For each integer $k\geq 3$, give an example of a finite group $G$ and a subgroup $H$ such that $|G|=k|H|$ and $H$ is not normal in $G$

For each integer $k\geq 3$, give an example of a finite group $G$ and a subgroup $H$ such that $|G|=k|H|$ and $H$ is not normal in $G$.

I think the case that $k=|G|$ is not considered here.

I have doubts about the way this question is asked; by Lagrange's theorem, not every $3\leq k<|G|$ can give a respective subgroup.

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The question gives us the power to construct the groups G and H, hence your concern that $k = |G|$ is misplaced. –  Calvin Lin Dec 30 '12 at 3:08