Cycling works primarily the glutes, quads, hamstrings, groin muscles, and gastrocnemius, which direct the force of the quads and glutes to the pedal.
Other muscles used include arm strength for tackling steep climbs, core stability to transfer force through the legs and pedals, and calves, ankles, and feet.
Power is most efficient when put into the downstroke, and it is important to optimize your position and cadence to use a different set of muscles. Strength and conditioning exercises can help efficiency on the bike and reduce the risk of injury.