Let G=Z(p1^e1) x ... x Z(pn^en) be any finite abelian group.

What are G's subgroups? I can get many subgroups by grouping the factors and multiplying them by constants, for example: If G=Z3 x Z9 x Z4 x Z8, then I can take H=3(Z3 x Z9) x (2Z4) x Z8. Do I get all subgroups that way? (I'm interested in all subgroups, not just up-to-isomorphism).

Which are the subgroups H in G for which G/H is primary cyclic?