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 > 

which is isomorphic to the multiplicative group generated by the 2x2 
rational matrices 

x = [ 1 1; 0 1 ],  y = [ 2 0; 0 1 ], 

and N = < x >. 

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