Divisors with Balanced Residues Mod 3

Find the number of odd positive integers $n < 1000$ such that if $S_n$ denotes the set of divisors of $n$, then exactly half of the elements of $S_n$ are divisible by 3, exactly a third of the elements of $S_n$ leave a remainder of 1 upon dividing by 3, and exactly a sixth of the elements of $S_n$ leave a remainder of 2 upon dividing by 3.