Parazoa (e.g., sponges), radially-symmetrical creatures (jellyfish, corals, etc.), acoelomates (flatworms, I think), pseudocoelomates (nematode parasites, round

Parazoa (e.g., sponges), radially-symmetrical creatures (jellyfish, corals, etc.), acoelomates (flatworms, I think), pseudocoelomates (nematode parasites, roundworms), echinodermata (sand dollars, sea urchins, etc), lophotrochozoans (molluscs, some worms, many more). A 19th century breakdown by Cuvier, influential for a long time, was: Vertebrates, arthropods, molluscs, radiates. (I don't know where worms fit.)