Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
374 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 84 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. M16×H(3,𝔽3) | C2 | semidirect |
| 3. #88538 | C2 | semidirect |
| 4. #88527 | C2 | semidirect |
| 5. C8×#272 | C2 | semidirect |
| 15. C8×H(3,𝔽3) | C22 | semidirect |
| 16. #6180 | C22 | semidirect |
| 19. C32×M16 | D3 | semidirect |
| 20. C32×M16 | D3 | semidirect |
| 52. C32×C8 | C2×D3 | semidirect |
| 53. C32×C8 | C2×D3 | semidirect |
| 59. #272 | M16 | direct (N × H) |
| 92. H(3,𝔽3) | C2×M16 | semidirect |
| 131. M16 | #272 | direct (N × H) |
| 157. C32 | D3×M16 | semidirect |
| 158. C32 | D3×M16 | semidirect |
| 167. C8 | C2×#272 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×#272 | 2 |
| M16×H(3,𝔽3) | 2 |
| #88538 | 2 |
| #88527 | 2 |
| C8×#272 | 2 |
| C3×D3×M16 | 3 |