Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
11164 conjugacy classes of subgroups; only the 6330 normal subgroups are shown.
This group has 11164 conjugacy classes of subgroups — too many to draw. Only its 6330 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C25×C32 | D3 |
| commutator, hypocenter | C3 | C24×C32×C4 |
| Frattini | C2 | C24×C32×D3 |
| Fitting, socle, prime radical | C25×C33 | C2 |
| cosocle | C2×C3 | C25×C32 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C25×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×C32×Dic3 | C2 | direct (N × H) |
| 4. C24×C3×Dic3 | C3 | direct (N × H) |
| 5. C24×C33 | C4 | semidirect |
| 7. C22×C32×Dic3 | C22 | direct (N × H) |
| 12. C23×C3×Dic3 | C2×C3 | direct (N × H) |
| 13. C23×C33 | C2×C4 | semidirect |
| 15. C2×C32×Dic3 | C23 | direct (N × H) |
| 17. C24×Dic3 | C32 | direct (N × H) |
| 20. C24×C32 | Dic3 | direct (N × H) |
| 21. C24×C32 | C3×C4 | semidirect |
| 25. C22×C3×Dic3 | C22×C3 | direct (N × H) |
| 26. C22×C33 | C22×C4 | semidirect |
| 28. C32×Dic3 | C24 | direct (N × H) |
| 33. C23×Dic3 | C2×C32 | direct (N × H) |
| 36. C23×C32 | C2×Dic3 | direct (N × H) |
| 37. C23×C32 | C2×C3×C4 | semidirect |
| 41. C2×C3×Dic3 | C23×C3 | direct (N × H) |
| 43. C2×C33 | C23×C4 | semidirect |
| 47. C24×C3 | C3×Dic3 | direct (N × H) |
| 48. C24×C3 | C32×C4 | semidirect |
| 52. C22×Dic3 | C22×C32 | direct (N × H) |
| 55. C22×C32 | C22×Dic3 | direct (N × H) |
| 56. C22×C32 | C22×C3×C4 | semidirect |
| 60. C3×Dic3 | C24×C3 | direct (N × H) |
| 63. C33 | C24×C4 | semidirect |
| 66. C23×C3 | C2×C3×Dic3 | direct (N × H) |
| 67. C23×C3 | C2×C32×C4 | semidirect |
| 71. C2×Dic3 | C23×C32 | direct (N × H) |
| 74. C2×C32 | C23×Dic3 | direct (N × H) |
| 75. C2×C32 | C23×C3×C4 | semidirect |
| 78. C24 | C32×Dic3 | direct (N × H) |
| 83. C22×C3 | C22×C3×Dic3 | direct (N × H) |
| 84. C22×C3 | C22×C32×C4 | semidirect |
| 88. Dic3 | C24×C32 | direct (N × H) |
| 90. C32 | C24×Dic3 | direct (N × H) |
| 91. C32 | C24×C3×C4 | semidirect |
| 92. C23 | C2×C32×Dic3 | direct (N × H) |
| 97. C2×C3 | C23×C3×Dic3 | direct (N × H) |
| 98. C2×C3 | C23×C32×C4 | semidirect |
| 101. C22 | C22×C32×Dic3 | direct (N × H) |
| 105. C3 | C24×C3×Dic3 | direct (N × H) |
| 106. C3 | C24×C32×C4 | semidirect |
| 107. C2 | C23×C32×Dic3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×C33 | 2 |
| C23×C32×Dic3 | 2 |
| C24×C32×C4 | 3 |
| C24×C3×Dic3 | 3 |