Subgroups of finite abelian groups.
Yes, this is true.
Theorem: Let $G$ be a finite abelian group. The following two statements hold.
(A) Each subgroup of $G$ is isomorphic to a quotient group of $G$,
(B) Each quotient group of $G$ is isomorphic to a subgroup of $G$.
For a proof see [this MSE question]( Is every quotient of a finite abelian group $G$ isomorphic to some subgroup of $G$?, which proves $(B)$; but also the same idea works for proving $(A)$. A further reference for the proofs is
L. Fuchs, Abelian Groups. Oxford 1960, page $53$.