Find two finite abelian groups of same order $G$ and $H$ such that $G \ncong H.$ Find two finite abelian groups of same order $G$ and $H$ such that $G \ncong H.$
I found groups $\mathbb{Z}_4$ and $V$ (Klein's group) that satisfy it, but would like more examples. I'm trying to use the result that $p$ is a prime number then all group of $p^2$ order is abelian.

The three abelian groups of order $8$, namely $G_1$, $G_2$ and $G_3$, belong to three different isotopy classes and hence there can't be two of them being isomorphic.
