Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
146 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C8×C211 | D8 |
| commutator | C8 | C22×C4×C211 |
| Frattini, cosocle | C2×C8 | C23×C211 |
| socle | C2×C211 | C4×D8 |
| prime radical | #508×C211 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C211 ◁ C2×C8×C211 ◁ C4×C8×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #443×C211 | C2 | semidirect |
| 3. #368×C211 | C2 | semidirect |
| 4. #359×C211 | C2 | semidirect |
| 5. C4×C16×C211 | C2 | semidirect |
| 8. Q32×C211 | C4 | semidirect |
| 9. SD32×C211 | C4 | semidirect |
| 10. D16×C211 | C4 | semidirect |
| 49. C211 | #2178 | direct (N × H) |
| 50. #2178 | C211 | direct (N × H) |
| 52. #443 | C2×C211 | semidirect |
| 53. #368 | C2×C211 | semidirect |
| 54. #359 | C2×C211 | semidirect |
| 55. C4×C16 | C2×C211 | semidirect |
| 58. Q32 | C4×C211 | semidirect |
| 59. SD32 | C4×C211 | semidirect |
| 60. D16 | C4×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #508×C211 | 2 |
| #443×C211 | 2 |
| #368×C211 | 2 |
| #359×C211 | 2 |
| C4×C16×C211 | 2 |
| #2178 | 211 |