Sure, under Linnaean taxonomy. But, well,
A) Linnaeus was a eugenecist so his scientific opinions are suspect and his morality is awful
B) he didn't know about evolution
C) he didn't know about prehistoric life
so his classification system? Sucks ass. It doesn't work anymore. It no longer reflects the diversity of life.
Instead, scientists - almost across the board, now - use Clades, or evolutionary relationships. No rankings, no hierarchies, just clades. It allows us to properly place prehistoric life, it removes our reliance on traits (which are almost always arbitrary) in classifying organisms, and allows us to communicate the history of life just by talking about their relationships.
So, for your own edification, here's the full classification of birds as we currently know it, from biggest to smallest:
Asgardians (Eukaryomorphans)
Eukaryota (note: Proteobacteria were added to an asgardian Eukaryote to form mitochondria)
ParaHoxozoa (Hox genes show up)
Bilateria (all bilateran animals)
Deuterostomia (Deuterostomes)
Gnathostomata (Jawed Vertebrates)
Osteichthyes (Bony Vertebrates)
Sarcopterygii (Lobe-Finned Fish)
Amniota (animals that lay amniotic eggs, or evolved from ones that did)
Sauropsida/Reptilia (reptiles sensu lato)
Sauria (reptiles sensu stricto)
Avemetatarsalia (Bird-line Archosaurs, birds sensu lato)
Ornithodira (Appearance of feathers, warm bloodedness)
Dinosauria (fully upright posture; All Dinosaurs)
Saurischia (bird like bones & lungs)
Theropoda (permanently bipedal group)
Maniraptoriformes (feathered wings on arms)
Paraves (fully sized winges, probable flighted ancestor)
Euornithes (wing configuration like modern birds)
Neornithes (modern birds, with fully modern bird beaks)
idk if this was a gotcha, trying to be helpful, or genuine confusion, but here you go.
all of this, ftr, is on wikipedia, and you could have looked it up yourself.
