The Lexwing is a massive airship owned by Lex Luthor.
The Joker came across the Lexwing in Lex Luthor's base, and was awed by its size, which was, according to him, twenty times bigger than Batman's Batwing. He assumed that it would be a hundred times as lethal. When Lex Luthor tried to kill Joker, he and Harley Quinn managed to subdue Mercy Graves and Luthor. After a new paint job coutsey of Harley, the pair fly the Lexwing over Metropolis, bringing Luthor along. Joker, in the desire to destroy everything Luthor built, started to fire missiles towards the city. Batman caught up in the Batwing and fired at the gigantic ship with little effect, and decided to break in by himself when Joker returns fire.
Batman attacked Joker inside the control room, sending him flying from his seat. Harley, knowing nothing about the ship, randomly pushed a button which caused it to go down in the middle of Metropolis. Superman lifted the Lexwing out of the way of a skyscraper. A punch from Batman made Joker drop his bag of explosive marbles, prompting Batman to grab Harley and tell Superman to get Luthor and escape the Lexwing before the detonation. The marbles blow up the plane, bringing down the Lexwing into the ocean, where it was destroyed in a huge explosion.