Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
1144 conjugacy classes of subgroups; only the 440 normal subgroups are shown.
This group has 1144 conjugacy classes of subgroups — too many to draw. Only its 440 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C211 | C23 |
| commutator | C22 | C23×C4×C211 |
| Frattini, cosocle | C23 | C24×C211 |
| prime radical | C25×C211 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #381×C211 | C2 | direct (N × H) |
| 4. C2×#95×C211 | C2 | semidirect |
| 7. C23×C4×C211 | C4 | semidirect |
| 13. #95×C211 | C22 | semidirect |
| 25. C22×C4×C211 | C2×C4 | semidirect |
| 59. C2×C211 | #381 | direct (N × H) |
| 63. C211 | C2×#381 | direct (N × H) |
| 64. C2×#381 | C211 | direct (N × H) |
| 67. #381 | C2×C211 | direct (N × H) |
| 68. C2×#95 | C2×C211 | semidirect |
| 71. C23×C4 | C4×C211 | semidirect |
| 77. #95 | C22×C211 | semidirect |
| 89. C22×C4 | C2×C4×C211 | semidirect |
| 123. C2 | #381×C211 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C4×C211 | 2 |
| C22×C22⋊C4×C211 | 2 |
| #381×C211 | 2 |
| C2×#95×C211 | 2 |
| C2×#381 | 211 |