I am interested in the following question:

Given a projective plane of order $n=2^a$, is its incidence matrix must contain the incidence matrix of the Fano plane? If not, is it true that for any $n$ of the form $n=2^a$ there exists a projective plane of this order whose incidence matrix contains the incidence matrix of the Fano plane?

The only thing that I managed to find is the following from "Fano configurations in subregular planes" (Fisher, 2010) "It is obvious that any translation plane of order $p^r$, for $p$ a prime, has a subplane of order $p$".