Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
212 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | C2×D8 |
| commutator | C8 | C22×C3×C8 |
| Frattini, cosocle | C4×C8 | C23×C3 |
| socle | C2×C3 | C8×D8 |
| prime radical | C3×#508 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ C2×C3×C16 ◁ C3×C4×C16 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#2170 | C2 | semidirect |
| 3. C3×#1373 | C2 | semidirect |
| 4. C3×#1335 | C2 | semidirect |
| 5. C3×#1311 | C2 | semidirect |
| 6. #12047 | C3 | direct (N × H) |
| 18. #2170 | C2×C3 | semidirect |
| 19. #1373 | C2×C3 | semidirect |
| 20. #1335 | C2×C3 | semidirect |
| 21. #1311 | C2×C3 | semidirect |
| 24. C3×Q32 | C8 | semidirect |
| 25. C3×SD32 | C8 | semidirect |
| 26. C3×D16 | C8 | semidirect |
| 67. Q32 | C3×C8 | semidirect |
| 68. SD32 | C3×C8 | semidirect |
| 69. D16 | C3×C8 | semidirect |
| 137. C3 | #12047 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#2178 | 2 |
| C3×#2170 | 2 |
| C3×#1373 | 2 |
| C3×#1335 | 2 |
| C3×#1311 | 2 |
| #12047 | 3 |