Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1573 conjugacy classes of subgroups; only the 257 normal subgroups are shown.
This group has 1573 conjugacy classes of subgroups — too many to draw. Only its 257 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 |
|---|---|---|
| 1. C23×C4×H(3,𝔽3) | C2 | semidirect |
| 3. C2×#88552 | C2 | semidirect |
| 4. C2×#88542 | C2 | semidirect |
| 5. #11720574 | C2 | semidirect |
| 14. C2×#6235 | C4 | semidirect |
| 19. C23×C32×C4 | D3 | semidirect |
| 34. C22×#272 | C2×C4 | semidirect |
| 38. C2×C4×H(3,𝔽3) | D4 | semidirect |
| 45. C2×#1084 | C2×C4 | semidirect |
| 64. C22×H(3,𝔽3) | C22⋊C4 | semidirect |
| 102. C2×C32×C4 | #67 | semidirect |
| 146. C22×C32 | (C2×D3)⋊C4 | semidirect |
| 161. C23×C4 | #272 | semidirect |
| 165. H(3,𝔽3) | #386 | semidirect |
| 265. C32 | #5213 | semidirect |
| 287. C2×C4 | #6235 | semidirect |
| 311. C22 | #88542 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C4×H(3,𝔽3) | 2 |
| C22×#6235 | 2 |
| C2×#88552 | 2 |
| C2×#88542 | 2 |
| #11720574 | 2 |
| C3×#5213 | 3 |