Help With Algebraic Properties of 12-sided Regular Polygon with Binary Colorable Vertices

Question

I'm fairly new to abstract algebra and I stumbled across this problem (in music, actually) involving dihedral symmetries of a dodecagon where the vertices are either filled or unfilled, like in this image: and I'm having trouble figuring out the structure that would describe the collection of groups of symmetries of such objects. Obviously $D_{24}$ is in it, so perhaps the collection of dihedral groups with order less than or equal to 24? How would I prove that? Do I call it a "collection" or something else, like a category?