Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1112 conjugacy classes of subgroups; only the 575 normal subgroups are shown.
This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 575 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 | C22×D3 | C24 |
| hypercenter | Q82 | D3 |
| hypocenter | C3 | C2×Q82 |
| 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 |
| 3. #5722 | C2 | semidirect |
| 5. Q8×Dic6 | C2 | semidirect |
| 21. Q82 | D3 | direct (N × H) |
| 27. D3×Q8 | Q8 | direct (N × H) |
| 52. C3×Q8 | C2×Q8 | semidirect |
| 60. Dic6 | C2×Q8 | semidirect |
| 86. Q8 | D3×Q8 | direct (N × H) |
| 95. D3 | Q82 | direct (N × H) |
| 103. C3 | C2×Q82 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×Q82 | 2 |
| D3×C4⋊Q8 | 2 |
| #5722 | 2 |
| C4×D3×Q8 | 2 |
| Q8×Dic6 | 2 |
| C2×Q82 | 3 |