Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
852 conjugacy classes of subgroups; only the 497 normal subgroups are shown.
This group has 852 conjugacy classes of subgroups — too many to draw. Only its 497 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #2863 | C2 | semidirect |
| 2. #2511 | C2 | semidirect |
| 4. Q8×C4⋊C4 | C2 | semidirect |
| 7. C42×Q8 | C2 | semidirect |
| 32. #126 | Q8 | direct (N × H) |
| 67. C4⋊C4 | C2×Q8 | semidirect |
| 73. C42 | C2×Q8 | semidirect |
| 79. Q8 | #126 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2863 | 2 |
| #2511 | 2 |
| #2497 | 2 |
| Q8×C4⋊C4 | 2 |
| Q8×C22⋊C4 | 2 |
| C4×#126 | 2 |
| C42×Q8 | 2 |