Finite Group DatabasebetaFinite Group DatabasebetaOrder 3712.
1620 conjugacy classes of subgroups; only the 573 normal subgroups are shown.
This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 573 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D29 |
| commutator, cosocle | C22×C29 | C25 |
| Frattini | C22 | C24×D29 |
| Fitting | C2×C4×Q8×C29 | C2 |
| socle | C23×C29 | C24 |
| hypercenter | C2×C4×Q8 | D29 |
| hypocenter | C29 | C2×#548 |
| LCS | G ▷ C22×C29 ▷ C29 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#90809 | C2 | semidirect |
| 2. C22×Q8×D29 | C2 | semidirect |
| 3. #14271825 | C2 | direct (N × H) |
| 4. C2×#11745237 | C2 | semidirect |
| 5. C2×#11745259 | C2 | semidirect |
| 6. C2×#11745257 | C2 | semidirect |
| 7. C2×C4×D116 | C2 | semidirect |
| 8. C2×C4×Q8×C29 | C2 | semidirect |
| 11. C2×#90809 | C22 | semidirect |
| 13. C2×Q8×D29 | C22 | semidirect |
| 19. C4×Q8×C29 | C22 | semidirect |
| 26. #11745259 | C22 | semidirect |
| 27. #11745257 | C22 | semidirect |
| 28. #11745237 | C22 | semidirect |
| 29. C4×D116 | C22 | semidirect |
| 32. C2×Q8×C29 | D4 | semidirect |
| 63. C22×D29 | #41 | semidirect |
| 66. Q8×C29 | C2×D4 | semidirect |
| 94. C2×D29 | C2×#41 | semidirect |
| 106. C2×C4×Q8 | D29 | semidirect |
| 108. C2×C29 | #548 | semidirect |
| 127. C4×Q8 | C2×D29 | semidirect |
| 135. C29 | C2×#548 | semidirect |
| 142. C2×Q8 | D116 | semidirect |
| 173. Q8 | C2×D116 | semidirect |
| 207. C2 | #14271825 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#90809 | 2 |
| C22×Q8×D29 | 2 |
| #14271825 | 2 |
| C2×#11745237 | 2 |
| C2×#11745259 | 2 |
| C2×#11745257 | 2 |
| C2×C4×D116 | 2 |
| C2×C4×Q8×C29 | 2 |
| C2×#548 | 29 |