One occupant was trapped and had to be cut from the taxi, which crashed around 2am, while paramedics treated five other occupants at the scene.
Police are trying to piece together how the crash happened, with questions over why six people were in the white sedan at the time.
Two 16-year-old girls and an 18-year-old woman were taken to the Royal Adelaide Hospital and a 14-year-old boy and 26-year-old man were taken to the Flinders Medical Centre.
The driver, a 20-year-old man, was also taken to the Flinders Medical Centre with minor injuries.
9News understands nobody suffered life-threatening injuries and that the driver was off-duty at the time of the smash.
South Australia Police said the driver was subjected to mandatory blood tests.