Here's a little algorithm using the music21 toolkit we developed at MIT. The #s refer to the number of semitones above C. It's a post-tonal context: enharmonic spelling is not taken into account:
>>> for i in range(1,5):
... for j in range(1+i, 9):
... c = chord.Chord([0, i, j])
... print "0,%d,%d: %s" % (i, j, c.commonName)
0,1,2: chromatic trimirror
0,1,3: phrygian trichord
0,1,4: major-minor trichord
0,1,5: incomplete major-seventh chord
0,1,6: tritone-fourth
0,1,7: tritone-fourth
0,1,8: incomplete major-seventh chord
0,2,3: minor trichord
0,2,4: whole-tone trichord
0,2,5: incomplete minor-seventh chord
0,2,6: incomplete dominant-seventh chord
0,2,7: quartal trichord
0,2,8: incomplete half-diminished seventh chord
0,3,4: major-minor trichord
0,3,5: incomplete dominant-seventh chord
0,3,6: diminished triad
0,3,7: minor triad
0,3,8: major triad
0,4,5: incomplete major-seventh chord
0,4,6: incomplete half-diminished seventh chord
0,4,7: major triad
0,4,8: augmented triad
0,3,8 is the major triad in first inversion. 0,2,5 it an incomplete minor seventh in third inversion. 0,1,8 is likewise a major-seventh in third inversion, while 0,4,5 is in second inversion. 0,2,6 is the dominant-seventh in third inversion minus the third, while 0,3,5 is in second inversion.
But most of these names are not standard. They come from: Larry Solomon's "The List of Chords, Their Properties and Use in Analysis," in Interface, Journal of New Music Research , 1982, v11/2. Online list: http://vladimir_ladma.sweb.cz/english/music/structs/mus_rot.htm