Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
924 conjugacy classes of subgroups; only the 368 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 368 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C211 | C23 |
| commutator | C22 | C23×C4×C211 |
| Frattini, cosocle | C23 | C24×C211 |
| socle | C23×C211 | C24 |
| prime radical | C2×#127×C211 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4×D4×C211 | C2 | semidirect |
| 3. #390×C211 | C2 | semidirect |
| 4. #384×C211 | C2 | semidirect |
| 5. C43×C211 | C2 | semidirect |
| 8. #127×C211 | C4 | direct (N × H) |
| 18. C2×D4×C211 | C2×C4 | semidirect |
| 27. C42×C211 | C2×C4 | semidirect |
| 28. C42×C211 | D4 | semidirect |
| 50. C4×C211 | C4×D4 | semidirect |
| 51. C4×C211 | #127 | direct (N × H) |
| 56. C211 | C4×#127 | direct (N × H) |
| 57. C4×#127 | C211 | direct (N × H) |
| 59. C2×C4×D4 | C2×C211 | semidirect |
| 60. #390 | C2×C211 | semidirect |
| 61. #384 | C2×C211 | semidirect |
| 62. C43 | C2×C211 | semidirect |
| 65. #127 | C4×C211 | direct (N × H) |
| 75. C2×D4 | C2×C4×C211 | semidirect |
| 84. C42 | C2×C4×C211 | semidirect |
| 85. C42 | D4×C211 | semidirect |
| 107. C4 | C4×D4×C211 | semidirect |
| 108. C4 | #127×C211 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#127×C211 | 2 |
| C2×C4×D4×C211 | 2 |
| #390×C211 | 2 |
| #384×C211 | 2 |
| C43×C211 | 2 |
| C4×#127 | 211 |