Show examples in which $U_1 \cong U_2$ but $G/U_1 \ncong G/U_2$, where $U_1, U_2 \triangleleft G$

Question

I have read the claim that the quotient groups of a group $G$ and two isomorphic groups need not be isomorphic. I find this strange, as to me two isomorphic groups are basically the same thing, except with some relabelling.

I have been trying to look for an example in the Klein-4 group, which is an easy one to find normal subgroups (being abelian), but unfortunately I have been able to find isomorphic maps between its subgroups.

Could you show me or point me to groups where I could find an illustration of this claim? I would also appreciate an intuitive explanation or a hint to prove how this can happen.

Answer 1

Let $G=\Bbb Z, U_1=2\Bbb Z, U_2=3\Bbb Z.$