The Eye Bigfin Reef Squid-Yellowstones Bay-Fethiye/Turkey Squids are members of the class Cephalopoda, subclass Coleoidea, order Teuthida, which has two major suborders, Myopsina and Oegopsina (including giant squid such as Architeuthis dux). Teuthida is the largest cephalopod order with around 300 species classified into 29 families.