Examples of non-abelian, non-isomorphic groups that have the same conjugacy class structure.

Question

What groups of the same order are not isomorphic but contain the same conjugacy class structure?

A bit more detailed question is: Are there examples where those groups are non-abelian?

The only (abelian) example I know is $Z_2 \times Z_2 \times Z_2$ and $Z_8$ which both have conjugacy classes of $$8 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1.$$

Answer 1

Example: the quaternion group $Q$ of order $8$ and the dihedral group $D_4$ of order $8$ have the same conjugacy class structure $(1+1+2+2+2)$, yet these non-abelian groups are not isomorphic.