Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
517 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 517 conjugacy classes of subgroups — too many to draw. Only its 110 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. C22×C4×H(3,𝔽3) | C2 | semidirect |
| 3. C2×C4×#272 | C2 | semidirect |
| 4. #88552 | C2 | semidirect |
| 5. #88542 | C2 | semidirect |
| 6. #88540 | C2 | semidirect |
| 7. C4×#1084 | C2 | semidirect |
| 14. #6235 | C4 | direct (N × H) |
| 19. C22×C32×C4 | D3 | semidirect |
| 28. C22×H(3,𝔽3) | C2×C4 | semidirect |
| 32. C2×#272 | C2×C4 | semidirect |
| 34. C4×H(3,𝔽3) | D4 | semidirect |
| 37. #1084 | C2×C4 | semidirect |
| 71. C22×C32 | C4×D3 | semidirect |
| 77. C32×C4 | #67 | semidirect |
| 82. H(3,𝔽3) | C4×D4 | semidirect |
| 110. C22×C4 | #272 | semidirect |
| 139. C32 | C4×#67 | semidirect |
| 165. C22 | C4×#272 | semidirect |
| 171. C4 | #6235 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×H(3,𝔽3) | 2 |
| C2×#6235 | 2 |
| C2×C4×#272 | 2 |
| #88552 | 2 |
| #88542 | 2 |
| #88540 | 2 |
| C4×#1084 | 2 |
| C3×C4×#67 | 3 |