Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
520 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 520 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 | C3×C4 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| socle | C2×C3 | #1999 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#3414 | C2 | semidirect |
| 2. C3×#3067 | C2 | semidirect |
| 3. C2×C3×#359 | C2 | semidirect |
| 4. C3×#2113 | C2 | semidirect |
| 5. C3×#1385 | C2 | semidirect |
| 6. #13312 | C3 | direct (N × H) |
| 22. #3414 | C2×C3 | semidirect |
| 23. #3067 | C2×C3 | semidirect |
| 24. C2×#359 | C2×C3 | semidirect |
| 25. #2113 | C2×C3 | semidirect |
| 26. #1385 | C2×C3 | semidirect |
| 267. C3 | #13312 | 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×#3067 | 2 |
| C2×C3×#359 | 2 |
| C3×#2113 | 2 |
| C3×#1385 | 2 |
| #13312 | 3 |