Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
174 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 174 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 |
| 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×D16×C211 | C2 | semidirect |
| 2. C22×C16×C211 | C2 | semidirect |
| 3. #466×C211 | C2 | semidirect |
| 4. #366×C211 | C2 | semidirect |
| 5. #357×C211 | C2 | semidirect |
| 27. C16×C211 | D4 | semidirect |
| 46. C22×C211 | D16 | semidirect |
| 57. C211 | #2215 | direct (N × H) |
| 58. #2215 | C211 | direct (N × H) |
| 59. C2×D16 | C2×C211 | semidirect |
| 60. C22×C16 | C2×C211 | semidirect |
| 61. #466 | C2×C211 | semidirect |
| 62. #366 | C2×C211 | semidirect |
| 63. #357 | C2×C211 | semidirect |
| 85. C16 | D4×C211 | semidirect |
| 104. C22 | D16×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D16×C211 | 2 |
| C22×C16×C211 | 2 |
| #466×C211 | 2 |
| #366×C211 | 2 |
| #357×C211 | 2 |
| #2215 | 211 |