Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
1132 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 1132 conjugacy classes of subgroups — too many to draw. Only its 238 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | D3×A4 |
| commutator, cosocle, hypocenter | C22×C3 | C22×C33 |
| Frattini | C1 | G |
| Fitting, socle | C23×C33 | C2×C3 |
| LCS | G ▷ C22×C3 | |
| UCS | C1 ◁ C2×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C33×A4 | C2 | semidirect |
| 2. C32×D3×A4 | C2 | direct (N × H) |
| 3. C23×C32×D3 | C3 | semidirect |
| 5. C2×C3×D3×A4 | C3 | direct (N × H) |
| 7. C33×A4 | C22 | semidirect |
| 8. C23×C33 | C2×C3 | semidirect |
| 12. C22×C32×D3 | C2×C3 | semidirect |
| 16. C2×C32×A4 | D3 | direct (N × H) |
| 17. C2×C32×A4 | C2×C3 | semidirect |
| 18. C3×D3×A4 | C2×C3 | direct (N × H) |
| 22. C23×C3×D3 | C32 | semidirect |
| 24. C2×D3×A4 | C32 | direct (N × H) |
| 28. C22×C33 | C22×C3 | semidirect |
| 35. C2×C32×D3 | A4 | direct (N × H) |
| 38. C32×A4 | C2×D3 | direct (N × H) |
| 39. C32×A4 | C22×C3 | semidirect |
| 46. C23×C32 | C3×D3 | semidirect |
| 47. C23×C32 | C2×C32 | semidirect |
| 51. C22×C3×D3 | C2×C32 | semidirect |
| 55. C2×C3×A4 | C3×D3 | direct (N × H) |
| 56. C2×C3×A4 | C2×C32 | semidirect |
| 57. D3×A4 | C2×C32 | direct (N × H) |
| 65. C2×C33 | C2×A4 | semidirect |
| 68. C32×D3 | C2×A4 | direct (N × H) |
| 70. C23×D3 | C33 | semidirect |
| 86. C22×C32 | C2×C3×D3 | semidirect |
| 87. C22×C32 | C22×C32 | semidirect |
| 94. C2×C3×D3 | C3×A4 | direct (N × H) |
| 97. C3×A4 | C2×C3×D3 | direct (N × H) |
| 98. C3×A4 | C22×C32 | semidirect |
| 103. C33 | C22×A4 | semidirect |
| 109. C23×C3 | C32×D3 | semidirect |
| 110. C23×C3 | C2×C33 | semidirect |
| 114. C22×D3 | C2×C33 | semidirect |
| 117. C2×A4 | C32×D3 | direct (N × H) |
| 132. C2×C32 | D3×A4 | direct (N × H) |
| 133. C2×C32 | C2×C3×A4 | semidirect |
| 136. C3×D3 | C2×C3×A4 | direct (N × H) |
| 152. C22×C3 | C2×C32×D3 | semidirect |
| 153. C22×C3 | C22×C33 | semidirect |
| 159. C2×D3 | C32×A4 | direct (N × H) |
| 161. A4 | C2×C32×D3 | direct (N × H) |
| 167. C32 | C2×D3×A4 | direct (N × H) |
| 168. C32 | C22×C3×A4 | semidirect |
| 173. C23 | C33×D3 | semidirect |
| 186. C2×C3 | C3×D3×A4 | direct (N × H) |
| 187. C2×C3 | C2×C32×A4 | semidirect |
| 189. D3 | C2×C32×A4 | direct (N × H) |
| 198. C22 | C2×C33×D3 | semidirect |
| 202. C3 | C2×C3×D3×A4 | direct (N × H) |
| 203. C3 | C22×C32×A4 | semidirect |
| 208. C2 | C32×D3×A4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C33×A4 | 2 |
| C32×D3×A4 | 2 |
| C23×C32×D3 | 3 |
| C22×C32×A4 | 3 |
| C2×C3×D3×A4 | 3 |
| C2×C33×D3 | 4 |