It is an old story. Just about any type of ship besides a Ohio Class Battleship or Nimitz class supercarrier is sinkable with two or three 250 Kg warheads. The issue of course is how to deliver those warheads. The chinese have some very quite (In electric mode) diesel/electric submarines that have penetrated CBG's before. It would be a suicide mission, since they would have no chance of surviving afterward, but it would be doable.
Sit at the bottom of a straight somewhere, wait until carrier is overhead, fire of torpedoes until the ASW craft spot you. Then maybe pray to your respective deities/party leaders. Fairly simple.
Also, mini-subs towing large charges/mines would work well while the vessel was in port.
I think war games in the 80's showed that so long as the bears fired enough "kitchen" cruise missiles, 1-2 was bound to get through out of 50 or so. Countermeasures have since improved, but so have cruise missiles.
Long story short, carriers are actually very vulnerable, assuming that you are willing to take heavy losses, and that your enemy is unsuspecting.
During wartime, different story. CBG commanders are not stupid enough to take their prized possession into enemy infested straights, and the local air dominance implied by the presence of the CBG means that getting 30-40 heavy bombers in range to fire a barrage of cruise missiles is probably not an option.
So, what to do? Well, you do what the USN did at midway, you set a trap. You provide a nice juicy target that looks mostly undefended, then you send in waves of fighters, bombers, and submarines to shock your way through most of the defenses.
Of course, the US had already broken Japanese Naval Codes, while the Japanese had not returned the favor. So, if you have information dominance, I think sinking a Carrier is doable. That or a surprise attack. Otherwise, you can assume that the CBG C.O is not going to risk his carrier unless he stands to make a substantial return.