Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
390 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 116 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 | C2×C22⋊C4 |
| commutator | C23 | C22×C4×C211 |
| Frattini, cosocle | C22×C4 | C23×C211 |
| prime radical | C2×#121×C211 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C211 ◁ C23×C4×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×M16×C211 | C2 | semidirect |
| 3. C2×#100×C211 | C2 | semidirect |
| 4. #407×C211 | C2 | semidirect |
| 5. #343×C211 | C2 | semidirect |
| 45. M16×C211 | D4 | semidirect |
| 97. C22×C211 | #100 | semidirect |
| 113. C211 | #1900 | direct (N × H) |
| 114. #1900 | C211 | direct (N × H) |
| 115. C22×M16 | C2×C211 | semidirect |
| 117. C2×#100 | C2×C211 | semidirect |
| 118. #407 | C2×C211 | semidirect |
| 119. #343 | C2×C211 | semidirect |
| 159. M16 | D4×C211 | semidirect |
| 211. C22 | #100×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×M16×C211 | 2 |
| C2×#121×C211 | 2 |
| C2×#100×C211 | 2 |
| #407×C211 | 2 |
| #343×C211 | 2 |
| #1900 | 211 |