Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
2618 conjugacy classes of subgroups.
The subgroup lattice has 514 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C3×#540 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#5535 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #83188 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #83040 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #82873 | C2 | 1 | 8 | yes | maximal, complemented |
| 6 | #82718 | C2 | 1 | 8 | yes | maximal, complemented |
| 7 | #82700 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #82676 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #82386 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #82378 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #82227 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #37298 | — | 3 | 1 | no | maximal, Sylow |
| 13 | C3×#540 | — | 2 | 8 | no | — |
| 14 | C2×C3×#127 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×C3×#125 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×C3×#123 | C22 | 1 | 4 | yes | — |
| 17 | C2×C3×#121 | C22 | 1 | 2 | yes | — |
| 18 | C2×C3×#121 | C22 | 1 | 4 | yes | — |
| 19 | C2×C3×C4×D4 | C22 | 1 | 2 | yes | — |
| 20 | C2×C3×#117 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#940 | C22 | 1 | 2 | yes | — |
| 22 | C2×#940 | C22 | 1 | 2 | yes | — |
| 23 | C2×#939 | C22 | 1 | 8 | yes | — |
| 24 | C2×#938 | C22 | 1 | 8 | yes | — |
| 25 | C2×#930 | C22 | 1 | 2 | yes | — |
| 26 | C2×#930 | C22 | 1 | 8 | yes | — |
| 27 | C2×#926 | C22 | 1 | 1 | yes | characteristic |
| 28 | C22×C4×Dic3 | C22 | 1 | 1 | yes | characteristic |
| 29 | #5535 | — | 2 | 8 | no | — |
| 30 | C2×#894 | C22 | 1 | 4 | yes | — |
| 31 | C2×#884 | C22 | 1 | 4 | yes | — |
| 32 | C2×#874 | C22 | 1 | 1 | yes | characteristic |
| 33 | C2×C4×D12 | C22 | 1 | 2 | yes | — |
| 34 | C2×#870 | C22 | 1 | 1 | yes | characteristic |
| 35 | #5223 | C22 | 1 | 8 | yes | — |
| 36 | #5223 | C22 | 1 | 8 | yes | — |
| 37 | #5221 | C22 | 1 | 2 | yes | — |
| 38 | #5221 | C22 | 1 | 2 | yes | — |
| 39 | #5210 | C22 | 1 | 8 | yes | — |
| 40 | #4994 | C22 | 1 | 8 | yes | — |
| 41 | #4989 | C22 | 1 | 8 | yes | — |
| 42 | #4988 | C22 | 1 | 2 | yes | — |
| 43 | #4988 | C22 | 1 | 2 | yes | — |
| 44 | #4984 | C22 | 1 | 4 | yes | — |
| 45 | #4977 | C22 | 1 | 2 | yes | — |
| 46 | #4975 | C22 | 1 | 2 | yes | — |
| 47 | #4961 | C22 | 1 | 8 | yes | — |
| 48 | #4960 | C22 | 1 | 8 | yes | — |
| 49 | #4957 | C22 | 1 | 8 | yes | — |
| 50 | #4950 | C22 | 1 | 4 | yes | — |
| 51 | #4948 | C22 | 1 | 8 | yes | — |
| 52 | C4×C4⋊Dic3 | C22 | 1 | 2 | yes | — |
| 53 | C4×C4⋊Dic3 | C22 | 1 | 2 | yes | — |
| 54 | C2×#540 | — | 3 | 1 | no | — |
| 55 | C2×#540 | D3 | 1 | 1 | yes | characteristic, complemented |
| 56 | #2856 | — | 3 | 2 | no | — |
| 57 | #2682 | — | 3 | 8 | no | — |
| 58 | #2675 | — | 3 | 8 | no | — |
| 59 | #2619 | — | 3 | 2 | no | — |
| 60 | #2566 | — | 3 | 2 | no | — |
| 61 | #2566 | — | 3 | 2 | no | — |
| 62 | #2541 | — | 3 | 2 | no | — |
| 63 | #2541 | — | 3 | 2 | no | — |
| 64 | #2296 | — | 3 | 1 | no | — |
| 65 | C22×C3×D4 | C23 | 1 | 2 | yes | — |
| 66 | C22×C3×D4 | C23 | 1 | 4 | yes | — |
| 67 | C22×C3×D4 | C23 | 1 | 4 | yes | — |
| 68 | C23×C3×C4 | C23 | 1 | 1 | yes | characteristic |
| 69 | C23×C3×C4 | C23 | 1 | 4 | yes | — |
| 70 | C22×#67 | C23 | 1 | 8 | yes | — |
| 71 | C22×#67 | C23 | 1 | 4 | yes | — |
| 72 | C22×#67 | C23 | 1 | 8 | yes | — |
| 73 | C23×Dic3 | C23 | 1 | 2 | yes | — |
| 74 | C22×D12 | C23 | 1 | 2 | yes | — |
| 75 | C22×D12 | C23 | 1 | 4 | yes | — |
| 76 | C22×C4×D3 | C23 | 1 | 4 | yes | — |
| 77 | C3×#127 | — | 2 | 4 | no | — |
| 78 | C3×#125 | — | 2 | 4 | no | — |
| 79 | C3×#123 | — | 2 | 16 | no | — |
| 80 | C3×#121 | — | 2 | 8 | no | — |
| 81 | C3×#121 | — | 2 | 16 | no | — |
| 82 | C3×C4×D4 | — | 2 | 8 | no | — |
| 83 | C3×#117 | D4 | 1 | 8 | yes | complemented |
| 84 | C2×C3×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 85 | C2×C3×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 86 | C2×C3×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 87 | C2×C3×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 88 | C2×C3×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 89 | C2×C3×C22⋊C4 | C23 | 1 | 2 | yes | — |
| 90 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 91 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 92 | C2×#217 | C23 | 1 | 8 | yes | — |
| 93 | C2×#217 | C23 | 1 | 8 | yes | — |
| 94 | C2×#217 | C23 | 1 | 2 | yes | — |
| 95 | #940 | — | 4 | 4 | no | — |
| 96 | #940 | — | 4 | 4 | no | — |
| 97 | #939 | — | 4 | 16 | no | — |
| 98 | #938 | — | 4 | 16 | no | — |
| 99 | #930 | — | 2 | 8 | no | — |
| 100 | #930 | — | 4 | 16 | no | — |
| 101 | C2×(C2×D3)⋊C4 | C23 | 1 | 4 | yes | — |
| 102 | C2×(C2×D3)⋊C4 | C23 | 1 | 8 | yes | — |
| 103 | C2×(C2×D3)⋊C4 | C23 | 1 | 4 | yes | — |
| 104 | C2×(C2×D3)⋊C4 | C23 | 1 | 8 | yes | — |
| 105 | #926 | — | 4 | 2 | no | — |
| 106 | C2×C4⋊Dic3 | C23 | 1 | 4 | yes | — |
| 107 | C2×C4⋊Dic3 | C23 | 1 | 4 | yes | — |
| 108 | C2×C4⋊Dic3 | C23 | 1 | 2 | yes | — |
| 109 | C2×C4⋊Dic3 | C23 | 1 | 2 | yes | — |
| 110 | C2×Dic3⋊C4 | C23 | 1 | 4 | yes | — |
| 111 | C2×Dic3⋊C4 | C23 | 1 | 8 | yes | — |
| 112 | C2×C4×Dic3 | — | 4 | 2 | no | — |
| 113 | C2×C4×Dic3 | C23 | 1 | 2 | yes | — |
| 114 | C2×C4×Dic3 | C23 | 1 | 2 | yes | — |
| 115 | C2×C4×Dic3 | C23 | 1 | 2 | yes | — |
| 116 | #894 | — | 2 | 16 | no | — |
| 117 | #884 | — | 2 | 16 | no | — |
| 118 | #874 | — | 2 | 4 | no | — |
| 119 | C4×D12 | — | 2 | 8 | no | — |
| 120 | #870 | — | 2 | 4 | no | — |
| 121 | C3⋊#95 | C23 | 1 | 4 | yes | — |
| 122 | C3⋊#95 | C23 | 1 | 4 | yes | — |
| 123 | C3⋊#95 | C23 | 1 | 8 | yes | — |
| 124 | #540 | — | 2 | 8 | no | — |
| 125 | #540 | — | 6 | 8 | no | — |
| 126 | C2×#127 | — | 3 | 2 | no | — |
| 127 | C2×#127 | — | 3 | 2 | no | — |
| 128 | C2×#127 | — | 3 | 1 | no | — |
| 129 | C2×#127 | C2×D3 | 1 | 1 | yes | characteristic |
| 130 | C2×#125 | — | 3 | 1 | no | — |
| 131 | C2×#125 | C2×D3 | 1 | 1 | yes | characteristic |
| 132 | C2×#123 | — | 3 | 4 | no | — |
| 133 | C2×#123 | C2×D3 | 1 | 4 | yes | — |
| 134 | C2×#121 | — | 3 | 2 | no | — |
| 135 | C2×#121 | — | 3 | 8 | no | — |
| 136 | C2×#121 | — | 3 | 8 | no | — |
| 137 | C2×#121 | — | 3 | 8 | no | — |
| 138 | C2×#121 | — | 3 | 4 | no | — |
| 139 | C2×#121 | C2×D3 | 1 | 2 | yes | — |
| 140 | C2×#121 | C2×D3 | 1 | 4 | yes | — |
| 141 | C2×C4×D4 | — | 3 | 2 | no | — |
| 142 | C2×C4×D4 | C2×D3 | 1 | 2 | yes | — |
| 143 | C2×#117 | — | 3 | 1 | no | — |
| 144 | C2×#117 | C2×D3 | 1 | 1 | yes | characteristic |
| 145 | C22×C42 | — | 3 | 1 | no | — |
| 146 | #396 | — | 3 | 8 | no | — |
| 147 | #396 | — | 3 | 8 | no | — |
| 148 | #394 | — | 3 | 8 | no | — |
| 149 | #392 | — | 3 | 8 | no | — |
| 150 | #392 | — | 3 | 8 | no | — |
| 151 | #392 | — | 3 | 8 | no | — |
| 152 | #390 | — | 3 | 2 | no | — |
| 153 | #390 | — | 3 | 2 | no | — |
| 154 | #390 | — | 3 | 2 | no | — |
| 155 | #390 | — | 3 | 2 | no | — |
| 156 | #390 | — | 3 | 8 | no | — |
| 157 | #389 | — | 3 | 4 | no | — |
| 158 | #385 | — | 3 | 8 | no | — |
| 159 | #385 | — | 3 | 8 | no | — |
| 160 | #385 | — | 3 | 4 | no | — |
| 161 | #384 | — | 3 | 2 | no | — |
| 162 | #382 | — | 3 | 2 | no | — |
| 163 | C4×C4⋊C4 | — | 3 | 2 | no | — |
| 164 | C4×C4⋊C4 | — | 3 | 2 | no | — |
| 165 | C24×C3 | C24 | 1 | 4 | yes | — |
| 166 | C24×C3 | C24 | 1 | 1 | yes | characteristic |
| 167 | C23×D3 | C24 | 1 | 4 | yes | — |
| 168 | C2×C3×D4 | — | 2 | 8 | no | — |
| 169 | C2×C3×D4 | — | 2 | 4 | no | — |
| 170 | C2×C3×D4 | — | 2 | 8 | no | — |
| 171 | C2×C3×D4 | — | 2 | 4 | no | — |
| 172 | C2×C3×D4 | — | 2 | 4 | no | — |
| 173 | C2×C3×D4 | — | 4 | 8 | no | — |
| 174 | C2×C3×D4 | — | 4 | 8 | no | — |
| 175 | C22×C3×C4 | — | 2 | 2 | no | — |
| 176 | C22×C3×C4 | — | 2 | 2 | no | — |
| 177 | C22×C3×C4 | — | 2 | 8 | no | — |
| 178 | C22×C3×C4 | — | 4 | 8 | no | — |
| 179 | C22×C3×C4 | C2×D4 | 1 | 4 | yes | — |
| 180 | C22×C3×C4 | C24 | 1 | 4 | yes | — |
| 181 | C22×C3×C4 | C24 | 1 | 2 | yes | — |
| 182 | C22×C3×C4 | C24 | 1 | 2 | yes | — |
| 183 | C22×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 184 | C22×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 185 | C2×#67 | — | 2 | 8 | no | — |
| 186 | C2×#67 | — | 2 | 16 | no | — |
| 187 | C2×#67 | — | 4 | 8 | no | — |
| 188 | C2×#67 | — | 4 | 8 | no | — |
| 189 | C2×#67 | — | 4 | 8 | no | — |
| 190 | C2×#67 | — | 4 | 8 | no | — |
| 191 | C2×#67 | — | 4 | 8 | no | — |
| 192 | C2×#67 | — | 4 | 8 | no | — |
| 193 | C22×Dic3 | — | 2 | 8 | no | — |
| 194 | C22×Dic3 | — | 4 | 2 | no | — |
| 195 | C22×Dic3 | C24 | 1 | 4 | yes | — |
| 196 | C22×Dic3 | C24 | 1 | 4 | yes | — |
| 197 | C22×Dic3 | C24 | 1 | 4 | yes | — |
| 198 | C2×D12 | — | 2 | 4 | no | — |
| 199 | C2×D12 | — | 2 | 4 | no | — |
| 200 | C2×D12 | — | 2 | 4 | no | — |
| 201 | C2×D12 | — | 2 | 8 | no | — |
| 202 | C2×D12 | — | 4 | 8 | no | — |
| 203 | C2×C4×D3 | — | 2 | 8 | no | — |
| 204 | C2×C4×D3 | — | 4 | 8 | no | — |
| 205 | C3×C4⋊C4 | — | 2 | 4 | no | — |
| 206 | C3×C4⋊C4 | — | 2 | 8 | no | — |
| 207 | C3×C4⋊C4 | C2×D4 | 1 | 8 | yes | — |
| 208 | C3×C22⋊C4 | — | 2 | 8 | no | — |
| 209 | C3×C22⋊C4 | — | 2 | 8 | no | — |
| 210 | C3×C22⋊C4 | C2×D4 | 1 | 8 | yes | — |
| 211 | C3×C42 | C2×D4 | 1 | 4 | yes | — |
| 212 | C3×C42 | C2×D4 | 1 | 4 | yes | — |
| 213 | #217 | — | 2 | 16 | no | — |
| 214 | #217 | — | 4 | 2 | no | — |
| 215 | #217 | — | 4 | 8 | no | — |
| 216 | (C2×D3)⋊C4 | — | 2 | 8 | no | — |
| 217 | (C2×D3)⋊C4 | — | 2 | 8 | no | — |
| 218 | (C2×D3)⋊C4 | — | 2 | 16 | no | — |
| 219 | (C2×D3)⋊C4 | — | 4 | 8 | no | — |
| 220 | C4⋊Dic3 | — | 2 | 4 | no | — |
| 221 | C4⋊Dic3 | — | 2 | 8 | no | — |
| 222 | C4⋊Dic3 | — | 2 | 8 | no | — |
| 223 | C4⋊Dic3 | — | 4 | 2 | no | — |
| 224 | Dic3⋊C4 | — | 2 | 8 | no | — |
| 225 | Dic3⋊C4 | — | 4 | 8 | no | — |
| 226 | C4×Dic3 | — | 2 | 4 | no | — |
| 227 | C4×Dic3 | — | 2 | 4 | no | — |
| 228 | C4×Dic3 | — | 4 | 2 | no | — |
| 229 | C22×D4 | — | 3 | 4 | no | — |
| 230 | C22×D4 | — | 3 | 8 | no | — |
| 231 | C22×D4 | — | 3 | 8 | no | — |
| 232 | C22×D4 | — | 3 | 2 | no | — |
| 233 | C22×D4 | — | 3 | 4 | no | — |
| 234 | C22×D4 | C22×D3 | 1 | 2 | yes | — |
| 235 | C22×D4 | C22×D3 | 1 | 4 | yes | — |
| 236 | C22×D4 | C22×D3 | 1 | 4 | yes | — |
| 237 | C23×C4 | — | 3 | 2 | no | — |
| 238 | C23×C4 | — | 3 | 4 | no | — |
| 239 | C23×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 240 | C23×C4 | C22×D3 | 1 | 4 | yes | — |
| 241 | #127 | — | 2 | 4 | no | — |
| 242 | #127 | — | 6 | 4 | no | — |
| 243 | #127 | — | 12 | 4 | no | — |
| 244 | #127 | — | 12 | 4 | no | — |
| 245 | #125 | — | 2 | 4 | no | — |
| 246 | #125 | — | 6 | 4 | no | — |
| 247 | #123 | — | 2 | 16 | no | — |
| 248 | #123 | — | 6 | 16 | no | — |
| 249 | #121 | — | 2 | 8 | no | — |
| 250 | #121 | — | 2 | 16 | no | — |
| 251 | #121 | — | 6 | 8 | no | — |
| 252 | #121 | — | 6 | 16 | no | — |
| 253 | #121 | — | 12 | 16 | no | — |
| 254 | #121 | — | 12 | 16 | no | — |
| 255 | #121 | — | 12 | 16 | no | — |
| 256 | C4×D4 | — | 2 | 8 | no | — |
| 257 | C4×D4 | — | 6 | 8 | no | — |
| 258 | #117 | — | 12 | 2 | no | — |
| 259 | #117 | #67 | 1 | 8 | yes | complemented |
| 260 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 261 | C2×C4⋊C4 | — | 3 | 4 | no | — |
| 262 | C2×C4⋊C4 | — | 3 | 4 | no | — |
| 263 | C2×C4⋊C4 | — | 3 | 4 | no | — |
| 264 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 265 | C2×C4⋊C4 | — | 3 | 8 | no | — |
| 266 | C2×C4⋊C4 | C22×D3 | 1 | 2 | yes | — |
| 267 | C2×C4⋊C4 | C22×D3 | 1 | 4 | yes | — |
| 268 | C2×C4⋊C4 | C22×D3 | 1 | 2 | yes | — |
| 269 | C2×C22⋊C4 | — | 3 | 8 | no | — |
| 270 | C2×C22⋊C4 | — | 3 | 4 | no | — |
| 271 | C2×C22⋊C4 | — | 3 | 8 | no | — |
| 272 | C2×C22⋊C4 | — | 3 | 4 | no | — |
| 273 | C2×C22⋊C4 | — | 3 | 2 | no | — |
| 274 | C2×C22⋊C4 | — | 3 | 8 | no | — |
| 275 | C2×C22⋊C4 | — | 3 | 8 | no | — |
| 276 | C2×C22⋊C4 | C22×D3 | 1 | 4 | yes | — |
| 277 | C2×C22⋊C4 | C22×D3 | 1 | 4 | yes | — |
| 278 | C2×C22⋊C4 | C22×D3 | 1 | 2 | yes | — |
| 279 | C2×C42 | — | 3 | 2 | no | — |
| 280 | C2×C42 | — | 3 | 2 | no | — |
| 281 | C2×C42 | — | 3 | 2 | no | — |
| 282 | C2×C42 | — | 12 | 2 | no | — |
| 283 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 284 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 285 | #95 | — | 3 | 4 | no | — |
| 286 | #95 | — | 3 | 4 | no | — |
| 287 | #95 | — | 3 | 8 | no | — |
| 288 | C23×C3 | — | 2 | 1 | no | — |
| 289 | C23×C3 | — | 2 | 4 | no | — |
| 290 | C23×C3 | — | 2 | 1 | no | — |
| 291 | C23×C3 | — | 2 | 4 | no | — |
| 292 | C23×C3 | — | 2 | 8 | no | — |
| 293 | C23×C3 | — | 2 | 8 | no | — |
| 294 | C23×C3 | — | 2 | 4 | no | — |
| 295 | C23×C3 | — | 2 | 4 | no | — |
| 296 | C23×C3 | C22×D4 | 1 | 2 | yes | — |
| 297 | C23×C3 | C25 | 1 | 1 | yes | characteristic |
| 298 | C22×D3 | — | 2 | 4 | no | — |
| 299 | C22×D3 | — | 2 | 8 | no | — |
| 300 | C22×D3 | — | 2 | 8 | no | — |
| 301 | C22×D3 | — | 2 | 4 | no | — |
| 302 | C22×D3 | — | 2 | 4 | no | — |
| 303 | C3×D4 | — | 4 | 8 | no | — |
| 304 | C3×D4 | — | 8 | 8 | no | — |
| 305 | C3×D4 | — | 8 | 8 | no | — |
| 306 | C2×C3×C4 | — | 2 | 1 | no | — |
| 307 | C2×C3×C4 | — | 2 | 4 | no | — |
| 308 | C2×C3×C4 | — | 2 | 4 | no | — |
| 309 | C2×C3×C4 | — | 2 | 1 | no | — |
| 310 | C2×C3×C4 | — | 2 | 8 | no | — |
| 311 | C2×C3×C4 | — | 2 | 4 | no | — |
| 312 | C2×C3×C4 | — | 2 | 8 | no | — |
| 313 | C2×C3×C4 | — | 8 | 8 | no | — |
| 314 | C2×C3×C4 | C22×D4 | 1 | 4 | yes | — |
| 315 | C2×C3×C4 | C22×D4 | 1 | 2 | yes | — |
| 316 | C2×C3×C4 | C22×D4 | 1 | 4 | yes | — |
| 317 | C2×C3×C4 | C22×D4 | 1 | 2 | yes | — |
| 318 | C2×C3×C4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 319 | C2×C3×C4 | 2−1+4 | 1 | 2 | yes | — |
| 320 | #67 | — | 8 | 16 | no | — |
| 321 | #67 | — | 8 | 8 | no | — |
| 322 | #67 | — | 8 | 16 | no | — |
| 323 | C2×Dic3 | — | 2 | 4 | no | — |
| 324 | C2×Dic3 | — | 2 | 4 | no | — |
| 325 | C2×Dic3 | — | 2 | 4 | no | — |
| 326 | C2×Dic3 | — | 2 | 4 | no | — |
| 327 | C2×Dic3 | — | 2 | 8 | no | — |
| 328 | C2×Dic3 | — | 2 | 4 | no | — |
| 329 | C2×Dic3 | — | 2 | 4 | no | — |
| 330 | C2×Dic3 | — | 4 | 8 | no | — |
| 331 | C2×Dic3 | H(5,𝔽2) | 1 | 8 | yes | — |
| 332 | D12 | — | 4 | 8 | no | — |
| 333 | D12 | — | 8 | 8 | no | — |
| 334 | C4×D3 | — | 8 | 8 | no | — |
| 335 | C24 | — | 3 | 4 | no | — |
| 336 | C24 | C23×D3 | 1 | 4 | yes | — |
| 337 | C24 | C23×D3 | 1 | 1 | yes | characteristic |
| 338 | C2×D4 | — | 2 | 8 | no | — |
| 339 | C2×D4 | — | 2 | 4 | no | — |
| 340 | C2×D4 | — | 2 | 4 | no | — |
| 341 | C2×D4 | — | 2 | 4 | no | — |
| 342 | C2×D4 | — | 2 | 8 | no | — |
| 343 | C2×D4 | — | 4 | 8 | no | — |
| 344 | C2×D4 | — | 4 | 8 | no | — |
| 345 | C2×D4 | — | 6 | 16 | no | — |
| 346 | C2×D4 | — | 6 | 8 | no | — |
| 347 | C2×D4 | — | 6 | 4 | no | — |
| 348 | C2×D4 | — | 6 | 4 | no | — |
| 349 | C2×D4 | — | 6 | 4 | no | — |
| 350 | C2×D4 | — | 6 | 8 | no | — |
| 351 | C2×D4 | — | 12 | 8 | no | — |
| 352 | C2×D4 | — | 12 | 8 | no | — |
| 353 | C2×D4 | — | 12 | 8 | no | — |
| 354 | C2×D4 | — | 12 | 8 | no | — |
| 355 | C2×D4 | — | 12 | 8 | no | — |
| 356 | C2×D4 | — | 12 | 8 | no | — |
| 357 | C2×D4 | — | 12 | 8 | no | — |
| 358 | C22×C4 | — | 2 | 2 | no | — |
| 359 | C22×C4 | — | 2 | 2 | no | — |
| 360 | C22×C4 | — | 2 | 8 | no | — |
| 361 | C22×C4 | — | 3 | 4 | no | — |
| 362 | C22×C4 | — | 3 | 4 | no | — |
| 363 | C22×C4 | — | 3 | 4 | no | — |
| 364 | C22×C4 | — | 4 | 8 | no | — |
| 365 | C22×C4 | — | 6 | 8 | no | — |
| 366 | C22×C4 | — | 6 | 8 | no | — |
| 367 | C22×C4 | — | 12 | 8 | no | — |
| 368 | C22×C4 | — | 12 | 2 | no | — |
| 369 | C22×C4 | C2×#67 | 1 | 4 | yes | — |
| 370 | C22×C4 | C23×D3 | 1 | 4 | yes | — |
| 371 | C22×C4 | C23×D3 | 1 | 2 | yes | — |
| 372 | C22×C4 | C23×D3 | 1 | 2 | yes | — |
| 373 | C22×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 374 | C22×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 375 | C4⋊C4 | — | 2 | 4 | no | — |
| 376 | C4⋊C4 | — | 2 | 8 | no | — |
| 377 | C4⋊C4 | — | 6 | 8 | no | — |
| 378 | C4⋊C4 | — | 6 | 8 | no | — |
| 379 | C4⋊C4 | — | 6 | 4 | no | — |
| 380 | C4⋊C4 | — | 6 | 8 | no | — |
| 381 | C4⋊C4 | — | 12 | 2 | no | — |
| 382 | C4⋊C4 | — | 12 | 8 | no | — |
| 383 | C4⋊C4 | C2×#67 | 1 | 8 | yes | — |
| 384 | C22⋊C4 | — | 2 | 8 | no | — |
| 385 | C22⋊C4 | — | 2 | 8 | no | — |
| 386 | C22⋊C4 | — | 6 | 16 | no | — |
| 387 | C22⋊C4 | — | 6 | 16 | no | — |
| 388 | C22⋊C4 | — | 6 | 8 | no | — |
| 389 | C22⋊C4 | — | 6 | 8 | no | — |
| 390 | C22⋊C4 | — | 12 | 8 | no | — |
| 391 | C22⋊C4 | — | 12 | 2 | no | — |
| 392 | C22⋊C4 | — | 12 | 8 | no | — |
| 393 | C22⋊C4 | C2×#67 | 1 | 8 | yes | — |
| 394 | C42 | — | 6 | 4 | no | — |
| 395 | C42 | — | 6 | 4 | no | — |
| 396 | C42 | — | 12 | 2 | no | — |
| 397 | C42 | C2×#67 | 1 | 4 | yes | — |
| 398 | C42 | C2×#67 | 1 | 4 | yes | — |
| 399 | C22×C3 | — | 2 | 2 | no | — |
| 400 | C22×C3 | — | 2 | 4 | no | — |
| 401 | C22×C3 | — | 4 | 2 | no | — |
| 402 | C22×C3 | — | 4 | 2 | no | — |
| 403 | C22×C3 | — | 4 | 4 | no | — |
| 404 | C22×C3 | — | 4 | 4 | no | — |
| 405 | C22×C3 | — | 4 | 4 | no | — |
| 406 | C22×C3 | — | 4 | 8 | no | — |
| 407 | C22×C3 | — | 4 | 8 | no | — |
| 408 | C22×C3 | C23×D4 | 1 | 1 | yes | characteristic |
| 409 | C22×C3 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 410 | C22×C3 | C2×H(5,𝔽2) | 1 | 4 | yes | — |
| 411 | C22×C3 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 412 | C2×D3 | — | 4 | 4 | no | — |
| 413 | C2×D3 | — | 4 | 4 | no | — |
| 414 | C2×D3 | — | 4 | 4 | no | — |
| 415 | C2×D3 | — | 4 | 8 | no | — |
| 416 | C2×D3 | — | 4 | 8 | no | — |
| 417 | C3×C4 | — | 2 | 2 | no | — |
| 418 | C3×C4 | — | 2 | 4 | no | — |
| 419 | C3×C4 | — | 2 | 4 | no | — |
| 420 | C3×C4 | — | 4 | 4 | no | — |
| 421 | C3×C4 | #537 | 1 | 4 | yes | — |
| 422 | Dic3 | — | 2 | 8 | no | — |
| 423 | Dic3 | — | 4 | 4 | no | — |
| 424 | Dic3 | — | 4 | 4 | no | — |
| 425 | C23 | — | 2 | 1 | no | — |
| 426 | C23 | — | 2 | 1 | no | — |
| 427 | C23 | — | 2 | 4 | no | — |
| 428 | C23 | — | 2 | 4 | no | — |
| 429 | C23 | — | 2 | 4 | no | — |
| 430 | C23 | — | 2 | 8 | no | — |
| 431 | C23 | — | 2 | 8 | no | — |
| 432 | C23 | — | 2 | 4 | no | — |
| 433 | C23 | — | 6 | 4 | no | — |
| 434 | C23 | — | 6 | 4 | no | — |
| 435 | C23 | — | 6 | 8 | no | — |
| 436 | C23 | — | 6 | 8 | no | — |
| 437 | C23 | — | 6 | 4 | no | — |
| 438 | C23 | C22×#67 | 1 | 2 | yes | — |
| 439 | C23 | C24×D3 | 1 | 1 | yes | characteristic |
| 440 | D4 | — | 4 | 8 | no | — |
| 441 | D4 | — | 8 | 8 | no | — |
| 442 | D4 | — | 8 | 8 | no | — |
| 443 | D4 | — | 12 | 8 | no | — |
| 444 | D4 | — | 24 | 8 | no | — |
| 445 | D4 | — | 24 | 16 | no | — |
| 446 | D4 | — | 24 | 8 | no | — |
| 447 | D4 | — | 24 | 16 | no | — |
| 448 | C2×C4 | — | 2 | 1 | no | — |
| 449 | C2×C4 | — | 2 | 1 | no | — |
| 450 | C2×C4 | — | 2 | 4 | no | — |
| 451 | C2×C4 | — | 2 | 4 | no | — |
| 452 | C2×C4 | — | 2 | 8 | no | — |
| 453 | C2×C4 | — | 2 | 8 | no | — |
| 454 | C2×C4 | — | 2 | 4 | no | — |
| 455 | C2×C4 | — | 3 | 8 | no | — |
| 456 | C2×C4 | — | 6 | 4 | no | — |
| 457 | C2×C4 | — | 6 | 4 | no | — |
| 458 | C2×C4 | — | 6 | 4 | no | — |
| 459 | C2×C4 | — | 6 | 8 | no | — |
| 460 | C2×C4 | — | 6 | 4 | no | — |
| 461 | C2×C4 | — | 6 | 4 | no | — |
| 462 | C2×C4 | — | 6 | 4 | no | — |
| 463 | C2×C4 | — | 8 | 8 | no | — |
| 464 | C2×C4 | — | 12 | 8 | no | — |
| 465 | C2×C4 | — | 24 | 8 | no | — |
| 466 | C2×C4 | #1009 | 1 | 2 | yes | — |
| 467 | C2×C4 | #1010 | 1 | 2 | yes | — |
| 468 | C2×C4 | C22×#67 | 1 | 4 | yes | — |
| 469 | C2×C4 | C22×#67 | 1 | 4 | yes | — |
| 470 | C2×C4 | C22×#67 | 1 | 2 | yes | — |
| 471 | C2×C4 | C22×#67 | 1 | 2 | yes | — |
| 472 | C2×C3 | — | 4 | 2 | no | — |
| 473 | C2×C3 | — | 8 | 4 | no | — |
| 474 | C2×C3 | C2×#537 | 1 | 1 | yes | characteristic |
| 475 | C2×C3 | #3498 | 1 | 2 | yes | — |
| 476 | C2×C3 | #3537 | 1 | 2 | yes | — |
| 477 | C2×C3 | #3548 | 1 | 2 | yes | — |
| 478 | D3 | — | 8 | 4 | no | — |
| 479 | C22 | — | 2 | 2 | no | — |
| 480 | C22 | — | 2 | 4 | no | — |
| 481 | C22 | — | 4 | 2 | no | — |
| 482 | C22 | — | 4 | 2 | no | — |
| 483 | C22 | — | 4 | 8 | no | — |
| 484 | C22 | — | 4 | 8 | no | — |
| 485 | C22 | — | 4 | 4 | no | — |
| 486 | C22 | — | 4 | 4 | no | — |
| 487 | C22 | — | 4 | 4 | no | — |
| 488 | C22 | — | 12 | 8 | no | — |
| 489 | C22 | — | 12 | 8 | no | — |
| 490 | C22 | — | 12 | 4 | no | — |
| 491 | C22 | — | 12 | 4 | no | — |
| 492 | C22 | — | 12 | 4 | no | — |
| 493 | C22 | C2×#1009 | 1 | 1 | yes | characteristic |
| 494 | C22 | C2×#1010 | 1 | 1 | yes | characteristic |
| 495 | C22 | D3×H(5,𝔽2) | 1 | 4 | yes | — |
| 496 | C22 | C23×#67 | 1 | 1 | yes | characteristic |
| 497 | C4 | — | 2 | 2 | no | — |
| 498 | C4 | — | 2 | 4 | no | — |
| 499 | C4 | — | 2 | 4 | no | — |
| 500 | C4 | — | 4 | 4 | no | — |
| 501 | C4 | — | 6 | 8 | no | — |
| 502 | C4 | — | 12 | 4 | no | — |
| 503 | C4 | — | 12 | 4 | no | — |
| 504 | C4 | #5833 | 1 | 4 | yes | — |
| 505 | C3 | #37298 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 506 | C2 | — | 4 | 2 | no | — |
| 507 | C2 | — | 8 | 4 | no | — |
| 508 | C2 | — | 24 | 4 | no | — |
| 509 | C2 | #86476 | 1 | 2 | yes | — |
| 510 | C2 | #86491 | 1 | 2 | yes | — |
| 511 | C2 | C2×#5833 | 1 | 1 | yes | characteristic |
| 512 | C2 | #87282 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator, socle, cosocle | C23×C3 | C25 |
| Fitting | C2×C3×#540 | C2 |
| hypercenter | C2×#540 | D3 |
| hypocenter | C3 | #37298 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#540 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#540 | C2 | semidirect |
| 2. C2×#5535 | C2 | semidirect |
| 3. #83188 | C2 | semidirect |
| 4. #83040 | C2 | semidirect |
| 5. #82873 | C2 | semidirect |
| 6. #82718 | C2 | semidirect |
| 7. #82700 | C2 | semidirect |
| 8. #82676 | C2 | semidirect |
| 9. #82386 | C2 | semidirect |
| 10. #82378 | C2 | semidirect |
| 11. #82227 | C2 | semidirect |
| 55. C2×#540 | D3 | semidirect |
| 83. C3×#117 | D4 | semidirect |
| 259. #117 | #67 | semidirect |
| 505. C3 | #37298 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#540 | 2 |
| C2×#5535 | 2 |
| #83188 | 2 |
| #83040 | 2 |
| #82873 | 2 |
| #82718 | 2 |
| #82700 | 2 |
| #82676 | 2 |
| #82386 | 2 |
| #82378 | 2 |
| #82227 | 2 |
| #37298 | 3 |