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 .

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.