Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
198 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 76 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 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C211 |
| prime radical | C2×#41×C211 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C211 ◁ C22×C4×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#106×C211 | C2 | semidirect |
| 4. #343×C211 | C2 | semidirect |
| 5. #341×C211 | C2 | semidirect |
| 63. C211 | #2079 | direct (N × H) |
| 64. #2079 | C211 | direct (N × H) |
| 65. C2×#106 | C2×C211 | semidirect |
| 68. #343 | C2×C211 | semidirect |
| 69. #341 | C2×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#106×C211 | 2 |
| #417×C211 | 2 |
| #408×C211 | 2 |
| #343×C211 | 2 |
| #341×C211 | 2 |
| #2079 | 211 |