A powershift transmission with a dual-clutch main transmission and a downstream planetary gearset transmission. The main transmission has first and second partial transmissions. Each partial transmission has an input shaft, an interlocking shifting element, an independent frictional powershift clutch, and a common output shaft. The downstream transmission has an input shaft which can be connected, via a third interlocking shifting element, to the common output shaft which is fixedly connected to a first planetary gearset element. The downstream transmission has an output shaft, which is connected to a second planetary gearset element. A third planetary gearset element can be coupled to a housing by a fourth interlocking shifting element or to the output shaft of the downstream transmission by a fifth interlocking shifting element. The common output shaft can be coupled by a sixth interlocking shifting element to the third planetary gearset element.