When there a subgroup isomorphic to $H\times K$ If $H$ and $K$ are subgroups of a finite group $G$, when can we say that there is a subgroup isomorphic to $H\times K$?
 A: 
Sufficient condition. If $H$ and $K$ are normal in $G$ and $H\cap K=\{1_G\}$, then $HK$ is a subgroup of $G$ isomorphic to $H\times K$.

Proof. Let $(h,k)\in H\times K$, one has $hk=kh$, since $hkh^{-1}h^{-1}\in H\cap K$. Therefore, $HK$ is a subgroup of $G$ and the following is a group homorphism: $$\varphi\colon\left\{\begin{array}{ccc}H\times K&\rightarrow&HK\\(h,k)&\mapsto&hk\end{array}\right.$$
Notice that by construction of $HK$, $\varphi$ is surjective and since $H\cap K=\{1_G\}$, $\varphi$ is injective. $\Box$
Remark. The fact that $HK$ is a subgroup of $G$ only requires that $H$ or $K$ is normal in $G$, but this a little bit tedious to establish.
A: We can say this in the obvious case that $G=H\times K$, or in the case that $G$ is an internal direct product of $H$ and $K$, i.e., both $H$ and $K$ are normal, and $H\cap K=1$.
For groups of small order we have already "counterexamples".
For example, $S_3$ has no non-trivial subgroups $H$ and $K$, such that $H\times K$ is again a subgroup. The only proper non-trivial subgroups of $S_3$ are $C_2$ and $C_3$, and clearly $C_2\times C_3$ is not a subgroup of $S_3$. 
