Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
336 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | #937 |
| commutator, cosocle | C3×C42 | C23 |
| Frattini | C42 | C22×D3 |
| Fitting | C3×#493 | C2 |
| socle | C22×C3 | #120 |
| hypercenter | #493 | D3 |
| hypocenter | C3 | #1619 |
| LCS | G ▷ C3×C42 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #493 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5717 | C2 | semidirect |
| 2. C3×#493 | C2 | semidirect |
| 3. #5072 | C2 | semidirect |
| 4. #4544 | C2 | semidirect |
| 5. #4459 | C2 | semidirect |
| 10. C3×#127 | C22 | semidirect |
| 15. C3×C4×C8 | C22 | semidirect |
| 17. C4⋊(C3⋊C8) | C22 | semidirect |
| 19. #493 | D3 | semidirect |
| 45. #127 | C2×D3 | semidirect |
| 51. C4×C8 | C2×D3 | semidirect |
| 68. D12 | D8 | semidirect |
| 143. C3 | #1619 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5717 | 2 |
| C3×#493 | 2 |
| #5072 | 2 |
| #4544 | 2 |
| #4459 | 2 |
| #1619 | 3 |