P-group embeddable in a maximal class group

Definition
Suppose $$p$$ is a prime number and $$P$$ is a finite $$p$$-group, i.e., $$P$$ is a group of prime power order where $$p$$ is the underlying prime. We say that $$P$$ is embeddable in a maximal class group if there exists a group of prime power order $$Q$$ containing $$P$$ such that $$Q$$ is a defining ingredient::maximal class group.