Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
924 conjugacy classes of subgroups; only the 289 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 289 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting, prime radical | C2×C3×#127 | C2 |
| socle, cosocle | C23×C3 | C24 |
| hypercenter | C2×#127 | D3 |
| hypocenter | C3 | C4×#127 |
| 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 |
|---|---|---|
| 2. C2×D4×Dic3 | C2 | semidirect |
| 3. #5221 | C2 | semidirect |
| 4. #4935 | C2 | semidirect |
| 5. C42×Dic3 | C2 | semidirect |
| 9. C3×#127 | C4 | semidirect |
| 24. C2×C3×D4 | C2×C4 | semidirect |
| 30. C3×C42 | C2×C4 | semidirect |
| 34. C4×Dic3 | D4 | semidirect |
| 37. #127 | Dic3 | direct (N × H) |
| 60. C2×D4 | C2×Dic3 | semidirect |
| 70. C42 | C2×Dic3 | semidirect |
| 77. C3×C4 | C4×D4 | semidirect |
| 80. Dic3 | #127 | direct (N × H) |
| 107. C4 | D4×Dic3 | semidirect |
| 108. C3 | C4×#127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#127 | 2 |
| C2×D4×Dic3 | 2 |
| #5221 | 2 |
| #4935 | 2 |
| C42×Dic3 | 2 |
| C4×#127 | 3 |