Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
266 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 266 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 | 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. C22×D8×C211 | C2 | semidirect |
| 2. C2×D16×C211 | C2 | semidirect |
| 3. #466×C211 | C2 | semidirect |
| 4. #357×C211 | C2 | semidirect |
| 5. C22⋊C16×C211 | C2 | semidirect |
| 10. C2×D8×C211 | C22 | semidirect |
| 14. C2×C16×C211 | C22 | semidirect |
| 15. #107×C211 | C22 | semidirect |
| 28. D8×C211 | D4 | semidirect |
| 63. C22×C211 | D16 | semidirect |
| 75. C211 | #2184 | direct (N × H) |
| 76. #2184 | C211 | direct (N × H) |
| 77. C22×D8 | C2×C211 | semidirect |
| 78. C2×D16 | C2×C211 | semidirect |
| 79. #466 | C2×C211 | semidirect |
| 80. #357 | C2×C211 | semidirect |
| 81. C22⋊C16 | C2×C211 | semidirect |
| 86. C2×D8 | C22×C211 | semidirect |
| 90. C2×C16 | C22×C211 | semidirect |
| 91. #107 | C22×C211 | semidirect |
| 104. D8 | D4×C211 | semidirect |
| 139. C22 | D16×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D8×C211 | 2 |
| C2×D16×C211 | 2 |
| #466×C211 | 2 |
| #357×C211 | 2 |
| C22⋊C16×C211 | 2 |
| #2184 | 211 |