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Fish are a paraphyletic group: that is, any clade containing all fish also contains the tetrapods, which are not fish. For this reason, groups such as the "Class Pisces" seen in older reference works are no longer used in formal classifications.
Fish are classified into the following major groups:

• Subclass Pteraspidomorphi (early jawless fish)
• Class Thelodonti
• Class Anaspida
• (unranked) Cephalaspidomorphi (early jawless fish)
  
  • (unranked) Hyperoartia or Petromyzontida
    
    • Petromyzontidae (lampreys)
    
    
  • Class Galeaspida
  • Class Pituriaspida
  • Class Osteostraci
  
  
• Infraphylum Gnathostomata (jawed vertebrates)
  
  • Class Placodermi (armoured fishes, extinct)
  • Class Chondrichthyes (cartilaginous fish)
  • Class Acanthodii (spiny sharks, extinct)
  • Superclass Osteichthyes (bony fish)
    
    • Class Actinopterygii (ray-finned fish)
      
      • Subclass Chondrostei
        
        • Order Acipenseriformes (sturgeons and paddlefishes)
        • Order Polypteriformes (reedfishes and bichirs).
        
        
      • Subclass Neopterygii
        
        • Infraclass Holostei (gars and bowfins)
        • Infraclass Teleostei (many orders of common fishes)
        
        
      
      
    • Class Sarcopterygii (lobe-finned fish)
      
      • Subclass Coelacanthimorpha (coelacanths)
      • Subclass Dipnoi (lungfish)
      
      
    
    
  
  

Some palaeontologists consider that Conodonta are chordates, and so regard them as primitive fish. For a fuller treatment of classification, see the vertebrate article.
The various fish groups taken together account for more than half of the known vertebrates. There are almost 28,000 known extant species of fish, of which almost 27,000 are bony fish, with the remainder being about 970 sharks, rays, and chimeras and about 108 hagfishes and lampreys.^[12] A third of all of these species are contained within the nine largest families; from largest to smallest, these families are Cyprinidae, Gobiidae, Cichlidae, Characidae, Loricariidae, Balitoridae, Serranidae, Labridae, and Scorpaenidae. On the other hand, about 64 families are monotypic, containing only one species. It is predicted that the eventual number of total extant species will be at least 32,500.
From Wikipedia