Distributive law for subgroups

Question

I believe that for subgroups $H,K,L\le G$ distributivity does not apply, but could someone give an example to illustrate this: $$ \langle H\cup K\rangle\cap L\neq\langle (H\cap L)\cup (K\cap L)\rangle $$

Apparently with subgroups $n\mathbb{Z}$ of $\mathbb{Z}$ equality is fulfilled.

Any hints would be appreciated.

Answer 1

Let $G = S_3$, and let $H$ and $L$ be generated by distinct $2$-cycles and $K$ be generated by a $3$-cycle. Then $\langle H \cup K\rangle \cap L = G \cap L = L$, whereas $\langle (H \cap L) \cup (K \cap L)\rangle = \langle 1 \cup 1\rangle = 1$.