Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
428 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 428 conjugacy classes of subgroups — too many to draw. Only its 176 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C211 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C211 |
| prime radical | C22×D4×C211 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #395×C211 | C2 | semidirect |
| 3. #393×C211 | C2 | semidirect |
| 4. #388×C211 | C2 | semidirect |
| 5. #382×C211 | C2 | semidirect |
| 57. C211 | #2785 | direct (N × H) |
| 58. #2785 | C211 | direct (N × H) |
| 59. #395 | C2×C211 | semidirect |
| 61. #393 | C2×C211 | semidirect |
| 62. #388 | C2×C211 | semidirect |
| 63. #382 | C2×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #395×C211 | 2 |
| #394×C211 | 2 |
| #393×C211 | 2 |
| #388×C211 | 2 |
| #382×C211 | 2 |
| #2785 | 211 |