Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
1000 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 224 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, Frattini, cosocle | C23 | C24×C211 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#120×C211 | C2 | semidirect |
| 2. #394×C211 | C2 | semidirect |
| 3. #390×C211 | C2 | semidirect |
| 4. C25×C211 | C22 | semidirect |
| 8. C2×C4⋊C4×C211 | C22 | semidirect |
| 9. C2×C22⋊C4×C211 | C22 | semidirect |
| 10. C2×C22⋊C4×C211 | C22 | semidirect |
| 11. C2×C42×C211 | C22 | semidirect |
| 12. #95×C211 | C22 | semidirect |
| 21. C2×D4×C211 | D4 | semidirect |
| 59. C211 | #2679 | direct (N × H) |
| 60. #2679 | C211 | direct (N × H) |
| 61. C2×#120 | C2×C211 | semidirect |
| 62. #394 | C2×C211 | semidirect |
| 63. #390 | C2×C211 | semidirect |
| 64. C25 | C22×C211 | semidirect |
| 68. C2×C4⋊C4 | C22×C211 | semidirect |
| 69. C2×C22⋊C4 | C22×C211 | semidirect |
| 70. C2×C22⋊C4 | C22×C211 | semidirect |
| 71. C2×C42 | C22×C211 | semidirect |
| 72. #95 | C22×C211 | semidirect |
| 81. C2×D4 | D4×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#120×C211 | 2 |
| #394×C211 | 2 |
| #390×C211 | 2 |
| #2679 | 211 |