Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
594 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 594 conjugacy classes of subgroups — too many to draw. Only its 122 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 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#509 | C2 | semidirect |
| 2. C3×#3150 | C2 | semidirect |
| 3. C2×C3×#360 | C2 | semidirect |
| 4. C3×#1972 | C2 | semidirect |
| 5. C3×#1377 | C2 | semidirect |
| 6. #13341 | C3 | direct (N × H) |
| 23. C2×#509 | C2×C3 | semidirect |
| 24. #3150 | C2×C3 | semidirect |
| 25. C2×#360 | C2×C3 | semidirect |
| 26. #1972 | C2×C3 | semidirect |
| 27. #1377 | C2×C3 | semidirect |
| 45. C2×C3×D8 | D4 | semidirect |
| 59. C3×M32 | D4 | semidirect |
| 142. C2×D8 | C3×D4 | semidirect |
| 156. M32 | C3×D4 | semidirect |
| 316. C3 | #13341 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#509 | 2 |
| C3×#3150 | 2 |
| C2×C3×#360 | 2 |
| C3×#1972 | 2 |
| C3×#1377 | 2 |
| #13341 | 3 |