Did you read Slip's response? You are missing $\varnothing$ and {1,5}, also the full set is {1,2,5}, not {1,2,3,4,5} ... 3 and 4 are not elements of the original set.
In general, if a set has n members, then it has $\displaystyle 2^n$ subsets, including the empty set and the entire set. Here n= 3 so there are $\displaystyle 2^3= 8$ subsets.