We know that $SU(2)$ is a double cover of $SO(3)$, such that $$SU(2)/Z_2=SO(3),$$ through a finite extension $N=Z_2$.
For other examples of simply-connected Lie groups such as $SU(2)$, $SU(N)$ or $E_8$,
(1) are there nontrivial extensions of $SU(2)$, $SU(3)$, $SU(N)$ or $E_8$?
(2) are there nontrivial finite extensions of $SU(2)$, $SU(3)$, $SU(N)$ or $E_8$?
Please provide examples or counter-statements saying why they do not exist.
