Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
728 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 728 conjugacy classes of subgroups — too many to draw. Only its 157 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, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×#127 | C2 |
| hypercenter | C2×#127 | D3 |
| hypocenter | C3 | #2619 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#127 | C2 | semidirect |
| 2. C2×#938 | C2 | semidirect |
| 3. C2×#872 | C2 | semidirect |
| 4. #5223 | C2 | semidirect |
| 5. #5222 | C2 | semidirect |
| 6. #5221 | C2 | semidirect |
| 7. #4936 | C2 | semidirect |
| 24. C2×#127 | D3 | semidirect |
| 47. C2×C4×D3 | D4 | semidirect |
| 48. C3×C42 | D4 | semidirect |
| 119. C42 | #67 | semidirect |
| 188. C3 | #2619 | 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×#938 | 2 |
| C2×#872 | 2 |
| #5223 | 2 |
| #5222 | 2 |
| #5221 | 2 |
| #4936 | 2 |
| #2619 | 3 |