| T | Name(s) | |G| | Parity | |CS9(G)| | Subfields | Other Representations | Resolvents |
|---|---|---|---|---|---|---|---|
| 1 | C(9)=9 | 9 = 32 | 1 | 9 | 3T1 | 3: 3T1 | |
| 2 | E(9)=3[x]3 | 9 = 32 | 1 | 9 | 3T1, 3T1, 3T1, 3T1 | 3: 3T1, 3T1, 3T1, 3T1 | |
| 3 | D(9)=9:2 | 18 = 2 · 32 | 1 | 1 | 3T2 | 18T5 | 2: 2T1 6: 3T2 |
| 4 | S(3)[x]3 | 18 = 2 · 32 | -1 | 3 | 3T1, 3T2 | 6T5, 18T3 | 2: 2T1 3: 3T1 6: 3T2, 6T1 |
| 5 | S(3)[1/2]S(3)=3^2:2 | 18 = 2 · 32 | 1 | 1 | 3T2, 3T2, 3T2, 3T2 | 18T4 | 2: 2T1 6: 3T2, 3T2, 3T2, 3T2 |
| 6 | 1/3[3^3]3 | 27 = 33 | 1 | 3 | 3T1 | 3: 3T1, 3T1, 3T1, 3T1 9: 9T2 | |
| 7 | E(9):3=[3^2]3 | 27 = 33 | 1 | 3 | 3T1 | 9T7, 9T7, 9T7 | 3: 3T1, 3T1, 3T1, 3T1 9: 9T2 |
| 8 | S(3)[x]S(3)=E(9):D_4 | 36 = 22 · 32 | -1 | 1 | 3T2, 3T2 | 6T9, 12T16, 18T9, 18T11, 18T11 | 2: 2T1, 2T1, 2T1 4: 4T2 6: 3T2, 3T2 12: 6T3, 6T3 |
| 9 | E(9):4 | 36 = 22 · 32 | 1 | 1 | 6T10, 6T10, 12T17, 12T17, 18T10 | 2: 2T1 4: 4T1 | |
| 10 | [3^2]S(3)_6 | 54 = 2 · 33 | 1 | 1 | 3T2 | 18T18 | 2: 2T1 3: 3T1 6: 3T2, 6T1 18: 6T5 |
| 11 | E(9):6=1/2[3^2:2]S(3) | 54 = 2 · 33 | 1 | 1 | 3T2 | 9T13, 18T20, 18T21, 18T22 | 2: 2T1 3: 3T1 6: 3T2, 6T1 18: 6T5 |
| 12 | [3^2]S(3) | 54 = 2 · 33 | -1 | 3 | 3T2 | 9T12, 9T12, 9T12, 18T24, 18T24, 18T24, 18T24 | 2: 2T1 6: 3T2, 3T2, 3T2, 3T2 18: 9T5 |
| 13 | E(9):D_6=[3^2:2]3=[1/2.S(3)^2]3 | 54 = 2 · 33 | -1 | 1 | 3T1 | 9T11, 18T20, 18T21, 18T22 | 2: 2T1 3: 3T1 6: 3T2, 6T1 18: 6T5 |
| 14 | M(9)=E(9):Q_8 | 72 = 23 · 32 | 1 | 1 | 12T47, 18T35, 18T35, 18T35 | 2: 2T1, 2T1, 2T1 4: 4T2 8: 8T5 | |
| 15 | E(9):8 | 72 = 23 · 32 | -1 | 1 | 12T46, 18T28 | 2: 2T1 4: 4T1 8: 8T1 | |
| 16 | E(9):D_8 | 72 = 23 · 32 | -1 | 1 | 6T13, 6T13, 12T34, 12T34, 12T35, 12T35, 12T36, 12T36, 18T34, 18T34, 18T36 | 2: 2T1, 2T1, 2T1 4: 4T2 8: 4T3 | |
| 17 | [3^3]3=3wr3 | 81 = 34 | 1 | 3 | 3T1 | 9T17, 9T17 | 3: 3T1, 3T1, 3T1, 3T1 9: 9T2 27: 9T7 |
| 18 | E(9):D_12=[3^2:2]S(3)=[1/2.S(3)^2]S(3) | 108 = 22 · 33 | -1 | 1 | 3T2 | 9T18, 18T51, 18T51, 18T55, 18T55, 18T56, 18T57, 18T57 | 2: 2T1, 2T1, 2T1 4: 4T2 6: 3T2, 3T2 12: 6T3, 6T3 36: 6T9 |
| 19 | E(9):2D_8 | 144 = 24 · 32 | -1 | 1 | 12T84, 18T68, 18T71, 18T73 | 2: 2T1, 2T1, 2T1 4: 4T2 8: 4T3 16: 8T8 | |
| 20 | [3^3]S(3)=3wrS(3) | 162 = 2 · 34 | -1 | 3 | 3T2 | 9T20, 9T20, 18T86, 18T86, 18T86 | 2: 2T1 3: 3T1 6: 3T2, 6T1 18: 6T5 54: 9T11 |
| 21 | 1/2.[3^3:2]S(3) | 162 = 2 · 34 | 1 | 1 | 3T2 | 9T21, 9T21, 18T88, 18T88, 18T88 | 2: 2T1 6: 3T2, 3T2, 3T2, 3T2 18: 9T5 54: 9T12 |
| 22 | [3^3:2]3 | 162 = 2 · 34 | -1 | 1 | 3T1 | 9T22, 9T22, 18T85, 18T85, 18T85 | 2: 2T1 3: 3T1 6: 3T2, 6T1 18: 6T5 54: 9T11 |
| 23 | E(9):2A_4 | 216 = 23 · 33 | 1 | 1 | 12T122 | 3: 3T1 12: 4T4 24: 8T12 | |
| 24 | [3^3:2]S(3) | 324 = 22 · 34 | -1 | 1 | 3T2 | 9T24, 9T24, 18T129, 18T129, 18T129, 18T136, 18T136, 18T136, 18T137, 18T137, 18T137 | 2: 2T1, 2T1, 2T1 4: 4T2 6: 3T2, 3T2 12: 6T3, 6T3 36: 6T9 108: 9T18 |
| 25 | [1/2.S(3)^3]3 | 324 = 22 · 34 | 1 | 1 | 3T1 | 12T132, 12T132, 12T133, 18T141, 18T141, 18T142, 18T143 | 3: 3T1 12: 4T4 |
| 26 | E(9):2S_4 | 432 = 24 · 33 | -1 | 1 | 12T157, 18T157 | 2: 2T1 6: 3T2 24: 4T5 48: 8T23 | |
| 27 | L(9)=PSL(2,8) | 504 = 23 · 32 · 7 | 1 | 1 | |||
| 28 | [S(3)^3]3=S(3)wr3 | 648 = 23 · 34 | -1 | 1 | 3T1 | 12T176, 18T197, 18T197, 18T198, 18T198, 18T202, 18T204, 18T206, 18T207 | 2: 2T1 3: 3T1 6: 6T1 12: 4T4 24: 6T6 |
| 29 | [1/2.S(3)^3]S(3) | 648 = 23 · 34 | -1 | 1 | 3T2 | 12T175, 18T219, 18T220, 18T223, 18T224 | 2: 2T1 6: 3T2 24: 4T5 |
| 30 | 1/2[S(3)^3]S(3) | 648 = 23 · 34 | 1 | 1 | 3T2 | 12T177, 12T177, 12T178, 18T217, 18T218, 18T221, 18T222 | 2: 2T1 6: 3T2 24: 4T5 |
| 31 | [S(3)^3]S(3)=S(3)wrS(3) | 1296 = 24 · 34 | -1 | 1 | 3T2 | 12T213, 18T300, 18T303, 18T311, 18T312, 18T314, 18T315, 18T319, 18T320 | 2: 2T1, 2T1, 2T1 4: 4T2 6: 3T2 12: 6T3 24: 4T5 48: 6T11 |
| 32 | L(9):3=P|L(2,8) | 1512 = 23 · 33 · 7 | 1 | 1 | 3: 3T1 | ||
| 33 | A9 | 181440 = 26 · 34 · 5 · 7 | 1 | 1 | |||
| 34 | S9 | 362880 = 27 · 34 · 5 · 7 | -1 | 1 | 18T887 | 2: 2T1 |