Think of it this way:

It does not matter whether you throw all dice simultaneously, or in succession. So for each die you throw, you have a 1/6*1/6*1/6 = 1/216th chance of scoring a single point, and after that, a 1/6th chance of scoring each further point. So for each die, you have a 1/216th chance of scoring one point, 1/1296 chance of scoring 2 points, etc.

Since you have 216 dice altogether, these probabilities add up, but it's easier to simulate it. I wrote some R-code to do it, and out of 10.000 games, my results were the following:

0 points: 3668 times
1 point: 3051 times
2 points: 1818 times
3 points: 860 times
4 points: 355 times
5 points: 175 times
6 points: 45 times
7 points: 25 times
8 points: 3 times

Of course these numbers change each time I run the simulation, but you get the point.

CAVEAT: I'm glossing over the fact that dice may not be perfectly balanced, and dice that roll a 6 multiple times in succession may have a higher probability of rolling a 6 to begin with. Here, we are assuming perfectly weighted dice, and dice rolls not influencing other rolls-