Beginning in the middle of the past century, the following classical problem attracted the attention of many mathematicians.
{\bf Problem 1.} {\it Describe the finite linear groups of small degree, i. e. finite subgroups in $GL_n(K)$ for every field $K$ and small $n$.}
One of the main problems of the modern post-classification finite group theory is the following problem.
{\bf Problem 2.} {\it Determine the maximal subgroups of finite almost simple groups, i. e. groups with nonabelian simple socle.}
Problems 1 and 2 are closely related. M. Aschbacher (1984) and P. Kleidman and M. Liebeck (1990) reduced the problem of describing the subgroup structure of finite classical groups to the study of the absolutely irreducible modular representations of finite quasisimple groups, i.~e. coverings of finite simple nonabelian groups. In his unpublished monograph P. Kleidman solved Problem 2 for finite classical almost simple groups of dimension $\leq 12$. In [Proc. conf. "The Atlas ten years on", to appear] we announced the classification of the conjugacy classes and the normalizers of absolutely irreducible quasisimple subgroups in $GL_n(q)$ for $13\leq n\leq 27$. Now we apply this classification to extending Kleidman's classification up to dimension 27. The main aim of our talk is to discuss the obtained results in this direction.
A new aspect of the old Problem 1 gives the following problem.
{\bf Problem 3.} {\it Describe the finite linear groups of small degree over local residue rings of $\bf Z$.}
The motivation is that linear groups over such rings are exactly automorphism groups of finite homocyclic $p$-groups. There is no publications at all concerning this natural and quite important problem. Let $p$ be a prime and $H = GL_n({\bf Z}/p^k{\bf Z})$, $k > 1$. Then $\bar H = H/O_p(H)\simeq GL_n(p)$. We will say that a subgroup $G$ of $\bar H$ is lifted to $H$ if the preimage of $G$ in $H$ is a splitting extension of $G$ by $O_p(H)$. Problem 3 is reduced essentually to classifying of subgroups of $GL_n(p)$ which are lifted to $H$. So one can use knowledge of subgroups of $GL_n(p)$. We will discuss also an application of our classification from above mentioned paper to solving Problem 3 for degree $\leq 20$ and $p > 3$. These results are obtained jointly with A. E. Zalesskii.