Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
368 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 88 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 | #99 |
| commutator | C2×C4 | C22×C4×C211 |
| Frattini, cosocle | C2×D4 | C23×C211 |
| prime radical | C2×#127×C211 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C211 ◁ C23×C211 ◁ C22×D4×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#99×C211 | C2 | semidirect |
| 3. #353×C211 | C2 | direct (N × H) |
| 5. C22×D4×C211 | C4 | semidirect |
| 8. #127×C211 | C4 | semidirect |
| 11. C2×C42×C211 | C4 | semidirect |
| 13. #99×C211 | C22 | semidirect |
| 25. C2×D4×C211 | C2×C4 | semidirect |
| 34. C42×C211 | C2×C4 | semidirect |
| 83. C2×C211 | #353 | direct (N × H) |
| 85. C211 | C2×#353 | direct (N × H) |
| 86. C2×#353 | C211 | direct (N × H) |
| 88. C2×#99 | C2×C211 | semidirect |
| 89. #353 | C2×C211 | direct (N × H) |
| 91. C22×D4 | C4×C211 | semidirect |
| 94. #127 | C4×C211 | semidirect |
| 97. C2×C42 | C4×C211 | semidirect |
| 99. #99 | C22×C211 | semidirect |
| 111. C2×D4 | C2×C4×C211 | semidirect |
| 120. C42 | C2×C4×C211 | semidirect |
| 169. C2 | #353×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×#99×C211 | 2 |
| #353×C211 | 2 |
| C2×#353 | 211 |