Is SO(3)/O isomorphic to SO(3)? Is SO(3)/O isomorphic to SO(3)? By O, I mean the octahedral rotation symmetry group. My intuition says that they are not. But I don't know how to disprove it

I don't think any such subgroup is normal, so the quotient isn't a group. Hence it's not a group isomorphic to $SO(3)$.