The cyclic group is one of the three Abelian groups of the five groups total of group order 8. Examples include the integers modulo 8 under addition () and the residue classes modulo 17 which have quadratic residues, i.e., under multiplication modulo 17. No modulo multiplication group is isomorphic to .
Its multiplication table is illustrated above.
The elements satisfy , four of them satisfy , and two satisfy .
Because the group is Abelian, each element is in its own conjugacy class. There are four subgroups: , , , and which, because the group is Abelian, are all normal. Since has normal subgroups other than the trivial subgroup and the entire group, it is not a simple group.