Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
952 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 952 conjugacy classes of subgroups — too many to draw. Only its 173 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 | #2431 |
| 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. C22×#237 | C2 | semidirect |
| 2. C2×C3×#127 | C2 | semidirect |
| 3. C2×#938 | C2 | semidirect |
| 4. C2×#936 | C2 | semidirect |
| 5. #5223 | C2 | semidirect |
| 6. #5221 | C2 | semidirect |
| 7. #5220 | C2 | semidirect |
| 8. C4×(C2×D3)⋊C4 | C2 | semidirect |
| 9. #4929 | C2 | semidirect |
| 35. C2×#127 | D3 | semidirect |
| 54. C2×C3×D4 | D4 | semidirect |
| 72. C2×C4×D3 | D4 | semidirect |
| 75. C2×Dic6 | D4 | semidirect |
| 130. C2×C3×C4 | #41 | semidirect |
| 167. C2×D4 | #67 | semidirect |
| 252. C2×C4 | #237 | semidirect |
| 288. C3 | #2431 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#237 | 2 |
| C2×C3×#127 | 2 |
| C2×#938 | 2 |
| C2×#936 | 2 |
| #5223 | 2 |
| #5221 | 2 |
| #5220 | 2 |
| C4×(C2×D3)⋊C4 | 2 |
| #4929 | 2 |
| #2431 | 3 |