Predominant group of arthropods, placed within the class 
Hexapoda along with the Protura
?, Collembola
?, and Diplura
?.  The only group other than vertebrates to have developed flight.
-  Order Archaeognatha?  (Jumping bristletails)
 -  Order Monura? - extinct
 -  Order Thysanura?  (Common bristletails: silverfish, firebrats)
 -  Pterygota?, the winged insects
-  Order Palaeodictyoptera? - extinct
 -  ...
 -  Order Ephemeroptera?  (Mayflies)
 -  Order Odonata?  (Dragonflies and damselflies)
 -  Neoptera?, insects with folding wings
-  Order Blattodea?  (Cockroaches)
 -  Order Mantodea?  (Mantids)
 -  Order Isoptera?  (Termites)
 -  Order Zoraptera?  (Zorapterans)
 -  Order Grylloblattodea?  (Ice Insects)
 -  Order Dermaptera?  (Earwigs)
 -  Order Plecoptera?  (Stoneflies)
 -  Order Orthoptera?  (Grasshoppers, crickets, and katydids)
 -  Order Phasmatodea?  (Walking sticks, timemas)
 -  Order Embioptera?  (Webspinners)
 -  Various extinct orders
 -  Hemipterodea?
-  Order Psocoptera?  (Booklice and barklice)
 -  Order Pthiraptera?  (Sucking and chewing lice)
 -  Order Hemiptera?  (True bugs, cicadas and kin - formerly Heteroptera + Homoptera)
 -  Order Thysanoptera?  (Thrips)
 
 -  Endopterygota?, insects with complete metamorphosis
-  Order Miomoptera? - extinct
 -  Order Megaloptera?  (Alderflies, dobsonflies, and fishflies)
 -  Order Raphidioptera?  (Snakeflies)
 -  Order Neuroptera?  (Net-veined insects)
 -  Order Coleoptera  (Beetles)
 -  Order Strepsiptera?  (Twisted-winged parasites)
 -  Order Mecoptera?  (Scorpionflies and kin)
 -  Order Siphonaptera?  (Fleas)
 -  Order Diptera?  (Flies)
 -  Order Trichoptera?  (Caddisflies)
 -  Order Lepidoptera?  (Butterflies and moths)
 -  Order Hymenoptera  (Bees, ants, wasps, sawflies, and kin)
 
 
 
 
Let's not overlook the (genus?) 
Nupedia, a kind of fly!  ;-)