Minimal proper quasifields with additional conditions

We investigate the finite semifields which are distributive quasifields, and finite near-fields which are associative quasifields. A quasifield $Q$ is said to be a minimal proper quasifield if any of its sub-quasifield $H\ne Q$ is a subfield. It turns out that there exists a minimal proper near-field such that its multiplicative group is a Miller–Moreno group. We obtain an algorithm for constructing a minimal proper near-field with the number of maximal subfields greater than fixed natural number. Thus, we find the answer to the question: Does there exist an integer $N$ such that the number of maximal subfields in arbitrary finite near-field is less than $N$? We prove that any semifield of order $p^4$ ($p$ be prime) is a minimal proper semifield.