Is there any nice reference in the literature which contains such a classification (say, for orders up to $30$, EDIT: $100$) with detailed proofs? It should only serve as a citation.
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This paper contains a classification of groups of order up to 30, with the exception of orders 16 and 24: Classification of Groups of Small Order by Michael Van Opstall (link, .ps file). It's not much, and I don't know anything about the paper other than what I've already written, but I think it's a good idea to have the link here in case someone stumbles on this page and doesn't come up with the idea of googling "classification of small groups".