The present baseline of FCC-ee considers 400 MHz single-cell cavities for the Z-pole machine, 400 MHz four-cell cavities for the W and H modes of operation, complemented by 800 MHz five-cell cavities for the tt operation. In this presentation different higher order mode (HOM) damping schemes are compared for the four operating scenarios of FCC-ee. The longitudinal and transverse impedance of the HOM-damped cavities are compared with an impedance limit set by synchrotron radiation. Studying one cavity is usually not sufficient to fully characterize the behavior of cavities in a module. In a module composed of several cavities, some modes could arise which are not present in a single cavity. Furthermore, there is a difference between the amount of power absorbed by the HOM couplers in a single cavity and cavities in a module due to the effect of end beam pipes. Therefore, the HOM power for a single-cavity and four-cavity module with different HOM damping schemes is calculated as well and the power handling capabilities required for the respective schemes is determined.