# $B,N,H$ be subgroups of $G$, is it true that $\langle B \cap H, N \cap H \rangle = \langle B,N \rangle \cap H$?

Let $G$ be a finite group, let $B,N,H$ be subgroups of $G$. I believe that $$\langle B \cap H, N \cap H \rangle = \langle B,N \rangle \cap H$$ but I do not find a satisfactory proof. I think this may be useful. I tried to use it but I did not manage to write a 100% satisfactory proof. Can someone help me with it?

-

This is false. Let $G=\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$, let $B=\langle (1,0)\rangle$, $N=\langle (0,1)\rangle$, and let $H=\langle(1,1)\rangle$. Then $$\langle B\cap H,N\cap H\rangle=\langle \{(0,0)\},\{(0,0)\}\rangle=\{(0,0)\}$$ but $$\langle B,N\rangle \cap H= G\cap H=H=\{(0,0),(1,1)\}$$