Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1104 conjugacy classes of subgroups; only the 309 normal subgroups are shown.
This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 309 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 | C22×C3 | C23×C4 |
| Fitting, prime radical | C23×C3×D4 | C2 |
| socle, cosocle | C23×C3 | C24 |
| hypercenter | C23×D4 | D3 |
| hypocenter | C3 | #2309 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C23×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#926 | C2 | semidirect |
| 3. #5247 | C2 | semidirect |
| 4. #5221 | C2 | semidirect |
| 8. C22×C3×D4 | C4 | semidirect |
| 12. C2×#217 | C22 | semidirect |
| 14. C2×C4⋊Dic3 | C22 | semidirect |
| 15. C2×C4×Dic3 | C22 | semidirect |
| 39. C22×D4 | Dic3 | semidirect |
| 58. C2×C3×C4 | C22⋊C4 | semidirect |
| 106. C2×C4 | #217 | semidirect |
| 127. C3 | #2309 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×D4 | 2 |
| C2×#926 | 2 |
| #5247 | 2 |
| #5221 | 2 |
| #2309 | 3 |