Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
800 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 213 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 | C24×C3 | C23 |
| hypercenter | C23×Q8 | D3 |
| hypocenter | C3 | #2434 |
| 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 |
|---|---|---|
| 2. C2×#944 | C2 | semidirect |
| 4. #5230 | C2 | semidirect |
| 5. #5229 | C2 | semidirect |
| 53. #217 | Q8 | semidirect |
| 97. C2×Dic3 | C2×Q8 | semidirect |
| 127. C22×C3 | C4⋊Q8 | semidirect |
| 170. C22 | #944 | semidirect |
| 181. C3 | #2434 | 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×#944 | 2 |
| C2×#924 | 2 |
| #5230 | 2 |
| #5229 | 2 |
| #5210 | 2 |
| C4×#217 | 2 |
| #2434 | 3 |