Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
351 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 351 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#494 | C2 | semidirect |
| 4. #1670 | C2 | semidirect |
| 5. #1669 | C2 | semidirect |
| 6. #1567 | C2 | semidirect |
| 7. #1549 | C2 | semidirect |
| 70. #41 | Q16 | semidirect |
| 116. Q8 | C2×Q16 | semidirect |
| 119. D4 | C2×Q16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#494 | 2 |
| #3079 | 2 |
| C8×#41 | 2 |
| #1670 | 2 |
| #1669 | 2 |
| #1567 | 2 |
| #1549 | 2 |