I am trying to better understand the cyclic groups and so far understand that a non-trivial subgroup is a subgroup that DOES NOT contain just the element identity alone, i.e. $\langle 6\rangle = \{6\}$ is a TRIVIAL subgroup of $\mathbb{Z}/6\mathbb{Z}$.
However, how would one go about finding the NON-TRIVIAL subgroups? Lets say for example cyclic group $G = (\mathbb{Z}/89\mathbb{Z})^\times$. The factors of $89-1=88$, are $\{1,2,4,8,11,22,44,88\}$ so these are the orders of the elements.
Now I think I found a few NON TRIVIAL subgroups, but I am not sure if they are correct $$ \langle 3\rangle = \{3,9,12,15,18,\ldots\}\\ \langle 4\rangle = \{4,8,12,16,\ldots\}\\ \langle 15\rangle = \{15,30,45,\ldots\} $$