To expand on an earlier answer, I will plagiarize an answer provided by Derek Holt in a sci.math thread:
A "standard" example of that is the Baumslag-Solitar group
G = < x,y | y x y^-1 = x^2 >$$G = \langle x,y \mid y x y^{-1} = x^2 \rangle$$
which is isomorphic to the multiplicative group generated by the 2x2$2\times2$ rational matrices
x = [ 1 1; 0 1 ], y = [ 2 0; 0 1 ],$$x = \left[\begin{array}{cc} 1& 1\\ 0& 1\end{array} \right]\qquad y = \left[\begin{array}{cc} 2& 0\\ 0& 1\end{array} \right]$$
and N = < x >$N = \langle x \rangle$.
Then x N x^-1 = N$x N x^{-1} = N$ and y N y^-1 < N$y N y^{-1} < N$, but N$N$ is not normal in G$G$ because y^-1 N y$y^{-1} N y$ is not contained in N$N$.
End of plagiarism.
Perhaps you can enhance this example to your purpose. Gerhard "Ask Me About System Design" Paseman, 2009.01.15
