Remark:

The result is false if the hypothesis of normality of $H$ is dropped. For example consider a cube in $\mathbb{R}^3$ with center at the origin and $H$ be the subgroup of $G = SO(3, \mathbb{R})$ that map the cube to itself. Then $H$ is the symmetric group on four letters (proof?). Clearly $H$ is not in the center of $G$.