Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
186 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 72 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 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C211 |
| prime radical | #127×C211 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C211 ◁ C42×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #498×C211 | C2 | semidirect |
| 2. #494×C211 | C2 | semidirect |
| 3. #487×C211 | C2 | semidirect |
| 7. #322×C211 | C2 | semidirect |
| 68. C211 | #1720 | direct (N × H) |
| 69. #1720 | C211 | direct (N × H) |
| 70. #498 | C2×C211 | semidirect |
| 71. #494 | C2×C211 | semidirect |
| 72. #487 | C2×C211 | semidirect |
| 76. #322 | C2×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #498×C211 | 2 |
| #494×C211 | 2 |
| #487×C211 | 2 |
| #486×C211 | 2 |
| #322×C211 | 2 |
| #1720 | 211 |