Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
788 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 788 conjugacy classes of subgroups — too many to draw. Only its 177 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2856 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C4×Q8 | C2 | semidirect |
| 2. C2×#460 | C2 | semidirect |
| 3. C2×#448 | C2 | semidirect |
| 5. #1880 | C2 | semidirect |
| 6. #1874 | C2 | semidirect |
| 7. #1803 | C2 | semidirect |
| 8. #1797 | C2 | semidirect |
| 40. C22×Q8 | D4 | semidirect |
| 65. C4×Q8 | D4 | semidirect |
| 132. C22×C4 | SD16 | semidirect |
| 182. Q8 | #121 | semidirect |
| 236. C22 | #460 | semidirect |
| 251. C4 | #448 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×Q8 | 2 |
| C2×#460 | 2 |
| C2×#448 | 2 |
| #2856 | 2 |
| #1880 | 2 |
| #1874 | 2 |
| #1803 | 2 |
| #1797 | 2 |