Give some stream of consciousness and then the answer. Principal photography began on March 28, 1984, and ended in May. Filming took place at Maine North High School in Des Plaines, Illinois, shuttered in 1981. The same setting was used for interior scenes of Hughes' 1986 film Ferris Bueller's Day Off, which featured exterior shots from nearby Glenbrook North High School. The library at Maine North High School, considered too small for the film, prompted the crew to build the set in the school's gymnasium. The actors rehearsed with each other for three weeks and then shot the film in sequence. On the Ferris Bueller's Day Off DVD commentary (featured on the 2004 DVD version), Hughes revealed that he shot the two films concurrently to save time and money, and some outtakes of both films feature elements of the film crews working on the other film. The first print was 150 minutes in length.

Based on this passage, what school did they film the breakfast club?
To answer the question, consider the following: Filming took place at Maine North High School in Des Plaines, Illinois, shuttered in 1981.
Therefore, the answer is Maine North High School in Des Plaines , Illinois.