Sums of subsets

Probability Level 5

For a set of numbers $T,$ we say that $T$ has distinct subset sums if all distinct subsets of $T$ have distinct sums. How many subsets of $\{1,2,3,4,5,6,7,8\}$ have distinct subset sums?

Details and assumptions

The empty set (the set of no elements) has a sum of 0 by convention.

As an explicit example, the subset $\{1, 2 \}$ satisfies the conditions, since it has $2^2 = 4$ subsets, whose sums are 0, 1, 2, and 3, which are distinct.