Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
852 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 852 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 | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C23×C3×Q8 | C2 |
| prime radical | C2×#930 | C2 |
| hypercenter | C23×Q8 | D3 |
| hypocenter | C3 | #2436 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C23×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C3×Q8 | C2 | semidirect |
| 2. C2×#947 | C2 | semidirect |
| 4. #5231 | C2 | semidirect |
| 5. #5229 | C2 | semidirect |
| 6. #5213 | C2 | semidirect |
| 7. C4×#217 | C2 | semidirect |
| 26. C23×Q8 | D3 | semidirect |
| 40. C2×C3×Q8 | D4 | semidirect |
| 110. C2×Q8 | #67 | semidirect |
| 132. C22×C3 | #124 | semidirect |
| 186. C22 | #947 | semidirect |
| 196. C3 | #2436 | 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×#947 | 2 |
| C2×#930 | 2 |
| #5231 | 2 |
| #5229 | 2 |
| #5213 | 2 |
| C4×#217 | 2 |
| #2436 | 3 |