Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
19068 conjugacy classes of subgroups; only the 3611 normal subgroups are shown.
This group has 19068 conjugacy classes of subgroups — too many to draw. Only its 3611 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C2×C3⋊(C3⋊D3) |
| commutator, cosocle | C2×C33 | C25 |
| Frattini | C2 | C24×C3⋊(C3⋊D3) |
| Fitting | C25×C33 | C2 |
| socle | C24×C33 | C22 |
| hypercenter | C25 | C3⋊(C3⋊D3) |
| hypocenter | C33 | C23×D4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C24 ◁ C25 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C25×C33 | C2 | semidirect |
| 2. C24×C3⋊(C3⋊D3) | C2 | semidirect |
| 3. C22×#6312 | C2 | direct (N × H) |
| 4. C23×C3⋊(C3⋊Dic3) | C2 | semidirect |
| 8. C24×C33 | C22 | semidirect |
| 10. C23×C3⋊(C3⋊D3) | C22 | semidirect |
| 11. C2×#6312 | C22 | direct (N × H) |
| 12. C22×C3⋊(C3⋊Dic3) | C22 | semidirect |
| 13. C25×C32 | D3 | semidirect |
| 18. C23×C33 | D4 | semidirect |
| 20. C23×C33 | C23 | semidirect |
| 22. C22×C3⋊(C3⋊D3) | C23 | semidirect |
| 23. #6312 | C23 | direct (N × H) |
| 24. C2×C3⋊(C3⋊Dic3) | C23 | semidirect |
| 28. C24×C32 | C2×D3 | semidirect |
| 34. C22×C33 | C2×D4 | semidirect |
| 36. C22×C33 | C24 | semidirect |
| 38. C2×C3⋊(C3⋊D3) | C24 | semidirect |
| 39. C3⋊(C3⋊Dic3) | C24 | semidirect |
| 40. C25×C3 | C3⋊D3 | semidirect |
| 45. C23×C32 | #67 | semidirect |
| 47. C23×C32 | C22×D3 | semidirect |
| 54. C2×C33 | C22×D4 | semidirect |
| 59. C24×C3 | C2×C3⋊D3 | semidirect |
| 65. C22×C32 | C2×#67 | semidirect |
| 67. C22×C32 | C23×D3 | semidirect |
| 72. C25 | C3⋊(C3⋊D3) | semidirect |
| 75. C33 | C23×D4 | semidirect |
| 77. C23×C3 | #638 | semidirect |
| 79. C23×C3 | C22×C3⋊D3 | semidirect |
| 85. C2×C32 | C22×#67 | semidirect |
| 92. C24 | C2×C3⋊(C3⋊D3) | semidirect |
| 96. C22×C3 | C2×#638 | semidirect |
| 98. C22×C3 | C23×C3⋊D3 | semidirect |
| 102. C32 | C23×#67 | semidirect |
| 106. C23 | #6312 | direct (N × H) |
| 108. C23 | C22×C3⋊(C3⋊D3) | semidirect |
| 112. C2×C3 | C22×#638 | semidirect |
| 118. C22 | C2×#6312 | direct (N × H) |
| 120. C22 | C23×C3⋊(C3⋊D3) | semidirect |
| 122. C3 | C23×#638 | semidirect |
| 125. C2 | C22×#6312 | 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 |
| C24×C3⋊(C3⋊D3) | 2 |
| C22×#6312 | 2 |
| C23×C3⋊(C3⋊Dic3) | 2 |
| C23×#638 | 3 |