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
$$G = \langle x,y \mid y x y^{-1} = x^2 \rangle$$
which is isomorphic to the multiplicative group generated by the $2\times2$ rational matrices
$$x = \left[\begin{array}{cc} 1& 1\\ 0& 1\end{array} \right]\qquad y = \left[\begin{array}{cc} 2& 0\\ 0& 1\end{array} \right]$$
and $N = \langle x \rangle$.
Then $x N x^{-1} = N$ and $y N y^{-1} < N$, but $N$ is not normal in $G$ because $y^{-1} N y$ is not contained in $N$.
End of plagiarism.
Perhaps you can enhance this example to your purpose. Gerhard "Ask Me About System Design" Paseman, 2009.01.15
