Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
572 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 572 conjugacy classes of subgroups — too many to draw. Only its 224 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C211 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C211 |
| socle | C22×C211 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C211 ◁ C2×C42×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#135×C211 | C2 | semidirect |
| 2. #532×C211 | C2 | semidirect |
| 3. #495×C211 | C2 | semidirect |
| 4. #494×C211 | C2 | semidirect |
| 5. #493×C211 | C2 | semidirect |
| 6. #492×C211 | C2 | semidirect |
| 7. C2×C4×C8×C211 | C2 | semidirect |
| 34. C2×C8×C211 | D4 | semidirect |
| 66. C4×C211 | #135 | semidirect |
| 72. C211 | #3147 | direct (N × H) |
| 73. #3147 | C211 | direct (N × H) |
| 74. C2×#135 | C2×C211 | semidirect |
| 75. #532 | C2×C211 | semidirect |
| 76. #495 | C2×C211 | semidirect |
| 77. #494 | C2×C211 | semidirect |
| 78. #493 | C2×C211 | semidirect |
| 79. #492 | C2×C211 | semidirect |
| 80. C2×C4×C8 | C2×C211 | semidirect |
| 107. C2×C8 | D4×C211 | semidirect |
| 139. C4 | #135×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#135×C211 | 2 |
| #532×C211 | 2 |
| #495×C211 | 2 |
| #494×C211 | 2 |
| #493×C211 | 2 |
| #492×C211 | 2 |
| C2×C4×C8×C211 | 2 |
| #3147 | 211 |