Question: if $G=\mathbb{Z_2}\oplus\mathbb{Z_2}\oplus...\oplus\mathbb{Z_2}$($n$ copies where $n≥3$) then number of subgroups of $G$ which are isomorphic to $\mathbb{Z_2}\oplus\mathbb{Z_2}$ is ?
My attempt: when $G=\mathbb{Z_2}\oplus\mathbb{Z_2}\oplus\mathbb{Z_2}$ I saw there are $7$ distinct subgroups which are isomorphic to $\mathbb{Z_2}\oplus\mathbb{Z_2}$. But for this, I calculated each of them by hand.They are $<(0,0,1),(0,1,0)>, <(0,0,1),(1,0,0)>, <(0,0,1),(1,1,0)>, <(0,1,0),(1,0,0)>, <(0,1,0),(1,1,1)>, <(1,1,0),(0,1,1)>, <(1,1,1),(0,1,1)>$ where $<a,b>$ means subgroup generated by $a$ and $b$.
Now when $n≥4$ how to find number of such subgroups? and which are they? Please help
Any two distinct non-zero elements of $G$ generate a subgroup isomorphic to $\mathbb Z_2\oplus\mathbb Z_2$. Each such subgroup is generated by any pair of its non-zero elements. Thus there are
$$ \frac{\binom{2^n-1}2}{\binom32}=\frac{\left(2^n-1\right)\left(2^n-2\right)}6 $$
such subgroups.