The next clade down, they are all extant and extinct turtles (and stem-turtles) that have a completely developed shell. In other words, they have both a carapace (top part of the shell) and a plastron (bottom part of the shell). Notable groups in Testudinata are:
A lot of the information we have starts to diverge in Testudinata, so many of the groups listed above could be renamed, have duplicate members, or really anything... It is all a bunch of speculation really. I will try to clear things up in their designated links.
Now are you really just here for the extant turtle species? Then look no further than the next clade down, Testudines!