Finite Group DatabasebetaFinite Group DatabasebetaOrder 1664.
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×D13 |
| commutator, cosocle | C22×C13 | C25 |
| Frattini | C22 | C24×D13 |
| Fitting | C2×C4×Q8×C13 | C2 |
| socle | C23×C13 | C24 |
| hypercenter | C2×C4×Q8 | D13 |
| hypocenter | C13 | C2×#548 |
| LCS | G ▷ C22×C13 ▷ C13 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#6131 | C2 | semidirect |
| 2. C22×Q8×D13 | C2 | semidirect |
| 3. C2×C4×Q8×C13 | C2 | semidirect |
| 4. #11719281 | C2 | direct (N × H) |
| 5. C2×#88151 | C2 | semidirect |
| 6. C2×#88149 | C2 | semidirect |
| 7. C2×#88129 | C2 | semidirect |
| 8. C2×C4×D52 | C2 | semidirect |
| 11. C2×#6131 | C22 | semidirect |
| 13. C2×Q8×D13 | C22 | semidirect |
| 19. C4×Q8×C13 | C22 | semidirect |
| 26. #88151 | C22 | semidirect |
| 27. #88149 | C22 | semidirect |
| 28. #88129 | C22 | semidirect |
| 29. C4×D52 | C22 | semidirect |
| 32. C2×Q8×C13 | D4 | semidirect |
| 64. C22×D13 | #41 | semidirect |
| 67. Q8×C13 | C2×D4 | semidirect |
| 89. C2×C4×Q8 | D13 | semidirect |
| 102. C2×D13 | C2×#41 | semidirect |
| 121. C4×Q8 | C2×D13 | semidirect |
| 129. C2×C13 | #548 | semidirect |
| 141. C2×Q8 | D52 | semidirect |
| 162. C13 | C2×#548 | semidirect |
| 173. Q8 | C2×D52 | semidirect |
| 207. C2 | #11719281 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#6131 | 2 |
| C22×Q8×D13 | 2 |
| C2×C4×Q8×C13 | 2 |
| #11719281 | 2 |
| C2×#88151 | 2 |
| C2×#88149 | 2 |
| C2×#88129 | 2 |
| C2×C4×D52 | 2 |
| C2×#548 | 13 |