Simple compact special unitary Lie group and dicyclic group Dic$_{k}$=Q$_{4k}$

Question

Given a simple compact Lie group, SU($n$) group, for $n=2,3,4,...$,

Can we determine what are the allowed dicyclic group Dic$_{k}$ (=generalized quaternion groups Q$_{4k}$ (of order $k=2,4,8,...$) realized as a subgroup embedded into SU($n$)?

What is the restriction of $k$ for given SU($n$) of

$n=2?$

$n=3?$

$n=4?$

$n=5?$

(BONUS but not necessary: Are there some general rules one can use to determine any $n$ what is the allowed $k$?)