It is shown that the commutative binary Knuth semifield planes of order
$2^{n}$, for $n=5k~$\ or $7k$ and $k$ odd, their transposes and
transpose-duals admit subplanes of order $2^{2}$. In addition, many of the
Kantor commutative semifield planes of order $2^{5k}$ or $2^{7k}$ also admit
subplanes of order $4$.
Furthermore, a large number of maximal partial spreads of order $p^{k}$ and
deficiency at least $p^{k}-p^{k-1}$ or translation planes of order $p^{k}$ are
constructed using direct sums of matrix spreads sets of different dimensions.
Given any translation plane $\pi_{0}$ of order $p^{d}$, there is either a
proper maximal partial spread of order $p^{c+d}$ whose associated translation
net contains a subplane of order $p^{d}$ isomorphic to $\pi_{0}$ or there is a
translation plane of order $p^{c+d}$ admitting a subplane of order $p^{d}$.
Other than the semifield planes mentioned above and a few sporadic planes of
even order, there are no other known translation planes of order $p^{c+d}$
admitting a subplane of order $p^{d}$, where $d$ does not divide $c$.