Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
232 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C211 | #102 |
| commutator | C8 | C22×C4×C211 |
| Frattini, cosocle | C2×C8 | C23×C211 |
| prime radical | C2×#135×C211 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C211 ◁ C22×C4×C211 ◁ C22×C8×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×M32×C211 | C2 | semidirect |
| 3. C2×#106×C211 | C2 | semidirect |
| 4. #360×C211 | C2 | direct (N × H) |
| 8. #135×C211 | C4 | semidirect |
| 10. C2×Q16×C211 | C4 | semidirect |
| 12. C2×D8×C211 | C4 | semidirect |
| 16. M32×C211 | C22 | semidirect |
| 19. #106×C211 | C22 | semidirect |
| 31. Q16×C211 | C2×C4 | semidirect |
| 35. D8×C211 | C2×C4 | semidirect |
| 83. C2×C211 | #360 | direct (N × H) |
| 85. C211 | C2×#360 | direct (N × H) |
| 86. C2×#360 | C211 | direct (N × H) |
| 88. C2×M32 | C2×C211 | semidirect |
| 89. C2×#106 | C2×C211 | semidirect |
| 90. #360 | C2×C211 | direct (N × H) |
| 94. #135 | C4×C211 | semidirect |
| 96. C2×Q16 | C4×C211 | semidirect |
| 98. C2×D8 | C4×C211 | semidirect |
| 102. M32 | C22×C211 | semidirect |
| 105. #106 | C22×C211 | semidirect |
| 117. Q16 | C2×C4×C211 | semidirect |
| 121. D8 | C2×C4×C211 | semidirect |
| 169. C2 | #360×C211 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#135×C211 | 2 |
| C2×M32×C211 | 2 |
| C2×#106×C211 | 2 |
| #360×C211 | 2 |
| C2×#360 | 211 |