Subnormal versus quasinormal subgroups Let $G$ be a group and $H$ a subgroup.
$H$ is subnormal if it exists a finite normal chain from $H$ to $G$.
$H$ is quasinormal if $HS=SH$ for all subgroup $S$ of $G$.   
If $G$ is a finite group, then every quasinormal subgroup is subnormal.
What about the converse :   

Question: Is there a subnormal subgroup which is not quasinormal ?


For an explicit example, take $\;H:=\{(1)\,,\,(12)(34)\}\le S_4\;$ . Then this subgroup is subnormal since
$$H\lhd \{(1)\,,\,(12)(34)\,,\,(13)(24)\,,\,(14)(23)\}\lhd A_4\lhd S_4$$
but it is not quasinormal since if we take $\;K:=\langle(123)\rangle=\{(1)\,,\,(123)\,,\,(132)\}\;$ then
$$HK=\{(1)\,,\,(123)\,,\,(132)\,,\,(12)(34)\,,\,(243)\,,\,(143)\}$$
whereas
$$KH=\{(1)\,,\,(12)(34)\,,\,(123)\,,\,(134)\,,\,(132)\,,\,(234)\}\neq HK$$
