Class Anthozoa -> Subclass Hexacorallia -> Order Zoanthidia
Zoanthids belong to the same Subclass as stony corals and corallimorphs. They share 6-fold symmetry but lack a calcium-based skeleton.
The marketplace for Zoanthids has produced some rather colorful names for different morphs. While some may seem silly and over-the-top, I have warmed up to them. There are discrepencies where a single zoa may have several "made up" names. Still, I find it helpful distinguishing the different color morphs. For example, if someone asks about a zoanthid with a green skirt, red face, and purple center, there are several that fit this description. In contrast, if someone were to ask about an "Eagle Eye Zoa" I know what it is!