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

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)

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

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

Cluster

A group of similar things or people positioned or occurring closely together. Clusters of creamy-white flowers.

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The Oxford Dictionary of English, edited by Angus Stevenson and published in 2010

With this definition you could borrow "supercluster" from astronomy:

Supercluster

A giant cluster of galaxies, a cluster consisting of a number of smaller clusters.

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Collins English Dictionary — Complete and Unabridged, 14th edition

Either that or "supergroup" which literally means a group of groups works. "Superset" also works, but is more of a mathematical term.