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 $2$?
Best Answer
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.