Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
1104 conjugacy classes of subgroups; only the 360 normal subgroups are shown.
This group has 1104 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 | C24 |
| commutator | C22 | C23×C4×C211 |
| Frattini, cosocle | C23 | C24×C211 |
| prime radical | C23×D4×C211 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#117×C211 | C2 | semidirect |
| 3. #390×C211 | C2 | semidirect |
| 4. #379×C211 | C2 | semidirect |
| 7. C22×D4×C211 | C4 | semidirect |
| 11. C2×C4⋊C4×C211 | C22 | semidirect |
| 13. C2×C22⋊C4×C211 | C22 | semidirect |
| 14. C2×C42×C211 | C22 | semidirect |
| 45. C2×C4×C211 | C22⋊C4 | semidirect |
| 66. C211 | #2309 | direct (N × H) |
| 67. #2309 | C211 | direct (N × H) |
| 69. C2×#117 | C2×C211 | semidirect |
| 70. #390 | C2×C211 | semidirect |
| 71. #379 | C2×C211 | semidirect |
| 74. C22×D4 | C4×C211 | semidirect |
| 78. C2×C4⋊C4 | C22×C211 | semidirect |
| 80. C2×C22⋊C4 | C22×C211 | semidirect |
| 81. C2×C42 | C22×C211 | semidirect |
| 112. C2×C4 | C22⋊C4×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D4×C211 | 2 |
| C2×#117×C211 | 2 |
| #390×C211 | 2 |
| #379×C211 | 2 |
| #2309 | 211 |