Question

Spontaneously breaking $SO(3)$ symmetry without $\mathbb Z_2 \times \mathbb Z_2$?

Answer 1

A state with cubic symmetry would be Z2×Z2Z2×Z2symmetric but not SO(3)SO(3) symmetric 

Answer 2

For v2 of the question, atoms with isotropic interactions can form a solid with cubic symmetry, although this may not be simplest case to analyse.