Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1032 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 1032 conjugacy classes of subgroups — too many to draw. Only its 178 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. C3×#64319 | C2 | semidirect |
| 2. C3×Q8×#157 | C2 | semidirect |
| 3. C3×#64231 | C2 | semidirect |
| 4. C3×#64230 | C2 | semidirect |
| 5. C3×C4×GO+(4,𝔽2) | C2 | semidirect |
| 6. #10596525 | C3 | direct (N × H) |
| 19. #64319 | C2×C3 | semidirect |
| 20. Q8×#157 | C2×C3 | semidirect |
| 21. #64231 | C2×C3 | semidirect |
| 22. #64230 | C2×C3 | semidirect |
| 23. C4×GO+(4,𝔽2) | C2×C3 | semidirect |
| 29. C33×Q8 | D4 | semidirect |
| 31. C3×#634 | D4 | semidirect |
| 62. C3×#157 | #41 | semidirect |
| 80. C32×Q8 | C3×D4 | semidirect |
| 84. #634 | C3×D4 | semidirect |
| 135. #157 | C3×#41 | semidirect |
| 150. C33 | #550 | semidirect |
| 156. C3×Q8 | GO+(4,𝔽2) | semidirect |
| 224. C32 | C3×#550 | semidirect |
| 227. Q8 | C3×GO+(4,𝔽2) | semidirect |
| 260. C3 | #10596525 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#64319 | 2 |
| C3×Q8×#157 | 2 |
| C3×#64231 | 2 |
| C3×#64230 | 2 |
| C3×C4×GO+(4,𝔽2) | 2 |
| #10596525 | 3 |
| C3×#550 | 9 |