Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
5872 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.
This group has 5872 conjugacy classes of subgroups — too many to draw. Only its 3221 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C2×C3 | C26 |
| Frattini | C2 | C25×D3 |
| Fitting | C22×C3×#41 | C2 |
| socle | C23×C3 | C24 |
| hypercenter | C22×#41 | D3 |
| hypocenter | C3 | C22×2−1+4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×#41 | C2 | semidirect |
| 2. C2×#1010 | C2 | direct (N × H) |
| 3. C22×D3×Q8 | C2 | semidirect |
| 4. C22×#237 | C2 | semidirect |
| 5. C22×#235 | C2 | semidirect |
| 6. C23×Dic6 | C2 | semidirect |
| 8. C2×C3×#41 | C22 | semidirect |
| 14. #1010 | C22 | direct (N × H) |
| 15. C2×D3×Q8 | C22 | semidirect |
| 16. C2×#237 | C22 | semidirect |
| 17. C2×#235 | C22 | semidirect |
| 22. C22×Dic6 | C22 | semidirect |
| 26. C22×#41 | D3 | semidirect |
| 31. C3×#41 | C23 | semidirect |
| 40. D3×Q8 | C23 | semidirect |
| 41. #237 | C23 | semidirect |
| 42. #235 | C23 | semidirect |
| 47. C2×Dic6 | C23 | semidirect |
| 51. C2×#41 | C2×D3 | semidirect |
| 84. #41 | C22×D3 | semidirect |
| 100. C22×C3 | 2−1+4 | semidirect |
| 126. C2×C3 | C2×2−1+4 | semidirect |
| 132. C22 | #1010 | direct (N × H) |
| 138. C3 | C22×2−1+4 | semidirect |
| 141. C2 | C2×#1010 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×#41 | 2 |
| C2×#1010 | 2 |
| C22×D3×Q8 | 2 |
| C22×#237 | 2 |
| C22×#235 | 2 |
| C23×Dic6 | 2 |
| C22×2−1+4 | 3 |