A Tokyo Elevator puzzle

The following ‘Elevator puzzle’ is taken from ‘The Tokyo Puzzles’ by Kojon Fujimura, first published in Japanese in 1969 and subsequently translated into English. An eight-storey building is equipped with three elevators. All elevators call at both top and bottom floors and four floors in between. It is possible to find a configuration of these ‘in between’ floors so that it is possible to travel between any two floors by riding a single elevator. The question is, if each lift called at three floors in between top and bottom, how many lifts would be required such that any two floors are connected by a single elevator?