"Can I put ... alto mouthpiece in the ... dishwasher?" No, for two reasons. The detergent is extremely harsh and incompatible with the hard rubber most woodwind mouthpieces are made of (even metal mouthpieces are affected). The water temperature in a dishwasher is generally raised by heating elements, often high enough to partly melt a mouthpiece, which even very hot tap water has been known to do.
Wash the mouthpiece with warm water and ordinary hand-wash dish soap. Don't push or pull anything into the mouthpiece requiring force to get through, be very cautious with the tip, and avoid scratching the inside. Rinse well, then spray with Sterisol germicide or 99% alcohol (propyl, isopropyl, or ethyl), wait a minute, and wipe dry.