Amycus Carrow was the last known DoDA professor. It was never stated who taught the class after the war.
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He doesn't officially become a professor but JK Rowling does mention that he goes to Defence Against the Dark Arts lessons in Hogwarts to give lectures. He actually becomes an Auror.
Defense against the dark arts
Arithmancy. Hermione takes this class, but Harry does not so we never actually meet Professor Vector.
First year students take flying lessons which is taught my Madam Hooch.Students are required to take the following class in their first five years at Hogwarts:Transfiguration taught by Professor McGongallCharms taught by Professor FlitwickHerbology taught by Professor SproutPotions taught by Professor SnapeHistory of Magic taught by Professor BinnsAstronomy taught by Professor SinistraDefence Against the Dark Arts taught by Professor Qurriel (first year), Professor Lockhart (second year), Professor Lupin (third year), Fake Professor Moody i.e Barty Crouch jr (fourth year) and Professor Umbridge (fifth year).In third to fifth year along with the subjects listed above Harry Potter studied:Divination taught by Professor TrewlaneyCare of Magical Creatures taught by Professor HagridIn sixth year students are allowed to drop and choose whatever subjects they please. Harry took the following subjects in sixth year (and seventh had he returned for it):Potions taught by Professor SlughornCharmsDefence Against the Dark Arts taught by Professor Snape (sixth year), Professor Carrow (seventh year, turned into 'Dark Arts')TransfigurationHerbology
Philosopher's (Sorcerer's) Stone: Quirinus Quirrel.Chamber of Secrets: Gilderoy Lockhart.Prisoner of Azkaban: Remus Lupin.Goblet of Fire: Barty Crouch Jr disguised as Alastor Mad-Eye Moody.Order of the Phoenix: Dolores UmbridgeHalf-Blood Prince: Severus SnapeDeathly Hallows: Amycus Carrow (taught Dark Arts, not defence)