Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
392 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 184 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 | #532×C211 | C2 |
| 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 |
|---|---|---|
| 1. #533×C211 | C2 | semidirect |
| 3. #491×C211 | C2 | semidirect |
| 4. #490×C211 | C2 | semidirect |
| 5. #489×C211 | C2 | semidirect |
| 6. #416×C211 | C2 | semidirect |
| 7. C2×#97×C211 | C2 | semidirect |
| 78. C211 | #3135 | direct (N × H) |
| 79. #3135 | C211 | direct (N × H) |
| 80. #533 | C2×C211 | semidirect |
| 82. #491 | C2×C211 | semidirect |
| 83. #490 | C2×C211 | semidirect |
| 84. #489 | C2×C211 | semidirect |
| 85. #416 | C2×C211 | semidirect |
| 86. C2×#97 | C2×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #533×C211 | 2 |
| #532×C211 | 2 |
| #491×C211 | 2 |
| #490×C211 | 2 |
| #489×C211 | 2 |
| #416×C211 | 2 |
| C2×#97×C211 | 2 |
| #3135 | 211 |