Finite quotients of 3-manifold groups

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.