Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1606 conjugacy classes of subgroups; only the 235 normal subgroups are shown.
This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 235 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×#5525 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #74119 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C3×#1820 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #71594 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #69537 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #68970 | C2 | 1 | 8 | yes | maximal, complemented |
| 7 | #21750 | — | 3 | 1 | no | maximal, Sylow |
| 8 | C22×#235 | C22 | 1 | 1 | yes | characteristic |
| 9 | C23×D12 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C3×#117 | C22 | 1 | 1 | yes | characteristic |
| 11 | #5525 | — | 2 | 8 | no | — |
| 12 | C2×#895 | C4 | 1 | 4 | yes | complemented |
| 13 | C2×#882 | C22 | 1 | 4 | yes | — |
| 14 | C2×D3×C22⋊C4 | C22 | 1 | 4 | yes | — |
| 15 | C2×#875 | C4 | 1 | 2 | yes | complemented |
| 16 | C2×#874 | C4 | 1 | 2 | yes | complemented |
| 17 | C3×#381 | C22 | 1 | 2 | yes | — |
| 18 | #5211 | C22 | 1 | 2 | yes | — |
| 19 | #4955 | C22 | 1 | 4 | yes | — |
| 20 | C2×#806 | C22 | 1 | 8 | yes | — |
| 21 | #4665 | — | 2 | 8 | no | — |
| 22 | C3×#342 | C22 | 1 | 4 | yes | — |
| 23 | #4526 | C22 | 1 | 4 | yes | — |
| 24 | C2×#535 | — | 3 | 1 | no | — |
| 25 | #2308 | — | 3 | 2 | no | — |
| 26 | #1898 | — | 3 | 8 | no | — |
| 27 | #1820 | — | 3 | 1 | no | — |
| 28 | #1820 | D3 | 1 | 1 | yes | characteristic, complemented |
| 29 | #1782 | — | 3 | 2 | no | — |
| 30 | C24×D3 | C23 | 1 | 2 | yes | — |
| 31 | C23×C3×C4 | C23 | 1 | 1 | yes | characteristic |
| 32 | C22×#67 | C23 | 1 | 2 | yes | — |
| 33 | C2×#235 | — | 2 | 4 | no | — |
| 34 | C2×#235 | — | 4 | 4 | no | — |
| 35 | C22×D12 | — | 4 | 4 | no | — |
| 36 | C22×D12 | C2×C4 | 1 | 2 | yes | — |
| 37 | C22×D12 | C2×C4 | 1 | 2 | yes | — |
| 38 | C22×D12 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 39 | C22×D12 | C2×C4 | 1 | 8 | yes | complemented |
| 40 | C22×D12 | D4 | 1 | 8 | yes | — |
| 41 | C22×C4×D3 | — | 2 | 4 | no | — |
| 42 | C22×C4×D3 | C2×C4 | 1 | 2 | yes | complemented |
| 43 | C22×Dic6 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 44 | C3×#117 | — | 2 | 4 | no | — |
| 45 | C2×C3×C4⋊C4 | C2×C4 | 1 | 2 | yes | complemented |
| 46 | C2×C3×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 47 | C2×C3×C22⋊C4 | C23 | 1 | 2 | yes | — |
| 48 | C2×C3×C42 | C2×C4 | 1 | 2 | yes | complemented |
| 49 | C2×#217 | C23 | 1 | 4 | yes | — |
| 50 | C2×(C2×D3)⋊C4 | — | 2 | 4 | no | — |
| 51 | C2×(C2×D3)⋊C4 | — | 2 | 8 | no | — |
| 52 | C2×(C2×D3)⋊C4 | C2×C4 | 1 | 8 | yes | complemented |
| 53 | #895 | — | 2 | 16 | no | — |
| 54 | #882 | — | 2 | 8 | no | — |
| 55 | #882 | — | 2 | 8 | no | — |
| 56 | D3×C22⋊C4 | — | 4 | 8 | no | — |
| 57 | #875 | — | 2 | 8 | no | — |
| 58 | #874 | — | 2 | 8 | no | — |
| 59 | C3×#95 | — | 2 | 4 | no | — |
| 60 | C3⋊#95 | — | 2 | 4 | no | — |
| 61 | #806 | — | 4 | 8 | no | — |
| 62 | C22×#41 | — | 3 | 1 | no | — |
| 63 | C23×D4 | — | 3 | 1 | no | — |
| 64 | #535 | — | 6 | 8 | no | — |
| 65 | C2×#127 | — | 3 | 2 | no | — |
| 66 | C2×#124 | — | 3 | 2 | no | — |
| 67 | C2×#121 | — | 3 | 4 | no | — |
| 68 | C2×#120 | — | 3 | 4 | no | — |
| 69 | C2×#117 | C2×D3 | 1 | 1 | yes | characteristic |
| 70 | C22×C22⋊C4 | — | 3 | 4 | no | — |
| 71 | C2×#99 | — | 3 | 8 | no | — |
| 72 | #386 | — | 6 | 8 | no | — |
| 73 | #381 | — | 3 | 2 | no | — |
| 74 | #381 | C2×D3 | 1 | 2 | yes | — |
| 75 | #379 | — | 3 | 4 | no | — |
| 76 | #342 | — | 3 | 4 | no | — |
| 77 | #342 | C2×D3 | 1 | 4 | yes | — |
| 78 | C24×C3 | C24 | 1 | 1 | yes | characteristic |
| 79 | C23×D3 | — | 2 | 8 | no | — |
| 80 | C23×D3 | — | 8 | 4 | no | — |
| 81 | C23×D3 | C22×C4 | 1 | 2 | yes | — |
| 82 | C23×D3 | C22×C4 | 1 | 4 | yes | — |
| 83 | C23×D3 | C2×D4 | 1 | 4 | yes | — |
| 84 | C23×D3 | C2×D4 | 1 | 4 | yes | — |
| 85 | C22×C3×C4 | — | 2 | 4 | no | — |
| 86 | C22×C3×C4 | — | 2 | 4 | no | — |
| 87 | C22×C3×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C22×C3×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 89 | C22×C3×C4 | C22×C4 | 1 | 4 | yes | — |
| 90 | C22×C3×C4 | C2×D4 | 1 | 4 | yes | — |
| 91 | C2×#67 | — | 2 | 4 | no | — |
| 92 | C2×#67 | — | 2 | 8 | no | — |
| 93 | C22×Dic3 | — | 2 | 4 | no | — |
| 94 | C22×Dic3 | C22×C4 | 1 | 2 | yes | — |
| 95 | #235 | — | 4 | 16 | no | — |
| 96 | C2×D12 | — | 2 | 2 | no | — |
| 97 | C2×D12 | — | 2 | 16 | no | — |
| 98 | C2×D12 | — | 4 | 4 | no | — |
| 99 | C2×D12 | — | 4 | 4 | no | — |
| 100 | C2×D12 | — | 4 | 8 | no | — |
| 101 | C2×D12 | — | 4 | 8 | no | — |
| 102 | C2×D12 | — | 4 | 2 | no | — |
| 103 | C2×D12 | — | 8 | 8 | no | — |
| 104 | C2×D12 | C22⋊C4 | 1 | 8 | yes | — |
| 105 | C2×D12 | C22⋊C4 | 1 | 8 | yes | — |
| 106 | C2×C4×D3 | — | 2 | 4 | no | — |
| 107 | C2×C4×D3 | — | 4 | 4 | no | — |
| 108 | C2×C4×D3 | — | 4 | 8 | no | — |
| 109 | C2×C4×D3 | — | 8 | 8 | no | — |
| 110 | C2×Dic6 | — | 2 | 2 | no | — |
| 111 | C2×Dic6 | — | 4 | 2 | no | — |
| 112 | C3×C4⋊C4 | — | 2 | 4 | no | — |
| 113 | C3×C22⋊C4 | — | 2 | 4 | no | — |
| 114 | C3×C22⋊C4 | — | 2 | 8 | no | — |
| 115 | C3×C42 | — | 2 | 4 | no | — |
| 116 | #217 | — | 2 | 8 | no | — |
| 117 | (C2×D3)⋊C4 | — | 2 | 16 | no | — |
| 118 | (C2×D3)⋊C4 | — | 4 | 8 | no | — |
| 119 | (C2×D3)⋊C4 | — | 8 | 8 | no | — |
| 120 | C25 | — | 3 | 2 | no | — |
| 121 | C2×#41 | — | 6 | 4 | no | — |
| 122 | C2×#41 | — | 12 | 4 | no | — |
| 123 | C22×Q8 | — | 3 | 1 | no | — |
| 124 | C22×D4 | — | 3 | 2 | no | — |
| 125 | C22×D4 | — | 3 | 2 | no | — |
| 126 | C22×D4 | — | 3 | 2 | no | — |
| 127 | C22×D4 | — | 3 | 1 | no | — |
| 128 | C22×D4 | — | 3 | 8 | no | — |
| 129 | C22×D4 | — | 3 | 8 | no | — |
| 130 | C22×D4 | — | 12 | 4 | no | — |
| 131 | C23×C4 | — | 3 | 2 | no | — |
| 132 | C23×C4 | — | 6 | 4 | no | — |
| 133 | C23×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 134 | #127 | — | 6 | 8 | no | — |
| 135 | #124 | — | 6 | 8 | no | — |
| 136 | #121 | — | 6 | 16 | no | — |
| 137 | #120 | — | 6 | 8 | no | — |
| 138 | #120 | — | 6 | 8 | no | — |
| 139 | #117 | — | 2 | 4 | no | — |
| 140 | C2×C4⋊C4 | C4×D3 | 1 | 2 | yes | complemented |
| 141 | C2×C22⋊C4 | — | 3 | 8 | no | — |
| 142 | C2×C22⋊C4 | — | 3 | 4 | no | — |
| 143 | C2×C22⋊C4 | — | 6 | 4 | no | — |
| 144 | C2×C22⋊C4 | — | 6 | 8 | no | — |
| 145 | C2×C22⋊C4 | — | 12 | 8 | no | — |
| 146 | C2×C22⋊C4 | C22×D3 | 1 | 4 | yes | — |
| 147 | C2×C22⋊C4 | C22×D3 | 1 | 2 | yes | — |
| 148 | C2×C42 | C4×D3 | 1 | 2 | yes | complemented |
| 149 | #99 | — | 12 | 8 | no | — |
| 150 | #95 | — | 2 | 4 | no | — |
| 151 | #95 | — | 6 | 4 | no | — |
| 152 | C23×C3 | — | 4 | 2 | no | — |
| 153 | C23×C3 | C23×C4 | 1 | 1 | yes | characteristic |
| 154 | C23×C3 | C22×D4 | 1 | 2 | yes | — |
| 155 | C23×C3 | H(5,𝔽2) | 1 | 4 | yes | — |
| 156 | C22×D3 | — | 2 | 4 | no | — |
| 157 | C22×D3 | — | 2 | 4 | no | — |
| 158 | C22×D3 | — | 2 | 8 | no | — |
| 159 | C22×D3 | — | 2 | 8 | no | — |
| 160 | C22×D3 | — | 2 | 2 | no | — |
| 161 | C22×D3 | — | 2 | 4 | no | — |
| 162 | C22×D3 | — | 4 | 8 | no | — |
| 163 | C22×D3 | — | 4 | 8 | no | — |
| 164 | C22×D3 | — | 4 | 8 | no | — |
| 165 | C22×D3 | — | 4 | 4 | no | — |
| 166 | C22×D3 | — | 8 | 16 | no | — |
| 167 | C22×D3 | C2×C22⋊C4 | 1 | 8 | yes | — |
| 168 | C2×C3×C4 | — | 2 | 2 | no | — |
| 169 | C2×C3×C4 | — | 2 | 2 | no | — |
| 170 | C2×C3×C4 | — | 2 | 8 | no | — |
| 171 | C2×C3×C4 | — | 2 | 4 | no | — |
| 172 | C2×C3×C4 | — | 4 | 4 | no | — |
| 173 | C2×C3×C4 | — | 4 | 8 | no | — |
| 174 | C2×C3×C4 | — | 4 | 4 | no | — |
| 175 | C2×C3×C4 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 176 | C2×C3×C4 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 177 | #67 | — | 4 | 8 | no | — |
| 178 | C2×Dic3 | — | 2 | 2 | no | — |
| 179 | C2×Dic3 | — | 2 | 4 | no | — |
| 180 | C2×Dic3 | — | 4 | 8 | no | — |
| 181 | C2×Dic3 | — | 4 | 4 | no | — |
| 182 | D12 | — | 4 | 8 | no | — |
| 183 | D12 | — | 4 | 8 | no | — |
| 184 | D12 | — | 4 | 4 | no | — |
| 185 | D12 | — | 8 | 16 | no | — |
| 186 | C4×D3 | — | 4 | 8 | no | — |
| 187 | C4×D3 | — | 8 | 16 | no | — |
| 188 | Dic6 | — | 4 | 4 | no | — |
| 189 | C24 | — | 3 | 4 | no | — |
| 190 | C24 | — | 3 | 4 | no | — |
| 191 | C24 | — | 3 | 4 | no | — |
| 192 | C24 | — | 3 | 2 | no | — |
| 193 | C24 | — | 6 | 8 | no | — |
| 194 | C24 | — | 24 | 4 | no | — |
| 195 | C24 | C23×D3 | 1 | 1 | yes | characteristic |
| 196 | #41 | — | 12 | 16 | no | — |
| 197 | C2×Q8 | — | 6 | 2 | no | — |
| 198 | C2×Q8 | — | 12 | 2 | no | — |
| 199 | C2×D4 | — | 3 | 8 | no | — |
| 200 | C2×D4 | — | 3 | 8 | no | — |
| 201 | C2×D4 | — | 6 | 16 | no | — |
| 202 | C2×D4 | — | 6 | 4 | no | — |
| 203 | C2×D4 | — | 6 | 8 | no | — |
| 204 | C2×D4 | — | 6 | 2 | no | — |
| 205 | C2×D4 | — | 12 | 8 | no | — |
| 206 | C2×D4 | — | 12 | 8 | no | — |
| 207 | C2×D4 | — | 12 | 4 | no | — |
| 208 | C2×D4 | — | 12 | 4 | no | — |
| 209 | C2×D4 | — | 12 | 2 | no | — |
| 210 | C2×D4 | — | 24 | 8 | no | — |
| 211 | C22×C4 | — | 2 | 4 | no | — |
| 212 | C22×C4 | — | 2 | 4 | no | — |
| 213 | C22×C4 | — | 3 | 2 | no | — |
| 214 | C22×C4 | — | 6 | 4 | no | — |
| 215 | C22×C4 | — | 6 | 4 | no | — |
| 216 | C22×C4 | — | 12 | 4 | no | — |
| 217 | C22×C4 | — | 12 | 8 | no | — |
| 218 | C22×C4 | — | 24 | 8 | no | — |
| 219 | C22×C4 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 220 | C22×C4 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 221 | C22×C4 | C2×C4×D3 | 1 | 4 | yes | — |
| 222 | C22×C4 | D3×D4 | 1 | 4 | yes | — |
| 223 | C4⋊C4 | — | 2 | 4 | no | — |
| 224 | C22⋊C4 | — | 2 | 4 | no | — |
| 225 | C22⋊C4 | — | 2 | 8 | no | — |
| 226 | C22⋊C4 | — | 6 | 16 | no | — |
| 227 | C22⋊C4 | — | 6 | 8 | no | — |
| 228 | C22⋊C4 | — | 12 | 8 | no | — |
| 229 | C22⋊C4 | — | 24 | 8 | no | — |
| 230 | C42 | — | 2 | 4 | no | — |
| 231 | C22×C3 | — | 2 | 4 | no | — |
| 232 | C22×C3 | — | 2 | 2 | no | — |
| 233 | C22×C3 | — | 4 | 4 | no | — |
| 234 | C22×C3 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 235 | C22×C3 | #518 | 1 | 2 | yes | — |
| 236 | C22×C3 | #534 | 1 | 4 | yes | — |
| 237 | C2×D3 | — | 2 | 8 | no | — |
| 238 | C2×D3 | — | 2 | 16 | no | — |
| 239 | C2×D3 | — | 4 | 8 | no | — |
| 240 | C2×D3 | — | 4 | 8 | no | — |
| 241 | C2×D3 | — | 4 | 2 | no | — |
| 242 | C2×D3 | — | 4 | 4 | no | — |
| 243 | C2×D3 | — | 8 | 8 | no | — |
| 244 | C2×D3 | — | 8 | 8 | no | — |
| 245 | C2×D3 | — | 8 | 4 | no | — |
| 246 | C3×C4 | — | 2 | 2 | no | — |
| 247 | C3×C4 | — | 2 | 2 | no | — |
| 248 | C3×C4 | — | 4 | 4 | no | — |
| 249 | C3×C4 | — | 8 | 4 | no | — |
| 250 | Dic3 | — | 4 | 2 | no | — |
| 251 | Dic3 | — | 8 | 4 | no | — |
| 252 | C23 | — | 3 | 8 | no | — |
| 253 | C23 | — | 4 | 2 | no | — |
| 254 | C23 | — | 6 | 4 | no | — |
| 255 | C23 | — | 6 | 4 | no | — |
| 256 | C23 | — | 6 | 8 | no | — |
| 257 | C23 | — | 6 | 8 | no | — |
| 258 | C23 | — | 6 | 2 | no | — |
| 259 | C23 | — | 6 | 4 | no | — |
| 260 | C23 | — | 12 | 8 | no | — |
| 261 | C23 | — | 12 | 8 | no | — |
| 262 | C23 | — | 12 | 8 | no | — |
| 263 | C23 | — | 12 | 4 | no | — |
| 264 | C23 | — | 24 | 16 | no | — |
| 265 | C23 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 266 | C23 | C2×D3×D4 | 1 | 2 | yes | — |
| 267 | C23 | #1009 | 1 | 4 | yes | — |
| 268 | Q8 | — | 12 | 4 | no | — |
| 269 | D4 | — | 12 | 8 | no | — |
| 270 | D4 | — | 12 | 8 | no | — |
| 271 | D4 | — | 12 | 4 | no | — |
| 272 | D4 | — | 12 | 8 | no | — |
| 273 | D4 | — | 24 | 16 | no | — |
| 274 | C2×C4 | — | 2 | 8 | no | — |
| 275 | C2×C4 | — | 2 | 2 | no | — |
| 276 | C2×C4 | — | 2 | 2 | no | — |
| 277 | C2×C4 | — | 2 | 4 | no | — |
| 278 | C2×C4 | — | 4 | 4 | no | — |
| 279 | C2×C4 | — | 4 | 8 | no | — |
| 280 | C2×C4 | — | 4 | 4 | no | — |
| 281 | C2×C4 | — | 6 | 2 | no | — |
| 282 | C2×C4 | — | 6 | 4 | no | — |
| 283 | C2×C4 | — | 12 | 8 | no | — |
| 284 | C2×C4 | — | 12 | 8 | no | — |
| 285 | C2×C4 | — | 12 | 4 | no | — |
| 286 | C2×C4 | — | 24 | 16 | no | — |
| 287 | C2×C4 | D3×C22⋊C4 | 1 | 2 | yes | — |
| 288 | C2×C4 | D3×C22⋊C4 | 1 | 2 | yes | — |
| 289 | C2×C3 | — | 2 | 4 | no | — |
| 290 | C2×C3 | — | 2 | 2 | no | — |
| 291 | C2×C3 | #2308 | 1 | 1 | yes | characteristic |
| 292 | C2×C3 | #2883 | 1 | 2 | yes | — |
| 293 | D3 | — | 4 | 8 | no | — |
| 294 | D3 | — | 8 | 2 | no | — |
| 295 | C22 | — | 2 | 4 | no | — |
| 296 | C22 | — | 2 | 2 | no | — |
| 297 | C22 | — | 4 | 4 | no | — |
| 298 | C22 | — | 6 | 16 | no | — |
| 299 | C22 | — | 6 | 8 | no | — |
| 300 | C22 | — | 12 | 8 | no | — |
| 301 | C22 | — | 12 | 8 | no | — |
| 302 | C22 | — | 12 | 4 | no | — |
| 303 | C22 | — | 12 | 2 | no | — |
| 304 | C22 | — | 24 | 8 | no | — |
| 305 | C22 | — | 24 | 8 | no | — |
| 306 | C22 | — | 24 | 4 | no | — |
| 307 | C22 | C2×D3×C22⋊C4 | 1 | 1 | yes | characteristic |
| 308 | C22 | #5521 | 1 | 2 | yes | — |
| 309 | C22 | #5653 | 1 | 4 | yes | — |
| 310 | C4 | — | 2 | 2 | no | — |
| 311 | C4 | — | 2 | 2 | no | — |
| 312 | C4 | — | 4 | 4 | no | — |
| 313 | C4 | — | 8 | 4 | no | — |
| 314 | C4 | — | 12 | 2 | no | — |
| 315 | C4 | — | 24 | 4 | no | — |
| 316 | C3 | #21750 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 317 | C2 | — | 2 | 4 | no | — |
| 318 | C2 | — | 2 | 2 | no | — |
| 319 | C2 | — | 12 | 8 | no | — |
| 320 | C2 | — | 24 | 2 | no | — |
| 321 | C2 | #74119 | 1 | 1 | yes | characteristic |
| 322 | C2 | #78694 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #74119 | C2 | semidirect |
| 3. C3×#1820 | C2 | semidirect |
| 4. #71594 | C2 | semidirect |
| 5. #69537 | C2 | semidirect |
| 6. #68970 | C2 | semidirect |
| 12. C2×#895 | C4 | semidirect |
| 15. C2×#875 | C4 | semidirect |
| 16. C2×#874 | C4 | semidirect |
| 28. #1820 | D3 | semidirect |
| 38. C22×D12 | C2×C4 | semidirect |
| 39. C22×D12 | C2×C4 | semidirect |
| 42. C22×C4×D3 | C2×C4 | semidirect |
| 43. C22×Dic6 | C2×C4 | semidirect |
| 45. C2×C3×C4⋊C4 | C2×C4 | semidirect |
| 48. C2×C3×C42 | C2×C4 | semidirect |
| 52. C2×(C2×D3)⋊C4 | C2×C4 | semidirect |
| 140. C2×C4⋊C4 | C4×D3 | semidirect |
| 148. C2×C42 | C4×D3 | semidirect |
| 316. C3 | #21750 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#5525 | 2 |
| #74119 | 2 |
| C3×#1820 | 2 |
| #71594 | 2 |
| #69537 | 2 |
| #68970 | 2 |
| #21750 | 3 |