Want to clarify whether I am correct or not, $\Phi(G) \subseteq \Phi(H)$? I Want you to clarify whether I am correct or not regarding following question. I will be thankful to you for telling me if I am wrong:
Let $G$ be finite group and $\Phi(G)$ denotes its frattini subgroup. Let $H$ be a subgroup of $G$ such that $\Phi(G) \subseteq H$. Since $\Phi(G)$ is group of all non generators of $G$ therefore it also non generators for $H$. This implies that $\Phi(G) \subseteq \Phi(H)$. 
regards
Steve

No. Take $G$ to be the dihedral group of order $8$. The Frattini subgroup is the unique normal subgroup of order $2$. But $G$ has two subgroups of order $4$ that are elementary abelian (and as such, they have trivial Frattini subgroup) and yet they contain $\Phi(G)$. 
