Prickly Pear cactus (genus Opuntia)

Kingdom - Plantae (plants)
Phylum - Tracheophyta (vascular plants)
Division - Magnoliophyta = Anthophyta] (angioperms)
Class - Dicotyledonae = Magnoliopsida (dicots)
Subclass - Caryophyllidae
Order - Caryophyllales
Family - Cactaceae (cacti)
Genus - About 100 genera and 1000-2000 species divided into three tribes

• Tribe
  • Pereskieae - two genera, Pereskia and Maihuenia. They have woody stems, leaves are persistent or semideciduous, areoles are spiny and woolly at the leaf axils, flowers are pendulous, and the fruit is fleshy.
  • Opunitieae - genera Opuntia (prickly pear cactus), Pereskiopsis, Pterocactus, Quiabentia, Tacinga. Fleshy, branched plants, areoles may or may not have spines but always have fine hairs (glochids), leaves are flat or cylindrical, spines are usually straight, the fruit is a seedy berry.
  • Cactoidea - about 100 genera (most cacti are in this tribe).
    Subtribes of Cactoidea
    • Browninginae - genera: Armatocereus, Browningia, Calymnanthium, Jasminocereus, Neoraimondia, Pseudoacanthocereus Stetsonia
    • Cactinae - genera: Ariocarpus, Astrophytum, Aztekium, Coryphantha, Echinocactus, Echinofossulocactus, Epithelantha, Escobaria, Ferocactus, Geohintonia, Leuchtenbergia, Lophophora, Mammillaria, Neolloydia, Obregonia, Ortegocactus, Pediocactus, Pelecyphora, Sclerocactus, Strombocactus, Thelocactus
    • Echinocereinae - genera: Acanthocereus, Echinocereus, Harrisia, Leptocereus, Peniocereus
    • Cereinae/Cerinae - genera: Arrojadoa, Austrocephalocereus, Brasilicereus, Cereus, Cipocereus, Coleocephalocereus, Melocactus, Micranthocereus, Pilosocereus, Stephanocereus
    • Hylocereinae - genera: Disocactus, Epiphyllum, Hylocereus, Selenicereus, Weberocereus
    • Notocactereinae - genera: Austrocactus, Blossfeldia, Copiapoa, Corryocactus, Eriosyce, Eulychnia, Frailea, Parodia, Uebelmannia, Yavia
    • Pachycereinae - genera: Bergerocactus, Carnegiea (saguaro), Cephalocereus, Escontria, Myrtillocactus, Neobuxbaumia, Pachycereus, Polaskia, Stenocereus
    • Rhipsalidinae - genera: Hatiora, Lepismium, Rhipsalis, Hatiora, Schlumbergera
    • Trichocereinae - genera: Acanthocalycium, Brachycereus, Cleistocactus, Cintia, Denmoza, Discocactus, Echinopsis, Espostoa, Facheiroa, Gymnocalycium, Haageocereus, Lasiocereus, Leocereus, Matucana, Mila, Neowerdermannia, Oreocereus, Oroya, Pygmaeocereus, Rebutia, Samaipaticereus, Weberbauerocereus.

Note: The classification of cacti is controversial.

References:
Anderson, E.F., The Cactus Family, 2001, The Timber Press.