Firstly, the ones I know through either myself or a partner being a member:
LGBT+ LibDems (SAO) Affectionately known as Plus, the artist formerly known as DELGA is the group within the lib dems that campaigns for the rights of the entire alphabet soup of gender and sexuality minorities. The reason we are not DELGA anymore is that we felt the need to represent all of the alphabet soup was more pressing than retaining the words "Gay Action" in our name*. You do not have to be a gender or sexuality minority person to join, just in favour of fighting for our rights.
HSLD (AO) Humanist and Secularist Lib Dems are comprised of people of all religions and none. We fight to prevent the intrusion of religion (any religion) into places where religion should not be going. We reckon that if one religion intrudes, that is bad for both non-religious people and people who believe in other religions than the one intruding. HSLD are definitely high speed low drag.
Lib Dem History Group (unofficial) I don't actually contribute to Lib Dem History Group, but I devour the Journal of Liberal History, which you get delivered to your door by being a member.
ALDC (SAO) The Association of Liberal Democrat Councillors is absolutely stuffed full of experts on effective campaign techniques, and they run amazing training events. They also run election alerts, so that everyone knows when a by election is going to happen. They are the engine of the famous LD by election juggernaut. Despite the name you do not have to be a local Councillor to join - I never have been - and in fact they have recently started referring to themselves as the association of LD councillors and campaigners, but they haven't added an extra C yet...
Secondly, the groups who cause waves within the party:
The SLF (unofficial). They're justified, and they're ancient and they drive an ice cream van. No, sorry, that's the KLF. The SLF are a huge grouping of people who label themselves social liberals. There are very few people within the party who wouldn't call themselves social liberals, and of those who wouldn't it's because the term social liberal has come to refer to the left of the party, rather than a set of beliefs. Therefore the SLF tends to lean left merely by virtue of the people who are prepared to join it. I lean a lot further left than most of the SLF, but I have never joined because I see them as a pointless splinter group, in the same way that the People's Front of Judea look upon the Judean People's Front**. The numbers of people who HAVE joined show that I am in something of a minority in that opinion; I'm OK with that. Most likely to be found claiming to both represent the left of the party and to represent the entirety of party opinion at the same time.
Liberal Reform (unofficial) If the K, sorry, SLF are the left-leaning splinter group, Liberal Reform are the right leaning ones. Most likely to be found containing people who disdain "identity politics" (while adopting the identity of a Liberal Reform member) and who blithely refer to themselves as Classical Liberals despite having never read Mill and being unaware that he referred to himself as a socialist. If the SLF are the Judean People's Front, these guys are the Popular Front, sitting down the front looking grumpy and flipping the bird at everyone else. The SLF are their deadly rivals. You can tell this by the frequency with which members of both say things like "I don't know why people think we're enemies, we should totally be working together!". Liberal Reform are most likely to be found spouting managementspeak bollocks, and failing to understand that actually, being poor is quite hard.
The Carbohydrate Groupings (extremely unofficial and urine-extracting) The Carbohydrate Groupings were mostly specifically formed to take the piss out of the SLF and Liberal Reform. Lib Dem Biscuits are a young and thrusting group of new people who want to make sure we all have something to dunk in our tea. Lib Dem Friends of Cake disapprove of what they call "the biscuity entryists". Lib Dem Tarts and Lib Dem Friends of Pie disdain both cake AND biscuits and fight with each other as to whether pastry cases should have lids or not, and who is the best at outrageous innuendo. Paddy Ashdown famously endorsed Lib Dem Friends of Cake on national television. IMHO all four of them are more useful to Liberalism and more fun than the SLF or Liberal Reform. Your mileage may vary.
Green Lib Dems (AO). I'm not going to take the piss out of this lot because they do a lot of good work, and I honestly keep meaning to get around to joining them.
Lib Dem Women (SAO). I was a member of one of the two groups that merged to form this, and I left at merger. I was happy as a member of the Campaign For Gender Balance, because they didn't specify any genders to balance, and thus were not exclusionary. Lib Dem Women have a tendency to believe in the gender binary, and as I am implacably opposed to the gender binary we're just never going to get along. I have a lot of admiration for many of their members though.
Liberal Democrat Christian Forum (AO) - mostly to be found piling in to the hall at conference to defeat any motion or amendment which restricts faith schools.
EMLD (SAO) Ethnic Minority Lib Dems campaign for the rights of BAME people. Like Green Lib Dems, these are people who do huge amounts of great work, and I'm not going to take the piss.
There are lots more groupings within the lib dems, of varying degrees of power and influence, and this list is only the ones I have noticed having an impact. Ones I would like to see having more impact on policy include ALDES, if only because they have the best logo, and LDMHA. Inevitably, i will get comments recommending different groups, or arguing with what I've put here. That's fine, we're all Liberals :)
* I am still sad that we don't have Gay Action in our name any more, but I feel, as a bi person, that the sacrifice was worth it.