Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
276 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 59 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 | #3414 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #3150 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #2143 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #1975 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #1375 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#135 | — | 2 | 1 | no | — |
| 7 | C2×#135 | C22 | 1 | 2 | yes | — |
| 8 | C2×#135 | C22 | 1 | 1 | yes | characteristic |
| 9 | #532 | — | 2 | 1 | no | — |
| 10 | #510 | — | 2 | 2 | no | — |
| 11 | #510 | — | 2 | 2 | no | — |
| 12 | #509 | — | 2 | 2 | no | — |
| 13 | #509 | — | 2 | 2 | no | — |
| 14 | #508 | — | 2 | 4 | no | — |
| 15 | C2×M32 | C22 | 1 | 2 | yes | — |
| 16 | #497 | — | 2 | 2 | no | — |
| 17 | #496 | — | 2 | 2 | no | — |
| 18 | C2×#108 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#104 | — | 2 | 1 | no | — |
| 20 | #408 | — | 2 | 1 | no | — |
| 21 | C2×#97 | C22 | 1 | 1 | yes | characteristic |
| 22 | #359 | — | 2 | 4 | no | — |
| 23 | C2×#41 | — | 2 | 2 | no | — |
| 24 | C2×#41 | — | 2 | 1 | no | — |
| 25 | #135 | — | 2 | 2 | no | — |
| 26 | #135 | — | 2 | 2 | no | — |
| 27 | #135 | — | 2 | 2 | no | — |
| 28 | #135 | — | 2 | 2 | no | — |
| 29 | #135 | — | 2 | 2 | no | — |
| 30 | #135 | — | 2 | 2 | no | — |
| 31 | #135 | — | 4 | 2 | no | — |
| 32 | #135 | — | 4 | 2 | no | — |
| 33 | C2×Q16 | — | 2 | 1 | no | — |
| 34 | C2×Q16 | D4 | 1 | 2 | yes | — |
| 35 | C2×Q16 | D4 | 1 | 1 | yes | characteristic, complemented |
| 36 | C2×SD16 | — | 2 | 2 | no | — |
| 37 | C2×SD16 | — | 2 | 2 | no | — |
| 38 | C2×SD16 | — | 2 | 1 | no | — |
| 39 | C2×D8 | — | 2 | 1 | no | — |
| 40 | C2×D8 | D4 | 1 | 2 | yes | — |
| 41 | C2×D8 | D4 | 1 | 1 | yes | characteristic, complemented |
| 42 | C2×M16 | — | 2 | 1 | no | — |
| 43 | C22×C8 | D4 | 1 | 1 | yes | characteristic |
| 44 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 45 | C4⋊Q8 | — | 2 | 1 | no | — |
| 46 | #127 | — | 2 | 1 | no | — |
| 47 | #124 | — | 2 | 2 | no | — |
| 48 | #121 | — | 4 | 2 | no | — |
| 49 | C4×D4 | — | 4 | 2 | no | — |
| 50 | C2×C42 | D4 | 1 | 1 | yes | characteristic |
| 51 | Q32 | — | 2 | 4 | no | — |
| 52 | SD32 | — | 2 | 4 | no | — |
| 53 | SD32 | — | 2 | 4 | no | — |
| 54 | D16 | — | 2 | 4 | no | — |
| 55 | M32 | — | 2 | 2 | no | — |
| 56 | M32 | — | 2 | 2 | no | — |
| 57 | C2×C16 | D4 | 1 | 4 | yes | complemented |
| 58 | #108 | — | 2 | 2 | no | — |
| 59 | #104 | — | 4 | 2 | no | — |
| 60 | C22⋊C8 | — | 4 | 2 | no | — |
| 61 | #97 | — | 2 | 2 | no | — |
| 62 | #41 | — | 4 | 1 | no | — |
| 63 | #41 | — | 4 | 2 | no | — |
| 64 | #41 | — | 4 | 2 | no | — |
| 65 | #41 | — | 4 | 1 | no | — |
| 66 | #41 | — | 4 | 1 | no | — |
| 67 | #41 | — | 4 | 2 | no | — |
| 68 | #41 | — | 4 | 2 | no | — |
| 69 | #41 | — | 4 | 1 | no | — |
| 70 | C2×Q8 | — | 2 | 2 | no | — |
| 71 | C2×Q8 | — | 2 | 1 | no | — |
| 72 | C2×D4 | — | 2 | 2 | no | — |
| 73 | C2×D4 | — | 2 | 1 | no | — |
| 74 | C2×D4 | — | 4 | 2 | no | — |
| 75 | C2×D4 | — | 4 | 2 | no | — |
| 76 | C2×D4 | — | 4 | 1 | no | — |
| 77 | C22×C4 | — | 2 | 1 | no | — |
| 78 | C22×C4 | — | 4 | 1 | no | — |
| 79 | C22×C4 | — | 4 | 2 | no | — |
| 80 | C22×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 81 | Q16 | — | 2 | 2 | no | — |
| 82 | Q16 | — | 2 | 2 | no | — |
| 83 | Q16 | — | 2 | 2 | no | — |
| 84 | Q16 | — | 4 | 2 | no | — |
| 85 | SD16 | — | 4 | 4 | no | — |
| 86 | SD16 | — | 4 | 2 | no | — |
| 87 | SD16 | — | 4 | 2 | no | — |
| 88 | SD16 | — | 4 | 2 | no | — |
| 89 | D8 | — | 2 | 2 | no | — |
| 90 | D8 | — | 2 | 2 | no | — |
| 91 | D8 | — | 2 | 2 | no | — |
| 92 | D8 | — | 4 | 2 | no | — |
| 93 | M16 | — | 4 | 1 | no | — |
| 94 | M16 | — | 4 | 1 | no | — |
| 95 | C2×C8 | — | 2 | 1 | no | — |
| 96 | C2×C8 | — | 2 | 1 | no | — |
| 97 | C2×C8 | — | 4 | 1 | no | — |
| 98 | C2×C8 | D8 | 1 | 2 | yes | — |
| 99 | C2×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 101 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 102 | C2×C8 | #41 | 1 | 1 | yes | characteristic |
| 103 | C4⋊C4 | — | 4 | 2 | no | — |
| 104 | C22⋊C4 | — | 4 | 4 | no | — |
| 105 | C42 | — | 2 | 1 | no | — |
| 106 | C42 | D8 | 1 | 2 | yes | — |
| 107 | C16 | — | 2 | 4 | no | — |
| 108 | C23 | — | 4 | 2 | no | — |
| 109 | C23 | — | 4 | 1 | no | — |
| 110 | C23 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 111 | Q8 | — | 4 | 2 | no | — |
| 112 | Q8 | — | 4 | 1 | no | — |
| 113 | Q8 | — | 4 | 2 | no | — |
| 114 | Q8 | — | 4 | 1 | no | — |
| 115 | D4 | — | 4 | 2 | no | — |
| 116 | D4 | — | 4 | 1 | no | — |
| 117 | D4 | — | 4 | 2 | no | — |
| 118 | D4 | — | 4 | 1 | no | — |
| 119 | D4 | — | 8 | 4 | no | — |
| 120 | D4 | — | 8 | 1 | no | — |
| 121 | D4 | — | 8 | 2 | no | — |
| 122 | D4 | — | 8 | 2 | no | — |
| 123 | D4 | — | 8 | 1 | no | — |
| 124 | C2×C4 | — | 2 | 2 | no | — |
| 125 | C2×C4 | — | 4 | 2 | no | — |
| 126 | C2×C4 | — | 4 | 1 | no | — |
| 127 | C2×C4 | — | 8 | 1 | no | — |
| 128 | C2×C4 | — | 8 | 1 | no | — |
| 129 | C2×C4 | — | 8 | 2 | no | — |
| 130 | C2×C4 | — | 8 | 1 | no | — |
| 131 | C2×C4 | — | 8 | 2 | no | — |
| 132 | C2×C4 | #120 | 1 | 1 | yes | characteristic |
| 133 | C2×C4 | #121 | 1 | 1 | yes | characteristic |
| 134 | C2×C4 | #127 | 1 | 1 | yes | characteristic |
| 135 | C2×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 136 | C2×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 137 | C2×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 138 | C8 | — | 2 | 2 | no | — |
| 139 | C8 | — | 8 | 1 | no | — |
| 140 | C8 | #120 | 1 | 2 | yes | — |
| 141 | C8 | #121 | 1 | 2 | yes | — |
| 142 | C22 | — | 4 | 1 | no | — |
| 143 | C22 | — | 8 | 4 | no | — |
| 144 | C22 | — | 8 | 2 | no | — |
| 145 | C22 | — | 8 | 1 | no | — |
| 146 | C22 | — | 8 | 2 | no | — |
| 147 | C22 | #447 | 1 | 1 | yes | characteristic |
| 148 | C22 | #459 | 1 | 1 | yes | characteristic |
| 149 | C22 | #496 | 1 | 1 | yes | characteristic |
| 150 | C4 | — | 4 | 2 | no | — |
| 151 | C4 | — | 8 | 2 | no | — |
| 152 | C4 | — | 8 | 1 | no | — |
| 153 | C4 | #392 | 1 | 1 | yes | characteristic |
| 154 | C4 | #447 | 1 | 1 | yes | characteristic |
| 155 | C4 | #459 | 1 | 1 | yes | characteristic |
| 156 | C4 | #493 | 1 | 1 | yes | characteristic |
| 157 | C2 | — | 2 | 1 | no | — |
| 158 | C2 | — | 2 | 1 | no | — |
| 159 | C2 | — | 2 | 1 | no | — |
| 160 | C2 | — | 16 | 2 | no | — |
| 161 | C2 | — | 16 | 1 | no | — |
| 162 | C2 | #2011 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #493 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2011 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3414 | C2 | semidirect |
| 2. #3150 | C2 | semidirect |
| 3. #2143 | C2 | semidirect |
| 4. #1975 | C2 | semidirect |
| 5. #1375 | C2 | semidirect |
| 35. C2×Q16 | D4 | semidirect |
| 41. C2×D8 | D4 | semidirect |
| 57. C2×C16 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3414 | 2 |
| #3150 | 2 |
| #2143 | 2 |
| #1975 | 2 |
| #1375 | 2 |