Noun
subplane (plural subplanes)
(mathematics) A plane making up part of another plane.
Fano subplanes A Fano subplane is a subplane isomorphic to PG(2,2), the unique projective plane of order 2. If you consider a quadrangle (a set of 4 points no three collinear) in this plane, the points determine six of the lines of the plane. Source: Internet
Every point of the plane lies on a line of a Baer subplane and every line of the plane contains a point of the Baer subplane. Source: Internet
An open question is: Does every non-desarguesian plane contain a Fano subplane? Source: Internet