See 'Mushroom Corals - The Corallimorpharians' in the Articles section.
FYI: The taxonomy information that has existed in this Order of organisms for a long time has been confusing to say the least. Nevertheless, 2003 revisions by noted British marine biologist and diver Vincent B. Hargreaves Ph.D. (Doctorate in Marine Biology with a double major in Ichthyology, and Marine Invertebrate Zoology) have resulted in a new rearrangement of many of the species known or thought to be in this Order. It is now said to encompass 3 families (Corallimorphidae, Discosomatidae, and Ricordeidae) containing 9 genera with 59 species.
Furthermore, changes have continue and as of 2017, if affecting those species mentioned here, please contact me with those and I'll update them as necessary.