  We prove that given any finite abelian group $A$ and any irreducible
$3$-manifold $M$ with empty or toroidal boundary which is not a graph manifold
there exists a finite cover $M' \to M$ so that $A$ is a direct factor in
$H_1(M',\mathbb{Z})$. This generalizes results of Sun and of Friedl-Herrmann.
