Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
450 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 450 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×#3713 |
| commutator, Fitting, cosocle | C33×C8 | C23 |
| Frattini, hypercenter | C8 | D33 |
| socle | C2×C33 | C2×D8 |
| hypocenter | C33 | #509 |
| LCS | G ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#63807 | C2 | semidirect |
| 2. C3×D3×D24 | C2 | semidirect |
| 3. #11722068 | C2 | semidirect |
| 4. #11722007 | C2 | semidirect |
| 5. #11722004 | C2 | semidirect |
| 6. #11722001 | C2 | semidirect |
| 7. C3×#63545 | C2 | semidirect |
| 13. C3×#3746 | C22 | semidirect |
| 16. C32×D24 | C22 | semidirect |
| 20. C3⋊(C3⋊(C3⋊C16)) | C22 | semidirect |
| 21. C3×C3⋊(C3⋊C16) | C22 | semidirect |
| 26. #63807 | D3 | semidirect |
| 27. D3×D24 | D3 | semidirect |
| 42. #63545 | D3 | semidirect |
| 69. #3746 | C2×D3 | semidirect |
| 84. C3×D24 | C2×D3 | semidirect |
| 85. C3×D24 | C2×D3 | semidirect |
| 99. C3⋊(C3⋊C16) | C2×D3 | semidirect |
| 213. D24 | D32 | semidirect |
| 268. C33 | #509 | semidirect |
| 384. C32 | #4911 | semidirect |
| 385. C32 | #5147 | semidirect |
| 386. C32 | #5193 | semidirect |
| 433. C3 | #10590630 | semidirect |
| 434. C3 | #10591024 | semidirect |
| 435. C3 | #10591031 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#63807 | 2 |
| C3×D3×D24 | 2 |
| #11722068 | 2 |
| #11722007 | 2 |
| #11722004 | 2 |
| #11722001 | 2 |
| C3×#63545 | 2 |
| #10591031 | 3 |
| #10591024 | 3 |
| #10590630 | 3 |