On the orbit decomposition of group of symmetries of a cube on the unit sphere

Question

Consider the cube to be the set $K = [-1,1]^3 \subset \mathbb{R}^3$. The group of symmetries of the cube is the subgroup of $O_3 = ( A \in M_{3,3}\mathbb{R} | AA^T = I)$ that preserves the cube i.e. $Av \in K$ for all $v \in K$. Let us call a 'orbit decomposition' of a group $G$ acting on a set $X$ to be partition of $X$ by the conjugacy-classes (in $G$) of the point stabilizers. For the action of $K$ on the unit sphere $S^2 \in \mathbb{R}^3$ describe the orbit decomposition. Give the answer as a list of (conjugacy classes) of subgroups of $O_3$ followed by the subsets of $S^2$ where the point stabilizer group is realized.