Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
456 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 456 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×#3714 |
| commutator, Fitting, cosocle | C33×C8 | C23 |
| Frattini, hypercenter | C8 | D33 |
| socle | C2×C33 | C2×D8 |
| prime radical | C3×#63807 | C2 |
| hypocenter | C33 | #510 |
| 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. D3×#3746 | C2 | semidirect |
| 3. #11722069 | C2 | semidirect |
| 4. #11722015 | C2 | semidirect |
| 5. #11722009 | C2 | semidirect |
| 6. C3×#63548 | C2 | semidirect |
| 7. C3×#63543 | C2 | semidirect |
| 44. #63548 | D3 | semidirect |
| 45. #63543 | D3 | semidirect |
| 274. C33 | #510 | semidirect |
| 387. C32 | #4909 | semidirect |
| 388. C32 | #4915 | semidirect |
| 389. C32 | #5195 | semidirect |
| 427. C3 | #10590239 | semidirect |
| 428. C3 | #10590638 | semidirect |
| 429. C3 | #10591028 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D3×#3746 | 2 |
| C3×#63807 | 2 |
| #11722069 | 2 |
| #11722015 | 2 |
| #11722009 | 2 |
| C3×#63548 | 2 |
| C3×#63543 | 2 |
| #10591028 | 3 |
| #10590638 | 3 |
| #10590239 | 3 |