Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1492 conjugacy classes of subgroups; only the 553 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 553 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 | C2×C3×C4×Q8 | C2 |
| hypercenter | C2×C4×Q8 | D3 |
| hypocenter | C3 | #3497 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#1007 | C2 | semidirect |
| 2. C2×C3×C4×Q8 | C2 | semidirect |
| 3. #5577 | C2 | semidirect |
| 4. #5576 | C2 | semidirect |
| 5. Q8×D12 | C2 | semidirect |
| 6. #5510 | C2 | semidirect |
| 7. #5477 | C2 | semidirect |
| 33. C2×C4×Q8 | D3 | semidirect |
| 34. C2×C3×Q8 | D4 | semidirect |
| 94. C2×Q8 | D12 | semidirect |
| 121. C3×C4 | 2−1+4 | semidirect |
| 156. C4 | #1007 | semidirect |
| 157. C3 | #3497 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1007 | 2 |
| C2×C3×C4×Q8 | 2 |
| #5577 | 2 |
| #5576 | 2 |
| Q8×D12 | 2 |
| #5510 | 2 |
| #5477 | 2 |
| #3497 | 3 |