How many subgroups of order $p^2$ are in $\mathbb{Zp \times Zp \times Zp}$

Question

I have two ideas to develop this problem, the first to use the Euler function, since Zp is cyclic. And the other is to use the canonical projection. I appreciate your ideas

Answer 1

Hint: Your group is a three-dimensional vector space over the $p$-element field. A subgroup of a vector space over such a field is always a vector subspace.