Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
3444 conjugacy classes of subgroups; only the 2268 normal subgroups are shown.
This group has 3444 conjugacy classes of subgroups — too many to draw. Only its 2268 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C33 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C33 |
| prime radical | C24×C33 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C33 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C33×#126 | C2 | direct (N × H) |
| 2. C2×C33×C4⋊C4 | C2 | semidirect |
| 4. C2×C33×C42 | C2 | semidirect |
| 5. C2×C32×#126 | C3 | direct (N × H) |
| 9. C33×C4⋊C4 | C22 | semidirect |
| 11. C33×C42 | C22 | semidirect |
| 12. C32×#126 | C2×C3 | direct (N × H) |
| 13. C2×C32×C4⋊C4 | C2×C3 | semidirect |
| 15. C2×C32×C42 | C2×C3 | semidirect |
| 23. C2×C3×#126 | C32 | direct (N × H) |
| 27. C32×C4⋊C4 | C22×C3 | semidirect |
| 29. C32×C42 | C22×C3 | semidirect |
| 36. C3×#126 | C2×C32 | direct (N × H) |
| 37. C2×C3×C4⋊C4 | C2×C32 | semidirect |
| 39. C2×C3×C42 | C2×C32 | semidirect |
| 47. C2×#126 | C33 | direct (N × H) |
| 49. C2×C33 | #126 | direct (N × H) |
| 54. C3×C4⋊C4 | C22×C32 | semidirect |
| 56. C3×C42 | C22×C32 | semidirect |
| 63. #126 | C2×C33 | direct (N × H) |
| 64. C2×C4⋊C4 | C2×C33 | semidirect |
| 66. C2×C42 | C2×C33 | semidirect |
| 67. C33 | C2×#126 | direct (N × H) |
| 76. C2×C32 | C3×#126 | direct (N × H) |
| 81. C4⋊C4 | C22×C33 | semidirect |
| 83. C42 | C22×C33 | semidirect |
| 90. C32 | C2×C3×#126 | direct (N × H) |
| 99. C2×C3 | C32×#126 | direct (N × H) |
| 107. C3 | C2×C32×#126 | direct (N × H) |
| 109. C2 | C33×#126 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×#126 | 2 |
| C2×C33×C4⋊C4 | 2 |
| C2×C33×C22⋊C4 | 2 |
| C2×C33×C42 | 2 |
| C2×C32×#126 | 3 |