Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
4116 conjugacy classes of subgroups; only the 891 normal subgroups are shown.
This group has 4116 conjugacy classes of subgroups — too many to draw. Only its 891 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 | C2×C33×C4⋊C4 | C2 |
| hypercenter | C2×C4⋊C4 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #537 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#89076 | C2 | semidirect |
| 2. C2×C33×C4⋊C4 | C2 | semidirect |
| 3. #11723971 | C2 | semidirect |
| 4. #11723938 | C2 | semidirect |
| 5. #11723936 | C2 | semidirect |
| 19. C2×C32×C4⋊C4 | D3 | semidirect |
| 26. C2×C33×C4 | D4 | semidirect |
| 58. C2×C3×C4⋊C4 | C3⋊D3 | semidirect |
| 65. C2×C32×C4 | D12 | semidirect |
| 105. C2×C4⋊C4 | C3⋊(C3⋊D3) | semidirect |
| 106. C33 | #537 | semidirect |
| 110. C2×C3×C4 | C4⋊(C3⋊D3) | semidirect |
| 145. C32 | #5510 | semidirect |
| 155. C2×C4 | C4⋊(C3⋊(C3⋊D3)) | semidirect |
| 173. C3 | #10595981 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#89076 | 2 |
| C2×C33×C4⋊C4 | 2 |
| #11723971 | 2 |
| #11723938 | 2 |
| #11723936 | 2 |
| #10595981 | 3 |