Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
204 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 204 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 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C211 |
| 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. #493×C211 | C2 | semidirect |
| 2. #459×C211 | C2 | semidirect |
| 3. C8×D4×C211 | C2 | semidirect |
| 4. #335×C211 | C2 | semidirect |
| 5. #331×C211 | C2 | semidirect |
| 41. D4×C211 | D8 | semidirect |
| 51. C8×C211 | D8 | semidirect |
| 70. C211 | #1667 | direct (N × H) |
| 71. #1667 | C211 | direct (N × H) |
| 72. #493 | C2×C211 | semidirect |
| 73. #459 | C2×C211 | semidirect |
| 74. C8×D4 | C2×C211 | semidirect |
| 75. #335 | C2×C211 | semidirect |
| 76. #331 | C2×C211 | semidirect |
| 112. D4 | D8×C211 | semidirect |
| 122. C8 | D8×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #493×C211 | 2 |
| #459×C211 | 2 |
| C8×D4×C211 | 2 |
| #335×C211 | 2 |
| #331×C211 | 2 |
| #1667 | 211 |