# Subplanes of Finite Projective Planes

If a finite projective plane $\pi_1$ of order $m$ contains, as a sub plane, a finite projective plane $\pi_2$ of order $n$, then $m \geq n^2$ with equality holding only in the case of a Baer sub plane. Otherwise $m \geq n^2 + n$. (This is a theorem of Bruck that can be found in Hall's Group Theory book, I believe.) My question is, how nearly can this bound be achieved?

For example, we have cases for $m=9$ and $n=2$, where $\pi_1$ is one of the non-Desaurgian planes of order $9$, and $\pi_2$ is the Fano plane. Is anything known about how close we can come to the bound in the general case? Has the bound been improved?

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• Not sure about the bound in general but there are some results about existence of sub-planes that are known. For example, combinatorics.org/ojs/index.php/eljc/article/view/v18i1p2. For details on small planes you can also refer to uwyo.edu/moorhouse/pub/planes – Anurag Jan 12 '15 at 12:48
• In general an algebraic projective plane of order $p^e$ contains algebraic subplanes of order $p^f$ for all factors $f|e$, so in particular $n=2$ is seen already for $m=8$. I think it's known that the only projective plane of order $7$ is algebraic, and that one does not contain the Fano plane. – Noam D. Elkies Apr 23 '19 at 2:38