Cockatoos belong to the family Cacatuidae and belong to the order Psittacidae which includes the true parrot and large New Zealand Parrot families. The genera and subgenera of Cockatoos as relates to their color, are as follows:

Genus Nymphicus
Cockatiel, Nymphicus hollandicus (Kerr, 1792)
Genus Probosciger
Palm Cockatoo, Probosciger aterrimus
Genus Calyptorhynchus
Subgenus Calyptorhynchus – black-and-red cockatoos
Red-tailed Black Cockatoo, Calyptorhynchus banksii
Glossy Black Cockatoo, Calyptorhynchus lathami (Temminck, 1807)
Subgenus Zanda – black-and-yellow/white cockatoos
Yellow-tailed Black Cockatoo, Calyptorhynchus funereus (Shaw, 1794)
Short-billed Black Cockatoo, Calyptorhynchus latirostris Carnaby, 1948
Long-billed Black Cockatoo, Calyptorhynchus baudinii Lear, 1832
Genus Callocephalon
Gang-gang Cockatoo, Callocephalon fimbriatum (Grant, 1803)
Genus Eolophus
Galah, Eolophus roseicapilla (Vieillot, 1817)
Genus Lophochroa
Major Mitchell's Cockatoo, Lophochroa leadbeateri leadbeateri
Western Major Mitchell's Cockatoo,
Lophochroa leadbeateri mollis
Genus Cacatua
Subgenus Cacatua – true white cockatoos
Yellow-crested Cockatoo Cacatua sulphurea
Abbott's Cockatoo, Cacatua sulphurea abbotti
Timor Cockatoo, Cacatua sulphurea parvula
Citron-crested Cockatoo, Cacatua sulphurea citrinocristata
Sulphur-crested Cockatoo, Cacatua galerita
Triton Cockatoo, Cacatua galerita triton
Eleonora Cockatoo, Cacatua galerita eleonora
Fitzroy Cockatoo, Cacatua galerita fitzroyi
Greater Sulphur-crested Cockatoo, Cacatua galerita galerita
Blue-eyed Cockatoo, Cacatua ophthalmica
White Cockatoo, Cacatua alba
Salmon-crested Cockatoo, Cacatua moluccensis

Subgenus Licmetis – corellas
Long-billed Corella, Cacatua (Licmetis) tenuirostris
Western Corella, Cacatua (Licmetis) pastinator
Muir's Corella, Cacatua (Licmetis) pastinator pastinator
Butler's Corella, Cacatua (Licmetis) pastinator butleri
Little Corella, Cacatua (Licmetis) sanguinea
Tanimbar Corella (also Goffin's Cockatoo), Cacatua goffiniana
Solomons Cockatoo, Cacatua (Licmetis) ducorpsii
Red-vented Cockatoo, Cacatua (Licmetis) haematuropygia