Most general Gauge Lie group in a Yang-Mills theory

The following is just an example where you can use gauge groups other than $SU(N)$.

While the discovery of the Higgs was and still remains a huge step towards our better understanding of particle physics there still exists the question of whether or not the Higgs is an elementary particle or a bound state of a strongly coupled sector in higher energies. Note that the latter possibility is still not excluded by the LHC data.

The key component of such a model is a strongly sector that triggers $chi SB$ in the fermion sector which resembles a lot QCD. This breaking has to generate at least four Nambu-Goldstone bosons.

The above is one of the requirements. There is another that the gauge group description has to be an asymptotically free theory for a certain choice of hypercolours and hyperflavours. In this context, the prefix hyper is to just denote the difference from the $SU(3)$ charges.

A final requirement was the existence of composite top partners.

Combining the above there was a group theoretic approach that discussed all possible scenarios for a composite Higgs model in this paper

As you can see from the above, the models that contain all the fermions in a single representation of the gauge group are described by the exceptional groups $F_4$ and $G_2$. There are also models with matter in two and more representations with a symplectic gauge group and a special orthogonal. The usual $SU(N)$ also appears in these theories.

The exceptional groups $E_{6,7,8}$ have applications in string theory and related topics thereof; the study of superconformal fixed points for example. Not sure if you are interested in these examples.

Hope this helps a bit!