Finite Group DatabasebetaFinite Group DatabasebetaOrder 1408.
688 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 688 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×D11 |
| commutator, socle, cosocle | C23×C11 | C24 |
| Fitting | C22×C42×C11 | C2 |
| prime radical | C2×#66983 | C2 |
| hypercenter | C22×C42 | D11 |
| hypocenter | C11 | #2854 |
| LCS | G ▷ C23×C11 ▷ C11 | |
| UCS | C1 ◁ C23 ◁ C22×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C42×C11 | C2 | semidirect |
| 3. C2×#66982 | C2 | semidirect |
| 4. #10622826 | C2 | semidirect |
| 5. #10622825 | C2 | semidirect |
| 6. #10622553 | C2 | semidirect |
| 7. #10622552 | C2 | semidirect |
| 8. #10622541 | C2 | semidirect |
| 36. C22×C4×C11 | D4 | semidirect |
| 92. C22×C42 | D11 | semidirect |
| 116. C4×C11 | #123 | semidirect |
| 159. C22×C4 | D44 | semidirect |
| 174. C11 | #2854 | semidirect |
| 227. C4 | #66982 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C42×C11 | 2 |
| C2×#66983 | 2 |
| C2×#66982 | 2 |
| #10622826 | 2 |
| #10622825 | 2 |
| #10622553 | 2 |
| #10622552 | 2 |
| #10622541 | 2 |
| #2854 | 11 |