Finite Group DatabasebetaFinite Group DatabasebetaOrder 1664.
1492 conjugacy classes of subgroups; only the 553 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 553 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 | C2×C4×Q8×C13 | C2 |
| hypercenter | C2×C4×Q8 | D13 |
| hypocenter | C13 | #3497 |
| LCS | G ▷ C22×C13 ▷ C13 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#88254 | C2 | semidirect |
| 2. C2×C4×Q8×C13 | C2 | semidirect |
| 3. #11719282 | C2 | semidirect |
| 4. #11719281 | C2 | semidirect |
| 5. Q8×D52 | C2 | semidirect |
| 6. #11719215 | C2 | semidirect |
| 7. #11719182 | C2 | semidirect |
| 26. C2×Q8×C13 | D4 | semidirect |
| 71. C2×C4×Q8 | D13 | semidirect |
| 82. C4×C13 | 2−1+4 | semidirect |
| 111. C2×Q8 | D52 | semidirect |
| 128. C13 | #3497 | semidirect |
| 157. C4 | #88254 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#88254 | 2 |
| C2×C4×Q8×C13 | 2 |
| #11719282 | 2 |
| #11719281 | 2 |
| Q8×D52 | 2 |
| #11719215 | 2 |
| #11719182 | 2 |
| #3497 | 13 |