Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
470 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 470 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 | C2×C3 | #1893 |
| commutator | C2×C8 | C22×C3×C4 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C3×#3060 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#3407 | C2 | semidirect |
| 3. C3×#2145 | C2 | semidirect |
| 4. C3×#1812 | C2 | semidirect |
| 5. C3×#1386 | C2 | semidirect |
| 6. #13077 | C3 | direct (N × H) |
| 13. C3×#470 | C4 | semidirect |
| 14. C3×#469 | C4 | semidirect |
| 22. #3407 | C2×C3 | semidirect |
| 24. #2145 | C2×C3 | semidirect |
| 25. #1812 | C2×C3 | semidirect |
| 26. #1386 | C2×C3 | semidirect |
| 49. C3×M32 | D4 | semidirect |
| 65. #470 | C3×C4 | semidirect |
| 66. #469 | C3×C4 | semidirect |
| 133. M32 | C3×D4 | semidirect |
| 271. C3 | #13077 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3407 | 2 |
| C3×#3060 | 2 |
| C3×#2145 | 2 |
| C3×#1812 | 2 |
| C3×#1386 | 2 |
| #13077 | 3 |