Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
4368 conjugacy classes of subgroups; only the 731 normal subgroups are shown.
This group has 4368 conjugacy classes of subgroups — too many to draw. Only its 731 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C33×C4×D4 | C2 |
| hypercenter | C4×D4 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #550 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#89076 | C2 | semidirect |
| 2. C33×C4×D4 | C2 | semidirect |
| 3. #11723981 | C2 | semidirect |
| 4. #11723971 | C2 | semidirect |
| 5. #11723934 | C2 | semidirect |
| 6. #11723925 | C2 | semidirect |
| 7. #11723915 | C2 | semidirect |
| 8. C4×#6308 | C2 | semidirect |
| 28. C32×C4×D4 | D3 | semidirect |
| 51. #6308 | D4 | semidirect |
| 80. C33×C4 | #41 | semidirect |
| 84. C3×C4×D4 | C3⋊D3 | semidirect |
| 140. C32×C4 | #235 | semidirect |
| 151. C4×D4 | C3⋊(C3⋊D3) | semidirect |
| 152. C33 | #550 | semidirect |
| 201. C3×C4 | #3828 | semidirect |
| 205. C32 | #5553 | semidirect |
| 239. C4 | #89076 | semidirect |
| 241. C3 | #10596024 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#89076 | 2 |
| C33×C4×D4 | 2 |
| #11723981 | 2 |
| #11723971 | 2 |
| #11723934 | 2 |
| #11723925 | 2 |
| #11723915 | 2 |
| C4×#6308 | 2 |
| #10596024 | 3 |