Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
756 conjugacy classes of subgroups; only the 340 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 340 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C211 | C22×C22⋊C4 |
| commutator | C22 | C23×C4×C211 |
| Frattini, cosocle | C2×C4 | C24×C211 |
| prime radical | C2×H(5,𝔽2)×C211 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C211 ◁ C2×#41×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #568×C211 | C2 | semidirect |
| 3. #413×C211 | C2 | semidirect |
| 4. C2×#100×C211 | C2 | semidirect |
| 12. C2×M16×C211 | C22 | semidirect |
| 13. #101×C211 | C22 | semidirect |
| 14. #100×C211 | C22 | semidirect |
| 61. C211 | #2888 | direct (N × H) |
| 62. #2888 | C211 | direct (N × H) |
| 64. #568 | C2×C211 | semidirect |
| 65. #413 | C2×C211 | semidirect |
| 66. C2×#100 | C2×C211 | semidirect |
| 74. C2×M16 | C22×C211 | semidirect |
| 75. #101 | C22×C211 | semidirect |
| 76. #100 | C22×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(5,𝔽2)×C211 | 2 |
| #568×C211 | 2 |
| #413×C211 | 2 |
| C2×#100×C211 | 2 |
| #2888 | 211 |