Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
992 conjugacy classes of subgroups; only the 360 normal subgroups are shown.
This group has 992 conjugacy classes of subgroups — too many to draw. Only its 360 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 | C22⋊C4 |
| commutator | C22 | C23×C4×C211 |
| Frattini, cosocle | C2×C4 | C24×C211 |
| prime radical | C23×D4×C211 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C211 ◁ C23×C4×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×M16×C211 | C2 | semidirect |
| 3. C2×#100×C211 | C2 | direct (N × H) |
| 11. C2×M16×C211 | C22 | semidirect |
| 13. #100×C211 | C22 | direct (N × H) |
| 30. M16×C211 | C23 | semidirect |
| 57. C22×C211 | #100 | direct (N × H) |
| 66. C2×C211 | C2×#100 | direct (N × H) |
| 68. C211 | C22×#100 | direct (N × H) |
| 69. C22×#100 | C211 | direct (N × H) |
| 71. C22×M16 | C2×C211 | semidirect |
| 72. C2×#100 | C2×C211 | direct (N × H) |
| 80. C2×M16 | C22×C211 | semidirect |
| 82. #100 | C22×C211 | direct (N × H) |
| 99. M16 | C23×C211 | semidirect |
| 126. C22 | #100×C211 | direct (N × H) |
| 135. C2 | C2×#100×C211 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D4×C211 | 2 |
| C22×M16×C211 | 2 |
| C2×#100×C211 | 2 |
| C22×#100 | 211 |