Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
840 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 840 conjugacy classes of subgroups — too many to draw. Only its 189 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, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#380 | C2 |
| prime radical | C22×#67 | C22 |
| hypercenter | #380 | D3 |
| hypocenter | C3 | #2409 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #380 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Dic6 | C2 | semidirect |
| 3. C2×#897 | C2 | semidirect |
| 4. C3×#380 | C2 | semidirect |
| 5. #4973 | C2 | semidirect |
| 7. #4950 | C2 | semidirect |
| 8. #4673 | C2 | semidirect |
| 9. #4656 | C2 | semidirect |
| 40. #380 | D3 | semidirect |
| 50. C2×#67 | Q8 | semidirect |
| 64. C2×Dic6 | D4 | semidirect |
| 104. C22×D3 | C2×Q8 | semidirect |
| 124. C2×Dic3 | C2×Q8 | semidirect |
| 164. C22×C3 | C22⋊Q8 | semidirect |
| 232. C22 | #897 | semidirect |
| 245. C3 | #2409 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Dic6 | 2 |
| C2×C4×#67 | 2 |
| C2×#897 | 2 |
| C3×#380 | 2 |
| #4973 | 2 |
| #4958 | 2 |
| #4950 | 2 |
| #4673 | 2 |
| #4656 | 2 |
| #2409 | 3 |