Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
356 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 224 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C211 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C211 |
| socle | C22×C211 | C22×D4 |
| prime radical | C23×C211 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C211 ◁ C2×C42×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #500×C211 | C2 | semidirect |
| 3. #499×C211 | C2 | semidirect |
| 4. #498×C211 | C2 | semidirect |
| 22. C2×C8×C211 | Q8 | semidirect |
| 55. C211 | #3160 | direct (N × H) |
| 56. #3160 | C211 | direct (N × H) |
| 58. #500 | C2×C211 | semidirect |
| 59. #499 | C2×C211 | semidirect |
| 60. #498 | C2×C211 | semidirect |
| 78. C2×C8 | Q8×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #533×C211 | 2 |
| #500×C211 | 2 |
| #499×C211 | 2 |
| #498×C211 | 2 |
| #427×C211 | 2 |
| C2×C4×C8×C211 | 2 |
| #3160 | 211 |