Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1056 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 185 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×#41 | C2 |
| hypercenter | C22×#41 | D3 |
| hypocenter | C3 | #2404 |
| 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 |
|---|---|---|
| 1. C22×C3×#41 | C2 | semidirect |
| 2. C2×#937 | C2 | semidirect |
| 3. C22×(C2×D3)⋊C4 | C2 | semidirect |
| 4. #5231 | C2 | semidirect |
| 5. #5223 | C2 | semidirect |
| 6. #5220 | C2 | semidirect |
| 7. #5213 | C2 | semidirect |
| 8. #5211 | C2 | semidirect |
| 13. C22×C3×Q8 | C22 | semidirect |
| 15. C22×C3×D4 | C22 | semidirect |
| 17. C23×C3×C4 | C22 | semidirect |
| 21. C2×#217 | C22 | semidirect |
| 27. C3⋊#95 | C22 | semidirect |
| 28. C3⋊#95 | C22 | semidirect |
| 29. C3⋊#95 | C22 | semidirect |
| 30. C22×#41 | D3 | semidirect |
| 52. C2×C3×D4 | D4 | semidirect |
| 57. C22×C3×C4 | D4 | semidirect |
| 76. C22×Q8 | C2×D3 | semidirect |
| 79. C22×D4 | C2×D3 | semidirect |
| 82. C23×C4 | C2×D3 | semidirect |
| 111. C22×D3 | #41 | semidirect |
| 148. C2×D4 | #67 | semidirect |
| 157. C22×C4 | #67 | semidirect |
| 259. C3 | #2404 | 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×#937 | 2 |
| C22×(C2×D3)⋊C4 | 2 |
| #5231 | 2 |
| #5223 | 2 |
| #5220 | 2 |
| #5213 | 2 |
| #5211 | 2 |
| #2404 | 3 |