Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
418 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 113 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 | #3069 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #3068 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #2070 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #2069 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #2019 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #2017 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #1858 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #575 | C22 | 1 | 3 | yes | — |
| 9 | C2×#137 | C22 | 1 | 3 | yes | — |
| 10 | C2×AGL(1,ℤ/8) | C22 | 1 | 3 | yes | — |
| 11 | C22×SD16 | C22 | 1 | 1 | yes | characteristic |
| 12 | #568 | C22 | 1 | 3 | yes | — |
| 13 | #473 | — | 2 | 6 | no | — |
| 14 | #472 | — | 2 | 6 | no | — |
| 15 | #471 | — | 2 | 6 | no | — |
| 16 | #471 | — | 2 | 6 | no | — |
| 17 | #430 | C22 | 1 | 3 | yes | — |
| 18 | #430 | C22 | 1 | 3 | yes | — |
| 19 | #413 | C22 | 1 | 3 | yes | — |
| 20 | C2×#101 | C22 | 1 | 3 | yes | — |
| 21 | C2×#100 | C22 | 1 | 3 | yes | — |
| 22 | #408 | C22 | 1 | 3 | yes | — |
| 23 | #341 | C22 | 1 | 3 | yes | — |
| 24 | #338 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×#41 | C23 | 1 | 3 | yes | — |
| 26 | C22×Q8 | C23 | 1 | 1 | yes | characteristic |
| 27 | C22×D4 | C23 | 1 | 1 | yes | characteristic |
| 28 | #137 | — | 2 | 3 | no | — |
| 29 | #137 | — | 2 | 3 | no | — |
| 30 | #137 | — | 2 | 6 | no | — |
| 31 | #137 | — | 4 | 3 | no | — |
| 32 | AGL(1,ℤ/8) | — | 2 | 3 | no | — |
| 33 | AGL(1,ℤ/8) | — | 2 | 3 | no | — |
| 34 | AGL(1,ℤ/8) | — | 2 | 6 | no | — |
| 35 | AGL(1,ℤ/8) | — | 4 | 3 | no | — |
| 36 | #135 | — | 4 | 3 | no | — |
| 37 | C2×Q16 | — | 2 | 3 | no | — |
| 38 | C2×SD16 | — | 2 | 3 | no | — |
| 39 | C2×SD16 | — | 2 | 3 | no | — |
| 40 | C2×SD16 | — | 4 | 3 | no | — |
| 41 | C2×D8 | — | 2 | 3 | no | — |
| 42 | #131 | — | 2 | 6 | no | — |
| 43 | #131 | — | 2 | 6 | no | — |
| 44 | C2×M16 | — | 4 | 3 | no | — |
| 45 | C2×M16 | C23 | 1 | 3 | yes | — |
| 46 | C2×M16 | C23 | 1 | 3 | yes | — |
| 47 | C2×M16 | C23 | 1 | 3 | yes | — |
| 48 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 49 | #108 | — | 2 | 6 | no | — |
| 50 | #108 | — | 2 | 6 | no | — |
| 51 | #101 | — | 2 | 6 | no | — |
| 52 | #101 | D4 | 1 | 6 | yes | — |
| 53 | #100 | — | 2 | 6 | no | — |
| 54 | #100 | D4 | 1 | 6 | yes | — |
| 55 | C22⋊C8 | — | 4 | 3 | no | — |
| 56 | C24 | — | 2 | 1 | no | — |
| 57 | #41 | — | 2 | 3 | no | — |
| 58 | #41 | — | 2 | 3 | no | — |
| 59 | #41 | — | 2 | 3 | no | — |
| 60 | #41 | — | 2 | 3 | no | — |
| 61 | C2×Q8 | — | 2 | 3 | no | — |
| 62 | C2×Q8 | — | 2 | 3 | no | — |
| 63 | C2×Q8 | C2×D4 | 1 | 3 | yes | — |
| 64 | C2×D4 | — | 2 | 3 | no | — |
| 65 | C2×D4 | — | 2 | 3 | no | — |
| 66 | C2×D4 | — | 2 | 3 | no | — |
| 67 | C2×D4 | C2×D4 | 1 | 3 | yes | — |
| 68 | C22×C4 | — | 2 | 3 | no | — |
| 69 | C22×C4 | — | 2 | 1 | no | — |
| 70 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 71 | Q16 | — | 4 | 6 | no | — |
| 72 | Q16 | — | 4 | 3 | no | — |
| 73 | SD16 | — | 4 | 6 | no | — |
| 74 | SD16 | — | 4 | 6 | no | — |
| 75 | SD16 | — | 4 | 3 | no | — |
| 76 | SD16 | — | 4 | 3 | no | — |
| 77 | SD16 | — | 4 | 4 | no | — |
| 78 | D8 | — | 4 | 6 | no | — |
| 79 | D8 | — | 4 | 3 | no | — |
| 80 | M16 | — | 2 | 3 | no | — |
| 81 | M16 | — | 2 | 3 | no | — |
| 82 | M16 | — | 2 | 3 | no | — |
| 83 | M16 | — | 2 | 3 | no | — |
| 84 | M16 | — | 4 | 6 | no | — |
| 85 | M16 | C2×D4 | 1 | 6 | yes | — |
| 86 | M16 | C2×D4 | 1 | 6 | yes | — |
| 87 | C2×C8 | — | 2 | 3 | no | — |
| 88 | C2×C8 | — | 2 | 3 | no | — |
| 89 | C2×C8 | — | 2 | 3 | no | — |
| 90 | C2×C8 | — | 2 | 3 | no | — |
| 91 | C2×C8 | — | 4 | 6 | no | — |
| 92 | C23 | — | 2 | 3 | no | — |
| 93 | C23 | — | 4 | 3 | no | — |
| 94 | C23 | — | 16 | 1 | no | — |
| 95 | C23 | 2−1+4 | 1 | 1 | yes | characteristic |
| 96 | Q8 | — | 2 | 3 | no | — |
| 97 | Q8 | — | 2 | 3 | no | — |
| 98 | Q8 | — | 4 | 3 | no | — |
| 99 | Q8 | — | 4 | 1 | no | — |
| 100 | D4 | — | 2 | 3 | no | — |
| 101 | D4 | — | 2 | 3 | no | — |
| 102 | D4 | — | 4 | 3 | no | — |
| 103 | D4 | — | 4 | 3 | no | — |
| 104 | D4 | — | 4 | 3 | no | — |
| 105 | D4 | — | 4 | 1 | no | — |
| 106 | C2×C4 | — | 2 | 3 | no | — |
| 107 | C2×C4 | — | 4 | 3 | no | — |
| 108 | C2×C4 | — | 4 | 3 | no | — |
| 109 | C2×C4 | — | 4 | 3 | no | — |
| 110 | C2×C4 | C22×D4 | 1 | 3 | yes | — |
| 111 | C2×C4 | H(5,𝔽2) | 1 | 3 | yes | — |
| 112 | C8 | — | 2 | 6 | no | — |
| 113 | C8 | — | 2 | 6 | no | — |
| 114 | C8 | — | 4 | 3 | no | — |
| 115 | C8 | — | 4 | 1 | no | — |
| 116 | C22 | — | 4 | 1 | no | — |
| 117 | C22 | — | 4 | 3 | no | — |
| 118 | C22 | — | 8 | 3 | no | — |
| 119 | C22 | — | 8 | 1 | no | — |
| 120 | C22 | — | 16 | 3 | no | — |
| 121 | C22 | #537 | 1 | 3 | yes | — |
| 122 | C4 | — | 4 | 3 | no | — |
| 123 | C4 | — | 8 | 1 | no | — |
| 124 | C4 | D42 | 1 | 3 | yes | — |
| 125 | C4 | #560 | 1 | 1 | yes | characteristic |
| 126 | C2 | — | 2 | 3 | no | — |
| 127 | C2 | — | 8 | 3 | no | — |
| 128 | C2 | — | 16 | 1 | no | — |
| 129 | C2 | #2675 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2675 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3069 | C2 | semidirect |
| 2. #3068 | C2 | semidirect |
| 3. #2070 | C2 | semidirect |
| 4. #2069 | C2 | semidirect |
| 5. #2019 | C2 | semidirect |
| 6. #2017 | C2 | semidirect |
| 7. #1858 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3069 | 2 |
| #3068 | 2 |
| #2070 | 2 |
| #2069 | 2 |
| #2019 | 2 |
| #2017 | 2 |
| #1858 | 2 |