Quotient of a group and subgroup

Consider a group $G$ and a normal subgroup $H$ .
Is it necessarily the case that the quotient group $G/H$ is isomorphic to a subgroup of $G$ ?

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Solution by Ancient Tree

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No, it is not necessarily the case. For example, with $G=\mathbb{Z}$ , the subgroups of $G$ are of the form $n\Z$ ; but the quotients $\Z /n\Z$ are not of this form since they are finite.

For a counter-example in the finite groups, it's a bit less trivial.