# Is there a way to classify groups up to isomorphism?

Is there a nice way to classify/decompose finite groups? The classification of finite simple groups lets us determine all possible composition factors in the composition series but there are nonisomorphic groups with the same composition factors in their respective composition series e.g. $C_{6}$ and $D_6$. If not, is there an analogous theory for a group to be 'completely reducible'?

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There has been a lot of research on this topic! I would recommend you look at the article

A millennium project: constructing small groups, Hans Ulrich Besche, Bettina Eick, E.A. O'Brien, Internat. J. Algebra Comput. 12 (5), 623-644, 2002.