  We study symmetry-protected topological (SPT) phases of matter in 2D
protected by symmetries acting on fractal subsystems of a certain type. Despite
the total symmetry group of such systems being subextensively large, we show
that only a small number of phases are actually realizable by local
Hamiltonians. Which phases are possible depends crucially on the spatial
structure of the symmetries, and we show that in many cases no non-trivial SPT
phases are possible at all. In cases where non-trivial SPT phases do exist, we
give an exhaustive enumeration of them in terms of their locality.
