Consider the finite $2$-groups containing cyclic subgroup of index $2$:

$C_{2^n}$, $C_{2^{n-1}}\times C_2$, $D_{2^n}$, $SD_{2^n}$, $QD_{2^n}$, $Q_{2^n}$.

Can every finite $2$ group be embedded in $H\times K$ where $H$, and $K$ are one of the groups in the list?