List all subgroups of $\Bbb Z_{12}$ List all subgroups of $\Bbb Z_{12}$

$\Bbb Z_{12}$ is cyclic so all its subgroups are also cyclic
$$\begin{aligned}
<1> &= Z_{12}
\\<2> &= \{ 0,2,4,6,8,10\}
\\<3> &= \{ 0,3,6,9\}
\\<4> &= \{ 0,4,8 \}
\\<6> &= \{ 6 ,0\}
\end{aligned}$$
are there more???

In general you have:

In the cyclic group of order $n$, there is exactly one subgroup for each divisor of $n$.

Since the divisors of $12$ are $1,2,3,4,6,12$, you have found all subgroups except the subgroup $\langle 12 \rangle$ corresponding to $12$, which is the trivial subgroup.
