Probability of Triples in Dice

I have been trying to compute the probability of throwing one triple of say sixes in a throw of 4 dice.

I figure that a triple of sixes with 3 dice is 1 (event) out of 216 (outcomes)--0.004629

I know that adding one dice to the throw, making 4 dice total, must increase the probability of throwing at least 3 sixes (a triple). However, when I do the computation, I come up with 4 out of 1296--0.003086

I am asking someone to show me the error in my ways so I can figure this out properly.

Thanks in advance.