Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
563 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 563 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#88434 | C2 | semidirect |
| 3. C4⋊((C3⋊(C3⋊C8))⋊C3) | C2 | semidirect |
| 4. C4×(C3⋊(C3⋊C8))⋊C3 | C2 | semidirect |
| 5. #11720323 | C2 | semidirect |
| 22. C4⋊(C3⋊(C3⋊C8)) | D3 | semidirect |
| 23. C4×C3⋊(C3⋊C8) | D3 | semidirect |
| 33. C4⋊#272 | C8 | semidirect |
| 38. (C3⋊(C3⋊C8))⋊C3 | D4 | semidirect |
| 68. C2×#272 | C2×C8 | semidirect |
| 70. C4×H(3,𝔽3) | C2×C8 | semidirect |
| 116. C3⋊(C3⋊C8) | D12 | semidirect |
| 175. C32×C4 | D3×C8 | semidirect |
| 176. C32×C4 | D3×C8 | semidirect |
| 189. H(3,𝔽3) | C8×D4 | semidirect |
| 289. C32 | C8×D12 | semidirect |
| 290. C32 | #4834 | semidirect |
| 324. C4 | #88434 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×C4⋊#272 | 2 |
| C2×#88434 | 2 |
| C4⋊((C3⋊(C3⋊C8))⋊C3) | 2 |
| C4×(C3⋊(C3⋊C8))⋊C3 | 2 |
| #11720323 | 2 |
| #10590329 | 3 |
| C3⋊C8×D12 | 3 |