A torus has a rotation symmetry along the axis, a sphere has "spherical" symmetry under rigid motions; doesn't SU(3) also have a symmetry?
The Gell-Mann matrices ( see https://en.wikipedia.org/wiki/Gell-Mann_matrices ), the generators of SU(3), have a kind of threefold symmetry; but which one exactly? For example, the matrices $\lambda_1, \lambda_4 \rm{\ and\ } \lambda_6$ and also $\lambda_2, \lambda_5 \rm{\ and\ } \lambda_7$ are related by a three-fold symmetry. When looking at the matrices, $Z_3$ is surely a symmetry. (And $Z_3$ is also the center of SU(3).) But it is so hard to think in 8 dimensions ...
Trying to make this idea more precise: What are the symmetries, respectively, of the Lie algebra, of the group, and of the manifold? Since I was looking at the Gell-Mann matrices, the symmetries of the Lie algebra would be most interesting of all.
As I found out in the meantime, the question makes no sense. It makes no sense to ask about the symmetry of a symmetry group. Z_3, the center of SU(3), is a subgroup of SU(3). But it is wrong to call it a symmetry of SU(3).
Sorry for the confusion.