I just found out about the Sons Of The Desert.
According to Wiki, they are “an international fraternal organization devoted to the lives and films of comedians Stan Laurel and Oliver Hardy.”
And they wear cool looking fez’ and chapters are called tents.
Share a “secret” organization you’ve heard about.
Do they have an Exhausted Ruler?
How the heck are the Sons of the Desert “secret” in any way? They’re a fan club. They have a wiki page! There must be thousands or tens of thousands of fan clubs you’re never heard of, with greatly varying levels of organization.
I was a member of the Marx Brotherhood, although that seems to have faded away. It’s “secret” in the sense that I’m sure you’re never heard of it. I didn’t even find out about it until years after it started and I am, somewhat obviously, a Marx Brothers fanatic.
One of our more troublesome patrons at the public library insisted on registering as Jeffrey T. Spaulding.
Well, there’s the Hermetic Order of the Golden Dawn, who served as the inspiration for Pterry’s Elucidated Brethren of the Ebon Night.
Side Note/Question: The Free Chelonians recognized each other by the visual sign of placing the palm of one hand flat on top of the clenched fist of the other hand (The Turtle Moves!). Is it ever stated what position the clenched fist is in (vertical, like giving Thumb’s Up? horizontal, like stabbing yourself in the chest?)
Anybody? I’d hate to do it wrong and get killed over something silly.
I don’t know how obvious it is. You have to have read Harpo Speaks!, a sixty year old autobiography, to get the connection.
The original Sons of the Desert were modeled after the Shriners/Masons, a fraternal organization with secrets only made known to members. So in that sense they are a “secret society.” I assume this is the sense in which the OP meant it.
“One morning I shot an elephant in my pyjamas. What he was doing in my pyjamas, I’ll never know.”
*Duck Cultists. *
I’ve said too much!
Or the seven zillion times people here have mentioned where the name comes from. I bet more Dopers know the source of Exapno Mapcase than of Cliff Edwards’ existence. And my name isn’t even spellable. 
I think you wildly overestimate how much people pay attention to things like that. I’ve often explained the source of my username, and while many people may remember, there’s lots I’m sure who don’t.
I was being hyperbolic. I even used a smilie.
Have you noticed how few people use smilies these days, BTW? Their use is way down.
My obscure, secret and somewhat terrifying orginization is F.E.A.R., the Federal Egg Answering Room, (which is, in reality, a front name for F.E.E.B.L.E. - the Free World Extra-Earthly Bodies Location and Extermination Centre), commandeered by an armpit fetishist (scroll ahead to 8:40)
I’ve always associated Colibri with that fantastic pure butane for blasting concentrates. 
I give you–The Black Chamber.
I don’t know how secret or obscure they are but I was kind of surprised when I came across a chapter house for the International Order of Odd Fellows.
I was surprised to find out there is a Federal Reserve Police force. Started in 2000. I found out because I saw one of their cars. They have 1000+ officers and 37 stations. They protect Fed facilities.
The most influential organization most people have never heard of:
The National Conference of Commissioners on Uniform State Laws. As the linked article states:
In other words, the NCCUSL promotes interstate cooperation on passing mutually uniform laws in a manner that keeps the participating states from being in violation of the federal Constitution’s Compact Clause.
This body has for decades made and promoted policy largely out of sight of the public. For example, the movement in the early twentieth century to suppress public carry of firearms with “May Issue” permit requirements was championed by the organization with “A Uniform Act to Regulate the Sale and Possession of Firearms”. State laws criminalizing drug possession in addition to federal laws were another example.
Does theChurch of the SubGenius qualify?
Well, yes. There is that.
(I was trying to sneak in a plug for Harpo Speaks!, one of the most enjoyable memoirs I’ve read)
There’s also the Church of the Flying Spaghetti Monster, although not much about it is secret.