Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
260 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C211 | #117 |
| commutator | C2×C4 | C22×C4×C211 |
| Frattini, cosocle | C22×C4 | C23×C211 |
| socle | C2×C211 | #380 |
| prime radical | C2×#41×C211 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C211 ◁ C2×M16×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #427×C211 | C2 | semidirect |
| 4. H(3,ℤ/4)×C211 | C2 | semidirect |
| 8. C2×M16×C211 | C4 | semidirect |
| 10. C22×C8×C211 | C4 | semidirect |
| 66. C211 | #1812 | direct (N × H) |
| 67. #1812 | C211 | direct (N × H) |
| 69. #427 | C2×C211 | semidirect |
| 71. H(3,ℤ/4) | C2×C211 | semidirect |
| 75. C2×M16 | C4×C211 | semidirect |
| 77. C22×C8 | C4×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#131×C211 | 2 |
| #427×C211 | 2 |
| #410×C211 | 2 |
| H(3,ℤ/4)×C211 | 2 |
| #1812 | 211 |