Questions of construction of quasifields with associative powers

The structure of finite quasi-fields with associative degrees is investigated. These are, above all, associative quasifields, called near-fields. These also include the Moufang quasifields which have loops of nonzero elements are, by definition, loops introduced by Ruth Moufang in 1935.
The paper presents the main definitions associated with quasifields. It is shown that identity element of any finite (right) quasifield $Q$ generates a simple subfield $P$, and $Q$ is always a one-sided module over $P$ , and a two-way — is not always. As a result, found new proof of a well-known statement: a simple subfield of a finite semifield always lies in the center. At the same time, the finite near-fields with a simple subfield that does not lie in the center. Famous the question of maximal subfields of finite quasifields is completely solved for a class of finite near-fields of order $p^r$ with prime numbers $p$ and $r$.
In solving the questions about maximal subfields and spectra of group orders of nonzero elements of finite Moufang quasifields, it is proposed to use the well-known analogues of the group-theoretic theorems of Lagrange and Sylow. All possible two-digit orders of the proper Moufang quasifields are listed.