Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
229 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 229 conjugacy classes of subgroups — too many to draw. Only its 45 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 | #3151 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #2126 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #1408 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | C2×AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 5 | #532 | C22 | 1 | 1 | yes | characteristic |
| 6 | C2×#127 | C4 | 1 | 1 | yes | characteristic, complemented |
| 7 | #496 | — | 2 | 2 | no | — |
| 8 | #493 | C22 | 1 | 2 | yes | — |
| 9 | #492 | C4 | 1 | 2 | yes | complemented |
| 10 | #410 | — | 2 | 2 | no | — |
| 11 | C4×M16 | C4 | 1 | 1 | yes | characteristic, complemented |
| 12 | #354 | — | 2 | 2 | no | — |
| 13 | #353 | C22 | 1 | 4 | yes | complemented |
| 14 | C2×#41 | D4 | 1 | 2 | yes | — |
| 15 | C22×D4 | — | 2 | 1 | no | — |
| 16 | C22×D4 | — | 4 | 1 | no | — |
| 17 | AGL(1,ℤ/8) | — | 4 | 4 | no | — |
| 18 | AGL(1,ℤ/8) | — | 4 | 4 | no | — |
| 19 | C2×SD16 | — | 2 | 4 | no | — |
| 20 | C2×D8 | — | 2 | 4 | no | — |
| 21 | C2×M16 | — | 2 | 1 | no | — |
| 22 | C4⋊Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | #127 | — | 2 | 1 | no | — |
| 24 | #127 | — | 4 | 1 | no | — |
| 25 | #127 | C2×C4 | 1 | 2 | yes | — |
| 26 | #127 | C23 | 1 | 1 | yes | characteristic |
| 27 | #124 | D4 | 1 | 2 | yes | — |
| 28 | #121 | — | 4 | 1 | no | — |
| 29 | C4×D4 | — | 4 | 1 | no | — |
| 30 | #117 | — | 4 | 2 | no | — |
| 31 | C2×C42 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | #99 | — | 2 | 4 | no | — |
| 33 | #99 | — | 4 | 2 | no | — |
| 34 | #97 | — | 2 | 1 | no | — |
| 35 | C4×C8 | C2×C4 | 1 | 2 | yes | — |
| 36 | C24 | — | 4 | 1 | no | — |
| 37 | #41 | — | 4 | 2 | no | — |
| 38 | #41 | — | 4 | 2 | no | — |
| 39 | C2×Q8 | C22⋊C4 | 1 | 2 | yes | — |
| 40 | C2×D4 | — | 2 | 2 | no | — |
| 41 | C2×D4 | — | 2 | 2 | no | — |
| 42 | C2×D4 | — | 4 | 2 | no | — |
| 43 | C2×D4 | — | 4 | 2 | no | — |
| 44 | C2×D4 | — | 4 | 1 | no | — |
| 45 | C2×D4 | — | 4 | 2 | no | — |
| 46 | C2×D4 | — | 4 | 2 | no | — |
| 47 | C2×D4 | — | 4 | 1 | no | — |
| 48 | C2×D4 | — | 8 | 4 | no | — |
| 49 | C2×D4 | C2×D4 | 1 | 2 | yes | — |
| 50 | C22×C4 | — | 2 | 1 | no | — |
| 51 | C22×C4 | — | 2 | 2 | no | — |
| 52 | C22×C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 53 | SD16 | — | 4 | 8 | no | — |
| 54 | D8 | — | 4 | 8 | no | — |
| 55 | M16 | — | 2 | 4 | no | — |
| 56 | C2×C8 | — | 2 | 2 | no | — |
| 57 | C4⋊C4 | — | 4 | 1 | no | — |
| 58 | C4⋊C4 | — | 8 | 2 | no | — |
| 59 | C22⋊C4 | — | 4 | 2 | no | — |
| 60 | C22⋊C4 | — | 4 | 4 | no | — |
| 61 | C42 | — | 2 | 1 | no | — |
| 62 | C42 | — | 8 | 2 | no | — |
| 63 | C42 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 64 | C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 65 | C23 | — | 2 | 2 | no | — |
| 66 | C23 | — | 4 | 2 | no | — |
| 67 | C23 | — | 8 | 1 | no | — |
| 68 | C23 | — | 8 | 1 | no | — |
| 69 | C23 | — | 8 | 4 | no | — |
| 70 | C23 | #99 | 1 | 1 | yes | characteristic |
| 71 | Q8 | — | 4 | 2 | no | — |
| 72 | D4 | — | 4 | 2 | no | — |
| 73 | D4 | — | 4 | 4 | no | — |
| 74 | D4 | — | 8 | 2 | no | — |
| 75 | D4 | — | 8 | 4 | no | — |
| 76 | D4 | — | 8 | 2 | no | — |
| 77 | D4 | — | 8 | 4 | no | — |
| 78 | D4 | — | 8 | 2 | no | — |
| 79 | C2×C4 | — | 2 | 1 | no | — |
| 80 | C2×C4 | — | 2 | 2 | no | — |
| 81 | C2×C4 | — | 2 | 1 | no | — |
| 82 | C2×C4 | — | 2 | 2 | no | — |
| 83 | C2×C4 | — | 4 | 2 | no | — |
| 84 | C2×C4 | — | 4 | 2 | no | — |
| 85 | C2×C4 | — | 4 | 2 | no | — |
| 86 | C2×C4 | — | 8 | 4 | no | — |
| 87 | C2×C4 | #99 | 1 | 1 | yes | characteristic |
| 88 | C2×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 89 | C8 | — | 2 | 4 | no | — |
| 90 | C22 | — | 2 | 1 | no | — |
| 91 | C22 | — | 4 | 1 | no | — |
| 92 | C22 | — | 4 | 2 | no | — |
| 93 | C22 | — | 8 | 2 | no | — |
| 94 | C22 | — | 8 | 4 | no | — |
| 95 | C22 | — | 8 | 2 | no | — |
| 96 | C22 | — | 16 | 2 | no | — |
| 97 | C22 | — | 16 | 2 | no | — |
| 98 | C22 | C2×#99 | 1 | 1 | yes | characteristic |
| 99 | C4 | — | 2 | 2 | no | — |
| 100 | C4 | — | 2 | 1 | no | — |
| 101 | C4 | — | 4 | 1 | no | — |
| 102 | C4 | — | 4 | 2 | no | — |
| 103 | C4 | — | 16 | 4 | no | — |
| 104 | C4 | #353 | 1 | 2 | yes | — |
| 105 | C2 | — | 2 | 1 | no | — |
| 106 | C2 | — | 4 | 1 | no | — |
| 107 | C2 | — | 8 | 2 | no | — |
| 108 | C2 | — | 16 | 2 | no | — |
| 109 | C2 | C2×#353 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#353 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #127 | C23 |
| prime radical | #3151 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2126 | C2 | semidirect |
| 3. #1408 | C2 | semidirect |
| 6. C2×#127 | C4 | semidirect |
| 9. #492 | C4 | semidirect |
| 11. C4×M16 | C4 | semidirect |
| 13. #353 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3151 | 2 |
| #2126 | 2 |
| #1408 | 2 |