Why are all highly composite numbers > 12 multiples of 12?

The highly composite numbers are:

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, etc.

My son (10) asked me this question while we were walking our dog. My first thought was that every HCN is probably a multiple of all of the smaller ones. Works in the sense that 12 is a multiple of 1 and of 2 and of 4 and of 6. But it almost immediately stops working. 36 is not a multiple of 24.

So, what happens at 12, so that every greater HCN is a multiple of 12, but stops happening at 24 so that larger HCNs aren't a multiple of 24?