Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
566 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 566 conjugacy classes of subgroups — too many to draw. Only its 125 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 | C2×#575 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #3414 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #3054 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #2191 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #2190 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #2145 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C2×#349 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×#41 | — | 2 | 1 | no | — |
| 9 | #575 | — | 2 | 4 | no | — |
| 10 | #575 | — | 2 | 2 | no | — |
| 11 | #575 | C22 | 1 | 4 | yes | — |
| 12 | C2×#137 | — | 2 | 2 | no | — |
| 13 | C2×AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 14 | C2×#135 | — | 2 | 1 | no | — |
| 15 | C2×#135 | C22 | 1 | 2 | yes | — |
| 16 | C2×#135 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | C22×M16 | C22 | 1 | 1 | yes | characteristic |
| 18 | #525 | — | 2 | 1 | no | — |
| 19 | #510 | — | 2 | 2 | no | — |
| 20 | #510 | C22 | 1 | 4 | yes | complemented |
| 21 | #509 | — | 2 | 2 | no | — |
| 22 | #509 | C22 | 1 | 4 | yes | complemented |
| 23 | #508 | — | 2 | 4 | no | — |
| 24 | C2×M32 | C22 | 1 | 2 | yes | complemented |
| 25 | #470 | C22 | 1 | 2 | yes | complemented |
| 26 | #469 | C22 | 1 | 2 | yes | complemented |
| 27 | #468 | — | 2 | 2 | no | — |
| 28 | #427 | C22 | 1 | 1 | yes | characteristic, complemented |
| 29 | #416 | — | 2 | 1 | no | — |
| 30 | #360 | C22 | 1 | 8 | yes | complemented |
| 31 | #349 | C22 | 1 | 4 | yes | complemented |
| 32 | C2×#41 | — | 2 | 2 | no | — |
| 33 | C2×#41 | — | 2 | 4 | no | — |
| 34 | C2×#41 | — | 2 | 4 | no | — |
| 35 | C2×#41 | — | 2 | 2 | no | — |
| 36 | C2×#41 | — | 2 | 1 | no | — |
| 37 | C2×#41 | — | 4 | 1 | no | — |
| 38 | C2×#41 | — | 4 | 2 | no | — |
| 39 | C2×#41 | — | 4 | 2 | no | — |
| 40 | C2×#41 | — | 4 | 1 | no | — |
| 41 | C22×Q8 | — | 2 | 1 | no | — |
| 42 | C22×D4 | — | 2 | 1 | no | — |
| 43 | C22×D4 | — | 4 | 1 | no | — |
| 44 | C23×C4 | — | 4 | 1 | no | — |
| 45 | C23×C4 | D4 | 1 | 1 | yes | characteristic |
| 46 | #137 | — | 2 | 8 | no | — |
| 47 | #137 | — | 2 | 8 | no | — |
| 48 | AGL(1,ℤ/8) | — | 2 | 8 | no | — |
| 49 | AGL(1,ℤ/8) | — | 2 | 8 | no | — |
| 50 | #135 | — | 2 | 4 | no | — |
| 51 | #135 | — | 2 | 4 | no | — |
| 52 | #135 | — | 2 | 2 | no | — |
| 53 | #135 | — | 2 | 2 | no | — |
| 54 | #135 | — | 2 | 2 | no | — |
| 55 | #135 | — | 2 | 2 | no | — |
| 56 | #135 | D4 | 1 | 4 | yes | complemented |
| 57 | #135 | C23 | 1 | 4 | yes | — |
| 58 | C2×Q16 | — | 2 | 1 | no | — |
| 59 | C2×Q16 | D4 | 1 | 2 | yes | complemented |
| 60 | C2×Q16 | C23 | 1 | 1 | yes | characteristic, complemented |
| 61 | C2×SD16 | — | 2 | 2 | no | — |
| 62 | C2×SD16 | — | 2 | 2 | no | — |
| 63 | C2×SD16 | — | 2 | 1 | no | — |
| 64 | C2×D8 | — | 2 | 1 | no | — |
| 65 | C2×D8 | D4 | 1 | 2 | yes | complemented |
| 66 | C2×D8 | C23 | 1 | 1 | yes | characteristic, complemented |
| 67 | C2×M16 | — | 2 | 2 | no | — |
| 68 | C2×M16 | — | 2 | 2 | no | — |
| 69 | C2×M16 | D4 | 1 | 2 | yes | — |
| 70 | C2×M16 | C23 | 1 | 2 | yes | — |
| 71 | C22×C8 | D4 | 1 | 1 | yes | characteristic |
| 72 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 73 | #123 | — | 4 | 1 | no | — |
| 74 | C22⋊Q8 | — | 2 | 2 | no | — |
| 75 | #121 | — | 2 | 2 | no | — |
| 76 | #120 | — | 4 | 1 | no | — |
| 77 | C4×D4 | — | 4 | 2 | no | — |
| 78 | #117 | — | 2 | 1 | no | — |
| 79 | Q32 | — | 2 | 4 | no | — |
| 80 | SD32 | — | 2 | 4 | no | — |
| 81 | SD32 | — | 2 | 4 | no | — |
| 82 | D16 | — | 2 | 4 | no | — |
| 83 | M32 | — | 2 | 2 | no | — |
| 84 | M32 | C23 | 1 | 4 | yes | complemented |
| 85 | C2×C16 | — | 2 | 2 | no | — |
| 86 | #107 | — | 2 | 1 | no | — |
| 87 | #106 | C23 | 1 | 2 | yes | complemented |
| 88 | #103 | — | 2 | 2 | no | — |
| 89 | #102 | — | 2 | 2 | no | — |
| 90 | C24 | — | 4 | 1 | no | — |
| 91 | C24 | D8 | 1 | 1 | yes | characteristic |
| 92 | #41 | — | 2 | 8 | no | — |
| 93 | #41 | — | 2 | 4 | no | — |
| 94 | #41 | — | 2 | 4 | no | — |
| 95 | #41 | — | 4 | 4 | no | — |
| 96 | #41 | — | 4 | 4 | no | — |
| 97 | #41 | — | 4 | 4 | no | — |
| 98 | #41 | — | 4 | 4 | no | — |
| 99 | #41 | — | 4 | 4 | no | — |
| 100 | #41 | — | 4 | 2 | no | — |
| 101 | #41 | — | 4 | 2 | no | — |
| 102 | #41 | — | 4 | 1 | no | — |
| 103 | #41 | — | 4 | 1 | no | — |
| 104 | #41 | — | 4 | 1 | no | — |
| 105 | #41 | — | 4 | 1 | no | — |
| 106 | C2×Q8 | — | 2 | 4 | no | — |
| 107 | C2×Q8 | — | 2 | 2 | no | — |
| 108 | C2×Q8 | — | 2 | 1 | no | — |
| 109 | C2×Q8 | — | 4 | 1 | no | — |
| 110 | C2×Q8 | — | 4 | 2 | no | — |
| 111 | C2×D4 | — | 2 | 4 | no | — |
| 112 | C2×D4 | — | 2 | 2 | no | — |
| 113 | C2×D4 | — | 2 | 1 | no | — |
| 114 | C2×D4 | — | 4 | 1 | no | — |
| 115 | C2×D4 | — | 4 | 2 | no | — |
| 116 | C2×D4 | — | 4 | 4 | no | — |
| 117 | C2×D4 | — | 4 | 2 | no | — |
| 118 | C2×D4 | — | 4 | 4 | no | — |
| 119 | C2×D4 | — | 4 | 2 | no | — |
| 120 | C2×D4 | — | 4 | 2 | no | — |
| 121 | C2×D4 | — | 4 | 1 | no | — |
| 122 | C22×C4 | — | 2 | 1 | no | — |
| 123 | C22×C4 | — | 2 | 1 | no | — |
| 124 | C22×C4 | — | 2 | 1 | no | — |
| 125 | C22×C4 | — | 2 | 1 | no | — |
| 126 | C22×C4 | — | 4 | 2 | no | — |
| 127 | C22×C4 | — | 4 | 4 | no | — |
| 128 | C22×C4 | — | 4 | 4 | no | — |
| 129 | C22×C4 | — | 4 | 1 | no | — |
| 130 | C22×C4 | — | 4 | 2 | no | — |
| 131 | C22×C4 | — | 4 | 1 | no | — |
| 132 | C22×C4 | D8 | 1 | 2 | yes | — |
| 133 | C22×C4 | D8 | 1 | 1 | yes | characteristic |
| 134 | C22×C4 | C2×D4 | 1 | 2 | yes | — |
| 135 | C22×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 136 | Q16 | — | 2 | 4 | no | — |
| 137 | Q16 | — | 2 | 2 | no | — |
| 138 | Q16 | — | 2 | 2 | no | — |
| 139 | Q16 | C2×D4 | 1 | 4 | yes | — |
| 140 | SD16 | — | 2 | 8 | no | — |
| 141 | SD16 | — | 2 | 4 | no | — |
| 142 | SD16 | — | 2 | 4 | no | — |
| 143 | SD16 | — | 4 | 2 | no | — |
| 144 | D8 | — | 2 | 4 | no | — |
| 145 | D8 | — | 2 | 2 | no | — |
| 146 | D8 | — | 2 | 2 | no | — |
| 147 | D8 | C2×D4 | 1 | 4 | yes | — |
| 148 | M16 | — | 2 | 4 | no | — |
| 149 | M16 | — | 2 | 4 | no | — |
| 150 | C2×C8 | — | 2 | 1 | no | — |
| 151 | C2×C8 | — | 2 | 1 | no | — |
| 152 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 153 | C2×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 154 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 155 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 156 | C2×C8 | C24 | 1 | 1 | yes | characteristic |
| 157 | C4⋊C4 | — | 2 | 2 | no | — |
| 158 | C4⋊C4 | — | 4 | 2 | no | — |
| 159 | C22⋊C4 | — | 4 | 1 | no | — |
| 160 | C22⋊C4 | — | 4 | 2 | no | — |
| 161 | C22⋊C4 | — | 4 | 2 | no | — |
| 162 | C42 | — | 4 | 1 | no | — |
| 163 | C16 | — | 2 | 4 | no | — |
| 164 | C23 | — | 2 | 1 | no | — |
| 165 | C23 | — | 2 | 1 | no | — |
| 166 | C23 | — | 4 | 4 | no | — |
| 167 | C23 | — | 4 | 2 | no | — |
| 168 | C23 | — | 4 | 1 | no | — |
| 169 | C23 | — | 8 | 1 | no | — |
| 170 | C23 | — | 8 | 4 | no | — |
| 171 | C23 | C2×D8 | 1 | 2 | yes | — |
| 172 | C23 | C2×D8 | 1 | 1 | yes | characteristic |
| 173 | Q8 | — | 2 | 4 | no | — |
| 174 | Q8 | — | 4 | 4 | no | — |
| 175 | Q8 | — | 4 | 2 | no | — |
| 176 | Q8 | — | 4 | 1 | no | — |
| 177 | Q8 | — | 4 | 1 | no | — |
| 178 | D4 | — | 2 | 4 | no | — |
| 179 | D4 | — | 4 | 4 | no | — |
| 180 | D4 | — | 4 | 8 | no | — |
| 181 | D4 | — | 4 | 4 | no | — |
| 182 | D4 | — | 4 | 4 | no | — |
| 183 | D4 | — | 4 | 1 | no | — |
| 184 | D4 | — | 4 | 1 | no | — |
| 185 | D4 | — | 4 | 2 | no | — |
| 186 | D4 | — | 8 | 4 | no | — |
| 187 | D4 | — | 8 | 1 | no | — |
| 188 | D4 | — | 8 | 1 | no | — |
| 189 | C2×C4 | — | 2 | 1 | no | — |
| 190 | C2×C4 | — | 2 | 1 | no | — |
| 191 | C2×C4 | — | 2 | 2 | no | — |
| 192 | C2×C4 | — | 2 | 2 | no | — |
| 193 | C2×C4 | — | 2 | 2 | no | — |
| 194 | C2×C4 | — | 2 | 2 | no | — |
| 195 | C2×C4 | — | 4 | 8 | no | — |
| 196 | C2×C4 | — | 4 | 4 | no | — |
| 197 | C2×C4 | — | 4 | 2 | no | — |
| 198 | C2×C4 | — | 4 | 4 | no | — |
| 199 | C2×C4 | — | 4 | 4 | no | — |
| 200 | C2×C4 | — | 4 | 2 | no | — |
| 201 | C2×C4 | — | 4 | 1 | no | — |
| 202 | C2×C4 | — | 8 | 1 | no | — |
| 203 | C2×C4 | — | 8 | 1 | no | — |
| 204 | C2×C4 | #120 | 1 | 1 | yes | characteristic |
| 205 | C2×C4 | #120 | 1 | 1 | yes | characteristic |
| 206 | C2×C4 | C2×D8 | 1 | 2 | yes | — |
| 207 | C2×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 208 | C2×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 209 | C2×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 210 | C2×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 211 | C8 | — | 2 | 2 | no | — |
| 212 | C8 | #120 | 1 | 2 | yes | — |
| 213 | C8 | C22×D4 | 1 | 2 | yes | — |
| 214 | C22 | — | 2 | 2 | no | — |
| 215 | C22 | — | 4 | 1 | no | — |
| 216 | C22 | — | 4 | 2 | no | — |
| 217 | C22 | — | 4 | 2 | no | — |
| 218 | C22 | — | 4 | 4 | no | — |
| 219 | C22 | — | 8 | 1 | no | — |
| 220 | C22 | — | 8 | 4 | no | — |
| 221 | C22 | — | 8 | 2 | no | — |
| 222 | C22 | — | 8 | 8 | no | — |
| 223 | C22 | — | 8 | 1 | no | — |
| 224 | C22 | #447 | 1 | 1 | yes | characteristic |
| 225 | C22 | #447 | 1 | 1 | yes | characteristic |
| 226 | C22 | C22×D8 | 1 | 1 | yes | characteristic |
| 227 | C4 | — | 2 | 2 | no | — |
| 228 | C4 | — | 4 | 4 | no | — |
| 229 | C4 | — | 8 | 2 | no | — |
| 230 | C4 | — | 8 | 1 | no | — |
| 231 | C4 | #447 | 1 | 1 | yes | characteristic |
| 232 | C4 | #447 | 1 | 1 | yes | characteristic |
| 233 | C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 234 | C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 235 | C2 | — | 2 | 1 | no | — |
| 236 | C2 | — | 2 | 1 | no | — |
| 237 | C2 | — | 2 | 1 | no | — |
| 238 | C2 | — | 4 | 2 | no | — |
| 239 | C2 | — | 8 | 4 | no | — |
| 240 | C2 | — | 16 | 1 | no | — |
| 241 | C2 | C2×#447 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C2×#447 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#575 | C2 | semidirect |
| 2. #3414 | C2 | semidirect |
| 3. #3054 | C2 | semidirect |
| 4. #2191 | C2 | semidirect |
| 5. #2190 | C2 | semidirect |
| 6. #2145 | C2 | semidirect |
| 7. C2×#349 | C2 | semidirect |
| 16. C2×#135 | C22 | semidirect |
| 20. #510 | C22 | semidirect |
| 22. #509 | C22 | semidirect |
| 24. C2×M32 | C22 | semidirect |
| 25. #470 | C22 | semidirect |
| 26. #469 | C22 | semidirect |
| 28. #427 | C22 | semidirect |
| 30. #360 | C22 | semidirect |
| 31. #349 | C22 | semidirect |
| 56. #135 | D4 | semidirect |
| 59. C2×Q16 | D4 | semidirect |
| 60. C2×Q16 | C23 | semidirect |
| 65. C2×D8 | D4 | semidirect |
| 66. C2×D8 | C23 | semidirect |
| 84. M32 | C23 | semidirect |
| 87. #106 | C23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#575 | 2 |
| #3414 | 2 |
| #3054 | 2 |
| #2191 | 2 |
| #2190 | 2 |
| #2145 | 2 |
| C2×#349 | 2 |