Many thousands of individual private landowners. The only main island which is close to being owned by a single interest is Sark, much of which is owned by the Barclay brothers (who entirely own the conjoined island of Brecqhou).
If you meant "which country owns the Channel Islands?", the answer is "none". Each is in effect an autonomous micro-state with its own government, such as the States of Jersey, but the patterns are not completely uniform. The CI are linked to the United Kingdom because the British monarch is also the Duke of Normandy, but the CI are not part of the UK, or of the European Union, and have independent fiscal and legal systems.