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Here is a simple example. As mentionned by Gjergji, this is a question about the number of conjugacy classes. Take $G=\frak A_4$, which has $3$ classes (the identity, the double transpositions and the $4$-cycles). Now take $H$ the subgroup spanned by a $4$-cycle. Because $|H|=4$ and $H$ is abelian, it has $4$ classes.
Edit. This is incorrect: $H$ is not a subgroup of $\frak A_4$, because a $4$-cycle is an odd permutation. I apologize. Note that I an accepted answer can not be deleted.
Here is a simple example. As mentionned by Gjergji, this is a question about the number of conjugacy classes. Take $G=\frak A_4$, which has $3$ classes (the identity, the double transpositions and the $4$-cycles). Now take $H$ the subgroup spanned by a $4$-cycle. Because $|H|=4$ and $H$ is abelian, it has $4$ classes.