Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1728 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 1728 conjugacy classes of subgroups — too many to draw. Only its 557 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C2×C3×#127 | C2 |
| socle | C23×C3 | C24 |
| hypercenter | C2×#127 | D3 |
| hypocenter | C3 | C2×#532 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#237 | C2 | semidirect |
| 2. C2×C3×#127 | C2 | semidirect |
| 3. C2×#938 | C2 | semidirect |
| 4. C2×#936 | C2 | semidirect |
| 5. C2×D4×Dic3 | C2 | semidirect |
| 6. #5716 | C2 | direct (N × H) |
| 7. C2×C42×D3 | C2 | semidirect |
| 8. C2×#869 | C2 | semidirect |
| 15. C2×#237 | C22 | semidirect |
| 19. C3×#127 | C22 | semidirect |
| 22. #938 | C22 | semidirect |
| 23. #936 | C22 | semidirect |
| 24. D4×Dic3 | C22 | semidirect |
| 28. C42×D3 | C22 | semidirect |
| 29. #869 | C22 | semidirect |
| 33. C2×#127 | D3 | semidirect |
| 55. C2×C4×D3 | D4 | semidirect |
| 75. #127 | C2×D3 | semidirect |
| 95. C2×C3×C4 | #41 | semidirect |
| 107. C4×D3 | C2×D4 | semidirect |
| 146. C3×C4 | C2×#41 | semidirect |
| 171. C2×C4 | #237 | semidirect |
| 175. C2×C3 | #532 | semidirect |
| 192. C4 | C2×#237 | semidirect |
| 193. C3 | C2×#532 | semidirect |
| 196. C2 | #5716 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#237 | 2 |
| C2×C3×#127 | 2 |
| C2×#938 | 2 |
| C2×#936 | 2 |
| C2×D4×Dic3 | 2 |
| #5716 | 2 |
| C2×C42×D3 | 2 |
| C2×#869 | 2 |
| C2×#532 | 3 |