Let $G$ be a finite $p$-group, $H_{1}$ and $H_{2}$ two proper subgroups of $G$. Write $C_{H_{1}}$ (resp. $C_{H_{2}}$) for the subgroup of $G$ generated by all the of the conjugacy class of $H_{1}$ (resp. $H_{2}$). Suppose that $H_{1} \subset H_{2}$ and $H_{1} \neq H_{2}$.
My question is as follows: dose $C_{H_{1}} \neq C_{H_{2}}$?
(Note that $C_H$ is better known as the "normal subgroup generated by $H$", or sometimes "normal closure of $H$")
As mentioned by several people, you always have $H\subset C_H$ and $C_H=C_{C_H}$. So taking $H_1=H$ and $H_2=C_H$ gives a counterexample as soon as $H$ is not a normal subgroup.