There are 49 winning lines.
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From a vertex there are $7$ other vertices that you can join to in order to make a winning line. $7 \times 8 = 56$ lines, but this counts each line from both ends, so there are $28$ 'vertex' winning lines.
From the middle of an edge there are $3$ other middles-of-edges that you can join to in order to make a winning line. $3 \times 12 = 36$ lines, but this counts each line from both ends so there are $18$ 'middle of edge' winning lines.
From the centre of each face there is one winning line, joining to the opposite face, so there are $3$ 'centre of face' winning lines.
So in total, there are $28 + 18 + 3 = 49$ winning lines.
There are $12$ edges on a cube so there are $12$ winning lines along edges.
There are $6$ faces on a cube, and $4$ winning lines that pass through the middle of each face, so there are $24$ winning lines through the middle of faces.
Finally we need to consider the winning lines that go through the centre cube:
vertex to opposite vertex: $4$
middle of edge to middle of opposite edge: $6$
middle of face to middle of opposite face: $3$
In total, there are $12 + 24 + 4 + 6 + 3 = 49$ winning lines.
On a plane there are $8$ winning lines.
In the cube, there are $3$ horizontal planes, so $8 \times 3 = 24$ winning lines.
There are also $3$ vertical planes going from left to right, but now with only $5$ new winning lines per plane, as the $3$ horizontal lines have already been counted. So $5 \times 3 = 15$ winning lines.
On the $3$ vertical planes going from front to back, we now only have $2$ new (diagonal) winning lines per plane. So $2 \times 3 = 6$ winning lines.
Finally, there are also diagonal planes to consider. There are $4$ winning lines going from corner to diagonally opposite corner.
In total, there are $24 + 15 + 6 + 4 = 49$ winning lines.