Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1116 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 1116 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 | C22×C3×C22⋊C4 | C2 |
| prime radical | C23×#67 | C2 |
| hypercenter | C22×C22⋊C4 | D3 |
| hypocenter | C3 | #2405 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C3×C22⋊C4 | C2 | semidirect |
| 3. C2×#929 | C2 | semidirect |
| 4. #5247 | C2 | semidirect |
| 5. #5213 | C2 | semidirect |
| 6. #4959 | C2 | semidirect |
| 7. #4955 | C2 | semidirect |
| 8. #4946 | C2 | semidirect |
| 12. C23×C3×C4 | C22 | semidirect |
| 13. C23×C3×C4 | C22 | semidirect |
| 21. C2×C3×C22⋊C4 | C22 | semidirect |
| 23. C2×#217 | C22 | semidirect |
| 24. C2×#217 | C22 | semidirect |
| 26. C2×(C2×D3)⋊C4 | C22 | semidirect |
| 27. C2×(C2×D3)⋊C4 | C22 | semidirect |
| 28. C2×Dic3⋊C4 | C22 | semidirect |
| 29. C3⋊#95 | C22 | semidirect |
| 32. C22×C22⋊C4 | D3 | semidirect |
| 82. C23×C4 | C2×D3 | semidirect |
| 83. C23×C4 | C2×D3 | semidirect |
| 93. C2×C22⋊C4 | C2×D3 | semidirect |
| 185. C22×C3 | #123 | semidirect |
| 266. C22 | #929 | semidirect |
| 276. C3 | #2405 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#67 | 2 |
| C22×C3×C22⋊C4 | 2 |
| C2×#929 | 2 |
| #5247 | 2 |
| #5213 | 2 |
| #4959 | 2 |
| #4955 | 2 |
| #4946 | 2 |
| #2405 | 3 |