Non isomorphic subgroups of cyclic groups

Question

let G be a cyclic group of order 12. find number of non isomorphic subgroups of G. How to approach such questions?

Answer 1

The results of cyclic groups are (e.g. https://en.wikipedia.org/wiki/Cyclic_group#Subgroups_and_notation):

All the subgroups of cyclic groups are cyclic, for every $n$ dividing the group order there is a cyclic subgroup of order $n$ and all cyclic groups of the same order are isomorphic.

So for a cyclic group of order 12 we have exactly 6 pairwise non isomorphic subgroups, which are isomorphic to cyclic groups of order $1,2,3,4,6,12$. This are all subgroups by Lagrange's theorem.

Answer 2

In a cyclic group of order $n$, there is exactly one subgroup of order $d$ for each divisor $d$ of $n$.

Groups of different order cannot be isomorphic.

Therefore, the number of non-isomorphic subgroups of $G$ is the number of divisors of $n$.