Subgroups in finite quasithln groups

A finite group $G$ is called quasithin if $m_p(M)\leqslant2$ for any 2-local subgroup $M$ in $G$ and any odd prime $p$. As usual, $m_p(X)$ denotes the $p$-rank of the group $X$. Let $\mathscr K$ denote the set of all known (at the present time) finite non-Abelian simple groups. A group $G$ is called a $\mathscr K$-group if each of its proper non-Abelian simple sections belongs to $\mathscr K$. The current state of the classification of finite simple groups points to the importance of studying simple quasithin $\mathscr K$-groups $G$. The structure of proper subgroups in such groups are investigated in this paper.
Moreover, a detailed study is made of the structure of 2-local subgroups in quasithin $\mathscr K$-groups whose 2-local 3-rank does not exceed 1. As an example of how the results can be applied, we examine the component case of a problem concerning quasithin groups of 2-local 3-rank at most 1.
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