanity. Just to be clear, she did this entirely through prerecorded messages left where certain people would find them. And she did it without the entirely benevolent first civilization conspirators noticing.
The Wanderers Library has a hilariously petty one. For a long time in the world of Elrich, elevators were the only method of going up. When stairs were invented, they posed a threat to the livelihood of the Elra Architectural Sub-Union, which made elevators. In order to discredit stairs, they loosen one step that the king will be going up. He falls to his death, thus making sure no one will use stairs again.
The Church Of The Sub Genius blames pretty much everything wrong with the world on "The CONSPIRACY" (which is apparently an acronym for "Cliques Of Normals Secretly Planning Insidious Rituals Aimed At Controlling You").