Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
374 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 64 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. Q8×#272 | C2 | semidirect |
| 3. SD16×H(3,𝔽3) | C2 | semidirect |
| 4. #88549 | C2 | semidirect |
| 5. #88546 | C2 | semidirect |
| 6. #88528 | C2 | semidirect |
| 7. C8×#272 | C2 | semidirect |
| 10. Q8×H(3,𝔽3) | C22 | semidirect |
| 17. #6229 | C22 | semidirect |
| 18. C8×H(3,𝔽3) | C22 | semidirect |
| 19. #6180 | C22 | semidirect |
| 22. C32×SD16 | D3 | semidirect |
| 38. C32×Q8 | C2×D3 | semidirect |
| 46. C32×C8 | C2×D3 | semidirect |
| 51. #272 | SD16 | direct (N × H) |
| 75. H(3,𝔽3) | C2×SD16 | semidirect |
| 107. SD16 | #272 | direct (N × H) |
| 125. C32 | D3×SD16 | semidirect |
| 129. Q8 | C2×#272 | semidirect |
| 136. C8 | C2×#272 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×#272 | 2 |
| D4×#272 | 2 |
| SD16×H(3,𝔽3) | 2 |
| #88549 | 2 |
| #88546 | 2 |
| #88528 | 2 |
| C8×#272 | 2 |
| C3×D3×SD16 | 3 |