Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
3666 conjugacy classes of subgroups; only the 667 normal subgroups are shown.
This group has 3666 conjugacy classes of subgroups — too many to draw. Only its 667 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C23×D3 |
| commutator, cosocle | C23×C3 | C25 |
| Frattini | C23 | C24×D3 |
| Fitting | C22×C3×#127 | C2 |
| socle | C24×C3 | C24 |
| hypercenter | C22×#127 | D3 |
| hypocenter | C3 | C2×#2818 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C22×#127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×#127 | C2 | semidirect |
| 2. C22×#876 | C2 | semidirect |
| 3. C2×#5223 | C2 | semidirect |
| 4. C2×#5222 | C2 | semidirect |
| 5. #83191 | C2 | direct (N × H) |
| 6. C2×#4937 | C2 | semidirect |
| 11. C2×C3×#127 | C22 | semidirect |
| 17. C2×#876 | C22 | semidirect |
| 18. #5223 | C22 | semidirect |
| 19. #5222 | C22 | semidirect |
| 21. #4937 | C22 | semidirect |
| 22. C22×#127 | D3 | semidirect |
| 39. C2×C3×C42 | D4 | semidirect |
| 52. C2×#127 | C2×D3 | semidirect |
| 84. C3×C42 | C2×D4 | semidirect |
| 106. C2×C42 | #67 | semidirect |
| 156. C42 | C2×#67 | semidirect |
| 195. C2×C3 | #2818 | semidirect |
| 213. C3 | C2×#2818 | semidirect |
| 216. C2 | #83191 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×#127 | 2 |
| C22×#876 | 2 |
| C2×#5223 | 2 |
| C2×#5222 | 2 |
| #83191 | 2 |
| C2×#4937 | 2 |
| C2×#2818 | 3 |