I don't know to find subgroups. $G=\mathbb Z_3\times\mathbb Z_5$
I don't know to find a subgroup.
Give me an example a subgroup of $G$ how to find that ?

Because $(3,5)=1$, $G$ is cuclic and $([1]_3,[1]_5)$ is a generator of $G$.
Now as I commented for every positive $n| |G|=15$ there exists a subgroup of $G$ the one that is generated by $([1]_3,[1]_5)^n$.So we have:
$1)$ for $n=1$:$<([1]_3,[1]_5)>=G$
$2)$ for $n=3$ :$<([0]_3,[3]_5)>=\{0\}\times\mathbb{Z_5}$
$3)$for $n=5$:$<([2]_3,[0]_5)>=\mathbb{Z_3}\times\{0\}$
$4)$for $n=15$:$<([0]_3,[0]_5)>=\{0,0\}$
