Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
304 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C211 | #386 |
| commutator | C2×C4 | C22×C4×C211 |
| Frattini, cosocle | C22×C4 | C23×C211 |
| prime radical | C2×#41×C211 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C211 ◁ C22×C4×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #419×C211 | C2 | semidirect |
| 5. C2×#101×C211 | C2 | semidirect |
| 6. #344×C211 | C2 | semidirect |
| 7. #338×C211 | C2 | semidirect |
| 12. C2×Q16×C211 | C4 | semidirect |
| 13. C2×SD16×C211 | C4 | semidirect |
| 107. C211 | #1902 | direct (N × H) |
| 108. #1902 | C211 | direct (N × H) |
| 112. #419 | C2×C211 | semidirect |
| 113. C2×#101 | C2×C211 | semidirect |
| 114. #344 | C2×C211 | semidirect |
| 115. #338 | C2×C211 | semidirect |
| 120. C2×Q16 | C4×C211 | semidirect |
| 121. C2×SD16 | C4×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#137×C211 | 2 |
| #568×C211 | 2 |
| #421×C211 | 2 |
| #419×C211 | 2 |
| C2×#101×C211 | 2 |
| #344×C211 | 2 |
| #338×C211 | 2 |
| #1902 | 211 |