I have to find all the subrings of $Z_8$. I've already deduced that since all subgrings are on subgroups of the additive group, that each must have zero in it, as it acts as the identity element in the additive group.
Obviously the original ring and the trivial ring are both subgroups. I easily found that $({[0]_8, [4]_8}, +, *)$ and $({[0]_8, [2]_8, [4]_8, [6]_8}, +, *)$ are also subgroups. And I think this is all of them, but I can't think of a way to be sure without ruling out all other options by trial error.
Am I missing any??