Finite Group DatabasebetaFinite Group DatabasebetaOrder 1664.
1736 conjugacy classes of subgroups; only the 581 normal subgroups are shown.
This group has 1736 conjugacy classes of subgroups — too many to draw. Only its 581 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×#550 |
| 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. C2×C4×Q8×C13 | C2 | semidirect |
| 3. #11719282 | C2 | direct (N × H) |
| 4. C2×#88149 | C2 | semidirect |
| 5. C2×#88128 | C2 | semidirect |
| 6. C2×C4×D52 | C2 | semidirect |
| 9. C2×#6131 | C22 | semidirect |
| 14. C4×Q8×C13 | C22 | semidirect |
| 19. #88149 | C22 | semidirect |
| 20. #88128 | C22 | semidirect |
| 21. C4×D52 | C22 | semidirect |
| 23. C2×Q8×C13 | D4 | semidirect |
| 48. Q8×C13 | C2×D4 | semidirect |
| 51. C2×C4×C13 | #41 | semidirect |
| 65. C2×C4×Q8 | D13 | semidirect |
| 77. C4×C13 | C2×#41 | semidirect |
| 88. C4×Q8 | C2×D13 | semidirect |
| 94. C2×C13 | #550 | semidirect |
| 101. C2×Q8 | D52 | semidirect |
| 119. C13 | C2×#550 | semidirect |
| 125. Q8 | C2×D52 | semidirect |
| 134. C2×C4 | #6131 | semidirect |
| 149. C4 | C2×#6131 | semidirect |
| 151. C2 | #11719282 | 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 |
| C2×C4×Q8×C13 | 2 |
| #11719282 | 2 |
| C2×#88149 | 2 |
| C2×#88128 | 2 |
| C2×C4×D52 | 2 |
| C2×#550 | 13 |