How many subgroups of order does the abelian group have?
A subgroup of an abelian group inherits commutativity, and we know the order of the subgroup is , so we can apply the fundamental theorem of finitely generated groups to see that the only groups of order are and . All that it amounts to is counting elements of order and in the two groups, and then I think you can take it from there to count how many subgroups of this order you could make out of these elements.
Hint: Theorem If d is a positive divisor of n, the number of elements of order d in a cyclic group of order n is where is Euler's totient function. Euler's totient function - Wikipedia, the free encyclopedia it is the number of positive integers less than and relatively prime to n.