Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
922 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 922 conjugacy classes of subgroups — too many to draw. Only its 195 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | D3×#127 |
| commutator, cosocle | C3×C42 | C24 |
| Frattini | C42 | C23×D3 |
| Fitting | C3×#1712 | C2 |
| socle | C22×C3 | C2×#127 |
| hypercenter | #1712 | D3 |
| hypocenter | C3 | #19603 |
| LCS | G ▷ C3×C42 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #1712 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #81782 | C2 | semidirect |
| 2. #81775 | C2 | semidirect |
| 3. C3×#1712 | C2 | semidirect |
| 4. #71275 | C2 | semidirect |
| 5. #71274 | C2 | semidirect |
| 6. #68065 | C2 | semidirect |
| 7. D3×C82 | C2 | semidirect |
| 28. #1712 | D3 | semidirect |
| 92. D3×C8 | D8 | semidirect |
| 130. C3×C8 | #135 | semidirect |
| 183. C8 | #912 | semidirect |
| 198. C3 | #19603 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #81782 | 2 |
| #81775 | 2 |
| C3×#1712 | 2 |
| #71275 | 2 |
| #71274 | 2 |
| #68065 | 2 |
| D3×C82 | 2 |
| #19603 | 3 |