Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
412 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C211 | C23 |
| commutator | C22 | C2×C42×C211 |
| Frattini, cosocle | C22×C4 | C23×C211 |
| socle | C23×C211 | C22×C4 |
| prime radical | C24×C211 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#97×C211 | C2 | semidirect |
| 4. #336×C211 | C2 | semidirect |
| 12. C22⋊C8×C211 | C4 | semidirect |
| 27. C2×C8×C211 | C2×C4 | semidirect |
| 56. C22×C211 | #97 | semidirect |
| 72. C211 | #1749 | direct (N × H) |
| 73. #1749 | C211 | direct (N × H) |
| 76. C2×#97 | C2×C211 | semidirect |
| 77. #336 | C2×C211 | semidirect |
| 85. C22⋊C8 | C4×C211 | semidirect |
| 100. C2×C8 | C2×C4×C211 | semidirect |
| 129. C22 | #97×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C42×C211 | 2 |
| C2×C22⋊C8×C211 | 2 |
| C2×#97×C211 | 2 |
| #336×C211 | 2 |
| #1749 | 211 |