Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
408 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 264 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C211 | C24 |
| commutator | C22 | C23×C4×C211 |
| Frattini, cosocle | C23 | C24×C211 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. #125×C211 | C4 | semidirect |
| 61. C211 | #2382 | direct (N × H) |
| 62. #2382 | C211 | direct (N × H) |
| 68. #125 | C4×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#125×C211 | 2 |
| #389×C211 | 2 |
| #387×C211 | 2 |
| #382×C211 | 2 |
| #2382 | 211 |