Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
154 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 154 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 | C2×C211 | #442 |
| commutator | C8 | C22×C4×C211 |
| Frattini, cosocle | C2×C8 | C23×C211 |
| prime radical | #509×C211 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C211 ◁ C2×C4×C211 ◁ #97×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #444×C211 | C2 | semidirect |
| 3. #442×C211 | C2 | semidirect |
| 4. #365×C211 | C2 | semidirect |
| 5. #361×C211 | C2 | semidirect |
| 6. #360×C211 | C2 | semidirect |
| 7. #347×C211 | C2 | semidirect |
| 12. SD32×C211 | C4 | semidirect |
| 13. D16×C211 | C4 | semidirect |
| 24. Q16×C211 | C2×C4 | semidirect |
| 28. D8×C211 | C2×C4 | semidirect |
| 38. C16×C211 | C2×C4 | semidirect |
| 68. C211 | AGL(1,ℤ/16) | direct (N × H) |
| 69. AGL(1,ℤ/16) | C211 | direct (N × H) |
| 71. #444 | C2×C211 | semidirect |
| 72. #442 | C2×C211 | semidirect |
| 73. #365 | C2×C211 | semidirect |
| 74. #361 | C2×C211 | semidirect |
| 75. #360 | C2×C211 | semidirect |
| 76. #347 | C2×C211 | semidirect |
| 81. SD32 | C4×C211 | semidirect |
| 82. D16 | C4×C211 | semidirect |
| 93. Q16 | C2×C4×C211 | semidirect |
| 97. D8 | C2×C4×C211 | semidirect |
| 107. C16 | C2×C4×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #509×C211 | 2 |
| #444×C211 | 2 |
| #442×C211 | 2 |
| #365×C211 | 2 |
| #361×C211 | 2 |
| #360×C211 | 2 |
| #347×C211 | 2 |
| AGL(1,ℤ/16) | 211 |