Number of Subgroups of $C_p \times C_p$ and $C_{p^2}$ for $p$ prime As the title says, I am interested in finding all subgroups of $C_p \times C_p$ and $C_{p^2}$ for $p$ prime.
We did not cover the Sylow-theorem so far in the lecture.
What I noticed so far: 
As $C_p$ is of order $p$, the elements can only have order $p$ or $1$ due to Lagrange's theorem. There is only one element of order 1, namely the neutral element. Hence there are $p^2-1$ elements in $C_p \times C_p$ with order $p$. 
Because $p$ is prime, $C_p$ is cyclic and hence there exists an element, call it $a$ which is of order p and generates the whole group. All other powers of $a$ are generating $C_p$ as well, so $(a^i,1)$ for $i=1 \dots p-1$, generate one subgroup that is not trivial. 
Using the same argumentation for the second factor, I conclude that there are $2$ nontrivial subgroups and $2$ trivial ones. 
Consider the case $p=2$ now.
$<(a,a)>$ is one as well, hence $5$ subgroups. But I am stuck with what happens for $(a^i, a^j)$ for $i \neq j$ and $i, j >0$ for general $p$ prime. Could you post some hints, please? 
For $C_{p^2}$, we know that there is at least one element, call it $c$. This generates the whole group. According to Lagrange, there can only be elements that are either of order $p^2, p$ or $1$. All $p^2$ elements will generate the whole group, so they are quite uninteresting. 
For $p=2$ again, $<2>$ is another subgroup, nontrival of order $p$. For this case, there are in total $3$ subgroups ($2$ trivial ones and $<2>$). I cannot find any meaningful generalisation. Any help is greatly appreciated.

the error is human; the author has not written p ^ 2-2 nowhere in the text, thank you.
For sub-Gropes not trivials of C_pxC_p there are exactly the number of element of order p divided by p-1 therefore p ^ 2-1 / p-1 = p + 1. therefore the nompbre of subgroups  (included trivial groups) is p + 2. Result depends only on the fact that p is a prime number, it is not linked to the particularity of the number p=2.
For subgroup generated by the elements indicated in the text just go to their determination and identify those that coincide or seek those who remain.
