does the subgroup of rotations of D12 has a subgroup of order 4? can you pick a suitable subset of the reflections in D12 that might correspond to the reflections of D4?

perhaps you might consider what happens if you regard D12 as a subgroup of S12, and then regard 1,2,3 as "being the same number", then 4,5,6 and "being a 2nd number", etc.

equivalently, taking a 12-gon with vertices 1 through 12, considering only rigid motions that permute vertices 1,4,7 and 10, the other vertices just "going along for the ride".