Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
662 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 662 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C2×C3×#493 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×SD32 | C2 | semidirect |
| 3. C2×C3×#357 | C2 | semidirect |
| 4. C3×#1969 | C2 | semidirect |
| 5. C3×#1374 | C2 | semidirect |
| 6. #13328 | C3 | direct (N × H) |
| 21. C22×SD32 | C2×C3 | semidirect |
| 23. C2×#357 | C2×C3 | semidirect |
| 24. #1969 | C2×C3 | semidirect |
| 25. #1374 | C2×C3 | semidirect |
| 32. C2×C3×Q16 | D4 | semidirect |
| 51. C2×C3×C16 | D4 | semidirect |
| 115. C2×Q16 | C3×D4 | semidirect |
| 134. C2×C16 | C3×D4 | semidirect |
| 161. C2×C3×C4 | SD32 | semidirect |
| 249. C2×C4 | C3×SD32 | semidirect |
| 282. C3 | #13328 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×SD32 | 2 |
| C2×C3×#493 | 2 |
| C2×C3×#357 | 2 |
| C3×#1969 | 2 |
| C3×#1374 | 2 |
| #13328 | 3 |