Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
172 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C211 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C211 |
| prime radical | #127×C211 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C211 ◁ C42×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #475×C211 | C2 | semidirect |
| 4. C8×Q8×C211 | C2 | semidirect |
| 5. #334×C211 | C2 | semidirect |
| 6. #332×C211 | C2 | semidirect |
| 46. Q8×C211 | SD16 | semidirect |
| 62. C8×C211 | SD16 | semidirect |
| 79. C211 | #1666 | direct (N × H) |
| 80. #1666 | C211 | direct (N × H) |
| 82. #475 | C2×C211 | semidirect |
| 84. C8×Q8 | C2×C211 | semidirect |
| 85. #334 | C2×C211 | semidirect |
| 86. #332 | C2×C211 | semidirect |
| 126. Q8 | SD16×C211 | semidirect |
| 142. C8 | SD16×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #492×C211 | 2 |
| #475×C211 | 2 |
| #460×C211 | 2 |
| C8×Q8×C211 | 2 |
| #334×C211 | 2 |
| #332×C211 | 2 |
| #331×C211 | 2 |
| #1666 | 211 |