Do finite indexes tell us about the finitely generated nature of the related group and subgroup? If a group and subgroup have a finite index, and we know that the subgroup is finitely generated, can we conclude that the group is also finitely generated?

Let $H=\langle h_1,\ldots, h_n\rangle$ and let $[G:H]=m$.  Suppose $g_1,\ldots,g_m$ is a complete set of coset representatives for $G/H$.  Since,
$$G=\bigcup_{k=1}^m g_kH$$
we can see that $G=\langle\{h_jg_k:1\le j\le n,1\le k\le m\}\rangle$.
