$U_8=\{1,3,5,7\}$ since this group has one element of order one, three elements of two order and no element of $4$ order .. so does the Klein group.
Both $U_8$ and the Klein group are non cyclic groups whose every proper subgroup is cyclic, so the Klein group is isomorphic to U(8)?
There are only two groups of order four: (1) the cyclic group and (2) the Klein group.
As all elements of $U(8)$ are of order $2$, $U(8)$ is indeed isomorphic as a group to the Klein group.
The key argument is that there is no other groups of order four than the two mentioned above.