Q:

What is a good word that means "group of groups?"

If you have a bunch of groups of different things, and you group the groups into a bigger group, what do you call the bigger group?

A:

You can use the word supergroup to refer to a group of subgroups.  The super- prefix is the opposite of the sub- prefix.
Wiktionary defines supergroup as:

Any group composed of other groups (in any of several contexts)

A:

You could also use the term "metagroup" - it would mean groups about groups in the same way that "metadata" means data about data.
http://en.wikipedia.org/wiki/Meta#About_.28its_own_category.29

A:

I'd say collection, if you want to avoid ~group terminology.
Other words could be bundle, cluster, pack, package or any of their synonyms.
http://www.thesaurus.com/browse/bundle

