18. C.

Notice that, the sum of the numbers on each face is equal, and the total sum of the numbers is
$$
1+2+3+4+5+6+7+8=36 \text {. }
$$

Therefore, the sum of the four numbers on each face is 18, and the sum of the numbers at the two vertices of non-coplanar parallel edges (opposite edges) is equal.

Consider the extreme case: If 1 and 8 are not on the same edge, then 1 and 8 are on opposite edges.

Let the other two vertices on the opposite edges be filled with $x, y(x, y \in \{2,3, \cdots, 7\})$.
Thus, $1+x=8+y$
$\Rightarrow 7=8-1=x-y \leqslant 7-2=5$.
Contradiction.
Therefore, 1 and 8 must be on the same edge.
Thus, the sum of the numbers at the two vertices on the other edge is also 9.
Hence, the pairs of numbers on the edges are $1-8$, $2-7,3-6,4-5$.

Since the sum of the four numbers on each face is 36, $(8,2,3,5)$ is on one face, and $(1,7,6,4)$ is on another face, as shown in Figure 6.
18. Label each vertex of a cube with one of the integers $1 \sim 8$, with each integer used only once, and the sum of the four numbers on each face being equal. Then there are ( ) different arrangements (if two arrangements can be made to coincide by rotation, they are considered the same arrangement).
(A) 1
(B) 3
(C) 6
(D) 12
(E) 24