Finite Group DatabasebetaFinite Group DatabasebetaOrder 10368.
1078 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #72213199 |
| commutator, hypocenter | C34 | C2×C82 |
| Frattini | C2 | C2×#72213199 |
| Fitting, socle | C22×C34 | C4×C8 |
| prime radical | C22×C3⋊(C3⋊(C3⋊D3)) | C42 |
| cosocle | #13049409 | C23 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #72220215 | C2 | direct (N × H) |
| 13. C2×#10620193 | C8 | semidirect |
| 15. C2×#10620192 | C8 | semidirect |
| 18. C2×C3⋊(C3⋊#622) | C8 | semidirect |
| 19. C2×C32×#622 | C8 | semidirect |
| 23. #13049399 | C8 | semidirect |
| 32. #10620193 | C2×C8 | semidirect |
| 33. #10620192 | C2×C8 | semidirect |
| 38. C3⋊(C3⋊#622) | C2×C8 | semidirect |
| 39. C32×#622 | C2×C8 | semidirect |
| 77. C2×C34 | C82 | semidirect |
| 79. C3⋊(C3⋊(C3⋊D3)) | C82 | semidirect |
| 92. C2×#622 | AGL(1,𝔽9) | semidirect |
| 116. C34 | C2×C82 | semidirect |
| 144. #622 | C2×AGL(1,𝔽9) | semidirect |
| 224. C2×C32 | C8×AGL(1,𝔽9) | semidirect |
| 266. C32 | C2×C8×AGL(1,𝔽9) | semidirect |
| 301. C2 | #72220215 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#72220201 | 2 |
| #72220215 | 2 |
| C2×#72213723 | 2 |
| C2×C8×AGL(1,𝔽9) | 9 |