A solution was offered by Andrew (no surname or school given).

Andrew has tried to explain his reasoning and break the problem down, although I think you will find that there are some more winning lines. Can you find them? A good start Andrew, thank you.

There are 6 distinct winning lines, which cannot be obtained one from any other by rotation or reflection.

These are arranged as follows:

A) The edge of the cube.
B) The diagonal of the face of the cube
C) The middle line of a face of the cube, this means a line that joins two points situated at the middle of two sides opposed to each other, on one face of the cube.
D) The diagonal of the middle plane of the cube
E) The middle line of the middle plane
F) The main diagonal of the cube

Another problem of interest could be to count how many lines are of each type:

A) there are 12 edges of the cube
B) there are 6 faces, and 2 diagonals of each face, consequently 12 lines
C) there are 6 faces, two middle lines for each, 12 lines
D) there are two middle planes, 4 diagonal lines
E) there are two middle planes, middle lines
F) there are two main diagonals

There are 44 winning lines.