Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
260 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 260 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 | C2×C8×C211 | C23 |
| commutator | C22 | C22×C8×C211 |
| Frattini, cosocle | C2×C8 | C23×C211 |
| socle | C22×C211 | C22×C8 |
| prime radical | C24×C211 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×M32×C211 | C2 | semidirect |
| 3. C22⋊C16×C211 | C2 | semidirect |
| 11. C2×C16×C211 | C22 | semidirect |
| 48. C22×C211 | M32 | semidirect |
| 58. C211 | #2112 | direct (N × H) |
| 59. #2112 | C211 | direct (N × H) |
| 61. C2×M32 | C2×C211 | semidirect |
| 62. C22⋊C16 | C2×C211 | semidirect |
| 70. C2×C16 | C22×C211 | semidirect |
| 107. C22 | M32×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C8×C211 | 2 |
| C2×M32×C211 | 2 |
| C22⋊C16×C211 | 2 |
| #2112 | 211 |