Finite Group DatabasebetaFinite Group DatabasebetaOrder 13504.
250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 250 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 | C22⋊Q8×C211 | C2 |
| hypercenter | C22⋊Q8 | D211 |
| hypocenter | C211 | #538 |
| LCS | G ▷ C22×C211 ▷ C211 | |
| UCS | C1 ◁ C22 ◁ C22⋊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. #20269962 | C2 | semidirect |
| 4. #20269952 | C2 | semidirect |
| 5. #20269951 | C2 | semidirect |
| 6. #20269947 | C2 | semidirect |
| 7. (C22⋊D211)⋊C4 | C2 | semidirect |
| 8. C2×(C2×D211)⋊C4 | C2 | semidirect |
| 9. C22⋊Q8×C211 | C2 | semidirect |
| 13. C2×D844 | C22 | semidirect |
| 16. C2×C4×D211 | C22 | semidirect |
| 17. C4⋊Dic211 | C22 | semidirect |
| 18. Dic211⋊C4 | C22 | semidirect |
| 23. (C2×D211)⋊C4 | C22 | semidirect |
| 24. (C2×D211)⋊C4 | C22 | semidirect |
| 25. C4×Dic211 | C22 | semidirect |
| 26. C2×Q8×C211 | C22 | semidirect |
| 28. C4⋊C4×C211 | C22 | semidirect |
| 29. C4⋊C4×C211 | C22 | semidirect |
| 74. C211 | #538 | semidirect |
| 79. C22⋊Q8 | D211 | semidirect |
| 86. C2×Q8 | C2×D211 | semidirect |
| 96. C4⋊C4 | C2×D211 | semidirect |
| 97. C4⋊C4 | C2×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 |
| #20269962 | 2 |
| #20269952 | 2 |
| #20269951 | 2 |
| #20269947 | 2 |
| (C22⋊D211)⋊C4 | 2 |
| C2×(C2×D211)⋊C4 | 2 |
| C22⋊Q8×C211 | 2 |
| #538 | 211 |