Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1068 conjugacy classes of subgroups; only the 529 normal subgroups are shown.
This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 529 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×Q82 | C2 |
| prime radical | C2×#239 | C22 |
| hypercenter | Q82 | D3 |
| hypocenter | C3 | Q8×#41 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×Q82 | C2 | semidirect |
| 2. #5730 | C2 | semidirect |
| 3. #5728 | C2 | semidirect |
| 4. Q8×D12 | C2 | semidirect |
| 6. C4×D3×Q8 | C2 | semidirect |
| 7. #5570 | C2 | semidirect |
| 28. Q82 | D3 | semidirect |
| 38. #239 | Q8 | direct (N × H) |
| 77. C3×Q8 | C2×Q8 | semidirect |
| 78. C3×Q8 | #41 | semidirect |
| 85. D12 | C2×Q8 | semidirect |
| 88. C4×D3 | C2×Q8 | semidirect |
| 124. Q8 | D3×Q8 | semidirect |
| 125. Q8 | #239 | direct (N × H) |
| 150. C3 | Q8×#41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×Q82 | 2 |
| #5730 | 2 |
| #5728 | 2 |
| Q8×D12 | 2 |
| C4×#239 | 2 |
| C4×D3×Q8 | 2 |
| #5570 | 2 |
| Q8×#41 | 3 |