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Choose three representative vectors, with nonzero coordinate scale suppressed. Their stabilizer subgroups and fixed-point subspaces of a group action areHere ${\kappa }_x{r}_z$ exchanges the first two coordinates and ${\kappa }_y{r}_x$ exchanges the last two. On the first line the stabilizer subgroup is every signed permutation of the two zero coordinates; fixing it forces those coordinates to vanish. On the second line it exchanges the equal coordinates and independently reflects the zero coordinate, forcing $x=y,z=0$. On the third line all coordinate permutations fix the vector, forcing $x=y=z$.

The three subgroups have different orders and hence cannot be conjugate. Their group orbits contain respectively six face-axis, twelve edge-axis and eight vertex-axis directions. Each fixed-point subspace of a group action is one-dimensional, so the equivariant branching lemma guarantees three distinct symmetry types of bifurcating equilibria. The axial isotropy types of the full cube symmetry group are precisely these: a vector with several distinct nonzero absolute coordinate values has a separate freely variable coordinate block for each such value in the fixed-point subspace of a group action of its stabilizer subgroup, giving dimension greater than one.