Only if the filter's manufacturer permits that exact method. Different filter types and finishes have different care requirements, and some are not washable.
Understanding this situation
A greasy metal-looking filter is not automatically compatible with dishwasher detergent, while non-washable media should not be processed at all. This guide does not offer extractor repair or installation services.
A useful next step
Follow the filter's own instructions and consider the dishwasher's permitted load arrangement. Do not mix unknown heavily contaminated components with ordinary dishes as an experiment. If residue or damage results, report the actual item and treatment.
Manuals and supporting guidance
Model-specific instructions take priority. These references support the practical principles discussed here; they do not diagnose your appliance.