Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
29593 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | 2−1+6 | C2 | 1 | 64 | yes | maximal, direct product factor |
| 2 | C2×#585 | C2 | 1 | 36 | yes | maximal, complemented |
| 3 | C22×2−1+4 | C2 | 1 | 27 | yes | maximal, complemented |
| 4 | #585 | C22 | 1 | 1152 | yes | complemented |
| 5 | C2×2−1+4 | C22 | 1 | 216 | yes | complemented |
| 6 | C2×2−1+4 | C22 | 1 | 864 | yes | complemented |
| 7 | C2×H(5,𝔽2) | C22 | 1 | 120 | yes | complemented |
| 8 | C22×#41 | C22 | 1 | 270 | yes | complemented |
| 9 | C23×Q8 | C22 | 1 | 45 | yes | complemented |
| 10 | 2−1+4 | C23 | 1 | 3456 | yes | complemented |
| 11 | H(5,𝔽2) | C23 | 1 | 1920 | yes | complemented |
| 12 | C2×#41 | C23 | 1 | 720 | yes | — |
| 13 | C2×#41 | C23 | 1 | 4320 | yes | complemented |
| 14 | C22×Q8 | C23 | 1 | 270 | yes | — |
| 15 | C22×Q8 | C23 | 1 | 720 | yes | complemented |
| 16 | C22×D4 | C23 | 1 | 270 | yes | — |
| 17 | C23×C4 | C23 | 1 | 135 | yes | — |
| 18 | C24 | C24 | 1 | 45 | yes | — |
| 19 | #41 | C24 | 1 | 5760 | yes | — |
| 20 | C2×Q8 | C24 | 1 | 120 | yes | — |
| 21 | C2×Q8 | C24 | 1 | 2160 | yes | — |
| 22 | C2×D4 | C24 | 1 | 216 | yes | — |
| 23 | C2×D4 | C24 | 1 | 2160 | yes | — |
| 24 | C22×C4 | C24 | 1 | 270 | yes | — |
| 25 | C22×C4 | C24 | 1 | 1080 | yes | — |
| 26 | C23 | — | 4 | 90 | no | — |
| 27 | C23 | C25 | 1 | 27 | yes | — |
| 28 | C23 | C25 | 1 | 180 | yes | — |
| 29 | Q8 | C25 | 1 | 480 | yes | — |
| 30 | D4 | C25 | 1 | 864 | yes | — |
| 31 | C2×C4 | C25 | 1 | 1080 | yes | — |
| 32 | C2×C4 | C25 | 1 | 36 | yes | — |
| 33 | C22 | — | 2 | 54 | no | — |
| 34 | C22 | — | 4 | 180 | no | — |
| 35 | C22 | C26 | 1 | 54 | yes | — |
| 36 | C22 | C26 | 1 | 1 | yes | characteristic |
| 37 | C4 | C26 | 1 | 72 | yes | — |
| 38 | C2 | — | 2 | 54 | no | — |
| 39 | C2 | 2−1+6 | 1 | 2 | yes | direct product factor |
| 40 | C2 | C27 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C26 |
| commutator, Frattini, cosocle | C2 | C27 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. 2−1+6 | C2 | direct (N × H) |
| 2. C2×#585 | C2 | semidirect |
| 3. C22×2−1+4 | C2 | semidirect |
| 4. #585 | C22 | semidirect |
| 5. C2×2−1+4 | C22 | semidirect |
| 6. C2×2−1+4 | C22 | semidirect |
| 7. C2×H(5,𝔽2) | C22 | semidirect |
| 8. C22×#41 | C22 | semidirect |
| 9. C23×Q8 | C22 | semidirect |
| 10. 2−1+4 | C23 | semidirect |
| 11. H(5,𝔽2) | C23 | semidirect |
| 13. C2×#41 | C23 | semidirect |
| 15. C22×Q8 | C23 | semidirect |
| 39. C2 | 2−1+6 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| 2−1+6 | 2 |
| C2×#585 | 2 |
| C22×2−1+4 | 2 |