KAGRA is an underground interferometric gravitational wave detector which is currently being commissioned. This detector relies on high-performance vibration-isolation systems to suspend its key optical components. These suspensions come in four different configurations, of which the type-B is used for the beam splitter and signal recycling mirrors. The type-B suspension comprises the payload, three geometric anti-spring filters for vertical isolation and one inverted pendulum for horizontal isolation. The payload comprises the optic, its marionette and their recoil masses, which hold local displacement sensors and coil magnet actuators used for damping the resonant modes of oscillation of the suspension itself. The beam splitter version has a modified lower section to accommodate a wider optical component. The payload is also equipped with an optical lever, used to monitor and control the position of the suspended optics from the ground. All four suspensions have now been installed in vacuum chambers. We describe the mechanical, electrical and control design, and the measured performance compared to requirements.