Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
816 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 816 conjugacy classes of subgroups — too many to draw. Only its 181 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, prime radical | C22×C3×#41 | C2 |
| hypercenter | C22×#41 | D3 |
| hypocenter | C3 | #2616 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#924 | C2 | semidirect |
| 3. #5230 | C2 | semidirect |
| 4. #5222 | C2 | semidirect |
| 5. #5212 | C2 | semidirect |
| 128. C3 | #2616 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×#41 | 2 |
| C2×#924 | 2 |
| #5230 | 2 |
| #5222 | 2 |
| #5212 | 2 |
| #2616 | 3 |