Quaternion, Dihedral groups and A-D-E classification

Question

$\bullet$ What is the role of Quaternion group $H$ and dihedral groups $D_n$ in A-D-E classification?

$\bullet$ Is Quaternion group $H$ in $A$ (special linear Lie algebra of traceless operators) or not?

$\bullet$ Is dihedral group $D_n$ in $D$ (even special orthogonal Lie algebra of even dimensional skew-symmetric operators) or not?