Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
846 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 846 conjugacy classes of subgroups — too many to draw. Only its 167 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 | #79055 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C2×#4711 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #78491 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C3×#1838 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #70527 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #68948 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #68942 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #68922 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #28177 | — | 3 | 1 | no | maximal, Sylow |
| 10 | C2×#908 | C22 | 1 | 4 | yes | — |
| 11 | C22×D24 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#901 | C22 | 1 | 2 | yes | — |
| 13 | #5477 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×C4×D12 | C22 | 1 | 2 | yes | — |
| 15 | C3×#423 | C22 | 1 | 2 | yes | — |
| 16 | C2×C3×#97 | C22 | 1 | 1 | yes | characteristic |
| 17 | #5133 | C22 | 1 | 4 | yes | — |
| 18 | C2×#823 | C22 | 1 | 4 | yes | — |
| 19 | C2×#821 | C22 | 1 | 2 | yes | — |
| 20 | #5110 | C22 | 1 | 2 | yes | — |
| 21 | C2×#817 | C22 | 1 | 1 | yes | characteristic |
| 22 | #4924 | C22 | 1 | 1 | yes | characteristic |
| 23 | #4844 | — | 2 | 4 | no | — |
| 24 | #4843 | — | 2 | 2 | no | — |
| 25 | #4841 | — | 2 | 2 | no | — |
| 26 | #4711 | — | 2 | 4 | no | — |
| 27 | #4706 | — | 2 | 4 | no | — |
| 28 | C3×#341 | C22 | 1 | 2 | yes | — |
| 29 | C3×#341 | C22 | 1 | 2 | yes | — |
| 30 | #4554 | C22 | 1 | 4 | yes | — |
| 31 | #3036 | — | 3 | 2 | no | — |
| 32 | C2×#442 | — | 3 | 1 | no | — |
| 33 | #2932 | — | 3 | 1 | no | — |
| 34 | #2079 | — | 3 | 2 | no | — |
| 35 | #2018 | — | 3 | 2 | no | — |
| 36 | #1907 | — | 3 | 4 | no | — |
| 37 | #1838 | D3 | 1 | 1 | yes | characteristic, complemented |
| 38 | #1800 | — | 3 | 2 | no | — |
| 39 | C2×#235 | C23 | 1 | 2 | yes | — |
| 40 | C22×D12 | C23 | 1 | 2 | yes | — |
| 41 | C22×C4×D3 | — | 2 | 2 | no | — |
| 42 | C2×C3×M16 | C23 | 1 | 4 | yes | — |
| 43 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 44 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 45 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 46 | #930 | — | 2 | 2 | no | — |
| 47 | C2×(C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 48 | C2×C4⋊Dic3 | C23 | 1 | 2 | yes | — |
| 49 | C2×#208 | C23 | 1 | 2 | yes | — |
| 50 | #908 | — | 2 | 8 | no | — |
| 51 | #908 | — | 4 | 4 | no | — |
| 52 | C2×D24 | — | 2 | 4 | no | — |
| 53 | C2×D24 | — | 2 | 2 | no | — |
| 54 | C2×D24 | — | 2 | 4 | no | — |
| 55 | C2×#204 | — | 2 | 4 | no | — |
| 56 | #901 | — | 4 | 4 | no | — |
| 57 | C2×ΓL(1,𝔽25) | — | 2 | 2 | no | — |
| 58 | C2×D3×C8 | — | 2 | 2 | no | — |
| 59 | #875 | D4 | 1 | 2 | yes | — |
| 60 | #874 | D4 | 1 | 1 | yes | characteristic |
| 61 | C4×D12 | — | 2 | 2 | no | — |
| 62 | C4×D12 | — | 2 | 2 | no | — |
| 63 | C4×D12 | — | 2 | 2 | no | — |
| 64 | C4×D12 | — | 2 | 2 | no | — |
| 65 | C4×D12 | — | 2 | 2 | no | — |
| 66 | #869 | D4 | 1 | 1 | yes | characteristic |
| 67 | C3×C4⋊C8 | — | 2 | 2 | no | — |
| 68 | C3×C4⋊C8 | — | 2 | 2 | no | — |
| 69 | C3×#97 | D4 | 1 | 4 | yes | — |
| 70 | #823 | — | 4 | 4 | no | — |
| 71 | #821 | — | 2 | 4 | no | — |
| 72 | #821 | — | 2 | 4 | no | — |
| 73 | (C2×D3)⋊C8 | — | 2 | 4 | no | — |
| 74 | #817 | — | 2 | 2 | no | — |
| 75 | #816 | — | 2 | 4 | no | — |
| 76 | C4⋊(C3⋊C8) | — | 2 | 2 | no | — |
| 77 | C2×AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 78 | C22×D8 | — | 3 | 1 | no | — |
| 79 | C2×#131 | — | 3 | 2 | no | — |
| 80 | #532 | — | 3 | 1 | no | — |
| 81 | C2×C4×D4 | — | 3 | 2 | no | — |
| 82 | #463 | — | 6 | 4 | no | — |
| 83 | #461 | — | 6 | 2 | no | — |
| 84 | #459 | — | 6 | 2 | no | — |
| 85 | #442 | — | 6 | 4 | no | — |
| 86 | #435 | — | 6 | 4 | no | — |
| 87 | C2×#106 | — | 3 | 1 | no | — |
| 88 | #423 | — | 3 | 1 | no | — |
| 89 | #423 | C2×D3 | 1 | 2 | yes | — |
| 90 | #417 | — | 3 | 4 | no | — |
| 91 | C2×#102 | — | 3 | 2 | no | — |
| 92 | C2×#100 | — | 3 | 4 | no | — |
| 93 | #408 | — | 3 | 2 | no | — |
| 94 | C2×#97 | C2×D3 | 1 | 1 | yes | characteristic |
| 95 | #343 | — | 3 | 4 | no | — |
| 96 | #341 | C2×D3 | 1 | 2 | yes | — |
| 97 | #341 | C2×D3 | 1 | 2 | yes | — |
| 98 | C23×D3 | — | 2 | 2 | no | — |
| 99 | C22×C3×C4 | #41 | 1 | 2 | yes | — |
| 100 | C22×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 101 | C2×#67 | — | 2 | 2 | no | — |
| 102 | C22×Dic3 | — | 2 | 2 | no | — |
| 103 | #235 | — | 2 | 4 | no | — |
| 104 | #235 | — | 2 | 4 | no | — |
| 105 | C2×D12 | — | 2 | 2 | no | — |
| 106 | C2×D12 | — | 2 | 2 | no | — |
| 107 | C2×D12 | — | 2 | 2 | no | — |
| 108 | C2×D12 | — | 2 | 2 | no | — |
| 109 | C2×D12 | — | 2 | 2 | no | — |
| 110 | C2×D12 | — | 2 | 2 | no | — |
| 111 | C2×D12 | C2×D4 | 1 | 2 | yes | — |
| 112 | C2×D12 | #41 | 1 | 4 | yes | — |
| 113 | C2×C4×D3 | — | 2 | 2 | no | — |
| 114 | C2×C4×D3 | — | 4 | 2 | no | — |
| 115 | C2×C4×D3 | — | 4 | 2 | no | — |
| 116 | C2×C4×D3 | — | 8 | 4 | no | — |
| 117 | C2×Dic6 | C2×D4 | 1 | 2 | yes | — |
| 118 | C3×M16 | — | 2 | 4 | no | — |
| 119 | C3×M16 | — | 2 | 4 | no | — |
| 120 | C2×C3×C8 | — | 2 | 2 | no | — |
| 121 | C2×C3×C8 | — | 2 | 2 | no | — |
| 122 | C2×C3×C8 | — | 2 | 4 | no | — |
| 123 | C2×C3×C8 | C2×D4 | 1 | 2 | yes | — |
| 124 | C2×C3×C8 | C2×D4 | 1 | 2 | yes | — |
| 125 | C3×C42 | C2×D4 | 1 | 2 | yes | — |
| 126 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 127 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 128 | (C2×D3)⋊C4 | — | 2 | 4 | no | — |
| 129 | (C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 130 | (C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 131 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 132 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 133 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 134 | #208 | — | 2 | 4 | no | — |
| 135 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 136 | D24 | — | 2 | 8 | no | — |
| 137 | D24 | — | 4 | 8 | no | — |
| 138 | #204 | — | 4 | 8 | no | — |
| 139 | ΓL(1,𝔽25) | — | 4 | 4 | no | — |
| 140 | D3×C8 | — | 4 | 4 | no | — |
| 141 | C2×#41 | — | 3 | 2 | no | — |
| 142 | C22×D4 | — | 3 | 2 | no | — |
| 143 | C23×C4 | — | 6 | 2 | no | — |
| 144 | AGL(1,ℤ/8) | — | 6 | 8 | no | — |
| 145 | AGL(1,ℤ/8) | — | 12 | 4 | no | — |
| 146 | C2×SD16 | — | 6 | 4 | no | — |
| 147 | C2×D8 | — | 6 | 4 | no | — |
| 148 | C2×D8 | — | 6 | 4 | no | — |
| 149 | C2×D8 | — | 6 | 2 | no | — |
| 150 | #131 | — | 12 | 4 | no | — |
| 151 | C2×M16 | — | 3 | 2 | no | — |
| 152 | C2×M16 | — | 6 | 2 | no | — |
| 153 | C2×M16 | C22×D3 | 1 | 4 | yes | — |
| 154 | C22×C8 | — | 6 | 2 | no | — |
| 155 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 156 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 157 | C4⋊Q8 | — | 3 | 1 | no | — |
| 158 | #127 | — | 3 | 1 | no | — |
| 159 | #124 | — | 3 | 2 | no | — |
| 160 | #121 | — | 6 | 2 | no | — |
| 161 | C4×D4 | — | 6 | 2 | no | — |
| 162 | C4×D4 | — | 6 | 2 | no | — |
| 163 | C4×D4 | — | 6 | 2 | no | — |
| 164 | C4×D4 | — | 6 | 2 | no | — |
| 165 | C4×D4 | — | 6 | 2 | no | — |
| 166 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 167 | C2×C22⋊C4 | — | 6 | 2 | no | — |
| 168 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 169 | #106 | — | 6 | 2 | no | — |
| 170 | C4⋊C8 | — | 2 | 2 | no | — |
| 171 | C4⋊C8 | — | 2 | 2 | no | — |
| 172 | C4⋊C8 | — | 6 | 2 | no | — |
| 173 | #103 | — | 6 | 4 | no | — |
| 174 | #102 | — | 6 | 4 | no | — |
| 175 | #102 | — | 6 | 4 | no | — |
| 176 | #100 | — | 12 | 4 | no | — |
| 177 | C22⋊C8 | — | 6 | 4 | no | — |
| 178 | #97 | D12 | 1 | 4 | yes | — |
| 179 | C23×C3 | C2×#41 | 1 | 1 | yes | characteristic |
| 180 | C22×D3 | — | 2 | 2 | no | — |
| 181 | C22×D3 | — | 2 | 4 | no | — |
| 182 | C22×D3 | — | 4 | 2 | no | — |
| 183 | C22×D3 | — | 4 | 2 | no | — |
| 184 | C22×D3 | — | 8 | 2 | no | — |
| 185 | C22×D3 | — | 8 | 2 | no | — |
| 186 | C2×C3×C4 | — | 4 | 2 | no | — |
| 187 | C2×C3×C4 | #121 | 1 | 2 | yes | — |
| 188 | C2×C3×C4 | #121 | 1 | 2 | yes | — |
| 189 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 190 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 191 | C2×C3×C4 | C2×#41 | 1 | 2 | yes | — |
| 192 | C2×C3×C4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 193 | #67 | — | 4 | 4 | no | — |
| 194 | C2×Dic3 | — | 2 | 2 | no | — |
| 195 | C2×Dic3 | — | 4 | 2 | no | — |
| 196 | C2×Dic3 | — | 8 | 2 | no | — |
| 197 | D12 | — | 2 | 4 | no | — |
| 198 | D12 | — | 2 | 4 | no | — |
| 199 | D12 | — | 2 | 4 | no | — |
| 200 | D12 | — | 4 | 2 | no | — |
| 201 | D12 | — | 4 | 2 | no | — |
| 202 | D12 | — | 8 | 2 | no | — |
| 203 | C4×D3 | — | 4 | 2 | no | — |
| 204 | C4×D3 | — | 4 | 2 | no | — |
| 205 | C4×D3 | — | 8 | 4 | no | — |
| 206 | C4×D3 | — | 8 | 4 | no | — |
| 207 | Dic6 | — | 2 | 4 | no | — |
| 208 | C3×C8 | — | 2 | 2 | no | — |
| 209 | C3×C8 | — | 2 | 2 | no | — |
| 210 | C3×C8 | — | 4 | 4 | no | — |
| 211 | C3⋊C8 | — | 4 | 2 | no | — |
| 212 | C24 | — | 6 | 2 | no | — |
| 213 | #41 | — | 6 | 4 | no | — |
| 214 | #41 | — | 6 | 4 | no | — |
| 215 | C2×Q8 | — | 3 | 2 | no | — |
| 216 | C2×D4 | — | 3 | 2 | no | — |
| 217 | C2×D4 | — | 3 | 4 | no | — |
| 218 | C2×D4 | — | 6 | 2 | no | — |
| 219 | C2×D4 | — | 6 | 2 | no | — |
| 220 | C2×D4 | — | 6 | 2 | no | — |
| 221 | C2×D4 | — | 6 | 2 | no | — |
| 222 | C2×D4 | — | 6 | 2 | no | — |
| 223 | C2×D4 | — | 6 | 2 | no | — |
| 224 | C2×D4 | — | 6 | 2 | no | — |
| 225 | C22×C4 | — | 6 | 2 | no | — |
| 226 | C22×C4 | — | 6 | 2 | no | — |
| 227 | C22×C4 | — | 12 | 2 | no | — |
| 228 | C22×C4 | — | 12 | 2 | no | — |
| 229 | C22×C4 | — | 24 | 4 | no | — |
| 230 | C22×C4 | #239 | 1 | 2 | yes | — |
| 231 | C22×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 232 | SD16 | — | 12 | 8 | no | — |
| 233 | D8 | — | 6 | 8 | no | — |
| 234 | D8 | — | 12 | 8 | no | — |
| 235 | M16 | — | 2 | 4 | no | — |
| 236 | M16 | — | 2 | 4 | no | — |
| 237 | M16 | — | 6 | 4 | no | — |
| 238 | M16 | — | 12 | 4 | no | — |
| 239 | C2×C8 | — | 2 | 2 | no | — |
| 240 | C2×C8 | — | 2 | 2 | no | — |
| 241 | C2×C8 | — | 2 | 4 | no | — |
| 242 | C2×C8 | — | 6 | 2 | no | — |
| 243 | C2×C8 | — | 12 | 4 | no | — |
| 244 | C2×C8 | C2×D12 | 1 | 2 | yes | — |
| 245 | C2×C8 | C2×D12 | 1 | 2 | yes | — |
| 246 | C4⋊C4 | — | 6 | 2 | no | — |
| 247 | C4⋊C4 | — | 6 | 2 | no | — |
| 248 | C4⋊C4 | — | 6 | 2 | no | — |
| 249 | C22⋊C4 | — | 6 | 4 | no | — |
| 250 | C22⋊C4 | — | 12 | 2 | no | — |
| 251 | C22⋊C4 | — | 12 | 2 | no | — |
| 252 | C42 | C2×D12 | 1 | 2 | yes | — |
| 253 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 254 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 255 | C22×C3 | — | 2 | 2 | no | — |
| 256 | C22×C3 | C2×#121 | 1 | 1 | yes | characteristic |
| 257 | C22×C3 | #553 | 1 | 2 | yes | — |
| 258 | C2×D3 | — | 4 | 2 | no | — |
| 259 | C2×D3 | — | 4 | 4 | no | — |
| 260 | C2×D3 | — | 4 | 4 | no | — |
| 261 | C2×D3 | — | 8 | 4 | no | — |
| 262 | C2×D3 | — | 8 | 2 | no | — |
| 263 | C2×D3 | — | 8 | 2 | no | — |
| 264 | C2×D3 | — | 8 | 4 | no | — |
| 265 | C3×C4 | — | 2 | 2 | no | — |
| 266 | C3×C4 | — | 2 | 2 | no | — |
| 267 | C3×C4 | #525 | 1 | 1 | yes | characteristic |
| 268 | C3×C4 | #535 | 1 | 1 | yes | characteristic |
| 269 | C3×C4 | #546 | 1 | 2 | yes | — |
| 270 | Dic3 | — | 4 | 2 | no | — |
| 271 | Dic3 | — | 8 | 2 | no | — |
| 272 | C23 | — | 6 | 2 | no | — |
| 273 | C23 | — | 6 | 4 | no | — |
| 274 | C23 | — | 12 | 2 | no | — |
| 275 | C23 | — | 12 | 2 | no | — |
| 276 | C23 | — | 24 | 2 | no | — |
| 277 | C23 | — | 24 | 2 | no | — |
| 278 | C23 | C2×#239 | 1 | 1 | yes | characteristic |
| 279 | Q8 | — | 6 | 4 | no | — |
| 280 | D4 | — | 6 | 4 | no | — |
| 281 | D4 | — | 6 | 4 | no | — |
| 282 | D4 | — | 6 | 4 | no | — |
| 283 | D4 | — | 12 | 4 | no | — |
| 284 | D4 | — | 12 | 2 | no | — |
| 285 | D4 | — | 12 | 2 | no | — |
| 286 | D4 | — | 24 | 2 | no | — |
| 287 | C2×C4 | — | 4 | 2 | no | — |
| 288 | C2×C4 | — | 6 | 2 | no | — |
| 289 | C2×C4 | — | 12 | 2 | no | — |
| 290 | C2×C4 | — | 12 | 2 | no | — |
| 291 | C2×C4 | — | 12 | 2 | no | — |
| 292 | C2×C4 | — | 24 | 4 | no | — |
| 293 | C2×C4 | — | 24 | 4 | no | — |
| 294 | C2×C4 | — | 24 | 2 | no | — |
| 295 | C2×C4 | #895 | 1 | 2 | yes | — |
| 296 | C2×C4 | #895 | 1 | 2 | yes | — |
| 297 | C2×C4 | C22×D12 | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | C2×D3×D4 | 1 | 1 | yes | characteristic |
| 299 | C2×C4 | D3×#41 | 1 | 2 | yes | — |
| 300 | C2×C4 | #1009 | 1 | 2 | yes | — |
| 301 | C8 | — | 2 | 2 | no | — |
| 302 | C8 | — | 2 | 2 | no | — |
| 303 | C8 | — | 4 | 4 | no | — |
| 304 | C8 | — | 12 | 2 | no | — |
| 305 | C2×C3 | — | 2 | 2 | no | — |
| 306 | C2×C3 | #2535 | 1 | 1 | yes | characteristic |
| 307 | C2×C3 | #3068 | 1 | 2 | yes | — |
| 308 | D3 | — | 8 | 2 | no | — |
| 309 | D3 | — | 8 | 4 | no | — |
| 310 | C22 | — | 2 | 2 | no | — |
| 311 | C22 | — | 12 | 4 | no | — |
| 312 | C22 | — | 12 | 2 | no | — |
| 313 | C22 | — | 12 | 4 | no | — |
| 314 | C22 | — | 24 | 4 | no | — |
| 315 | C22 | — | 24 | 4 | no | — |
| 316 | C22 | — | 24 | 2 | no | — |
| 317 | C22 | — | 24 | 2 | no | — |
| 318 | C22 | C2×#895 | 1 | 1 | yes | characteristic |
| 319 | C22 | #5695 | 1 | 2 | yes | — |
| 320 | C4 | — | 2 | 2 | no | — |
| 321 | C4 | — | 2 | 2 | no | — |
| 322 | C4 | — | 12 | 2 | no | — |
| 323 | C4 | — | 24 | 2 | no | — |
| 324 | C4 | #5524 | 1 | 1 | yes | characteristic |
| 325 | C4 | #5525 | 1 | 1 | yes | characteristic |
| 326 | C4 | #5676 | 1 | 2 | yes | — |
| 327 | C3 | #28177 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 328 | C2 | — | 2 | 2 | no | — |
| 329 | C2 | — | 24 | 2 | no | — |
| 330 | C2 | — | 24 | 4 | no | — |
| 331 | C2 | #74239 | 1 | 1 | yes | characteristic |
| 332 | C2 | #79261 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #79055 | C2 | semidirect |
| 2. C2×#4711 | C2 | semidirect |
| 3. #78491 | C2 | semidirect |
| 4. C3×#1838 | C2 | semidirect |
| 5. #70527 | C2 | semidirect |
| 6. #68948 | C2 | semidirect |
| 7. #68942 | C2 | semidirect |
| 8. #68922 | C2 | semidirect |
| 37. #1838 | D3 | semidirect |
| 327. C3 | #28177 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #79055 | 2 |
| C2×#4711 | 2 |
| #78491 | 2 |
| C3×#1838 | 2 |
| #70527 | 2 |
| #68948 | 2 |
| #68942 | 2 |
| #68922 | 2 |
| #28177 | 3 |