Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
274 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C211 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C211 |
| prime radical | C22×D4×C211 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C211 ◁ C2×C42×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#107×C211 | C2 | semidirect |
| 2. C2×#106×C211 | C2 | semidirect |
| 6. #382×C211 | C2 | semidirect |
| 7. #336×C211 | C2 | semidirect |
| 99. C211 | #2092 | direct (N × H) |
| 100. #2092 | C211 | direct (N × H) |
| 101. C2×#107 | C2×C211 | semidirect |
| 102. C2×#106 | C2×C211 | semidirect |
| 106. #382 | C2×C211 | semidirect |
| 107. #336 | C2×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#107×C211 | 2 |
| C2×#106×C211 | 2 |
| C2×#102×C211 | 2 |
| #390×C211 | 2 |
| #382×C211 | 2 |
| #336×C211 | 2 |
| #2092 | 211 |