There is a well-known classification of finite abelian groups into products of cyclic groups.
What about finite abelian group schemes, where we may put in the qualifiers “affine”, “etale”, or “connected” if it helps?
There are some easy examples showing that the theory is richer than that of groups:
One might hope to obtain more finite abelian group schemes from abelian varieties, but over fields of characteristic $p$, the $p$-torsion of an abelian variety is a product of the finite group schemes already mentioned. We also know that any affine etale finite abelian group scheme becomes constant after base change.
Are there other (fundamentally different) examples of finite abelian group schemes, and is there some sort of classification of them all?