Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
552 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #493 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| socle | C2×C3 | #2011 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C2×C3×#97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#3414 | C2 | semidirect |
| 2. C3×#3150 | C2 | semidirect |
| 3. C3×#2143 | C2 | semidirect |
| 4. C3×#1975 | C2 | semidirect |
| 5. C3×#1375 | C2 | semidirect |
| 6. #13340 | C3 | direct (N × H) |
| 24. #3414 | C2×C3 | semidirect |
| 25. #3150 | C2×C3 | semidirect |
| 26. #2143 | C2×C3 | semidirect |
| 27. #1975 | C2×C3 | semidirect |
| 28. #1375 | C2×C3 | semidirect |
| 41. C2×C3×Q16 | D4 | semidirect |
| 47. C2×C3×D8 | D4 | semidirect |
| 63. C2×C3×C16 | D4 | semidirect |
| 143. C2×Q16 | C3×D4 | semidirect |
| 149. C2×D8 | C3×D4 | semidirect |
| 165. C2×C16 | C3×D4 | semidirect |
| 320. C3 | #13340 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3414 | 2 |
| C3×#3150 | 2 |
| C3×#2143 | 2 |
| C3×#1975 | 2 |
| C3×#1375 | 2 |
| #13340 | 3 |