Finding a specific homomorphism as an example of a theorem

I am trying to construct an example of a theorem from my Algebra text.

Quote:

If $\displaystyle f: G \to H$ is a homomorphism of groups and N is a normal subgroup of G contained in the kernel of f, then there is...

At this point all I am trying to do is come up with a non-trivial example of the requirements of the theorem, that is to say a group N such that N is not simply the kernel of f. I started with D4 but can't think of a good group to make a homomorphism with. I might not be thinking big enough...

Thanks!

-Dan