Prove that the quaternions and the Dihedral group are non-isomorphic groups of order 8.
The dihedral group of order 8 can be seen as the group of symmetries of the square. There are two diagonal symetries, which are involutions, therefore (with your notation) contains at least 2 elements of order 2; that is sufficient to prove what you want.
has exactly 5 elements of order 2 (two diagonal symmetries, 2 lateral symmetries and one central symmetry) and 2 elements of order 4 (one rotation of and its inverse).