Is there an "easy" method to finding subgroups of finitely generated abelian groups using the First Isomorphism Theorem? I seem to remember something like this but I can't quite get it. For example, the subgroups of [itex]G=Z_2\oplus Z[/itex] are easy...you only have [itex]0\oplus nZ[/itex] and [itex]Z_2\oplus nZ[/itex] for [itex]n\geq 0.[/itex] But if you have a different group, say [itex]G=Z_6\oplus Z_4[/itex], it's possible the subgroups aren't of the form [itex]<a>\oplus<b>[/itex] correct? Like <(2,2)>. How would you describe all the subgroups? I can do it by brute force..I'm looking for an quick easier asnwer if one exists...even in only some situations EDIT: maybe this makes more sense if I only need to know subgroups of a specific index?