Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
460 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 200 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 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C211 |
| prime radical | C2×#41×C211 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C211 ◁ C2×#41×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q16×C211 | C2 | semidirect |
| 4. #470×C211 | C2 | semidirect |
| 5. #467×C211 | C2 | semidirect |
| 6. #427×C211 | C2 | semidirect |
| 7. #419×C211 | C2 | semidirect |
| 88. C211 | #3059 | direct (N × H) |
| 89. #3059 | C211 | direct (N × H) |
| 90. C22×Q16 | C2×C211 | semidirect |
| 93. #470 | C2×C211 | semidirect |
| 94. #467 | C2×C211 | semidirect |
| 95. #427 | C2×C211 | semidirect |
| 96. #419 | C2×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q16×C211 | 2 |
| C2×#131×C211 | 2 |
| #536×C211 | 2 |
| #470×C211 | 2 |
| #467×C211 | 2 |
| #427×C211 | 2 |
| #419×C211 | 2 |
| #3059 | 211 |