Example 1 Choose several colors from 6 different given colors to color a cube, with each face colored in one color and ensuring that any two adjacent faces sharing an edge are of different colors. How many different coloring schemes are there (if one of the two colored cubes can be rotated to match the coloring state of the other, the original coloring schemes are considered the same)?
Explanation: Clearly, the maximum number of colors used is 6, and the minimum is 3. Below, we use the grouping method to count.

(1) When using 6 colors, each color is used to paint one face. According to the problem, the first color can be rotated to the top. At this point, the bottom is one of the other 5 colors, giving 5 different coloring methods. The remaining 4 colors are used to paint the front, back, left, and right faces. We can choose one of these colors to paint the front face, and the back face can be one of the remaining 3 colors, giving 3 different coloring methods. The remaining 2 colors are used to paint the left and right faces, giving 2 different coloring methods. Therefore, the total number of different coloring methods is \(5 \times 3 \times 2 = 30\).

(2) When using 5 colors, we select 5 colors from 6, and then choose one of these 5 colors to paint two opposite faces, which can be rotated to the top and bottom, giving \(6 \times 5 = 30\) different coloring methods. The remaining 4 colors are used to paint the 4 side faces. We choose one of these colors to paint the front face, and the back face can be one of the remaining 3 colors, giving 3 different coloring methods. The remaining 2 colors are used to paint the left and right faces. Unlike in (1), at this point, we can rotate 180 degrees around the axis through the centers of the front and back faces, so there is only 1 coloring method. Therefore, the total number of different coloring methods is 90.

(3) When using 4 colors, we select 4 colors from 6, and then choose 2 of these 4 colors to paint two opposite faces. The number of different coloring methods is \(\mathrm{C}_{6}^{4} \mathrm{C}_{4}^{2} = 90\).

(4) When using 3 colors, the number of different coloring methods is \(\mathrm{C}_{5}^{3} = 20\).

In summary, the total number of different coloring methods is 230.