Biggest noncommutative group $N$ where a group $G$ is normal.

Question

Given a finite group $G$, it is possible to obtain the biggest non commutative group $N$ such that $G\lhd N$ with $N\neq G$, $\vert N\vert < \infty$ and $N$ not a direct or semidirect product?

Answer 1

No. Given $G$ and any non-commutative finite group $H$, then $G\times H$ and $G\times H\times H$ and $G\times H\times H\times H$ etc. are all non-isomorphic groups whose orders tend to infinity that have the desired properties.

To address the edits:

In what sense do you mean "not a direct or semidirect product"? If you had an example $N$, what exactly in your imagined condition stops me from just taking the direct product of $N$ with literally any other finite group? Or from taking the holomorph of $N$? Do you want $N$ to be indecomposable? Do you want $N$ to not admit any non-trivial semi-direct product whatsoever, for any subgroups? Or only no semi-direct product expression involving $G$ as one of the subgroups?

The concept of "maximal such group" is unlikely to be well-defined, especially not within finite groups. I'm having trouble imagining a definition that would even guarantee a unique isomorphism class at that order. "Minimal" is more likely to have a meaningful answer, though you still have to be careful to specify what you mean--for much the same reasons, but "maximal" requires rather strong conditions to stop you from just constructing arbitrary towers of examples, whereas "minimal" tends to be much more intrinsically restrictive.

Answer 2

The answer is a resounding no. There are all manner of ways of "putting groups together" so-as to satisfy your normality requirement that have no restrictions on what those initial groups look like (esp. one of them could be monstrously large [pun intended]).

What you're interested in is fusion systems. It's technical but has elementary prerequisites. I suggest the following:

http://web.mat.bham.ac.uk/C.W.Parker/Fusion/fusion-intro.pdf

It is a way of "zippering" groups together along subsets that is actually quite beautiful and has some geometric applications, but I'm not the best person to give you many details, sadly!