Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
462 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 462 conjugacy classes of subgroups — too many to draw. Only its 114 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 |
| prime radical | C3×#3150 | C2 |
| 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 |
|---|---|---|
| 2. C2×C3×#360 | C2 | semidirect |
| 3. C3×#2139 | C2 | semidirect |
| 4. C3×#1944 | C2 | semidirect |
| 5. C3×#1377 | C2 | semidirect |
| 6. #13410 | C3 | direct (N × H) |
| 22. C2×#360 | C2×C3 | semidirect |
| 23. #2139 | C2×C3 | semidirect |
| 24. #1944 | C2×C3 | semidirect |
| 25. #1377 | C2×C3 | semidirect |
| 57. C3×#106 | D4 | semidirect |
| 148. #106 | C3×D4 | semidirect |
| 298. C3 | #13410 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3150 | 2 |
| C2×C3×#360 | 2 |
| C3×#2139 | 2 |
| C3×#1944 | 2 |
| C3×#1377 | 2 |
| #13410 | 3 |