Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
484 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 197 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×#407 | C2 |
| prime radical | C24×C3 | C2×C4 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #407 | D3 |
| hypocenter | C3 | #2920 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #407 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #5118 | C2 | semidirect |
| 5. Dic3×M16 | C2 | semidirect |
| 6. #4719 | C2 | semidirect |
| 7. #4718 | C2 | semidirect |
| 22. #815 | C22 | semidirect |
| 23. Dic3⋊C8 | C22 | semidirect |
| 24. C8×Dic3 | C22 | semidirect |
| 91. C2×Dic3 | M16 | semidirect |
| 180. C3 | #2920 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×Dic3 | 2 |
| C3×#407 | 2 |
| #5207 | 2 |
| #5118 | 2 |
| Dic3×M16 | 2 |
| #4719 | 2 |
| #4718 | 2 |
| #2920 | 3 |