This covers book clubs and movie-only clubs. There are also video-game-exclusive clubs, which are not included.
Liechtenstein cause a serious upset in the qualifying stages by
winning the group over China, the runners-up in 2010. Liechtenstein’s
team mascot is a gloomy, oversized Augurey called Hans who has his own
fan club.
I am certain this is, or was, on Pottermore at some point. Curiously, I cannot find the source on Pottermore. There is a "History of the Quidditch World Cup."
Hogwarts Gobstones Club
Gobstones is a game played with marbles that squirt foul-smelling liquid.
Gobstones is an ancient wizarding game that resembles marbles, the
principal difference being that every time a point is conceded, the
winning stone squirts a foul-smelling liquid into the loser's face.
Players start the game with fifteen small, round Gobstones each
(Gobstones are sold in sets of thirty) and the winner must capture all
of his opponent’s stones. Though most commonly (as the name implies)
made of stone, Gobstones may also be made of precious metals.
Hogwarts apparently has a Gobstones Club
'Does this mean they're going to shut down the Gobstones Club?' one of
them
asked his friend.
'I reckon you'll be OK with Gobstones,' Ron said darkly, making the
second- year jump.
Harry Potter and the Order of the Phoenix
Official Gobstone Club
Apparently the Ministry of Magic has a national Gobstones club.
'Level Seven, Department of Magical Games and Sports, incorporating
the British and Irish Quidditch League Headquarters, Official
Gobstones Club and Ludicrous Patents Office.'
Harry Potter and the Order of the Phoenix
Dueling Club
This was created by Gilderoy Lockhart in an attempt to instruct students in self-defense, prompted by a rash of unexplained attacks.
Lockhart waved an arm for silence and called “Gather round, gather
round! Can everyone see me? Can you all hear me? Excellent!
“Now, Professor Dumbledore has granted me permission to start this
little dueling club, to train you all in case you ever need to defend
yourselves as I myself have done on countless occasions — for full
details, see my published works.
Harry Potter and the Chamber of Secrets
It was essentially a failure, mainly due to the character of the individual teaching it.
Dumbledore's Army
This organization was created for the purpose of teaching self-defense, but was much more successful than the dueling club. Harry Potter was the leader, with Hermione Granger as the organizer.
'All in favour of the DA?' said Hermione bossily, kneeling up on her
cushion to count. That's a majority - motion passed!'
She pinned the piece of parchment with all of their signatures on it
on to the wall and wrote across the top in large letters:
DUMBLEDORE'S ARMY
Harry Potter and the Order of the Phoenix
Charms Club
"Vicky Frobisher and Geoffrey Hooper both flew better this evening, but
Hoopers a real whiner, he's always moaning about something or other,
and Vicky's involved in all sorts of societies. She admitted herself
that if training clashed with her Charms Club she'd put Charms first."
--Angelina Johnson, Harry Potter and the Order of the Phoenix
Hogwarts Choir
This is a movie-only group.

Astronomy Club
This club was taught by the Astronomy teacher, Professor Sinistra. This is a movie-only club, known only from a poster in the Gryffindor common room.
Slug Club
A group of students favored by Professor Slughorn for the advantages they could bring him.
The afternoon wore on with more anecdotes about illustrious wizards
Slughorn had taught, all of whom had been delighted to join what he
called the “Slug Club” at Hogwarts. Harry could not wait to leave, but
couldn’t see how to do so politely. Finally the train emerged from yet
another long misty stretch into a red sunset, and Slughorn looked
around, blinking in the twilight.
Harry Potter and the Half-Blood Prince
Slugs and Bugs Club

This is another movie-only club.
Chudley Cannons Fan Club

This is a club devoted to the notoriously unlucky Quidditch team the Chudley Cannons. Another film-only club.

Another club found only in the movies, presumably dedicating to racing rats.
Art Club for Gryffindor
This club is movie-only, and apparently did drawings of dragons.
