Finite Group DatabasebetaFinite Group DatabasebetaOrder 1664.
1068 conjugacy classes of subgroups; only the 529 normal subgroups are shown.
This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 529 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 | C24×D13 |
| commutator, socle, cosocle | C22×C13 | C25 |
| Fitting | Q82×C13 | C2 |
| prime radical | C2×#6131 | C22 |
| hypercenter | Q82 | D13 |
| hypocenter | C13 | Q8×#41 |
| LCS | G ▷ C22×C13 ▷ C13 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q82×C13 | C2 | semidirect |
| 2. #11719435 | C2 | semidirect |
| 3. #11719433 | C2 | semidirect |
| 4. Q8×D52 | C2 | semidirect |
| 6. C4×Q8×D13 | C2 | semidirect |
| 7. #11719275 | C2 | semidirect |
| 30. #6131 | Q8 | direct (N × H) |
| 51. Q8×C13 | C2×Q8 | semidirect |
| 52. Q8×C13 | #41 | semidirect |
| 59. D52 | C2×Q8 | semidirect |
| 62. C4×D13 | C2×Q8 | semidirect |
| 64. Q82 | D13 | semidirect |
| 125. C13 | Q8×#41 | semidirect |
| 129. Q8 | Q8×D13 | semidirect |
| 130. Q8 | #6131 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q82×C13 | 2 |
| #11719435 | 2 |
| #11719433 | 2 |
| Q8×D52 | 2 |
| C4×#6131 | 2 |
| C4×Q8×D13 | 2 |
| #11719275 | 2 |
| Q8×#41 | 13 |