the problem with this is that the suppression level will vary dramtically according to ammo selection, barrel length, action type, and suppressor design. it's really tough to make a call on this, especially since we'd need to know wether it was for a pistol of for a rifle.
if the weapon is shot with supersonic loads, the sonic crack will still be very loud, and suppression will not be very effective.
if the weapon has a very short barrel (pistol barrels 3 inches or shorter) the rounds may never become supersonic, but will lose a lot of downrange power from the slower velocities. this should be pretty quite though.
if the weapon has a ported barrel, to reduce supersonic ammo to subsonic levels, you'll lose precious velocity as well. this should be pretty quiet.
the best package for a quiet yet powerful .22 would be a bolt action shooting the Aguilla SSS round. those 60 grain bullets are very quiet, but hit with a lot of force, and are great for small game. you'll need a fifle with a 1 in 9 twist to properly stabilze the bullets. Aguilla SSS will feed in many semiautos, but because the case is so short, there will be some backblast from the chamber which will make the weapon uneccesarily loud. go with a maunally operated action.
if you want a quiet rifle but dont want the hassle of all the silencer crap, you could use a very long barreled .22 with aguailla SSS. if the barrel is long enough the powder will burn completely by the time the bullet exits the barrel. i imagine you'd need a barrel at least 24 inches long, and perhaps up to 28 inches (which will even silence some supersonic loads!)