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$C_p\times C_p$ has $1+1+(p-1)=p+1$ subgroups of order $p$ (all of which $\cong C_p$), so it has $1+(1+p)+1=3+p$ subgroups totally.

But how many subgroups in $C_p\times C_p \times C_p\times C_p$? (its seems hard to count subgroups of order $p^2$ and order $p^3$)

Furthermore, could we easily point out how many subgroups which isomorphic to $C_p\times C_{p^2}$ in group $C_{p^2}\times C_{p^2}\times C_{p^2}$?

I only figure out how to find all order-p subgroups in the elementary-$p$ group:

For example in $C_p\times C_p \times C_p$: $p^3=x(p-1)+1\Leftrightarrow x=(p^2+p+1)$, so it has $p^2+p+1=(p-1)^2+C_3^1(p-1)+C_3^1$ order-$p$ subgroups in total. So there are 3 types order-p subgroups:

$\{(x,1,1)| x\in C_p\}$, $\{(x,x^{\alpha},1)| x\in C_p\}$, $\{(x,x^{\alpha},x^{\beta})| x\in C_p\}$, where $\alpha$ and $\beta$ $\in Aut {C_p}$

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Subgroups of elementary abelian groups are also elementary abelian, so the problem in that context reduces to one of vector spaces: given a vector space $V$ of dimension $n$ over a finite field $\Bbb F_q$ (for us our $q$ is prime, but the question's difficulty is unchanged by generalizing), how many subspaces of dimension $d$ are there? This is a textbook problem - the answer is $q$-binomial coefficients. See the discussion in this question for instance. (Note this hints at finite sets being vector spaces over the field with one element, since $q$-binomial coeffs when $q=1$ are just binomial coeffs.)

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