I learned from MO Subgroups of a finite abelian group that the problem of enumerating subgroups (not up to isomorphism) of finite abelian groups is a difficult one.
Are there simple formulas if one restricts to low rank for the subgroups? For example, are there formulas for enumerating cyclic subgroups, or subgroups whose minimal number of generators is $2$?
I think that you can find the formulas that you are looking for in the paper "An arithmetic method of counting the subgroups of a finite abelian group" by Marius Tarnauceanu, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 53(101) (2010), no. 4, 373–386.
In particular, Theorem 4.3 seems relevant, but there are other results that might be of interest to your question.
The paper can be downloaded from the journal's website.
If $G$ is an abelian $p$-group and $p^k\le\exp(G)$, then the number of cyclic subgropups of order $p^k$ in $G$ is $$ {\rm c}_k(G)=\frac{|\Omega_k(G)-\Omega_{k-1}(G)|}{(p-1)p^{k-1}}. $$ If the type of $G$ is given, it is easy to compute $|\Omega_k(G)|$. The displayed formula is also suit for the regular $p$-groups. Computation of subgroups of given order is known for not very complicated $p$-groups, for example, for metacyclic $p$-groups (see \S 124 in the book of Berkovich-Janko; in that series a great number of counting theorems is proved, in particular, celebrated Kulakoff's theorem for all $p$).
