Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
672 conjugacy classes of subgroups; only the 280 normal subgroups are shown.
This group has 672 conjugacy classes of subgroups — too many to draw. Only its 280 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 | C25×C211 | C22 |
| 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. #388×C211 | C2 | semidirect |
| 3. #380×C211 | C2 | semidirect |
| 5. C4×C22⋊C4×C211 | C2 | semidirect |
| 9. C2×C4⋊C4×C211 | C22 | semidirect |
| 10. C2×C22⋊C4×C211 | C4 | semidirect |
| 15. C2×C42×C211 | C22 | semidirect |
| 16. #95×C211 | C22 | semidirect |
| 17. #95×C211 | C22 | semidirect |
| 81. C211 | #2312 | direct (N × H) |
| 82. #2312 | C211 | direct (N × H) |
| 84. #388 | C2×C211 | semidirect |
| 85. #380 | C2×C211 | semidirect |
| 87. C4×C22⋊C4 | C2×C211 | semidirect |
| 91. C2×C4⋊C4 | C22×C211 | semidirect |
| 92. C2×C22⋊C4 | C4×C211 | semidirect |
| 97. C2×C42 | C22×C211 | semidirect |
| 98. #95 | C22×C211 | semidirect |
| 99. #95 | C22×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C22⋊C4×C211 | 2 |
| #388×C211 | 2 |
| #380×C211 | 2 |
| #379×C211 | 2 |
| C4×C22⋊C4×C211 | 2 |
| #2312 | 211 |