Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
412 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 128 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 | C2×D4 |
| commutator | C2×C4 | C22×C4×C211 |
| Frattini, cosocle | C22×C4 | C23×C211 |
| prime radical | C2×#127×C211 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C211 ◁ C2×C42×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#102×C211 | C2 | semidirect |
| 3. C2×#97×C211 | C2 | semidirect |
| 4. #382×C211 | C2 | semidirect |
| 9. #127×C211 | C4 | semidirect |
| 27. C2×C8×C211 | D4 | semidirect |
| 65. C211 | #1961 | direct (N × H) |
| 66. #1961 | C211 | direct (N × H) |
| 68. C2×#102 | C2×C211 | semidirect |
| 69. C2×#97 | C2×C211 | semidirect |
| 70. #382 | C2×C211 | semidirect |
| 75. #127 | C4×C211 | semidirect |
| 93. C2×C8 | D4×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#127×C211 | 2 |
| C2×#102×C211 | 2 |
| C2×#97×C211 | 2 |
| #382×C211 | 2 |
| #1961 | 211 |