“Doper X is not a member of any public groups” it says as a default on most profiles I’ve checked.
Are any Dopers members of these so-called public groups? How many chocolate and unmeantionable acts involving goats does it take to find out more about these secret cliques?
Does anyone want to start a public group with me? Then we will be able to post things like:
*
“Let me tell you: people are starting to talk about you.” * 
Well, Doper X is a big jerk, and the mods wanted to make sure everyone knew he wouldn’t be tolerated in any public groups, lest we all swarm to Chicago and revolt, so they put that in all of our profiles.
/stupid joke answer
More seriously, it’s likely a feature offered by the board software (vBulletin), but turned off by the staff, most likely because the the server/bandwidth resources it may consume outweigh the benefits it’d bring us.
Not that you asked, but I assume the case is similar with “child forums,” which you’ll see as a search option in the Advanced Search page. It appears to be a valid option, but doesn’t actually do anything useful, since we don’t have any child forums.
With access to the VBB forums and the “back end” of another board, I’ve never been able to figure out what “public groups” are or why one might want to institute them.
Child forums are a useful gimmick that SDMB hasn’t found it necessary to institute. But let’s say that it is decided that Cafe Society would be easier navigated if it were broken into stuff on books, TV, movies, and music – so they start four forums, located on the menu under Cafe Society, in which all threads on these four topics are moved, and where you’re expected to start one – go to “Cafe Society/Music” to ask a question about what Bernie Taupin meant by a given line in “Someone Saved My Life Tonight,” Cafe Society/TV to discuss the latest episode of QAF or Big Brother, etc.
I suggested some time ago that with the plethora of questions about computers generally and spyware, adware, browser hijackers, and such in particular, GQ might well find it useful to have a separate child forum devoted to computer questions.
However, the policy of SDMB staff is to keep eight very active forums in which everything is distributed without the plethora of breakdowns possible, it being easier for a relatively small staff to manage them than 25 more “focused” forums. And I can understand and agree with this POV.
Makes perfect sense. Thanks Troy and **Polycarp. **
Just to make it nice and clear - “Public Groups” aren’t aren’t groups that the public can create themselves, just groups created by the administration that anyone can join freely. They also have private groups, where users request access, and a designated user grants or denies their entry. You can use these groups to designate access to certain forums, whether or not users can use features like private messaging and avatars, whether people can edit posts, and so on. Pretty handy. 
Doper X is not a member of any public group because Doper X is secretly Speed Doper’s older brother who was thought to have died in a posting accident long ago but is now back, hiding his identity to help Speed Doper post and to find out who tried to kill him in that posting accident long ago which forced him to hide his identity. Ha HAH!
So there is a Clique [sup]TM[/sup]
And we can conclude from this that Moriah needs to post more instead of watching ‘The Young and the Restless’!!
And we can conclude from this that Polycarp need to post less and watch more early '80s anime!! 
Hell, Speedy was on in the early seventies and may still be on AFAIK.
speedracer.com
“In 1967, the 52 episodes of the series were dubbed into English and syndicated nationally on television.”
Wow! I had no idea Speed Racer had been around for so long! I remember when it first came out (as far as I was aware) in my market. I must have been in first grade! All the kids were talking about it. I had a sense that it had been around for a few years in other parts of the country (I knew California got a lot of early anime) but not, I thought, before the late 70s (at the earliest). I’m surprised to 1967, because I thought it was strictly a Gen X phenomenon, which to me means formative years in the late seventies and eighties. (I was born in 1977, for the record.) Who knew Speed Racer preceded disco?