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