A group is subdirectly irreducible provided it has a least nontrivial normal subgroup. Subdirectly irreducible groups are also referred to as monolithic groups in the literature. Every simple group is sub-directly irreducible, but there are many subdirectly irreducible groups that are not simple.


Is there any classification for subdirectly irreducible groups?


Any comments or hints are highly appreciated.



Subdirectly irreducible groups
I don't think there is a classification of finite monolithic groups. Here is an extended comment. I am assuming the group $G$ is finite. The minimal normal subgroup has no characteristic subgroup and hence is isomorphic to $T^n$ where $T$ is a finite simple group. If $T$ is cyclic of prime order then the action by conjugation of $G$ on $N$ must be an irreducible representation, as Derek's comment indicates, since an invariant subspace would give a smaller normal subgroup. 


If $T$ is non-abelian the action of $G$ on $N$ by conjugation is faithful because the centralizer of $N$ is normal and hence contains $N$ or is trivial. 


If you assume that no proper subgroup of $G$ generates the same variety as $G$ then more can be said and you might consult the book of Hanna Neumann, Varieties of Groups. 


