The classification of finite simple group is a theorem describing all possible finite simple groups. The importance of simple groups is that they are, in a way, "irreducible" - they cannot be quotiented by any of their subgroup. They also come up in the composition series of a group, and hence it is sometimes said that every finite group is "made up" of finite simple groups. However, one cannot reconstruct the group from its composition series. I was therefore wondering whether some kind of classification of all finite groups is possible, which would give us means of constructing all groups? I know this is a rather vague question, but to give everyone some idea, here is the classification of finite abelian groups in the spirit of what I want:
Definition: Call a finite abelian group $G$ irreducible if it's not isomorphic to a direct product of two nontrivial groups.
Classification of finite irreducible groups: The finite irreducible groups are precisely the cyclic groups of prime power order.
Classification of finite abelian groups: Every finite abelian group can be written (uniquely) as a direct product of irreducible groups.
Long time ago I had a false belief that every finite group can be written as a direct product of simple groups, and together with CFSG this would provide a dream classification of all finite groups, but sadly this isn't true. Replacing this with a semidirect products doesn't give us a true statement either, and at this point I am out of ideas.
I realize that there might not be such a classification, and the difficulty in describing a groups of a given order (like $p^k$ with $p$ prime) only supports this. Nevertheless there still might be hope for a partial classification, like we have for simple groups or abelian groups, or maybe there is some description which is rather computationally infeasible.
