Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
3304 conjugacy classes of subgroups; only the 723 normal subgroups are shown.
This group has 3304 conjugacy classes of subgroups — too many to draw. Only its 723 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 | #555 |
| 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. C33×C4×D4 | C2 | semidirect |
| 2. #11723979 | C2 | semidirect |
| 3. #11723971 | C2 | semidirect |
| 4. #11723967 | C2 | semidirect |
| 5. #11723933 | C2 | semidirect |
| 6. #11723926 | C2 | semidirect |
| 7. C4×C4⋊(C3⋊(C3⋊D3)) | C2 | semidirect |
| 8. #11723910 | C2 | semidirect |
| 26. C32×C4×D4 | D3 | semidirect |
| 78. C3×C4×D4 | C3⋊D3 | semidirect |
| 140. C4×D4 | C3⋊(C3⋊D3) | semidirect |
| 143. C33 | #555 | semidirect |
| 192. C32 | #5563 | semidirect |
| 225. C3 | #10596034 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×C4×D4 | 2 |
| #11723979 | 2 |
| #11723971 | 2 |
| #11723967 | 2 |
| #11723933 | 2 |
| #11723926 | 2 |
| C4×C4⋊(C3⋊(C3⋊D3)) | 2 |
| #11723910 | 2 |
| #10596034 | 3 |