Theorem 1 of [LS] [Liebeck and Seitz - On the subgroup structure of exceptional groups](https://doi.org/10.1090/S0002-9947-98-02121-7) says:



> 
> Let $X = X(q)$ be a quasisimple group of Lie type in characteristic $p$, and suppose that $X < G$, where $G$ is a simple adjoint algebraic group of exceptional type, also in characteristic $p$. Assume that [$q$ is large]. Then … there is a closed connected subgroup $\overline X$ of $G$ containing $X$, such that every $X$-invariant subspace of the Lie algebra $L(G)$ is also $\overline X$-invariant ….
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[LS] emphasises that the condition is only on $q$, not on $p$; that is, it is OK for the characteristic $p$ to be small, as long as we take a large power of it (usually $q > 9$). Subgroups $X$ as described are called generic. The much more recent [Litterick - On non-generic finite subgroups of exceptional algebraic groups](https://doi.org/10.1090/memo/1207) justifies the restriction to non-generic groups by saying that the generic case is well understood, and points to [LS], which I take indirectly to be indicating that [LS] remains the state of the art.


What, if anything, is known about generic subgroups, in the above sense, for small $q$, especially in the spirit of the quoted Theorem 1?



Generic finite subgroups, associated to small finite fields, of reductive algebraic groups
The state of the art for subgroups $\operatorname{PSL}\_2(q)$, which constitute most of the cases where $q > 9$ in [LS], is now given by David Craven's Memoir [Maximal $\operatorname{PSL}\_2$ Subgroups of Exceptional Groups of Lie Type](https://doi.org/10.1090/memo/1355) ([arxiv version](https://arxiv.org/abs/1610.07469)). For maximal subgroups, complete information for groups of type $F\_4$ and $E\_6$ is given in another preprint [The Maximal Subgroups of the Exceptional Groups $F\_4(q)$, $E\_6(q)$ and $^2E\_6(q)$ and Related Almost Simple Groups](https://arxiv.org/abs/2103.04869) of Craven. The case where $X(q)$ has untwisted rank more than $\frac{1}{2}\operatorname{rank}(G)$ (but arbitrary $q$) is given by Liebeck and Seitz in [Maximal Subgroups of Large Rank in Exceptional Groups of Lie Type](https://doi.org/10.1112/S0024610704006179).


In all, this leaves a very small number of subgroup types; I'm not sure anyone has worried about the remaining cases explicitly.


