TITLE: $G$ be a finite group such that for any subgroup $H,K$ of $G$ , $HK$ is also a subgroup of $G$ , then is every subgroup of $G$ normal?
QUESTION [6 upvotes]: Let $G$ be a  finite group such that $HK=KH$ for any subgroups $H,K$ of $G$ . Then is every subgroup of $G$ normal ? 
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(to anyone thinking this question is not worth keeping: Here is the response from the OP to a comment from Tobias Kildetoft stating that the condition implies uniqueness of all Sylow subgroups, and hence that $G$ is nilpotent, JL)
If for a fixed prime $p$, $H,K$ are two different Sylow-p subgroups then $|H \cap K| < |H|=|K|$ and then the subgroup $HK$ is a $p$-subgroup with order $|H||K|/|H \cap K| > |H|$ , impossible ! Hence for a given prime $p$ , there is a unique Sylow-p subgroup . But I have no idea whether these line of arguments passes to all subgroups or not ...
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REPLY [7 votes]: The group $\langle x,y \mid x^{p^2} = y^p=1, y^{-1}xy=x^{p+1} \rangle$ of order $p^3$ is a counterexample for all odd primes $p$.
The only non-normal subgroups are the $p$ conjugates of $\langle y \rangle$, and the product of any two of these is the subgroup $\langle x^p,y \rangle$.
Added: Sorry, I can see now that this example has already been mentioned in comments by James, but it is completely buried in zillions of comments, so it should really be an answer.