  For $G$ and $H_1,\dots, H_n$ finite groups, does there exist a $3$-manifold
group with $G$ as a quotient but no $H_i$ as a quotient? We answer all such
questions in terms of the group cohomology of finite groups. We prove
non-existence with topological results generalizing the theory of
semicharacteristics. To prove existence of 3-manifolds with certain finite
quotients but not others, we use a probabilistic method, by first proving a
formula for the distribution of the (profinite completion of) the fundamental
group of a random 3-manifold in the Dunfield-Thurston model of random Heegaard
splittings as the genus goes to infinity. We believe this is the first
construction of a new distribution of random groups from its moments.
