Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1240 conjugacy classes of subgroups.
The subgroup lattice has 662 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #60646 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #33222 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #19655 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#1714 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #18651 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #18641 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #18631 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #18621 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #10172 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #10170 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #10148 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #10146 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 13 | #9862 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 14 | #9768 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 15 | #8185 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 16 | C2×#577 | — | 2 | 2 | no | — |
| 17 | #3499 | C22 | 1 | 1 | yes | characteristic |
| 18 | #3498 | C22 | 1 | 1 | yes | characteristic |
| 19 | #3227 | C22 | 1 | 1 | yes | characteristic |
| 20 | #3227 | C22 | 1 | 1 | yes | characteristic, complemented |
| 21 | #3206 | C22 | 1 | 1 | yes | characteristic |
| 22 | #3206 | C22 | 1 | 1 | yes | characteristic, complemented |
| 23 | C2×#500 | C22 | 1 | 1 | yes | characteristic, complemented |
| 24 | C2×#498 | C22 | 1 | 1 | yes | characteristic, complemented |
| 25 | #3154 | — | 2 | 4 | no | — |
| 26 | #3154 | — | 2 | 2 | no | — |
| 27 | #3152 | — | 2 | 1 | no | — |
| 28 | #3152 | C22 | 1 | 1 | yes | characteristic |
| 29 | #3151 | — | 2 | 1 | no | — |
| 30 | #3151 | C22 | 1 | 1 | yes | characteristic |
| 31 | #3147 | — | 2 | 1 | no | — |
| 32 | #3147 | C22 | 1 | 1 | yes | characteristic |
| 33 | C2×#492 | — | 2 | 1 | no | — |
| 34 | C2×#492 | C22 | 1 | 1 | yes | characteristic |
| 35 | C2×#492 | C22 | 1 | 1 | yes | characteristic |
| 36 | C2×#487 | C22 | 1 | 1 | yes | characteristic, complemented |
| 37 | C2×#486 | C22 | 1 | 1 | yes | characteristic |
| 38 | #3079 | C22 | 1 | 1 | yes | characteristic |
| 39 | #3079 | C22 | 1 | 1 | yes | characteristic |
| 40 | C4×#131 | C22 | 1 | 1 | yes | characteristic |
| 41 | #1997 | C22 | 1 | 1 | yes | characteristic, complemented |
| 42 | #1997 | C22 | 1 | 1 | yes | characteristic, complemented |
| 43 | #1952 | — | 2 | 2 | no | — |
| 44 | #1848 | C22 | 1 | 1 | yes | characteristic, complemented |
| 45 | #1816 | — | 2 | 2 | no | — |
| 46 | #1714 | — | 2 | 4 | no | — |
| 47 | #1714 | — | 2 | 1 | no | — |
| 48 | #1714 | — | 2 | 1 | no | — |
| 49 | #1707 | — | 2 | 1 | no | — |
| 50 | #1706 | — | 2 | 1 | no | — |
| 51 | #1654 | — | 2 | 2 | no | — |
| 52 | #1651 | — | 2 | 2 | no | — |
| 53 | #1642 | — | 2 | 2 | no | — |
| 54 | #1639 | — | 2 | 2 | no | — |
| 55 | #1637 | — | 2 | 2 | no | — |
| 56 | #1635 | — | 2 | 2 | no | — |
| 57 | #1631 | — | 2 | 2 | no | — |
| 58 | #1629 | — | 2 | 2 | no | — |
| 59 | #1546 | C22 | 1 | 1 | yes | characteristic, complemented |
| 60 | #1546 | C22 | 1 | 1 | yes | characteristic, complemented |
| 61 | #1545 | C22 | 1 | 1 | yes | characteristic |
| 62 | #1545 | C22 | 1 | 1 | yes | characteristic |
| 63 | #1492 | C22 | 1 | 1 | yes | characteristic |
| 64 | #1492 | C22 | 1 | 1 | yes | characteristic |
| 65 | #1492 | C22 | 1 | 1 | yes | characteristic, complemented |
| 66 | #1492 | C22 | 1 | 1 | yes | characteristic, complemented |
| 67 | #1476 | C22 | 1 | 1 | yes | characteristic |
| 68 | #1476 | C22 | 1 | 1 | yes | characteristic |
| 69 | #1450 | C22 | 1 | 1 | yes | characteristic, complemented |
| 70 | C2×#322 | C22 | 1 | 1 | yes | characteristic, complemented |
| 71 | #1278 | C22 | 1 | 1 | yes | characteristic, complemented |
| 72 | #1278 | C22 | 1 | 1 | yes | characteristic, complemented |
| 73 | C2×2−1+4 | — | 2 | 1 | no | — |
| 74 | C2×H(5,𝔽2) | — | 2 | 1 | no | — |
| 75 | #577 | — | 4 | 8 | no | — |
| 76 | #577 | — | 4 | 8 | no | — |
| 77 | C2×#137 | — | 2 | 4 | no | — |
| 78 | C2×#137 | — | 2 | 2 | no | — |
| 79 | C2×AGL(1,ℤ/8) | — | 2 | 4 | no | — |
| 80 | C2×AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 81 | C2×#135 | — | 2 | 4 | no | — |
| 82 | C2×#135 | — | 2 | 2 | no | — |
| 83 | C22×SD16 | — | 2 | 4 | no | — |
| 84 | C22×SD16 | — | 2 | 2 | no | — |
| 85 | C22×SD16 | — | 4 | 1 | no | — |
| 86 | C2×#131 | D4 | 1 | 2 | yes | — |
| 87 | #550 | — | 2 | 1 | no | — |
| 88 | #550 | — | 2 | 1 | no | — |
| 89 | D4×Q8 | — | 2 | 1 | no | — |
| 90 | D4×Q8 | — | 2 | 1 | no | — |
| 91 | #547 | — | 2 | 1 | no | — |
| 92 | #547 | — | 2 | 1 | no | — |
| 93 | #547 | — | 4 | 1 | no | — |
| 94 | #547 | — | 4 | 1 | no | — |
| 95 | D42 | — | 2 | 1 | no | — |
| 96 | D42 | — | 2 | 1 | no | — |
| 97 | D42 | — | 4 | 1 | no | — |
| 98 | D42 | — | 4 | 1 | no | — |
| 99 | #536 | — | 2 | 2 | no | — |
| 100 | #536 | — | 2 | 1 | no | — |
| 101 | #535 | — | 2 | 2 | no | — |
| 102 | #535 | — | 2 | 1 | no | — |
| 103 | #532 | — | 2 | 1 | no | — |
| 104 | #532 | — | 2 | 1 | no | — |
| 105 | #532 | C23 | 1 | 1 | yes | characteristic |
| 106 | #532 | C23 | 1 | 1 | yes | characteristic |
| 107 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 108 | C2×C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 109 | C2×C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 110 | C2×C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 111 | C2×#127 | — | 2 | 1 | no | — |
| 112 | C2×#127 | C23 | 1 | 1 | yes | characteristic |
| 113 | C4×#41 | C23 | 1 | 1 | yes | characteristic |
| 114 | #500 | — | 2 | 1 | no | — |
| 115 | #500 | — | 2 | 1 | no | — |
| 116 | #500 | — | 2 | 2 | no | — |
| 117 | #498 | — | 2 | 2 | no | — |
| 118 | #498 | — | 2 | 1 | no | — |
| 119 | #498 | — | 2 | 1 | no | — |
| 120 | #498 | — | 2 | 2 | no | — |
| 121 | #497 | — | 2 | 4 | no | — |
| 122 | #497 | — | 2 | 2 | no | — |
| 123 | #496 | — | 2 | 4 | no | — |
| 124 | #496 | — | 2 | 2 | no | — |
| 125 | #495 | — | 2 | 4 | no | — |
| 126 | #495 | — | 2 | 2 | no | — |
| 127 | #494 | — | 2 | 1 | no | — |
| 128 | #494 | D4 | 1 | 1 | yes | characteristic |
| 129 | #493 | — | 2 | 1 | no | — |
| 130 | #493 | D4 | 1 | 1 | yes | characteristic |
| 131 | #492 | — | 2 | 4 | no | — |
| 132 | #492 | — | 2 | 1 | no | — |
| 133 | #492 | — | 2 | 1 | no | — |
| 134 | #492 | — | 2 | 2 | no | — |
| 135 | #492 | — | 2 | 1 | no | — |
| 136 | #492 | — | 2 | 1 | no | — |
| 137 | #492 | — | 2 | 1 | no | — |
| 138 | #492 | — | 2 | 1 | no | — |
| 139 | #492 | — | 2 | 2 | no | — |
| 140 | #492 | D4 | 1 | 1 | yes | characteristic |
| 141 | #492 | D4 | 1 | 4 | yes | — |
| 142 | #492 | D4 | 1 | 1 | yes | characteristic |
| 143 | #487 | — | 2 | 2 | no | — |
| 144 | #487 | — | 2 | 1 | no | — |
| 145 | #487 | — | 2 | 1 | no | — |
| 146 | #487 | — | 2 | 2 | no | — |
| 147 | #486 | — | 2 | 2 | no | — |
| 148 | #486 | — | 2 | 1 | no | — |
| 149 | #486 | — | 2 | 1 | no | — |
| 150 | #486 | — | 2 | 2 | no | — |
| 151 | #481 | — | 4 | 1 | no | — |
| 152 | #481 | — | 4 | 1 | no | — |
| 153 | #477 | — | 2 | 1 | no | — |
| 154 | #477 | — | 2 | 1 | no | — |
| 155 | #476 | — | 2 | 1 | no | — |
| 156 | #476 | — | 2 | 1 | no | — |
| 157 | #476 | — | 2 | 2 | no | — |
| 158 | #476 | — | 2 | 2 | no | — |
| 159 | #475 | — | 2 | 1 | no | — |
| 160 | #475 | — | 2 | 1 | no | — |
| 161 | #474 | — | 2 | 1 | no | — |
| 162 | #474 | — | 2 | 1 | no | — |
| 163 | #461 | — | 2 | 2 | no | — |
| 164 | #461 | — | 2 | 2 | no | — |
| 165 | #448 | — | 4 | 1 | no | — |
| 166 | #448 | — | 4 | 1 | no | — |
| 167 | #431 | C23 | 1 | 1 | yes | characteristic |
| 168 | #431 | C23 | 1 | 1 | yes | characteristic |
| 169 | C2×#108 | — | 2 | 2 | no | — |
| 170 | #428 | — | 2 | 1 | no | — |
| 171 | #428 | — | 2 | 1 | no | — |
| 172 | C2×#107 | — | 2 | 2 | no | — |
| 173 | C2×#106 | — | 4 | 1 | no | — |
| 174 | #423 | C23 | 1 | 1 | yes | characteristic |
| 175 | #423 | C23 | 1 | 1 | yes | characteristic |
| 176 | #417 | — | 2 | 1 | no | — |
| 177 | #417 | — | 2 | 1 | no | — |
| 178 | C2×#103 | — | 4 | 1 | no | — |
| 179 | C2×#102 | — | 4 | 1 | no | — |
| 180 | #405 | — | 2 | 2 | no | — |
| 181 | #405 | D4 | 1 | 2 | yes | — |
| 182 | C4×M16 | — | 2 | 1 | no | — |
| 183 | C4×M16 | C23 | 1 | 1 | yes | characteristic |
| 184 | C2×C4×C8 | — | 2 | 1 | no | — |
| 185 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 186 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 187 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 188 | #343 | — | 2 | 2 | no | — |
| 189 | #343 | — | 2 | 2 | no | — |
| 190 | #332 | — | 2 | 2 | no | — |
| 191 | #332 | — | 2 | 2 | no | — |
| 192 | #331 | — | 2 | 2 | no | — |
| 193 | #331 | — | 2 | 2 | no | — |
| 194 | Q8⋊C8 | D4 | 1 | 2 | yes | — |
| 195 | Q8⋊C8 | D4 | 1 | 2 | yes | — |
| 196 | D4⋊C8 | — | 2 | 2 | no | — |
| 197 | D4⋊C8 | — | 2 | 2 | no | — |
| 198 | D4⋊C8 | — | 2 | 2 | no | — |
| 199 | D4⋊C8 | — | 2 | 2 | no | — |
| 200 | D4⋊C8 | D4 | 1 | 2 | yes | — |
| 201 | D4⋊C8 | D4 | 1 | 2 | yes | — |
| 202 | #322 | — | 2 | 2 | no | — |
| 203 | #322 | — | 2 | 1 | no | — |
| 204 | C82 | — | 2 | 1 | no | — |
| 205 | 2−1+4 | — | 4 | 4 | no | — |
| 206 | 2−1+4 | — | 4 | 4 | no | — |
| 207 | H(5,𝔽2) | — | 4 | 4 | no | — |
| 208 | H(5,𝔽2) | — | 4 | 4 | no | — |
| 209 | C2×#41 | — | 2 | 2 | no | — |
| 210 | C2×#41 | — | 2 | 2 | no | — |
| 211 | C2×#41 | — | 2 | 1 | no | — |
| 212 | C2×#41 | — | 2 | 1 | no | — |
| 213 | C2×#41 | — | 4 | 2 | no | — |
| 214 | C2×#41 | — | 4 | 1 | no | — |
| 215 | C2×#41 | — | 4 | 1 | no | — |
| 216 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 217 | C22×Q8 | — | 2 | 2 | no | — |
| 218 | C22×Q8 | — | 2 | 1 | no | — |
| 219 | C22×Q8 | — | 2 | 1 | no | — |
| 220 | C22×Q8 | — | 2 | 1 | no | — |
| 221 | C22×Q8 | — | 4 | 1 | no | — |
| 222 | C22×D4 | — | 2 | 2 | no | — |
| 223 | C22×D4 | — | 2 | 1 | no | — |
| 224 | C22×D4 | — | 2 | 1 | no | — |
| 225 | C22×D4 | — | 2 | 1 | no | — |
| 226 | C22×D4 | — | 4 | 1 | no | — |
| 227 | C22×D4 | — | 4 | 1 | no | — |
| 228 | C22×D4 | — | 4 | 2 | no | — |
| 229 | C22×D4 | — | 4 | 1 | no | — |
| 230 | #137 | — | 4 | 4 | no | — |
| 231 | #137 | — | 4 | 16 | no | — |
| 232 | #137 | — | 4 | 4 | no | — |
| 233 | AGL(1,ℤ/8) | — | 4 | 4 | no | — |
| 234 | AGL(1,ℤ/8) | — | 4 | 16 | no | — |
| 235 | AGL(1,ℤ/8) | — | 4 | 4 | no | — |
| 236 | #135 | — | 4 | 16 | no | — |
| 237 | #135 | — | 4 | 4 | no | — |
| 238 | #135 | — | 4 | 4 | no | — |
| 239 | C2×Q16 | — | 2 | 4 | no | — |
| 240 | C2×Q16 | — | 2 | 2 | no | — |
| 241 | C2×SD16 | — | 2 | 4 | no | — |
| 242 | C2×SD16 | — | 2 | 4 | no | — |
| 243 | C2×SD16 | — | 2 | 2 | no | — |
| 244 | C2×SD16 | — | 2 | 4 | no | — |
| 245 | C2×SD16 | — | 2 | 4 | no | — |
| 246 | C2×SD16 | — | 2 | 2 | no | — |
| 247 | C2×SD16 | — | 4 | 16 | no | — |
| 248 | C2×SD16 | — | 4 | 4 | no | — |
| 249 | C2×SD16 | — | 4 | 2 | no | — |
| 250 | C2×SD16 | — | 4 | 2 | no | — |
| 251 | C2×SD16 | — | 4 | 4 | no | — |
| 252 | C2×SD16 | — | 4 | 4 | no | — |
| 253 | C2×SD16 | — | 8 | 4 | no | — |
| 254 | C2×D8 | — | 2 | 4 | no | — |
| 255 | C2×D8 | — | 2 | 2 | no | — |
| 256 | #131 | — | 2 | 4 | no | — |
| 257 | #131 | SD16 | 1 | 8 | yes | complemented |
| 258 | C2×M16 | — | 2 | 2 | no | — |
| 259 | C2×M16 | — | 2 | 1 | no | — |
| 260 | C2×M16 | — | 2 | 1 | no | — |
| 261 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 262 | C22×C8 | — | 2 | 2 | no | — |
| 263 | C22×C8 | — | 2 | 1 | no | — |
| 264 | C22×C8 | — | 2 | 1 | no | — |
| 265 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 266 | C4⋊Q8 | — | 2 | 2 | no | — |
| 267 | C4⋊Q8 | — | 2 | 1 | no | — |
| 268 | C4⋊Q8 | — | 2 | 1 | no | — |
| 269 | C4⋊Q8 | — | 2 | 1 | no | — |
| 270 | C4⋊Q8 | — | 2 | 1 | no | — |
| 271 | C4⋊Q8 | — | 2 | 1 | no | — |
| 272 | C4⋊Q8 | — | 2 | 1 | no | — |
| 273 | C4⋊Q8 | — | 2 | 1 | no | — |
| 274 | C4⋊Q8 | — | 2 | 1 | no | — |
| 275 | C4⋊Q8 | — | 2 | 1 | no | — |
| 276 | C4⋊Q8 | — | 2 | 1 | no | — |
| 277 | C4⋊Q8 | — | 2 | 1 | no | — |
| 278 | C4⋊Q8 | — | 4 | 1 | no | — |
| 279 | C4⋊Q8 | — | 4 | 1 | no | — |
| 280 | C4⋊Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 281 | C4⋊Q8 | C2×D4 | 1 | 2 | yes | — |
| 282 | C4⋊Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 283 | #127 | — | 2 | 2 | no | — |
| 284 | #127 | — | 2 | 1 | no | — |
| 285 | #127 | — | 2 | 1 | no | — |
| 286 | #127 | — | 2 | 1 | no | — |
| 287 | #127 | — | 4 | 1 | no | — |
| 288 | #127 | — | 4 | 1 | no | — |
| 289 | #127 | C2×D4 | 1 | 1 | yes | characteristic |
| 290 | #127 | C2×D4 | 1 | 2 | yes | — |
| 291 | #127 | C2×D4 | 1 | 1 | yes | characteristic |
| 292 | #124 | — | 2 | 2 | no | — |
| 293 | #124 | — | 2 | 2 | no | — |
| 294 | #124 | — | 2 | 1 | no | — |
| 295 | #124 | — | 2 | 1 | no | — |
| 296 | #123 | — | 4 | 2 | no | — |
| 297 | #123 | — | 4 | 2 | no | — |
| 298 | #123 | — | 4 | 2 | no | — |
| 299 | C22⋊Q8 | — | 4 | 2 | no | — |
| 300 | C22⋊Q8 | — | 4 | 2 | no | — |
| 301 | C22⋊Q8 | — | 4 | 1 | no | — |
| 302 | C22⋊Q8 | — | 4 | 1 | no | — |
| 303 | #121 | — | 2 | 1 | no | — |
| 304 | #121 | — | 2 | 1 | no | — |
| 305 | #121 | — | 2 | 1 | no | — |
| 306 | #121 | — | 2 | 1 | no | — |
| 307 | #121 | — | 4 | 1 | no | — |
| 308 | #121 | — | 4 | 1 | no | — |
| 309 | #121 | — | 4 | 2 | no | — |
| 310 | #121 | — | 4 | 2 | no | — |
| 311 | #121 | — | 4 | 1 | no | — |
| 312 | #121 | — | 4 | 1 | no | — |
| 313 | #121 | — | 4 | 1 | no | — |
| 314 | #121 | — | 4 | 1 | no | — |
| 315 | #120 | — | 4 | 2 | no | — |
| 316 | #120 | — | 4 | 2 | no | — |
| 317 | #120 | — | 4 | 2 | no | — |
| 318 | C4×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 319 | C4×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 320 | C4×D4 | — | 2 | 1 | no | — |
| 321 | C4×D4 | — | 2 | 1 | no | — |
| 322 | C4×D4 | — | 2 | 1 | no | — |
| 323 | C4×D4 | — | 2 | 1 | no | — |
| 324 | C4×D4 | — | 2 | 1 | no | — |
| 325 | C4×D4 | — | 2 | 1 | no | — |
| 326 | C4×D4 | — | 4 | 1 | no | — |
| 327 | C4×D4 | — | 4 | 1 | no | — |
| 328 | C4×D4 | — | 4 | 1 | no | — |
| 329 | C4×D4 | — | 4 | 1 | no | — |
| 330 | C4×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 331 | C4×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 332 | #117 | — | 2 | 1 | no | — |
| 333 | #117 | C2×D4 | 1 | 1 | yes | characteristic |
| 334 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 335 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 336 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 337 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 338 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 339 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 340 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 341 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 342 | C2×C42 | — | 2 | 1 | no | — |
| 343 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 344 | #108 | — | 4 | 4 | no | — |
| 345 | #107 | — | 4 | 2 | no | — |
| 346 | #107 | — | 4 | 2 | no | — |
| 347 | #107 | — | 4 | 1 | no | — |
| 348 | #107 | — | 4 | 1 | no | — |
| 349 | #106 | — | 4 | 4 | no | — |
| 350 | #106 | — | 4 | 2 | no | — |
| 351 | #106 | — | 4 | 2 | no | — |
| 352 | #106 | — | 4 | 1 | no | — |
| 353 | #106 | — | 4 | 1 | no | — |
| 354 | C4⋊C8 | — | 2 | 1 | no | — |
| 355 | C4⋊C8 | — | 2 | 1 | no | — |
| 356 | C4⋊C8 | — | 2 | 1 | no | — |
| 357 | C4⋊C8 | — | 2 | 1 | no | — |
| 358 | C4⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 359 | C4⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 360 | #103 | — | 4 | 4 | no | — |
| 361 | #103 | — | 4 | 2 | no | — |
| 362 | #103 | — | 4 | 2 | no | — |
| 363 | #103 | — | 4 | 1 | no | — |
| 364 | #103 | — | 4 | 1 | no | — |
| 365 | #102 | — | 4 | 4 | no | — |
| 366 | #102 | — | 4 | 2 | no | — |
| 367 | #102 | — | 4 | 2 | no | — |
| 368 | #102 | — | 4 | 1 | no | — |
| 369 | #102 | — | 4 | 1 | no | — |
| 370 | C22⋊C8 | — | 2 | 1 | no | — |
| 371 | C22⋊C8 | — | 2 | 1 | no | — |
| 372 | #97 | — | 2 | 2 | no | — |
| 373 | #97 | — | 2 | 1 | no | — |
| 374 | C4×C8 | — | 2 | 2 | no | — |
| 375 | C4×C8 | — | 2 | 1 | no | — |
| 376 | C4×C8 | — | 2 | 1 | no | — |
| 377 | C4×C8 | — | 2 | 1 | no | — |
| 378 | C4×C8 | — | 2 | 1 | no | — |
| 379 | C4×C8 | — | 2 | 1 | no | — |
| 380 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 381 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 382 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 383 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 384 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 385 | C24 | — | 4 | 2 | no | — |
| 386 | C24 | — | 4 | 1 | no | — |
| 387 | #41 | — | 2 | 2 | no | — |
| 388 | #41 | — | 4 | 8 | no | — |
| 389 | #41 | — | 4 | 8 | no | — |
| 390 | #41 | — | 4 | 2 | no | — |
| 391 | #41 | — | 4 | 2 | no | — |
| 392 | #41 | — | 4 | 2 | no | — |
| 393 | #41 | — | 4 | 2 | no | — |
| 394 | #41 | — | 8 | 8 | no | — |
| 395 | #41 | — | 8 | 2 | no | — |
| 396 | #41 | — | 8 | 2 | no | — |
| 397 | #41 | — | 8 | 2 | no | — |
| 398 | #41 | — | 8 | 2 | no | — |
| 399 | #41 | C2×SD16 | 1 | 4 | yes | — |
| 400 | C2×Q8 | — | 2 | 1 | no | — |
| 401 | C2×Q8 | — | 2 | 2 | no | — |
| 402 | C2×Q8 | — | 2 | 1 | no | — |
| 403 | C2×Q8 | — | 4 | 2 | no | — |
| 404 | C2×Q8 | — | 4 | 8 | no | — |
| 405 | C2×Q8 | — | 4 | 1 | no | — |
| 406 | C2×Q8 | — | 4 | 1 | no | — |
| 407 | C2×Q8 | — | 4 | 1 | no | — |
| 408 | C2×Q8 | — | 4 | 1 | no | — |
| 409 | C2×Q8 | — | 4 | 1 | no | — |
| 410 | C2×Q8 | — | 4 | 2 | no | — |
| 411 | C2×Q8 | — | 4 | 2 | no | — |
| 412 | C2×Q8 | — | 8 | 2 | no | — |
| 413 | C2×Q8 | — | 8 | 2 | no | — |
| 414 | C2×Q8 | — | 8 | 2 | no | — |
| 415 | C2×Q8 | — | 8 | 2 | no | — |
| 416 | C2×Q8 | #127 | 1 | 1 | yes | characteristic |
| 417 | C2×D4 | — | 2 | 1 | no | — |
| 418 | C2×D4 | — | 2 | 1 | no | — |
| 419 | C2×D4 | — | 2 | 2 | no | — |
| 420 | C2×D4 | — | 2 | 1 | no | — |
| 421 | C2×D4 | — | 2 | 1 | no | — |
| 422 | C2×D4 | — | 2 | 1 | no | — |
| 423 | C2×D4 | — | 2 | 1 | no | — |
| 424 | C2×D4 | — | 2 | 1 | no | — |
| 425 | C2×D4 | — | 2 | 2 | no | — |
| 426 | C2×D4 | — | 2 | 2 | no | — |
| 427 | C2×D4 | — | 4 | 8 | no | — |
| 428 | C2×D4 | — | 4 | 2 | no | — |
| 429 | C2×D4 | — | 4 | 2 | no | — |
| 430 | C2×D4 | — | 4 | 2 | no | — |
| 431 | C2×D4 | — | 4 | 2 | no | — |
| 432 | C2×D4 | — | 4 | 2 | no | — |
| 433 | C2×D4 | — | 4 | 1 | no | — |
| 434 | C2×D4 | — | 4 | 1 | no | — |
| 435 | C2×D4 | — | 4 | 1 | no | — |
| 436 | C2×D4 | — | 4 | 1 | no | — |
| 437 | C2×D4 | — | 4 | 1 | no | — |
| 438 | C2×D4 | — | 4 | 2 | no | — |
| 439 | C2×D4 | — | 4 | 2 | no | — |
| 440 | C2×D4 | — | 8 | 2 | no | — |
| 441 | C2×D4 | — | 8 | 2 | no | — |
| 442 | C2×D4 | — | 8 | 8 | no | — |
| 443 | C2×D4 | — | 8 | 2 | no | — |
| 444 | C2×D4 | — | 8 | 2 | no | — |
| 445 | C2×D4 | — | 16 | 2 | no | — |
| 446 | C2×D4 | — | 16 | 2 | no | — |
| 447 | C2×D4 | #127 | 1 | 1 | yes | characteristic |
| 448 | C22×C4 | — | 2 | 1 | no | — |
| 449 | C22×C4 | — | 2 | 1 | no | — |
| 450 | C22×C4 | — | 2 | 1 | no | — |
| 451 | C22×C4 | — | 4 | 2 | no | — |
| 452 | C22×C4 | — | 4 | 2 | no | — |
| 453 | C22×C4 | — | 4 | 2 | no | — |
| 454 | C22×C4 | — | 4 | 1 | no | — |
| 455 | C22×C4 | — | 4 | 1 | no | — |
| 456 | C22×C4 | — | 4 | 1 | no | — |
| 457 | C22×C4 | — | 4 | 1 | no | — |
| 458 | C22×C4 | — | 4 | 1 | no | — |
| 459 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 460 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 461 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 462 | Q16 | — | 4 | 8 | no | — |
| 463 | Q16 | — | 4 | 4 | no | — |
| 464 | SD16 | — | 4 | 8 | no | — |
| 465 | SD16 | — | 4 | 8 | no | — |
| 466 | SD16 | — | 4 | 4 | no | — |
| 467 | SD16 | — | 4 | 8 | no | — |
| 468 | SD16 | — | 4 | 8 | no | — |
| 469 | SD16 | — | 4 | 4 | no | — |
| 470 | SD16 | — | 8 | 8 | no | — |
| 471 | SD16 | — | 16 | 2 | no | — |
| 472 | SD16 | — | 16 | 2 | no | — |
| 473 | D8 | — | 4 | 8 | no | — |
| 474 | D8 | — | 4 | 4 | no | — |
| 475 | M16 | — | 2 | 2 | no | — |
| 476 | M16 | — | 2 | 8 | no | — |
| 477 | M16 | — | 4 | 2 | no | — |
| 478 | M16 | — | 4 | 2 | no | — |
| 479 | M16 | C2×SD16 | 1 | 4 | yes | — |
| 480 | C2×C8 | — | 2 | 2 | no | — |
| 481 | C2×C8 | — | 2 | 8 | no | — |
| 482 | C2×C8 | — | 2 | 2 | no | — |
| 483 | C2×C8 | — | 2 | 2 | no | — |
| 484 | C2×C8 | — | 2 | 2 | no | — |
| 485 | C2×C8 | — | 2 | 2 | no | — |
| 486 | C2×C8 | — | 4 | 2 | no | — |
| 487 | C2×C8 | — | 4 | 2 | no | — |
| 488 | C2×C8 | — | 4 | 1 | no | — |
| 489 | C2×C8 | — | 4 | 1 | no | — |
| 490 | C2×C8 | #120 | 1 | 2 | yes | — |
| 491 | C2×C8 | #120 | 1 | 2 | yes | — |
| 492 | C2×C8 | C2×SD16 | 1 | 4 | yes | — |
| 493 | C2×C8 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 494 | C4⋊C4 | — | 2 | 1 | no | — |
| 495 | C4⋊C4 | — | 2 | 1 | no | — |
| 496 | C4⋊C4 | — | 2 | 2 | no | — |
| 497 | C4⋊C4 | — | 2 | 1 | no | — |
| 498 | C4⋊C4 | — | 2 | 1 | no | — |
| 499 | C4⋊C4 | — | 2 | 2 | no | — |
| 500 | C4⋊C4 | — | 2 | 1 | no | — |
| 501 | C4⋊C4 | — | 2 | 1 | no | — |
| 502 | C4⋊C4 | — | 4 | 2 | no | — |
| 503 | C4⋊C4 | — | 4 | 2 | no | — |
| 504 | C4⋊C4 | — | 4 | 1 | no | — |
| 505 | C4⋊C4 | — | 4 | 1 | no | — |
| 506 | C4⋊C4 | — | 4 | 1 | no | — |
| 507 | C4⋊C4 | — | 4 | 1 | no | — |
| 508 | C4⋊C4 | — | 4 | 1 | no | — |
| 509 | C4⋊C4 | — | 4 | 1 | no | — |
| 510 | C4⋊C4 | — | 4 | 1 | no | — |
| 511 | C4⋊C4 | — | 4 | 1 | no | — |
| 512 | C4⋊C4 | — | 4 | 1 | no | — |
| 513 | C4⋊C4 | — | 4 | 1 | no | — |
| 514 | C4⋊C4 | #127 | 1 | 1 | yes | characteristic |
| 515 | C4⋊C4 | #127 | 1 | 1 | yes | characteristic |
| 516 | C22⋊C4 | — | 2 | 1 | no | — |
| 517 | C22⋊C4 | — | 2 | 1 | no | — |
| 518 | C22⋊C4 | — | 2 | 1 | no | — |
| 519 | C22⋊C4 | — | 4 | 2 | no | — |
| 520 | C22⋊C4 | — | 4 | 2 | no | — |
| 521 | C22⋊C4 | — | 4 | 2 | no | — |
| 522 | C22⋊C4 | — | 4 | 2 | no | — |
| 523 | C22⋊C4 | — | 4 | 1 | no | — |
| 524 | C22⋊C4 | — | 4 | 1 | no | — |
| 525 | C22⋊C4 | — | 4 | 1 | no | — |
| 526 | C22⋊C4 | — | 4 | 1 | no | — |
| 527 | C42 | — | 2 | 1 | no | — |
| 528 | C42 | — | 2 | 1 | no | — |
| 529 | C42 | — | 2 | 1 | no | — |
| 530 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 531 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 532 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 533 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 534 | C23 | — | 2 | 1 | no | — |
| 535 | C23 | — | 4 | 1 | no | — |
| 536 | C23 | — | 4 | 1 | no | — |
| 537 | C23 | — | 4 | 2 | no | — |
| 538 | C23 | — | 8 | 4 | no | — |
| 539 | C23 | — | 8 | 2 | no | — |
| 540 | C23 | — | 16 | 1 | no | — |
| 541 | C23 | — | 16 | 1 | no | — |
| 542 | C23 | — | 16 | 2 | no | — |
| 543 | C23 | — | 16 | 2 | no | — |
| 544 | C23 | #535 | 1 | 1 | yes | characteristic |
| 545 | Q8 | — | 2 | 1 | no | — |
| 546 | Q8 | — | 4 | 2 | no | — |
| 547 | Q8 | — | 4 | 4 | no | — |
| 548 | Q8 | — | 4 | 2 | no | — |
| 549 | Q8 | — | 8 | 4 | no | — |
| 550 | Q8 | — | 8 | 2 | no | — |
| 551 | Q8 | — | 8 | 2 | no | — |
| 552 | Q8 | — | 8 | 1 | no | — |
| 553 | Q8 | — | 8 | 1 | no | — |
| 554 | Q8 | — | 8 | 2 | no | — |
| 555 | Q8 | — | 8 | 2 | no | — |
| 556 | Q8 | #492 | 1 | 2 | yes | — |
| 557 | D4 | — | 2 | 4 | no | — |
| 558 | D4 | — | 2 | 1 | no | — |
| 559 | D4 | — | 4 | 2 | no | — |
| 560 | D4 | — | 4 | 4 | no | — |
| 561 | D4 | — | 4 | 2 | no | — |
| 562 | D4 | — | 4 | 2 | no | — |
| 563 | D4 | — | 4 | 2 | no | — |
| 564 | D4 | — | 4 | 2 | no | — |
| 565 | D4 | — | 4 | 2 | no | — |
| 566 | D4 | — | 8 | 2 | no | — |
| 567 | D4 | — | 8 | 2 | no | — |
| 568 | D4 | — | 8 | 4 | no | — |
| 569 | D4 | — | 8 | 4 | no | — |
| 570 | D4 | — | 8 | 4 | no | — |
| 571 | D4 | — | 8 | 2 | no | — |
| 572 | D4 | — | 8 | 2 | no | — |
| 573 | D4 | — | 8 | 1 | no | — |
| 574 | D4 | — | 8 | 1 | no | — |
| 575 | D4 | — | 8 | 2 | no | — |
| 576 | D4 | — | 8 | 2 | no | — |
| 577 | D4 | — | 16 | 2 | no | — |
| 578 | D4 | — | 16 | 2 | no | — |
| 579 | D4 | #492 | 1 | 2 | yes | — |
| 580 | C2×C4 | — | 2 | 1 | no | — |
| 581 | C2×C4 | — | 2 | 4 | no | — |
| 582 | C2×C4 | — | 2 | 2 | no | — |
| 583 | C2×C4 | — | 2 | 2 | no | — |
| 584 | C2×C4 | — | 2 | 1 | no | — |
| 585 | C2×C4 | — | 2 | 1 | no | — |
| 586 | C2×C4 | — | 4 | 2 | no | — |
| 587 | C2×C4 | — | 4 | 2 | no | — |
| 588 | C2×C4 | — | 4 | 1 | no | — |
| 589 | C2×C4 | — | 4 | 2 | no | — |
| 590 | C2×C4 | — | 4 | 1 | no | — |
| 591 | C2×C4 | — | 8 | 4 | no | — |
| 592 | C2×C4 | — | 8 | 4 | no | — |
| 593 | C2×C4 | — | 8 | 4 | no | — |
| 594 | C2×C4 | — | 8 | 2 | no | — |
| 595 | C2×C4 | — | 8 | 1 | no | — |
| 596 | C2×C4 | — | 8 | 1 | no | — |
| 597 | C2×C4 | — | 8 | 1 | no | — |
| 598 | C2×C4 | — | 8 | 1 | no | — |
| 599 | C2×C4 | — | 8 | 1 | no | — |
| 600 | C2×C4 | — | 8 | 1 | no | — |
| 601 | C2×C4 | — | 16 | 1 | no | — |
| 602 | C2×C4 | — | 16 | 1 | no | — |
| 603 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 604 | C2×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 605 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 606 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 607 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 608 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 609 | C2×C4 | C22×SD16 | 1 | 2 | yes | — |
| 610 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 611 | C8 | — | 2 | 4 | no | — |
| 612 | C8 | — | 4 | 2 | no | — |
| 613 | C8 | — | 4 | 2 | no | — |
| 614 | C8 | — | 4 | 2 | no | — |
| 615 | C8 | — | 4 | 2 | no | — |
| 616 | C8 | — | 8 | 1 | no | — |
| 617 | C8 | — | 8 | 1 | no | — |
| 618 | C8 | #450 | 1 | 4 | yes | — |
| 619 | C8 | #450 | 1 | 4 | yes | — |
| 620 | C22 | — | 2 | 1 | no | — |
| 621 | C22 | — | 2 | 1 | no | — |
| 622 | C22 | — | 2 | 2 | no | — |
| 623 | C22 | — | 4 | 1 | no | — |
| 624 | C22 | — | 4 | 1 | no | — |
| 625 | C22 | — | 8 | 1 | no | — |
| 626 | C22 | — | 8 | 2 | no | — |
| 627 | C22 | — | 8 | 1 | no | — |
| 628 | C22 | — | 8 | 1 | no | — |
| 629 | C22 | — | 8 | 1 | no | — |
| 630 | C22 | — | 8 | 2 | no | — |
| 631 | C22 | — | 8 | 1 | no | — |
| 632 | C22 | — | 8 | 1 | no | — |
| 633 | C22 | — | 8 | 2 | no | — |
| 634 | C22 | — | 32 | 2 | no | — |
| 635 | C22 | — | 32 | 4 | no | — |
| 636 | C22 | #2433 | 1 | 1 | yes | characteristic |
| 637 | C22 | #3208 | 1 | 2 | yes | — |
| 638 | C4 | — | 2 | 2 | no | — |
| 639 | C4 | — | 2 | 1 | no | — |
| 640 | C4 | — | 2 | 1 | no | — |
| 641 | C4 | — | 4 | 1 | no | — |
| 642 | C4 | — | 4 | 1 | no | — |
| 643 | C4 | — | 8 | 1 | no | — |
| 644 | C4 | — | 8 | 2 | no | — |
| 645 | C4 | — | 8 | 1 | no | — |
| 646 | C4 | — | 16 | 1 | no | — |
| 647 | C4 | — | 16 | 1 | no | — |
| 648 | C4 | C2×#450 | 1 | 2 | yes | — |
| 649 | C4 | C2×#492 | 1 | 1 | yes | characteristic |
| 650 | C4 | #3151 | 1 | 1 | yes | characteristic |
| 651 | C4 | #3288 | 1 | 2 | yes | — |
| 652 | C4 | #3289 | 1 | 2 | yes | — |
| 653 | C2 | — | 2 | 2 | no | — |
| 654 | C2 | — | 4 | 2 | no | — |
| 655 | C2 | — | 16 | 1 | no | — |
| 656 | C2 | — | 16 | 1 | no | — |
| 657 | C2 | — | 16 | 2 | no | — |
| 658 | C2 | #25717 | 1 | 1 | yes | characteristic |
| 659 | C2 | #25807 | 1 | 1 | yes | characteristic |
| 660 | C2 | #25808 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #60646 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33222 | C2 | semidirect |
| 3. #19655 | C2 | semidirect |
| 4. C2×#1714 | C2 | semidirect |
| 5. #18651 | C2 | semidirect |
| 6. #18641 | C2 | semidirect |
| 7. #18631 | C2 | semidirect |
| 8. #18621 | C2 | semidirect |
| 9. #10172 | C2 | semidirect |
| 10. #10170 | C2 | semidirect |
| 11. #10148 | C2 | semidirect |
| 12. #10146 | C2 | semidirect |
| 13. #9862 | C2 | semidirect |
| 14. #9768 | C2 | semidirect |
| 15. #8185 | C2 | semidirect |
| 20. #3227 | C22 | semidirect |
| 22. #3206 | C22 | semidirect |
| 23. C2×#500 | C22 | semidirect |
| 24. C2×#498 | C22 | semidirect |
| 36. C2×#487 | C22 | semidirect |
| 41. #1997 | C22 | semidirect |
| 42. #1997 | C22 | semidirect |
| 44. #1848 | C22 | semidirect |
| 59. #1546 | C22 | semidirect |
| 60. #1546 | C22 | semidirect |
| 65. #1492 | C22 | semidirect |
| 66. #1492 | C22 | semidirect |
| 69. #1450 | C22 | semidirect |
| 70. C2×#322 | C22 | semidirect |
| 71. #1278 | C22 | semidirect |
| 72. #1278 | C22 | semidirect |
| 257. #131 | SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60646 | 2 |
| #33222 | 2 |
| #19655 | 2 |
| C2×#1714 | 2 |
| #18651 | 2 |
| #18641 | 2 |
| #18631 | 2 |
| #18621 | 2 |
| #10172 | 2 |
| #10170 | 2 |
| #10148 | 2 |
| #10146 | 2 |
| #9862 | 2 |
| #9768 | 2 |
| #8185 | 2 |