Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1252 conjugacy classes of subgroups; only the 230 normal subgroups are shown.
This group has 1252 conjugacy classes of subgroups — too many to draw. Only its 230 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#3549 | C2 | semidirect |
| 2. C3×#3151 | C2 | semidirect |
| 3. C3×#1655 | C2 | semidirect |
| 4. C3×#1525 | C2 | semidirect |
| 5. #19073 | C3 | direct (N × H) |
| 7. C3×H(3,𝔽4) | C22 | semidirect |
| 12. C2×C3×#127 | C22 | semidirect |
| 13. C3×#496 | C22 | semidirect |
| 16. C3×C4×M16 | C22 | semidirect |
| 17. C3×#329 | C22 | semidirect |
| 18. #3549 | C2×C3 | semidirect |
| 19. #3151 | C2×C3 | semidirect |
| 20. #1655 | C2×C3 | semidirect |
| 21. #1525 | C2×C3 | semidirect |
| 46. H(3,𝔽4) | C22×C3 | semidirect |
| 51. C2×#127 | C22×C3 | semidirect |
| 52. #496 | C22×C3 | semidirect |
| 55. C4×M16 | C22×C3 | semidirect |
| 56. #329 | C22×C3 | semidirect |
| 188. C3 | #19073 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3549 | 2 |
| C3×#3151 | 2 |
| C3×#1655 | 2 |
| C3×#1525 | 2 |
| #19073 | 3 |