What are the connected covers of $S^2 \times S^2$?

Question

I'm trying to understand the relation between covering maps and the fundamental group of a space. In this specific example I want to find the connected covers of $S^2 \times S^2$ but in general I'd like to know how to approach such a problem. I understand that a covering map induces a monomorphism of the fundamental groups, but I'm not sure if there needs to exist a covering space for every subgroup? Is that true? And if not necessarily, how would one even approach a problem of finding all covers?