Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
784 conjugacy classes of subgroups; only the 420 normal subgroups are shown.
This group has 784 conjugacy classes of subgroups — too many to draw. Only its 420 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C33×C4 | D8 |
| commutator | C8 | C2×C33×C4 |
| Frattini, cosocle | C2×C8 | C22×C33 |
| socle | C2×C33 | #102 |
| prime radical | C33×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C33×C4 ◁ C2×C33×C4 ◁ C2×C33×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C33×C16 | C2 | semidirect |
| 3. C33×#108 | C2 | semidirect |
| 4. C32×#359 | C3 | direct (N × H) |
| 13. C2×C32×C16 | C2×C3 | semidirect |
| 14. C32×#108 | C2×C3 | semidirect |
| 23. C3×#359 | C32 | direct (N × H) |
| 37. C2×C3×C16 | C2×C32 | semidirect |
| 38. C3×#108 | C2×C32 | semidirect |
| 47. #359 | C33 | direct (N × H) |
| 64. C2×C16 | C2×C33 | semidirect |
| 65. #108 | C2×C33 | semidirect |
| 66. C33 | #359 | direct (N × H) |
| 90. C32 | C3×#359 | direct (N × H) |
| 107. C3 | C32×#359 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×#135 | 2 |
| C2×C33×C16 | 2 |
| C33×#108 | 2 |
| C32×#359 | 3 |