Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
496 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#459 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #459 | D3 |
| hypocenter | C3 | #3035 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #459 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#912 | C2 | semidirect |
| 2. #5716 | C2 | semidirect |
| 3. C4×#237 | C2 | semidirect |
| 4. C3×#459 | C2 | semidirect |
| 5. #5153 | C2 | semidirect |
| 6. #5075 | C2 | semidirect |
| 7. #5019 | C2 | semidirect |
| 8. #4849 | C2 | semidirect |
| 9. D3×C4⋊C8 | C2 | semidirect |
| 10. #4772 | C2 | semidirect |
| 11. #4767 | C2 | semidirect |
| 53. #459 | D3 | semidirect |
| 69. #237 | D4 | semidirect |
| 223. C3×C4 | #135 | semidirect |
| 291. C4 | #912 | semidirect |
| 294. C3 | #3035 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#912 | 2 |
| #5716 | 2 |
| C4×#237 | 2 |
| C3×#459 | 2 |
| #5153 | 2 |
| #5075 | 2 |
| #5019 | 2 |
| #4849 | 2 |
| D3×C4⋊C8 | 2 |
| #4772 | 2 |
| #4767 | 2 |
| #3035 | 3 |