Finite Group DatabasebetaFinite Group DatabasebetaOrder 26816.
290 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 113 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 | C23×D419 |
| commutator, socle, cosocle | C22×C419 | C24 |
| Fitting | C4×Q8×C419 | C2 |
| hypercenter | C4×Q8 | D419 |
| hypocenter | C419 | #548 |
| LCS | G ▷ C22×C419 ▷ C419 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#20269412 | C2 | semidirect |
| 2. C2×Q8×D419 | C2 | semidirect |
| 3. #27211706 | C2 | semidirect |
| 4. #27211698 | C2 | semidirect |
| 5. #27211693 | C2 | semidirect |
| 6. C4×D1676 | C2 | semidirect |
| 7. C4×Q8×C419 | C2 | semidirect |
| 31. Q8×C419 | D4 | semidirect |
| 41. C2×D419 | #41 | semidirect |
| 52. C419 | #548 | semidirect |
| 59. C4×Q8 | D419 | semidirect |
| 78. Q8 | D1676 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#20269412 | 2 |
| C2×Q8×D419 | 2 |
| #27211706 | 2 |
| #27211698 | 2 |
| #27211693 | 2 |
| C4×D1676 | 2 |
| C4×Q8×C419 | 2 |
| #548 | 419 |