How many ways can a cube fit into a sphere through its vertices?

Question

I mean, we know that every cube has 8 vertices. Now imagine a sphere with a fixed radius. Cubes can have arbitrary sides, one way is this. that there are no vertices on the sphere and the entire cube is inside the sphere, the other case is that the cube has only one vertex on the points of the sphere and the other 7 are inside the sphere. Now comment on the rest of the states, i.e. 2, 3, 4, 5, 6, 7, and 8 vertices on the points of the sphere. Comment which ones are impossible? My idea is that states 3, 5, 6, and 7 are impossible Of course, it is difficult to explain because each of them has different states, for example, three vertices is impossible because either all three must be in the same plane, in which case a rectangular sphere will not be formed in the circular cross-section, or with two vertices in the same plane. be in another plane, which is also impossible because if two vertices are in the plane, it will eventually lead to 4 vertices on the points of the sphere.