Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
154 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C211 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C211 |
| prime radical | C23×C211 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C211 ◁ C22×C4×C211 ◁ C22×C8×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q32×C211 | C2 | semidirect |
| 4. #366×C211 | C2 | semidirect |
| 5. #358×C211 | C2 | semidirect |
| 42. C22×C211 | Q32 | semidirect |
| 53. C211 | #2216 | direct (N × H) |
| 54. #2216 | C211 | direct (N × H) |
| 55. C2×Q32 | C2×C211 | semidirect |
| 58. #366 | C2×C211 | semidirect |
| 59. #358 | C2×C211 | semidirect |
| 96. C22 | Q32×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q32×C211 | 2 |
| C22×C16×C211 | 2 |
| #467×C211 | 2 |
| #366×C211 | 2 |
| #358×C211 | 2 |
| #2216 | 211 |