Understanding Subgroups of $Z_n$ Specifically, $Z_8$

Question

I'm trying to determine the subgroup lattice for $Z_8$ which of course requires determining what the subgroups of $Z_8$ are.

I'm pretty sure they are:

$\langle1\rangle = \{0, 1, 2, 3, 4, 5, 6, 7\}$

$\langle2\rangle = \{0, 2, 4, 6\}$

$\langle4\rangle = \{0, 4\}$

and the trivial subgroup

$\langle0\rangle = \{0\}$

Are these all the subgroups of $Z_8$?

Second-guessing myself and thinking that

$\langle3\rangle = \{0, 1, 2, 3, 4, 5, 6, 7\}$ is also a subgroup as are $\langle5\rangle, \langle7\rangle$ for the same reason? Are they? Are they not?

Answer 1

$1,3,5$ and $7$ generate the whole group (because they are relatively prime to $8$).

You have all the subgroups there:

There are $\varphi (8)=4$ generators, mentioned above ($\varphi$ is Euler's totient function).

Other than that you have found $\langle 0\rangle, \langle 2\rangle, \langle 4\rangle $, and finally $\langle 6\rangle =\langle 2\rangle$.

That's all you have to check, because all the subgroups of a cyclic group are cyclic.

Answer 2

Generally, for the cyclic group $Z_n$, the unique subgroups are in a bijective correspondence with the divisors of $n$. Specifically, for every $k \mid n$, $\exists! H \leqslant G$ such that $|H| = k$. Therefore, you may conclude that the provided subgroups generated by 1, 2, 4, 8 (0) represent all subgroups of $Z_8$.