Finite Group DatabasebetaFinite Group Databasebeta← C2×C3⋊D3×PSU(3,𝔽2) #72228422
Order 2592.
1142 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 1142 conjugacy classes of subgroups — too many to draw. Only its 188 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊D3×PSU(3,𝔽2) |
| commutator, cosocle | C32×C3⋊D3 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C2×C34 | C2×Q8 |
| prime radical | C2×C3⋊D32 | C22 |
| hypocenter | C34 | C22×Q8 |
| LCS | G ▷ C32×C3⋊D3 ▷ C34 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#10620209 | C2 | semidirect |
| 2. C2×C32×PSU(3,𝔽2) | C2 | semidirect |
| 4. C3⋊D3×PSU(3,𝔽2) | C2 | direct (N × H) |
| 9. #10620209 | C22 | semidirect |
| 10. C32×PSU(3,𝔽2) | C22 | semidirect |
| 13. C2×C3×PSU(3,𝔽2) | D3 | semidirect |
| 16. C2×C3⋊(C3⋊(C3⋊D3)) | Q8 | semidirect |
| 17. C2×C32×C3⋊D3 | Q8 | semidirect |
| 19. C3⋊D32 | Q8 | semidirect |
| 30. C3×PSU(3,𝔽2) | C2×D3 | semidirect |
| 32. C2×C34 | C2×Q8 | semidirect |
| 33. C3⋊(C3⋊(C3⋊D3)) | C2×Q8 | semidirect |
| 34. C32×C3⋊D3 | C2×Q8 | semidirect |
| 35. C32×C3⋊D3 | C2×Q8 | semidirect |
| 40. C2×PSU(3,𝔽2) | C3⋊D3 | direct (N × H) |
| 59. C34 | C22×Q8 | semidirect |
| 67. PSU(3,𝔽2) | C2×C3⋊D3 | direct (N × H) |
| 75. C2×C33 | D3×Q8 | semidirect |
| 80. C3×C3⋊D3 | D3×Q8 | semidirect |
| 101. C2×C3⋊D3 | PSU(3,𝔽2) | direct (N × H) |
| 114. C33 | C2×D3×Q8 | semidirect |
| 129. C2×C32 | Q8×C3⋊D3 | semidirect |
| 130. C2×C32 | C2×PSU(3,𝔽2) | semidirect |
| 137. C3⋊D3 | Q8×C3⋊D3 | semidirect |
| 138. C3⋊D3 | C2×PSU(3,𝔽2) | direct (N × H) |
| 164. C32 | C2×Q8×C3⋊D3 | semidirect |
| 165. C32 | C22×PSU(3,𝔽2) | semidirect |
| 177. C2×C3 | D3×PSU(3,𝔽2) | semidirect |
| 193. C3 | C2×D3×PSU(3,𝔽2) | semidirect |
| 198. C2 | C3⋊D3×PSU(3,𝔽2) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#10620209 | 2 |
| C2×C32×PSU(3,𝔽2) | 2 |
| C2×C3⋊D3×#157 | 2 |
| C3⋊D3×PSU(3,𝔽2) | 2 |
| C2×D3×PSU(3,𝔽2) | 3 |
| C2×Q8×C3⋊D3 | 9 |