Let $G$ be a semi-simple linear algebraic group over the complex numbers, e.g. the special linear group. Can you find an example of a finite sub-group $H$ of $G$ which does not normalize any maximal torus? or any non-trivial torus? In other words, given $H$, is it possible to find a torus $T$ in $G$ such that $H$ is contained in the normalizer of $T$?



Finite groups normalizing a torus
Yes, take the symmetry group of a platonic solid in $SO(3)$. It acts irreducibly on the adjoint representation (which is the standard representation), hence the only connected subgroups it normalizes are the trivial group and all of $SO(3)$. Neither is a torus.


