Finite Group DatabasebetaFinite Group DatabasebetaOrder 13504.
216 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D211 |
| commutator, socle, cosocle | C22×C211 | C24 |
| Fitting | C4⋊Q8×C211 | C2 |
| prime radical | #20269952 | C2 |
| hypercenter | C4⋊Q8 | D211 |
| hypocenter | C211 | #542 |
| LCS | G ▷ C22×C211 ▷ C211 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20269972 | C2 | semidirect |
| 2. #20269966 | C2 | semidirect |
| 3. #20269965 | C2 | semidirect |
| 4. #20269962 | C2 | semidirect |
| 5. #20269957 | C2 | semidirect |
| 6. #20269953 | C2 | semidirect |
| 8. #20269942 | C2 | semidirect |
| 9. #20269939 | C2 | semidirect |
| 10. #20269934 | C2 | semidirect |
| 11. C4⋊Q8×C211 | C2 | semidirect |
| 73. C211 | #542 | semidirect |
| 75. C4⋊Q8 | D211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20269972 | 2 |
| #20269966 | 2 |
| #20269965 | 2 |
| #20269962 | 2 |
| #20269957 | 2 |
| #20269953 | 2 |
| #20269952 | 2 |
| #20269942 | 2 |
| #20269939 | 2 |
| #20269934 | 2 |
| C4⋊Q8×C211 | 2 |
| #542 | 211 |