Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
948 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 948 conjugacy classes of subgroups — too many to draw. Only its 185 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 | #2541 |
| 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×#940 | C2 | semidirect |
| 3. C2×#940 | C2 | semidirect |
| 4. C2×#939 | C2 | semidirect |
| 5. C2×D4×Dic3 | C2 | semidirect |
| 6. #5223 | C2 | semidirect |
| 7. #5221 | C2 | semidirect |
| 8. #4939 | C2 | semidirect |
| 9. C4×Dic3⋊C4 | C2 | semidirect |
| 35. C2×#127 | D3 | semidirect |
| 53. C2×C3×D4 | D4 | semidirect |
| 77. Dic3⋊C4 | D4 | semidirect |
| 79. C4×Dic3 | D4 | semidirect |
| 162. C2×D4 | #67 | semidirect |
| 200. C3×C4 | #121 | semidirect |
| 204. Dic3 | #127 | semidirect |
| 277. C4 | #939 | semidirect |
| 278. C3 | #2541 | 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×#940 | 2 |
| C2×#939 | 2 |
| C2×D4×Dic3 | 2 |
| #5223 | 2 |
| #5221 | 2 |
| #4939 | 2 |
| C4×Dic3⋊C4 | 2 |
| #2541 | 3 |