6. Among the 8 vertices, 12 midpoints of edges, 6 centers of faces, and 1 center of the cube, these 27 points, the number of "6-point coplanar groups" is ( ).
(A) 1320
(B) 1326
(C) 1332
(D) 1336
6.C.

As shown in Figure 6, let the plane containing the 6-point set be $\alpha$. We will count the number of 6-point sets in the same plane $\alpha$ as follows:
(1) If $\alpha$ is a side face (such as $ABCD$), there are 6 side faces, and each side face has 9 points, forming a total of $6 C_{9}^{6}$ 6-point sets.
(2) If $\alpha$ is a mid-section (such as $P_{9} P_{10} P_{11} P_{12}$), there are 3 mid-sections, and each mid-section has 9 points, forming a total of $3 C_{9}^{6}$ 6-point sets.
(3) If $\alpha$ is a diagonal face (such as $AC C_{1} A_{1}$), there are 6 diagonal faces, and each face has 9 points, forming a total of $6 C_{9}^{6}$ 6-point sets.
(4) If $\alpha$ is a central hexagonal oblique section (such as $P_{5} P_{6} P_{11} P_{3} P_{4} P_{9}$), it passes through the center $O$ of the body, and there are 4 such central hexagonal oblique sections. Each such $\alpha$ has 7 points, forming a total of $4 C_{7}^{6}$ 6-point sets.
(5) If $\alpha$ is a triangular oblique section (such as $A B_{1} C$), there are 8 such sections, and each such $\alpha$ has 6 points, forming a total of $8 C_{6}^{6}$ 6-point sets.
(6) If $\alpha$ is a rectangular oblique section (such as $A_{1} B_{1} P_{6} P_{8}$), there are $12 \times 2=24$ such sections, and each such $\alpha$ has 6 points, forming a total of $24 C_{6}^{6}$ 6-point sets.
(7) If $\alpha$ is a rectangular biased section (such as $P_{1} P_{2} P_{6} P_{5}$), there are 12 such sections, and each such $\alpha$ has 6 points, forming a total of $12 C_{6}^{6}$ 6-point sets. In summary, the number of 6-point sets in the same plane is $(6+3+6) C_{9}^{6}+28+8+24+12=1332$.