Minimum size of generating set of subgroup may be strictly greater than minimum size of generating set of group

Statement
It is possible to have a finite group $$G$$ and a subgroup $$H$$ such that the minimum size of generating set for $$H$$ is strictly greater than the minimum size of generating set for $$G$$.

Related facts

 * Schreier's lemma says that if the subgroup is a subgroup of finite index, then the minimum size of generating set for the subgroup is bounded by the product of the index of the subgroup and the minimum size of the generating set for the whole group (we can actually do a little better based on the lemma).
 * Minimum size of generating set of extension group is bounded by sum of minimum size of generating set of normal subgroup and quotient group
 * Minimum size of generating set of quotient group is at most minimum size of generating set of group

Facts used

 * 1) uses::Every finite group is a subgroup of a 2-generated group, i.e., every finite group is a subgroup of a finite group with a generating set of size two. This follows from Cayley's theorem and the fact the symmetric group on a finite set is 2-generated.
 * 2) There exist finite groups whose minimum size of generating set is strictly greater than 2. For instance, elementary abelian group:E8 has a minimum size of generating set equal to three, because it is a three-dimensional vector space over field:F2 and generating sets for it as a group are the same as generating sets for it as a vector space.

Proof
The proof follows directly by combining Facts (1) and (2).