The 8 valid combinations of 3 numbers that add to 15 are:

• 9 5 1
• 7 5 3
• 2 5 8
• 4 5 6
• 2 9 4
• 6 1 8
• 6 7 2
• 8 3 4

All 8 of those combinations need to appear in the square as a row, column or diagonal. The centre cell must appear in the middle row, middle column and both diagonals. So it must be a number that appears four times, and the only digit that does is 5. So 5 must be the centre.

Similarly, each of the corner pieces must form part of a row, a column and a diagonal. So each corner cell must be a number that appears 3 times. Those are the even numbers 2, 4, 6 and 8. That means the diagonals must be "2 5 8" and "4 5 6"

2 _ 4
_ 5 _
6 _ 8

That leaves the middle edge cells, each of which needs to appear in a row and a column. These are the odd numbers 1, 3, 7 and 9. So the middle row and column must be "9 5 1" and "7 5 3". Just insert these into the above such that the rows and columns add up correctly, leaving you with:

2 9 4
7 5 3
6 1 8

From that you can take the mirror images (horizontally and vertically) and the rotation of each.

Mirror images:

4 9 2 | 2 9 4
3 5 7 | 7 5 3
8 1 6 | 6 1 8
-------------
8 1 6 | 6 1 8
3 5 7 | 7 5 3
4 9 2 | 2 9 4

Rotations:

8 3 4 | 4 3 8
1 5 9 | 9 5 1
6 7 2 | 2 7 6
-------------
6 7 2 | 2 7 6
1 5 9 | 9 5 1
8 3 4 | 4 3 8