Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
524 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 524 conjugacy classes of subgroups — too many to draw. Only its 200 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 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C211 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C211 ◁ C2×#41×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×#131×C211 | C2 | semidirect |
| 3. #535×C211 | C2 | semidirect |
| 4. #469×C211 | C2 | semidirect |
| 5. #466×C211 | C2 | semidirect |
| 6. #427×C211 | C2 | semidirect |
| 7. #418×C211 | C2 | semidirect |
| 15. C2×M16×C211 | C22 | semidirect |
| 17. C22×C8×C211 | C22 | semidirect |
| 20. #121×C211 | C22 | semidirect |
| 23. #107×C211 | C22 | semidirect |
| 24. #106×C211 | C22 | semidirect |
| 25. #102×C211 | C22 | semidirect |
| 44. C2×C8×C211 | D4 | semidirect |
| 93. C211 | #3058 | direct (N × H) |
| 94. #3058 | C211 | direct (N × H) |
| 95. C22×D8 | C2×C211 | semidirect |
| 96. C2×#131 | C2×C211 | semidirect |
| 97. #535 | C2×C211 | semidirect |
| 98. #469 | C2×C211 | semidirect |
| 99. #466 | C2×C211 | semidirect |
| 100. #427 | C2×C211 | semidirect |
| 101. #418 | C2×C211 | semidirect |
| 109. C2×M16 | C22×C211 | semidirect |
| 111. C22×C8 | C22×C211 | semidirect |
| 114. #121 | C22×C211 | semidirect |
| 117. #107 | C22×C211 | semidirect |
| 118. #106 | C22×C211 | semidirect |
| 119. #102 | C22×C211 | semidirect |
| 138. C2×C8 | D4×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×#131×C211 | 2 |
| #535×C211 | 2 |
| #469×C211 | 2 |
| #466×C211 | 2 |
| #427×C211 | 2 |
| #418×C211 | 2 |
| #3058 | 211 |