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The Atlas of Finite Group Representations has information on the maximal subgroups of some particular finite simple groups. This is at least a handy place to start. See http://brauer.maths.qmul.ac. …

Here's a short answer, assuming only that $C$ is an abelian $p$-group and $\vert D \vert$ is relatively prime to $p$. Remember that $G \cong H\rtimes K$ iff normal subgroup $H$ has a complement in $G …

You can handle several of the groups on the list from your edit with a modification of the argument of Marty Isaacs. For the rest, you can find guidance as to what a counterexample would look like. …

I'm interested in writing GAP code to compute the inner automorphism group of a finite Lie algebra. (I'd like to be able to group conjugate subalgebras together.) I've had trouble finding good refer …

This has the content of a comment, but is somewhat longer and with more formatting than a comment allows, so I post it as an answer. Anyway: I tried the GAP computation for 2^4, and got results. Fo …

I accepted @Paul Levy's answer, but meanwhile I want to add some further extended comments, references, and experimental data. Perhaps this will be useful to someone else at some point. As Paul Levy …

A pretty complete and accessible description of this can be found in a survey article of Oliver King. In addition to the link, here's the citation info: King, Oliver H., The subgroup structure of fi …

You might find helpful the paper of Liebeck and Shalev "Classical Groups, Probabilistic Methods, and the (2,3)-Generation Problem". There, they show that there are three involutions that generate all …

Carlisle King recently posted an arXiv preprint which (claims to) show that every finite simple group is generated by an involution, together with another element of prime order. http://arxiv.org/abs/ …

Unless I'm mistaken, the group that you're constructing here is an extension of $\mathbb{Z}$ by $G$. It would cause less confusion if you reserved the notation $\mathbb{Z} \times G$ for the direct pr …

UPDATE: The original poster of the question, together with Mamta Balodi, have shown that the labeling I suggest below is an EL-labeling if and only if group (product) complements coincide with lattic …

In a somewhat different direction from Alireza: the conjecture is true for a large family of groups, including all abelian groups and many supersolvable groups. Let me start with the abelian case. P …

Are you running computer experiments to verify conjectures? If so, the GAP SmallGroups library will give you exactly what you want up to $n = 1023$. For example, the GAP commands n:=16;; G:=SmallGr …