Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2389 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 2389 conjugacy classes of subgroups — too many to draw. Only its 557 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | 2−1+6 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #3487 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #3027 | C2 | 1 | 12 | yes | maximal, complemented |
| 4 | #3025 | C2 | 1 | 12 | yes | maximal, complemented |
| 5 | #2884 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #585 | — | 2 | 12 | no | — |
| 7 | #585 | C22 | 1 | 12 | yes | — |
| 8 | C2×2−1+4 | — | 2 | 12 | no | — |
| 9 | C2×2−1+4 | C22 | 1 | 3 | yes | — |
| 10 | C22×#41 | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | #550 | — | 2 | 12 | no | — |
| 12 | #548 | — | 2 | 12 | no | — |
| 13 | #535 | C22 | 1 | 18 | yes | complemented |
| 14 | #525 | C22 | 1 | 18 | yes | complemented |
| 15 | #519 | C22 | 1 | 3 | yes | complemented |
| 16 | #458 | C22 | 1 | 16 | yes | complemented |
| 17 | B(4,𝔽2) | C22 | 1 | 48 | yes | complemented |
| 18 | #410 | C22 | 1 | 36 | yes | complemented |
| 19 | 2−1+4 | — | 2 | 48 | no | — |
| 20 | 2−1+4 | — | 2 | 36 | no | — |
| 21 | 2−1+4 | — | 2 | 24 | no | — |
| 22 | H(5,𝔽2) | — | 2 | 16 | no | — |
| 23 | H(5,𝔽2) | — | 2 | 36 | no | — |
| 24 | H(5,𝔽2) | C23 | 1 | 16 | yes | — |
| 25 | C2×#41 | — | 2 | 48 | no | — |
| 26 | C2×#41 | — | 2 | 12 | no | — |
| 27 | C2×#41 | — | 2 | 72 | no | — |
| 28 | C2×#41 | D4 | 1 | 12 | yes | complemented |
| 29 | C2×#41 | C23 | 1 | 18 | yes | — |
| 30 | C22×Q8 | — | 2 | 16 | no | — |
| 31 | C22×Q8 | D4 | 1 | 12 | yes | complemented |
| 32 | C22×Q8 | C23 | 1 | 1 | yes | characteristic |
| 33 | C22×D4 | C23 | 1 | 3 | yes | — |
| 34 | C23×C4 | C23 | 1 | 3 | yes | — |
| 35 | #127 | — | 2 | 18 | no | — |
| 36 | #124 | — | 2 | 18 | no | — |
| 37 | #123 | C23 | 1 | 12 | yes | — |
| 38 | C22⋊Q8 | — | 2 | 18 | no | — |
| 39 | #121 | — | 2 | 18 | no | — |
| 40 | #121 | — | 2 | 36 | no | — |
| 41 | #120 | C23 | 1 | 36 | yes | — |
| 42 | C4×Q8 | — | 2 | 12 | no | — |
| 43 | C4×D4 | — | 2 | 36 | no | — |
| 44 | #117 | C23 | 1 | 18 | yes | — |
| 45 | #99 | C23 | 1 | 48 | yes | — |
| 46 | C24 | C24 | 1 | 3 | yes | — |
| 47 | #41 | — | 2 | 144 | no | — |
| 48 | #41 | — | 2 | 48 | no | — |
| 49 | #41 | — | 2 | 48 | no | — |
| 50 | #41 | — | 2 | 48 | no | — |
| 51 | #41 | — | 2 | 72 | no | — |
| 52 | #41 | — | 4 | 16 | no | — |
| 53 | C2×Q8 | — | 2 | 72 | no | — |
| 54 | C2×Q8 | — | 2 | 48 | no | — |
| 55 | C2×Q8 | C2×D4 | 1 | 18 | yes | — |
| 56 | C2×Q8 | C2×D4 | 1 | 12 | yes | — |
| 57 | C2×D4 | — | 2 | 72 | no | — |
| 58 | C2×D4 | — | 2 | 48 | no | — |
| 59 | C2×D4 | — | 2 | 36 | no | — |
| 60 | C2×D4 | — | 2 | 18 | no | — |
| 61 | C2×D4 | C2×D4 | 1 | 18 | yes | — |
| 62 | C2×D4 | C24 | 1 | 12 | yes | — |
| 63 | C22×C4 | — | 2 | 48 | no | — |
| 64 | C22×C4 | — | 2 | 18 | no | — |
| 65 | C22×C4 | C2×D4 | 1 | 36 | yes | — |
| 66 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 67 | C22×C4 | C24 | 1 | 3 | yes | — |
| 68 | C4⋊C4 | — | 2 | 18 | no | — |
| 69 | C4⋊C4 | — | 2 | 12 | no | — |
| 70 | C22⋊C4 | — | 2 | 12 | no | — |
| 71 | C22⋊C4 | — | 2 | 36 | no | — |
| 72 | C22⋊C4 | C24 | 1 | 12 | yes | — |
| 73 | C42 | — | 2 | 18 | no | — |
| 74 | C23 | — | 2 | 16 | no | — |
| 75 | C23 | — | 2 | 9 | no | — |
| 76 | C23 | — | 8 | 3 | no | — |
| 77 | C23 | C22×D4 | 1 | 12 | yes | — |
| 78 | C23 | C25 | 1 | 1 | yes | characteristic |
| 79 | Q8 | — | 2 | 16 | no | — |
| 80 | Q8 | — | 2 | 36 | no | — |
| 81 | Q8 | #120 | 1 | 16 | yes | — |
| 82 | D4 | — | 2 | 48 | no | — |
| 83 | D4 | — | 2 | 24 | no | — |
| 84 | D4 | — | 2 | 36 | no | — |
| 85 | D4 | — | 4 | 12 | no | — |
| 86 | D4 | — | 8 | 36 | no | — |
| 87 | C2×C4 | — | 2 | 48 | no | — |
| 88 | C2×C4 | — | 2 | 72 | no | — |
| 89 | C2×C4 | — | 2 | 12 | no | — |
| 90 | C2×C4 | — | 2 | 3 | no | — |
| 91 | C2×C4 | — | 4 | 12 | no | — |
| 92 | C2×C4 | C22×D4 | 1 | 12 | yes | — |
| 93 | C2×C4 | C22×D4 | 1 | 18 | yes | — |
| 94 | C22 | — | 2 | 12 | no | — |
| 95 | C22 | — | 4 | 1 | no | — |
| 96 | C22 | — | 4 | 12 | no | — |
| 97 | C22 | — | 4 | 18 | no | — |
| 98 | C22 | — | 4 | 3 | no | — |
| 99 | C22 | — | 8 | 16 | no | — |
| 100 | C22 | C23×D4 | 1 | 3 | yes | — |
| 101 | C4 | — | 2 | 12 | no | — |
| 102 | C4 | — | 4 | 12 | no | — |
| 103 | C4 | — | 4 | 1 | no | — |
| 104 | C4 | C2×#120 | 1 | 12 | yes | — |
| 105 | C2 | — | 2 | 3 | no | — |
| 106 | C2 | — | 4 | 12 | no | — |
| 107 | C2 | — | 8 | 3 | no | — |
| 108 | C2 | C22×#120 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#120 |
| commutator, Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. 2−1+6 | C2 | semidirect |
| 2. #3487 | C2 | semidirect |
| 3. #3027 | C2 | semidirect |
| 4. #3025 | C2 | semidirect |
| 5. #2884 | C2 | semidirect |
| 10. C22×#41 | C22 | semidirect |
| 13. #535 | C22 | semidirect |
| 14. #525 | C22 | semidirect |
| 15. #519 | C22 | semidirect |
| 16. #458 | C22 | semidirect |
| 17. B(4,𝔽2) | C22 | semidirect |
| 18. #410 | C22 | semidirect |
| 28. C2×#41 | D4 | semidirect |
| 31. C22×Q8 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| 2−1+6 | 2 |
| #3487 | 2 |
| #3027 | 2 |
| #3025 | 2 |
| #2884 | 2 |