Types of 2-factorisations of <{1,2,3,4}>_{9}.

Label   2-factor
    A   C3C3C3
    B   C5C4
    C   C6C3
    D   C9

The following nine types do not exist:
    AAAB   AAAC   AAAD   AABB   AABC   AABD   AACD   
    ABCC   BBBB   

The following types exist:

AAAA
  (0,1,2)(3,4,5)(6,7,8)
  (0,3,6)(1,4,7)(2,5,8)
  (0,4,8)(1,5,6)(2,3,7)
  (0,5,7)(1,3,8)(2,4,6)
AACC
  (0,1,2)(3,4,5)(6,7,8)
  (0,3,6)(1,4,7)(2,5,8)
  (0,4,2,6,5,7)(1,3,8)
  (0,5,1,6,4,8)(2,3,7)
AADD
  (0,1,2)(3,4,5)(6,7,8)
  (0,3,6)(1,4,7)(2,5,8)
  (0,4,2,3,7,5,6,1,8)
  (0,5,1,3,8,4,6,2,7)
ABBB
  (0,1,2)(3,4,5)(6,7,8)
  (0,3,1,4,6)(2,7,5,8)
  (0,4,2,6,5)(1,7,3,8)
  (1,5,2,3,6)(0,7,4,8)
ABBC
  (0,1,2)(3,4,5)(6,7,8)
  (0,3,1,4,6)(2,7,5,8)
  (0,5,2,4,7)(1,6,3,8)
  (1,5,6,2,3,7)(0,4,8)
ABBD
  (0,1,2)(3,4,5)(6,7,8)
  (0,3,1,4,6)(2,7,5,8)
  (0,4,7,1,8)(2,3,6,5)
  (0,5,1,6,2,4,8,3,7)
ABCD
  (0,1,2)(3,4,5)(6,7,8)
  (0,3,1,4,6)(2,7,5,8)
  (0,5,1,7,4,8)(2,3,6)
  (0,4,2,5,6,1,8,3,7)
ABDD
  (0,1,2)(3,4,5)(6,7,8)
  (0,3,1,4,6)(2,7,5,8)
  (0,4,2,5,6,3,7,1,8)
  (0,5,1,6,2,3,8,4,7)
ACCC
  (0,1,2)(3,4,5)(6,7,8)
  (0,3,6,1,4,7)(2,5,8)
  (0,4,6,2,3,8)(1,5,7)
  (1,3,7,2,4,8)(0,5,6)
ACCD
  (0,1,2)(3,4,5)(6,7,8)
  (0,3,1,6,4,7)(2,5,8)
  (0,4,8,1,5,6)(2,3,7)
  (0,5,7,1,4,2,6,3,8)
ACDD
  (0,1,2)(3,4,5)(6,7,8)
  (0,3,1,6,4,7)(2,5,8)
  (0,4,1,5,6,2,7,3,8)
  (0,5,7,1,8,4,2,3,6)
ADDD
  (0,1,2)(3,4,5)(6,7,8)
  (0,3,1,4,6,2,7,5,8)
  (0,4,2,5,6,1,8,3,7)
  (0,5,1,7,4,8,2,3,6)
BBBC
  (0,1,2,3,4)(5,6,7,8)
  (0,2,4,5,7)(1,6,3,8)
  (0,3,7,1,5)(2,6,4,8)
  (1,3,5,2,7,4)(0,6,8)
BBBD
  (0,1,2,3,4)(5,6,7,8)
  (0,2,4,5,7)(1,3,6,8)
  (0,3,5,1,6)(2,7,4,8)
  (0,5,2,6,4,1,7,3,8)
BBCC
  (0,1,2,3,4)(5,6,7,8)
  (0,2,5,3,6)(1,7,4,8)
  (0,3,7,2,6,8)(1,4,5)
  (1,3,8,2,4,6)(0,5,7)
BBCD
  (0,1,2,3,4)(5,6,7,8)
  (0,2,4,5,7)(1,3,6,8)
  (0,5,2,7,3,8)(1,4,6)
  (0,3,5,1,7,4,8,2,6)
BBDD
  (0,1,2,3,4)(5,6,7,8)
  (0,2,4,5,7)(1,3,6,8)
  (0,3,5,1,6,2,7,4,8)
  (0,5,2,8,3,7,1,4,6)
BCCC
  (0,1,2,3,4)(5,6,7,8)
  (0,2,4,1,5,7)(3,6,8)
  (0,6,2,5,4,8)(1,3,7)
  (1,6,4,7,2,8)(0,3,5)
BCCD
  (0,1,2,3,4)(5,6,7,8)
  (0,2,4,1,5,7)(3,6,8)
  (0,5,2,6,4,8)(1,3,7)
  (0,3,5,4,7,2,8,1,6)
BCDD
  (0,1,2,3,4)(5,6,7,8)
  (0,2,4,1,5,7)(3,6,8)
  (0,3,5,2,6,4,7,1,8)
  (0,5,4,8,2,7,3,1,6)
BDDD
  (0,1,2,3,4)(5,6,7,8)
  (0,2,4,1,5,7,3,6,8)
  (0,3,5,2,6,4,8,1,7)
  (0,5,4,7,2,8,3,1,6)
CCCC
  (0,1,2,3,4,5)(6,7,8)
  (0,2,4,6,1,7)(3,5,8)
  (0,4,7,3,1,8)(2,5,6)
  (1,4,8,2,7,5)(0,3,6)
CCCD
  (0,1,2,3,4,5)(6,7,8)
  (0,2,4,6,1,7)(3,5,8)
  (0,3,7,2,5,6)(1,4,8)
  (0,4,7,5,1,3,6,2,8)
CCDD
  (0,1,2,3,4,5)(6,7,8)
  (0,2,4,6,1,7)(3,5,8)
  (0,3,1,4,7,5,6,2,8)
  (0,4,8,1,5,2,7,3,6)
CDDD
  (0,1,2,3,4,5)(6,7,8)
  (0,2,4,1,6,3,7,5,8)
  (0,3,1,5,6,4,8,2,7)
  (0,4,7,1,8,3,5,2,6)
DDDD
  (0,1,2,3,4,5,6,7,8)
  (0,2,4,1,3,6,8,5,7)
  (0,3,5,1,7,2,8,4,6)
  (0,4,7,3,8,1,6,2,5)