Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
398 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 90 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 | #393 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C3×#532 | C22 |
| LCS | G ▷ C2×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×#2605 | C2 | semidirect |
| 5. C3×#1300 | C2 | semidirect |
| 6. C3×#1288 | C2 | semidirect |
| 7. C3×#1285 | C2 | semidirect |
| 8. #10378 | C3 | direct (N × H) |
| 28. #2605 | C2×C3 | semidirect |
| 32. #1300 | C2×C3 | semidirect |
| 33. #1288 | C2×C3 | semidirect |
| 34. #1285 | C2×C3 | semidirect |
| 38. C3×C4⋊Q8 | Q8 | semidirect |
| 39. C3×#127 | Q8 | semidirect |
| 139. C4⋊Q8 | C3×Q8 | semidirect |
| 140. #127 | C3×Q8 | semidirect |
| 356. C3 | #10378 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#2605 | 2 |
| C3×#1980 | 2 |
| C3×#1514 | 2 |
| C3×#1505 | 2 |
| C3×#1300 | 2 |
| C3×#1288 | 2 |
| C3×#1285 | 2 |
| #10378 | 3 |