Let $G$ be a group, and $H_i$ be subgroups of $G$. What would be counterexamples for $H_1 (H_2 \cap H_3) = H_1 H_2 \cap H_1 H_3$ (here $H_i H_j$ means subset product)? I found on proofwiki that this does not hold when $H_i$ are just subsets, but nothing about subgroups (https://proofwiki.org/wiki/Product_of_Subset_with_Intersection and https://proofwiki.org/wiki/Product_of_Subset_with_Intersection/Equality_does_not_Hold).
I already checked subgroups of $\mathbb Z$, $C_{n}$, $D_n$, what when either $H_i$ is subgroup of another, or when either one is trivial group or whole $G$ and I am pretty much out of options. Any help will be appreciated.
Take $G$ to be the Klein 4-group and the $H_i$ to be the three distinct subgroups of order $2$. Then the LHS is $H_1$, while the RHS is $G$.