Finite Group DatabasebetaFinite Group DatabasebetaOrder 1664.
696 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D13 |
| commutator, socle, cosocle | C23×C13 | C24 |
| Fitting | C22×C42×C13 | C2 |
| prime radical | C2×#88184 | C2 |
| hypercenter | C22×C42 | D13 |
| hypocenter | C13 | #2850 |
| LCS | G ▷ C23×C13 ▷ C13 | |
| UCS | C1 ◁ C23 ◁ C22×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C42×C13 | C2 | semidirect |
| 3. C2×#88178 | C2 | semidirect |
| 4. C2×#88129 | C2 | semidirect |
| 5. #11718874 | C2 | semidirect |
| 6. #11718601 | C2 | semidirect |
| 7. #11718600 | C2 | semidirect |
| 8. #11718597 | C2 | semidirect |
| 9. #11718590 | C2 | semidirect |
| 56. C42×C13 | D4 | semidirect |
| 103. C22×C42 | D13 | semidirect |
| 113. C22×C13 | #124 | semidirect |
| 191. C42 | C22⋊D13 | semidirect |
| 192. C13 | #2850 | semidirect |
| 237. C22 | #88129 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C42×C13 | 2 |
| C2×#88184 | 2 |
| C2×#88178 | 2 |
| C2×#88129 | 2 |
| #11718874 | 2 |
| #11718601 | 2 |
| #11718600 | 2 |
| #11718597 | 2 |
| #11718590 | 2 |
| #2850 | 13 |