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?