Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
820 conjugacy classes of subgroups; only the 333 normal subgroups are shown.
This group has 820 conjugacy classes of subgroups — too many to draw. Only its 333 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 | C22×C3 | C23×C4 |
| Fitting | C23×C3×Q8 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C24×C3 | C23 |
| hypercenter | C23×Q8 | D3 |
| hypocenter | C3 | Q8×C22⋊C4 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C23×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×Q8×Dic3 | C2 | semidirect |
| 3. #5229 | C2 | semidirect |
| 8. C22×C3×Q8 | C4 | semidirect |
| 37. #217 | Q8 | direct (N × H) |
| 41. C22×Q8 | Dic3 | semidirect |
| 59. C3×Q8 | C22⋊C4 | semidirect |
| 70. C2×Dic3 | C2×Q8 | semidirect |
| 91. C22×C3 | C4×Q8 | semidirect |
| 108. Q8 | #217 | direct (N × H) |
| 130. C22 | Q8×Dic3 | semidirect |
| 140. C3 | Q8×C22⋊C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×Q8 | 2 |
| C2×Q8×Dic3 | 2 |
| #5229 | 2 |
| #5212 | 2 |
| C4×#217 | 2 |
| Q8×C22⋊C4 | 3 |