Finite Group DatabasebetaFinite Group DatabasebetaOrder 2592.
1142 conjugacy classes of subgroups; only the 243 normal subgroups are shown.
This group has 1142 conjugacy classes of subgroups — too many to draw. Only its 243 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 | #10620209 |
| commutator, cosocle | C32×C3⋊D3 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C22×C34 | Q8 |
| prime radical | C22×C32×C3⋊D3 | C22 |
| hypocenter | C34 | C22×Q8 |
| LCS | G ▷ C32×C3⋊D3 ▷ C34 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#10620209 | C2 | direct (N × H) |
| 8. #10620209 | C22 | direct (N × H) |
| 12. C22×C34 | Q8 | semidirect |
| 13. C2×C32×C3⋊D3 | Q8 | semidirect |
| 24. C2×C34 | C2×Q8 | semidirect |
| 25. C32×C3⋊D3 | C2×Q8 | semidirect |
| 35. C22×C33 | Dic6 | semidirect |
| 36. C2×C3×C3⋊D3 | Dic6 | semidirect |
| 43. C34 | C22×Q8 | semidirect |
| 55. C2×C33 | C2×Dic6 | semidirect |
| 56. C3×C3⋊D3 | C2×Dic6 | semidirect |
| 69. C22×C32 | #634 | semidirect |
| 70. C22×C32 | PSU(3,𝔽2) | semidirect |
| 71. C2×C3⋊D3 | #634 | semidirect |
| 82. C33 | C22×Dic6 | semidirect |
| 94. C2×C32 | C2×#634 | semidirect |
| 95. C2×C32 | C2×PSU(3,𝔽2) | semidirect |
| 96. C3⋊D3 | C2×#634 | semidirect |
| 107. C22×C3 | C3⋊PSU(3,𝔽2) | semidirect |
| 116. C32 | C22×#634 | semidirect |
| 117. C32 | C22×PSU(3,𝔽2) | semidirect |
| 126. C2×C3 | C2×C3⋊PSU(3,𝔽2) | semidirect |
| 131. C22 | #10620209 | direct (N × H) |
| 136. C3 | C22×C3⋊PSU(3,𝔽2) | semidirect |
| 139. C2 | C2×#10620209 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3⋊(C3⋊#157) | 2 |
| C22×C32×#157 | 2 |
| C2×#10620209 | 2 |
| C22×C3⋊PSU(3,𝔽2) | 3 |
| C22×#634 | 9 |