Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1672 conjugacy classes of subgroups.
The subgroup lattice has 595 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Q8×#137 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | Q8×AGL(1,ℤ/8) | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×Q8×SD16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | Q8×#131 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #33220 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #28700 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #28296 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #25492 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | #25490 | C2 | 1 | 3 | yes | maximal, complemented |
| 10 | #24858 | C2 | 1 | 3 | yes | maximal, complemented |
| 11 | #24856 | C2 | 1 | 3 | yes | maximal |
| 12 | Q8×#108 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 13 | Q8×#101 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 14 | Q8×#100 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 15 | C4×#471 | C2 | 1 | 3 | yes | maximal |
| 16 | Q8×#41 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×Q82 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×D4×Q8 | C22 | 1 | 1 | yes | characteristic |
| 19 | #3387 | — | 2 | 3 | no | — |
| 20 | #3386 | — | 2 | 3 | no | — |
| 21 | #3385 | — | 2 | 3 | no | — |
| 22 | #3384 | — | 2 | 3 | no | — |
| 23 | Q8×Q16 | — | 2 | 1 | no | — |
| 24 | Q8×SD16 | — | 2 | 1 | no | — |
| 25 | Q8×SD16 | — | 2 | 3 | no | — |
| 26 | Q8×SD16 | — | 2 | 3 | no | — |
| 27 | Q8×SD16 | — | 2 | 1 | no | — |
| 28 | Q8×SD16 | — | 2 | 1 | no | — |
| 29 | Q8×SD16 | — | 2 | 1 | no | — |
| 30 | Q8×D8 | — | 2 | 1 | no | — |
| 31 | #3165 | C22 | 1 | 3 | yes | — |
| 32 | #3165 | C22 | 1 | 3 | yes | — |
| 33 | C2×#498 | C22 | 1 | 3 | yes | — |
| 34 | #3079 | C22 | 1 | 3 | yes | — |
| 35 | #3079 | C22 | 1 | 3 | yes | — |
| 36 | #3078 | C22 | 1 | 3 | yes | — |
| 37 | #3077 | C22 | 1 | 3 | yes | — |
| 38 | C2×#475 | C22 | 1 | 3 | yes | — |
| 39 | C2×#476 | C22 | 1 | 3 | yes | — |
| 40 | C2×#471 | C22 | 1 | 1 | yes | characteristic |
| 41 | Q8×M16 | — | 2 | 1 | no | — |
| 42 | Q8×M16 | C22 | 1 | 1 | yes | characteristic |
| 43 | Q8×M16 | C22 | 1 | 1 | yes | characteristic |
| 44 | Q8×M16 | C22 | 1 | 1 | yes | characteristic |
| 45 | #2962 | — | 2 | 3 | no | — |
| 46 | #2962 | C22 | 1 | 3 | yes | — |
| 47 | #2960 | — | 2 | 3 | no | — |
| 48 | #2960 | C22 | 1 | 3 | yes | — |
| 49 | C2×C8×Q8 | — | 2 | 1 | no | — |
| 50 | C2×C8×Q8 | C22 | 1 | 1 | yes | characteristic |
| 51 | C4×#137 | C22 | 1 | 3 | yes | — |
| 52 | C4×AGL(1,ℤ/8) | C22 | 1 | 3 | yes | — |
| 53 | C2×C4×SD16 | C22 | 1 | 3 | yes | — |
| 54 | #2906 | C22 | 1 | 3 | yes | — |
| 55 | C4×#131 | C22 | 1 | 3 | yes | — |
| 56 | #2090 | C22 | 1 | 6 | yes | — |
| 57 | #2090 | C22 | 1 | 6 | yes | — |
| 58 | #2064 | C22 | 1 | 6 | yes | — |
| 59 | #2063 | C22 | 1 | 6 | yes | — |
| 60 | #1997 | C22 | 1 | 3 | yes | — |
| 61 | #1997 | C22 | 1 | 3 | yes | — |
| 62 | #1997 | C22 | 1 | 3 | yes | — |
| 63 | #1997 | C22 | 1 | 3 | yes | — |
| 64 | #1977 | C22 | 1 | 3 | yes | — |
| 65 | #1951 | C22 | 1 | 3 | yes | — |
| 66 | #1951 | C22 | 1 | 6 | yes | — |
| 67 | #1951 | C22 | 1 | 6 | yes | — |
| 68 | #1950 | C22 | 1 | 3 | yes | — |
| 69 | #1950 | C22 | 1 | 3 | yes | — |
| 70 | #1931 | C22 | 1 | 3 | yes | — |
| 71 | #1930 | C22 | 1 | 3 | yes | — |
| 72 | #1908 | C22 | 1 | 3 | yes | — |
| 73 | #1907 | C22 | 1 | 3 | yes | — |
| 74 | #1848 | C22 | 1 | 3 | yes | — |
| 75 | #1815 | C22 | 1 | 6 | yes | — |
| 76 | #1815 | C22 | 1 | 3 | yes | — |
| 77 | #1801 | C22 | 1 | 3 | yes | — |
| 78 | #1800 | C22 | 1 | 3 | yes | — |
| 79 | C4×#108 | C22 | 1 | 3 | yes | — |
| 80 | C4×#101 | C22 | 1 | 3 | yes | — |
| 81 | C4×#100 | C22 | 1 | 3 | yes | — |
| 82 | C23×Q8 | — | 2 | 1 | no | — |
| 83 | C2×#137 | C23 | 1 | 1 | yes | characteristic |
| 84 | C2×AGL(1,ℤ/8) | C23 | 1 | 1 | yes | characteristic |
| 85 | C22×SD16 | C23 | 1 | 1 | yes | characteristic |
| 86 | C2×#131 | C23 | 1 | 3 | yes | — |
| 87 | C2×#131 | C23 | 1 | 1 | yes | characteristic |
| 88 | Q82 | — | 2 | 1 | no | — |
| 89 | Q82 | — | 2 | 3 | no | — |
| 90 | Q82 | — | 2 | 1 | no | — |
| 91 | Q82 | — | 2 | 3 | no | — |
| 92 | Q82 | D4 | 1 | 1 | yes | characteristic |
| 93 | #557 | D4 | 1 | 3 | yes | — |
| 94 | #554 | — | 2 | 3 | no | — |
| 95 | #554 | D4 | 1 | 3 | yes | — |
| 96 | D4×Q8 | — | 2 | 1 | no | — |
| 97 | D4×Q8 | — | 2 | 1 | no | — |
| 98 | D4×Q8 | — | 2 | 1 | no | — |
| 99 | D4×Q8 | — | 2 | 3 | no | — |
| 100 | D4×Q8 | — | 2 | 3 | no | — |
| 101 | D4×Q8 | D4 | 1 | 1 | yes | characteristic |
| 102 | #533 | — | 2 | 3 | no | — |
| 103 | #533 | C23 | 1 | 3 | yes | — |
| 104 | C2×C4⋊Q8 | — | 2 | 3 | no | — |
| 105 | C2×C4⋊Q8 | C23 | 1 | 3 | yes | — |
| 106 | C2×C4⋊Q8 | C23 | 1 | 3 | yes | — |
| 107 | C2×C22⋊Q8 | — | 2 | 3 | no | — |
| 108 | C4×#41 | C23 | 1 | 3 | yes | — |
| 109 | C2×C4×Q8 | — | 2 | 1 | no | — |
| 110 | C2×C4×Q8 | — | 2 | 1 | no | — |
| 111 | C2×C4×Q8 | C23 | 1 | 3 | yes | — |
| 112 | C2×C4×Q8 | C23 | 1 | 1 | yes | characteristic |
| 113 | C2×C4×D4 | C23 | 1 | 3 | yes | — |
| 114 | #501 | — | 2 | 6 | no | — |
| 115 | #501 | — | 2 | 6 | no | — |
| 116 | #500 | — | 2 | 3 | no | — |
| 117 | #500 | — | 2 | 3 | no | — |
| 118 | #498 | — | 2 | 3 | no | — |
| 119 | #498 | — | 2 | 3 | no | — |
| 120 | #498 | — | 2 | 3 | no | — |
| 121 | #498 | — | 2 | 3 | no | — |
| 122 | #498 | — | 2 | 3 | no | — |
| 123 | #498 | — | 2 | 3 | no | — |
| 124 | #479 | — | 2 | 6 | no | — |
| 125 | #478 | — | 2 | 6 | no | — |
| 126 | #477 | — | 2 | 3 | no | — |
| 127 | #477 | — | 2 | 3 | no | — |
| 128 | #477 | — | 2 | 3 | no | — |
| 129 | #476 | — | 2 | 3 | no | — |
| 130 | #476 | — | 2 | 3 | no | — |
| 131 | #476 | — | 2 | 3 | no | — |
| 132 | #476 | — | 2 | 3 | no | — |
| 133 | #476 | — | 2 | 3 | no | — |
| 134 | #476 | — | 2 | 6 | no | — |
| 135 | #475 | — | 2 | 3 | no | — |
| 136 | #475 | — | 2 | 3 | no | — |
| 137 | #475 | — | 2 | 3 | no | — |
| 138 | #475 | — | 2 | 3 | no | — |
| 139 | #475 | — | 2 | 3 | no | — |
| 140 | #475 | — | 2 | 6 | no | — |
| 141 | #474 | — | 2 | 3 | no | — |
| 142 | #474 | — | 2 | 3 | no | — |
| 143 | #474 | — | 2 | 3 | no | — |
| 144 | #471 | Q8 | 1 | 8 | yes | direct product factor |
| 145 | #446 | — | 2 | 3 | no | — |
| 146 | #446 | — | 2 | 3 | no | — |
| 147 | #446 | — | 2 | 3 | no | — |
| 148 | #446 | — | 2 | 6 | no | — |
| 149 | #446 | — | 2 | 3 | no | — |
| 150 | #446 | — | 2 | 3 | no | — |
| 151 | #446 | — | 2 | 3 | no | — |
| 152 | C8×Q8 | — | 2 | 1 | no | — |
| 153 | C8×Q8 | — | 2 | 1 | no | — |
| 154 | C8×Q8 | — | 2 | 1 | no | — |
| 155 | C8×Q8 | — | 2 | 3 | no | — |
| 156 | C8×Q8 | — | 2 | 3 | no | — |
| 157 | C8×Q8 | — | 2 | 3 | no | — |
| 158 | C8×Q8 | D4 | 1 | 1 | yes | characteristic |
| 159 | C8×Q8 | D4 | 1 | 3 | yes | — |
| 160 | C8×Q8 | D4 | 1 | 3 | yes | — |
| 161 | C8×Q8 | D4 | 1 | 1 | yes | characteristic |
| 162 | #442 | — | 2 | 3 | no | — |
| 163 | #441 | — | 2 | 3 | no | — |
| 164 | #440 | — | 2 | 3 | no | — |
| 165 | #440 | — | 2 | 3 | no | — |
| 166 | C4×Q16 | — | 2 | 3 | no | — |
| 167 | C4×SD16 | — | 2 | 3 | no | — |
| 168 | C4×SD16 | — | 2 | 3 | no | — |
| 169 | C4×SD16 | — | 2 | 3 | no | — |
| 170 | C4×SD16 | — | 2 | 3 | no | — |
| 171 | C4×SD16 | — | 2 | 3 | no | — |
| 172 | C4×SD16 | — | 2 | 3 | no | — |
| 173 | C4×D8 | — | 2 | 3 | no | — |
| 174 | C2×#108 | C23 | 1 | 3 | yes | — |
| 175 | C2×#108 | C23 | 1 | 6 | yes | — |
| 176 | C2×#108 | C23 | 1 | 1 | yes | characteristic |
| 177 | #428 | C23 | 1 | 6 | yes | — |
| 178 | #428 | C23 | 1 | 6 | yes | — |
| 179 | #428 | C23 | 1 | 3 | yes | — |
| 180 | #428 | C23 | 1 | 3 | yes | — |
| 181 | C2×#106 | C23 | 1 | 6 | yes | — |
| 182 | C2×#106 | C23 | 1 | 3 | yes | — |
| 183 | #424 | — | 2 | 6 | no | — |
| 184 | #424 | C23 | 1 | 6 | yes | — |
| 185 | #423 | — | 2 | 3 | no | — |
| 186 | #423 | C23 | 1 | 3 | yes | — |
| 187 | #423 | C23 | 1 | 3 | yes | — |
| 188 | #423 | C23 | 1 | 3 | yes | — |
| 189 | C2×C4⋊C8 | — | 2 | 3 | no | — |
| 190 | C2×C4⋊C8 | C23 | 1 | 3 | yes | — |
| 191 | #419 | C23 | 1 | 3 | yes | — |
| 192 | #418 | C23 | 1 | 3 | yes | — |
| 193 | #417 | C23 | 1 | 3 | yes | — |
| 194 | #417 | C23 | 1 | 3 | yes | — |
| 195 | C2×#103 | C23 | 1 | 3 | yes | — |
| 196 | C2×#102 | C23 | 1 | 3 | yes | — |
| 197 | C2×#101 | C23 | 1 | 3 | yes | — |
| 198 | C2×#101 | C23 | 1 | 1 | yes | characteristic |
| 199 | C2×#100 | C23 | 1 | 3 | yes | — |
| 200 | C2×#100 | C23 | 1 | 1 | yes | characteristic |
| 201 | #405 | — | 2 | 3 | no | — |
| 202 | #405 | — | 2 | 3 | no | — |
| 203 | #405 | C23 | 1 | 3 | yes | — |
| 204 | #405 | C23 | 1 | 3 | yes | — |
| 205 | C4×M16 | — | 2 | 3 | no | — |
| 206 | C4×M16 | C23 | 1 | 3 | yes | — |
| 207 | C4×M16 | C23 | 1 | 3 | yes | — |
| 208 | C4×M16 | C23 | 1 | 3 | yes | — |
| 209 | C2×C4×C8 | — | 2 | 3 | no | — |
| 210 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 211 | #343 | C23 | 1 | 6 | yes | — |
| 212 | #343 | C23 | 1 | 6 | yes | — |
| 213 | #343 | C23 | 1 | 3 | yes | — |
| 214 | #343 | C23 | 1 | 3 | yes | — |
| 215 | #343 | C23 | 1 | 3 | yes | — |
| 216 | #343 | C23 | 1 | 3 | yes | — |
| 217 | #341 | C23 | 1 | 3 | yes | — |
| 218 | #341 | C23 | 1 | 3 | yes | — |
| 219 | C2×#41 | C24 | 1 | 1 | yes | characteristic |
| 220 | C22×Q8 | — | 2 | 1 | no | — |
| 221 | C22×Q8 | — | 2 | 3 | no | — |
| 222 | C22×Q8 | — | 2 | 3 | no | — |
| 223 | C22×Q8 | — | 4 | 3 | no | — |
| 224 | C22×Q8 | — | 4 | 1 | no | — |
| 225 | C22×Q8 | — | 8 | 4 | no | — |
| 226 | C22×Q8 | #41 | 1 | 1 | yes | characteristic |
| 227 | C22×Q8 | C24 | 1 | 1 | yes | characteristic |
| 228 | C22×D4 | C24 | 1 | 1 | yes | characteristic |
| 229 | C23×C4 | — | 2 | 3 | no | — |
| 230 | #137 | — | 2 | 2 | no | — |
| 231 | #137 | C2×Q8 | 1 | 4 | yes | complemented |
| 232 | AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 233 | AGL(1,ℤ/8) | C2×Q8 | 1 | 4 | yes | — |
| 234 | C2×Q16 | — | 2 | 1 | no | — |
| 235 | C2×SD16 | — | 2 | 1 | no | — |
| 236 | C2×SD16 | — | 2 | 1 | no | — |
| 237 | C2×SD16 | — | 2 | 1 | no | — |
| 238 | C2×SD16 | — | 2 | 1 | no | — |
| 239 | C2×SD16 | — | 2 | 2 | no | — |
| 240 | C2×SD16 | C2×Q8 | 1 | 4 | yes | complemented |
| 241 | C2×D8 | — | 2 | 1 | no | — |
| 242 | #131 | — | 2 | 2 | no | — |
| 243 | #131 | — | 2 | 6 | no | — |
| 244 | #131 | — | 4 | 3 | no | — |
| 245 | #131 | C2×Q8 | 1 | 4 | yes | complemented |
| 246 | C2×M16 | — | 2 | 1 | no | — |
| 247 | C2×M16 | — | 2 | 3 | no | — |
| 248 | C2×M16 | C24 | 1 | 3 | yes | — |
| 249 | C2×M16 | C24 | 1 | 3 | yes | — |
| 250 | C2×M16 | C24 | 1 | 3 | yes | — |
| 251 | C2×M16 | C24 | 1 | 1 | yes | characteristic |
| 252 | C2×M16 | C24 | 1 | 1 | yes | characteristic |
| 253 | C2×M16 | C24 | 1 | 1 | yes | characteristic |
| 254 | C22×C8 | — | 2 | 1 | no | — |
| 255 | C22×C8 | — | 2 | 3 | no | — |
| 256 | C22×C8 | C24 | 1 | 3 | yes | — |
| 257 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 258 | C4⋊Q8 | — | 2 | 3 | no | — |
| 259 | C4⋊Q8 | — | 2 | 3 | no | — |
| 260 | C4⋊Q8 | — | 2 | 3 | no | — |
| 261 | C4⋊Q8 | — | 2 | 3 | no | — |
| 262 | C4⋊Q8 | — | 2 | 3 | no | — |
| 263 | C4⋊Q8 | — | 2 | 3 | no | — |
| 264 | C4⋊Q8 | — | 2 | 3 | no | — |
| 265 | C4⋊Q8 | — | 2 | 3 | no | — |
| 266 | C4⋊Q8 | — | 2 | 3 | no | — |
| 267 | C4⋊Q8 | — | 2 | 3 | no | — |
| 268 | C4⋊Q8 | — | 4 | 3 | no | — |
| 269 | C4⋊Q8 | — | 4 | 6 | no | — |
| 270 | C4⋊Q8 | — | 4 | 3 | no | — |
| 271 | C4⋊Q8 | C2×D4 | 1 | 3 | yes | — |
| 272 | C4⋊Q8 | C2×D4 | 1 | 3 | yes | — |
| 273 | #125 | — | 2 | 6 | no | — |
| 274 | #125 | C2×D4 | 1 | 6 | yes | — |
| 275 | C22⋊Q8 | — | 2 | 3 | no | — |
| 276 | C22⋊Q8 | — | 2 | 3 | no | — |
| 277 | C22⋊Q8 | — | 2 | 3 | no | — |
| 278 | C22⋊Q8 | — | 2 | 3 | no | — |
| 279 | C22⋊Q8 | — | 2 | 6 | no | — |
| 280 | C22⋊Q8 | — | 4 | 3 | no | — |
| 281 | C22⋊Q8 | — | 4 | 6 | no | — |
| 282 | C22⋊Q8 | — | 4 | 3 | no | — |
| 283 | C4×Q8 | — | 2 | 1 | no | — |
| 284 | C4×Q8 | — | 2 | 3 | no | — |
| 285 | C4×Q8 | — | 2 | 3 | no | — |
| 286 | C4×Q8 | — | 2 | 3 | no | — |
| 287 | C4×Q8 | — | 2 | 3 | no | — |
| 288 | C4×Q8 | — | 2 | 3 | no | — |
| 289 | C4×Q8 | — | 2 | 3 | no | — |
| 290 | C4×Q8 | — | 2 | 3 | no | — |
| 291 | C4×Q8 | — | 4 | 3 | no | — |
| 292 | C4×Q8 | — | 4 | 1 | no | — |
| 293 | C4×Q8 | C2×D4 | 1 | 3 | yes | — |
| 294 | C4×Q8 | C2×D4 | 1 | 3 | yes | — |
| 295 | C4×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 296 | C4×Q8 | C2×D4 | 1 | 3 | yes | — |
| 297 | C4×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 298 | C4×Q8 | C2×D4 | 1 | 3 | yes | — |
| 299 | C4×D4 | — | 2 | 3 | no | — |
| 300 | C4×D4 | — | 2 | 3 | no | — |
| 301 | C4×D4 | — | 2 | 3 | no | — |
| 302 | C4×D4 | — | 2 | 3 | no | — |
| 303 | C4×D4 | — | 2 | 3 | no | — |
| 304 | C4×D4 | — | 2 | 3 | no | — |
| 305 | C4×D4 | C2×D4 | 1 | 3 | yes | — |
| 306 | C4×D4 | C2×D4 | 1 | 3 | yes | — |
| 307 | #117 | — | 2 | 3 | no | — |
| 308 | #117 | C24 | 1 | 3 | yes | — |
| 309 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 310 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 311 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 312 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 313 | C2×C4⋊C4 | #41 | 1 | 3 | yes | — |
| 314 | C2×C4⋊C4 | C24 | 1 | 3 | yes | — |
| 315 | C2×C4⋊C4 | C24 | 1 | 3 | yes | — |
| 316 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 317 | C2×C42 | — | 2 | 3 | no | — |
| 318 | C2×C42 | — | 2 | 3 | no | — |
| 319 | C2×C42 | C24 | 1 | 3 | yes | — |
| 320 | #108 | — | 2 | 6 | no | — |
| 321 | #108 | — | 4 | 6 | no | — |
| 322 | #108 | C2×Q8 | 1 | 4 | yes | complemented |
| 323 | #107 | — | 2 | 3 | no | — |
| 324 | #107 | — | 2 | 3 | no | — |
| 325 | #107 | — | 2 | 6 | no | — |
| 326 | #107 | — | 2 | 6 | no | — |
| 327 | #106 | — | 2 | 3 | no | — |
| 328 | #106 | — | 2 | 3 | no | — |
| 329 | #106 | — | 2 | 6 | no | — |
| 330 | #106 | — | 2 | 6 | no | — |
| 331 | #106 | — | 2 | 3 | no | — |
| 332 | #106 | — | 2 | 3 | no | — |
| 333 | #106 | — | 2 | 6 | no | — |
| 334 | #106 | — | 2 | 6 | no | — |
| 335 | C4⋊C8 | — | 2 | 6 | no | — |
| 336 | C4⋊C8 | — | 2 | 6 | no | — |
| 337 | C4⋊C8 | — | 2 | 3 | no | — |
| 338 | C4⋊C8 | — | 2 | 3 | no | — |
| 339 | C4⋊C8 | — | 2 | 3 | no | — |
| 340 | C4⋊C8 | — | 2 | 3 | no | — |
| 341 | C4⋊C8 | — | 2 | 3 | no | — |
| 342 | C4⋊C8 | — | 2 | 3 | no | — |
| 343 | C4⋊C8 | C2×D4 | 1 | 3 | yes | — |
| 344 | C4⋊C8 | C2×D4 | 1 | 3 | yes | — |
| 345 | C4⋊C8 | C2×D4 | 1 | 6 | yes | — |
| 346 | #103 | — | 2 | 3 | no | — |
| 347 | #103 | — | 2 | 3 | no | — |
| 348 | #103 | — | 2 | 3 | no | — |
| 349 | #103 | — | 2 | 3 | no | — |
| 350 | #103 | — | 2 | 3 | no | — |
| 351 | #103 | — | 2 | 3 | no | — |
| 352 | #102 | — | 2 | 3 | no | — |
| 353 | #102 | — | 2 | 3 | no | — |
| 354 | #102 | — | 2 | 3 | no | — |
| 355 | #102 | — | 2 | 3 | no | — |
| 356 | #102 | — | 2 | 3 | no | — |
| 357 | #102 | — | 2 | 3 | no | — |
| 358 | #101 | — | 4 | 3 | no | — |
| 359 | #101 | C2×Q8 | 1 | 4 | yes | complemented |
| 360 | #100 | — | 4 | 3 | no | — |
| 361 | #100 | C2×Q8 | 1 | 4 | yes | complemented |
| 362 | #97 | — | 2 | 3 | no | — |
| 363 | #97 | — | 2 | 3 | no | — |
| 364 | #97 | — | 2 | 3 | no | — |
| 365 | #97 | — | 2 | 3 | no | — |
| 366 | #97 | — | 2 | 3 | no | — |
| 367 | C4×C8 | — | 2 | 3 | no | — |
| 368 | C4×C8 | — | 2 | 3 | no | — |
| 369 | C4×C8 | — | 2 | 3 | no | — |
| 370 | C4×C8 | — | 2 | 3 | no | — |
| 371 | C4×C8 | — | 2 | 3 | no | — |
| 372 | C4×C8 | C2×D4 | 1 | 3 | yes | — |
| 373 | C4×C8 | C2×D4 | 1 | 3 | yes | — |
| 374 | C4×C8 | C2×D4 | 1 | 3 | yes | — |
| 375 | C4×C8 | C2×D4 | 1 | 3 | yes | — |
| 376 | C24 | — | 2 | 1 | no | — |
| 377 | #41 | — | 2 | 1 | no | — |
| 378 | #41 | — | 2 | 1 | no | — |
| 379 | #41 | C22×Q8 | 1 | 2 | yes | — |
| 380 | #41 | 2−1+4 | 1 | 2 | yes | — |
| 381 | C2×Q8 | — | 2 | 4 | no | — |
| 382 | C2×Q8 | — | 2 | 3 | no | — |
| 383 | C2×Q8 | — | 2 | 3 | no | — |
| 384 | C2×Q8 | — | 2 | 1 | no | — |
| 385 | C2×Q8 | — | 2 | 1 | no | — |
| 386 | C2×Q8 | — | 2 | 1 | no | — |
| 387 | C2×Q8 | — | 2 | 1 | no | — |
| 388 | C2×Q8 | — | 2 | 1 | no | — |
| 389 | C2×Q8 | — | 4 | 4 | no | — |
| 390 | C2×Q8 | — | 4 | 3 | no | — |
| 391 | C2×Q8 | — | 4 | 3 | no | — |
| 392 | C2×Q8 | — | 4 | 3 | no | — |
| 393 | C2×Q8 | — | 4 | 3 | no | — |
| 394 | C2×Q8 | — | 8 | 6 | no | — |
| 395 | C2×Q8 | — | 8 | 12 | no | — |
| 396 | C2×Q8 | — | 8 | 4 | no | — |
| 397 | C2×Q8 | — | 16 | 3 | no | — |
| 398 | C2×Q8 | #121 | 1 | 3 | yes | — |
| 399 | C2×Q8 | #121 | 1 | 1 | yes | characteristic |
| 400 | C2×Q8 | C22×D4 | 1 | 1 | yes | characteristic |
| 401 | C2×Q8 | C22×Q8 | 1 | 2 | yes | — |
| 402 | C2×D4 | — | 2 | 1 | no | — |
| 403 | C2×D4 | — | 2 | 1 | no | — |
| 404 | C2×D4 | — | 2 | 1 | no | — |
| 405 | C2×D4 | — | 2 | 1 | no | — |
| 406 | C2×D4 | — | 2 | 1 | no | — |
| 407 | C2×D4 | — | 2 | 1 | no | — |
| 408 | C2×D4 | C22×D4 | 1 | 1 | yes | characteristic |
| 409 | C2×D4 | C22×Q8 | 1 | 2 | yes | — |
| 410 | C22×C4 | — | 2 | 1 | no | — |
| 411 | C22×C4 | — | 2 | 1 | no | — |
| 412 | C22×C4 | — | 2 | 3 | no | — |
| 413 | C22×C4 | — | 2 | 3 | no | — |
| 414 | C22×C4 | — | 2 | 3 | no | — |
| 415 | C22×C4 | — | 2 | 3 | no | — |
| 416 | C22×C4 | — | 4 | 3 | no | — |
| 417 | C22×C4 | — | 4 | 3 | no | — |
| 418 | C22×C4 | — | 8 | 6 | no | — |
| 419 | C22×C4 | C2×#41 | 1 | 3 | yes | — |
| 420 | C22×C4 | C2×#41 | 1 | 3 | yes | — |
| 421 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 422 | Q16 | — | 2 | 4 | no | — |
| 423 | SD16 | — | 2 | 4 | no | — |
| 424 | SD16 | — | 2 | 4 | no | — |
| 425 | SD16 | — | 2 | 4 | no | — |
| 426 | SD16 | — | 2 | 4 | no | — |
| 427 | D8 | — | 2 | 4 | no | — |
| 428 | M16 | — | 2 | 1 | no | — |
| 429 | M16 | — | 2 | 1 | no | — |
| 430 | M16 | — | 2 | 1 | no | — |
| 431 | M16 | — | 2 | 4 | no | — |
| 432 | M16 | — | 2 | 3 | no | — |
| 433 | M16 | — | 2 | 3 | no | — |
| 434 | M16 | — | 2 | 3 | no | — |
| 435 | M16 | — | 2 | 3 | no | — |
| 436 | M16 | — | 2 | 3 | no | — |
| 437 | M16 | — | 2 | 3 | no | — |
| 438 | M16 | — | 4 | 6 | no | — |
| 439 | M16 | C22×Q8 | 1 | 2 | yes | — |
| 440 | M16 | C22×Q8 | 1 | 2 | yes | — |
| 441 | M16 | C22×Q8 | 1 | 2 | yes | — |
| 442 | C2×C8 | — | 2 | 1 | no | — |
| 443 | C2×C8 | — | 2 | 1 | no | — |
| 444 | C2×C8 | — | 2 | 1 | no | — |
| 445 | C2×C8 | — | 2 | 3 | no | — |
| 446 | C2×C8 | — | 2 | 3 | no | — |
| 447 | C2×C8 | — | 2 | 4 | no | — |
| 448 | C2×C8 | — | 2 | 3 | no | — |
| 449 | C2×C8 | — | 2 | 3 | no | — |
| 450 | C2×C8 | — | 2 | 3 | no | — |
| 451 | C2×C8 | — | 4 | 6 | no | — |
| 452 | C2×C8 | C22×D4 | 1 | 3 | yes | — |
| 453 | C2×C8 | C22×D4 | 1 | 3 | yes | — |
| 454 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 455 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 456 | C2×C8 | C22×Q8 | 1 | 2 | yes | — |
| 457 | C2×C8 | 2−1+4 | 1 | 2 | yes | — |
| 458 | C4⋊C4 | — | 2 | 6 | no | — |
| 459 | C4⋊C4 | — | 2 | 3 | no | — |
| 460 | C4⋊C4 | — | 2 | 3 | no | — |
| 461 | C4⋊C4 | — | 2 | 3 | no | — |
| 462 | C4⋊C4 | — | 2 | 3 | no | — |
| 463 | C4⋊C4 | — | 2 | 6 | no | — |
| 464 | C4⋊C4 | — | 2 | 3 | no | — |
| 465 | C4⋊C4 | — | 2 | 3 | no | — |
| 466 | C4⋊C4 | — | 2 | 3 | no | — |
| 467 | C4⋊C4 | — | 2 | 3 | no | — |
| 468 | C4⋊C4 | — | 4 | 3 | no | — |
| 469 | C4⋊C4 | — | 4 | 3 | no | — |
| 470 | C4⋊C4 | — | 4 | 3 | no | — |
| 471 | C4⋊C4 | — | 4 | 3 | no | — |
| 472 | C4⋊C4 | — | 4 | 3 | no | — |
| 473 | C4⋊C4 | — | 4 | 3 | no | — |
| 474 | C4⋊C4 | #121 | 1 | 3 | yes | — |
| 475 | C4⋊C4 | #121 | 1 | 6 | yes | — |
| 476 | C4⋊C4 | #121 | 1 | 3 | yes | — |
| 477 | C4⋊C4 | C22×D4 | 1 | 3 | yes | — |
| 478 | C4⋊C4 | C22×D4 | 1 | 3 | yes | — |
| 479 | C22⋊C4 | — | 2 | 3 | no | — |
| 480 | C22⋊C4 | — | 2 | 3 | no | — |
| 481 | C22⋊C4 | — | 2 | 3 | no | — |
| 482 | C22⋊C4 | — | 4 | 3 | no | — |
| 483 | C22⋊C4 | — | 4 | 3 | no | — |
| 484 | C42 | — | 2 | 3 | no | — |
| 485 | C42 | — | 2 | 3 | no | — |
| 486 | C42 | — | 2 | 3 | no | — |
| 487 | C42 | — | 4 | 3 | no | — |
| 488 | C42 | — | 4 | 3 | no | — |
| 489 | C42 | C22×D4 | 1 | 3 | yes | — |
| 490 | C42 | C22×D4 | 1 | 3 | yes | — |
| 491 | C42 | C22×D4 | 1 | 3 | yes | — |
| 492 | C42 | C22×D4 | 1 | 3 | yes | — |
| 493 | C23 | — | 2 | 1 | no | — |
| 494 | C23 | — | 2 | 2 | no | — |
| 495 | C23 | — | 4 | 1 | no | — |
| 496 | C23 | — | 4 | 1 | no | — |
| 497 | C23 | — | 8 | 1 | no | — |
| 498 | C23 | — | 8 | 1 | no | — |
| 499 | C23 | C22×#41 | 1 | 1 | yes | characteristic |
| 500 | Q8 | — | 2 | 1 | no | — |
| 501 | Q8 | — | 2 | 6 | no | — |
| 502 | Q8 | — | 2 | 2 | no | — |
| 503 | Q8 | — | 2 | 2 | no | — |
| 504 | Q8 | — | 4 | 1 | no | — |
| 505 | Q8 | — | 4 | 1 | no | — |
| 506 | Q8 | — | 4 | 6 | no | — |
| 507 | Q8 | — | 8 | 3 | no | — |
| 508 | Q8 | — | 8 | 6 | no | — |
| 509 | Q8 | — | 8 | 6 | no | — |
| 510 | Q8 | — | 16 | 3 | no | — |
| 511 | Q8 | — | 16 | 3 | no | — |
| 512 | Q8 | #471 | 1 | 4 | yes | direct product factor |
| 513 | Q8 | D4×Q8 | 1 | 2 | yes | — |
| 514 | D4 | — | 2 | 2 | no | — |
| 515 | D4 | — | 2 | 2 | no | — |
| 516 | D4 | — | 2 | 2 | no | — |
| 517 | D4 | — | 2 | 2 | no | — |
| 518 | D4 | — | 2 | 1 | no | — |
| 519 | D4 | — | 4 | 1 | no | — |
| 520 | D4 | — | 4 | 1 | no | — |
| 521 | D4 | D4×Q8 | 1 | 2 | yes | — |
| 522 | C2×C4 | — | 2 | 1 | no | — |
| 523 | C2×C4 | — | 2 | 2 | no | — |
| 524 | C2×C4 | — | 2 | 2 | no | — |
| 525 | C2×C4 | — | 2 | 6 | no | — |
| 526 | C2×C4 | — | 2 | 3 | no | — |
| 527 | C2×C4 | — | 2 | 3 | no | — |
| 528 | C2×C4 | — | 2 | 2 | no | — |
| 529 | C2×C4 | — | 4 | 3 | no | — |
| 530 | C2×C4 | — | 4 | 1 | no | — |
| 531 | C2×C4 | — | 4 | 1 | no | — |
| 532 | C2×C4 | — | 4 | 6 | no | — |
| 533 | C2×C4 | — | 4 | 3 | no | — |
| 534 | C2×C4 | — | 4 | 3 | no | — |
| 535 | C2×C4 | — | 8 | 3 | no | — |
| 536 | C2×C4 | — | 8 | 6 | no | — |
| 537 | C2×C4 | — | 8 | 6 | no | — |
| 538 | C2×C4 | — | 8 | 3 | no | — |
| 539 | C2×C4 | — | 8 | 3 | no | — |
| 540 | C2×C4 | C2×#121 | 1 | 3 | yes | — |
| 541 | C2×C4 | C2×#121 | 1 | 3 | yes | — |
| 542 | C2×C4 | C2×#121 | 1 | 3 | yes | — |
| 543 | C2×C4 | C2×#121 | 1 | 3 | yes | — |
| 544 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 545 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 546 | C2×C4 | C23×Q8 | 1 | 1 | yes | characteristic |
| 547 | C2×C4 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 548 | C2×C4 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 549 | C2×C4 | #585 | 1 | 1 | yes | characteristic |
| 550 | C8 | — | 2 | 2 | no | — |
| 551 | C8 | — | 2 | 2 | no | — |
| 552 | C8 | — | 2 | 2 | no | — |
| 553 | C8 | — | 2 | 3 | no | — |
| 554 | C8 | — | 2 | 3 | no | — |
| 555 | C8 | — | 4 | 3 | no | — |
| 556 | C8 | — | 4 | 3 | no | — |
| 557 | C8 | — | 4 | 3 | no | — |
| 558 | C8 | D4×Q8 | 1 | 2 | yes | — |
| 559 | C8 | D4×Q8 | 1 | 2 | yes | — |
| 560 | C22 | — | 2 | 1 | no | — |
| 561 | C22 | — | 2 | 1 | no | — |
| 562 | C22 | — | 2 | 2 | no | — |
| 563 | C22 | — | 4 | 1 | no | — |
| 564 | C22 | — | 4 | 1 | no | — |
| 565 | C22 | — | 4 | 2 | no | — |
| 566 | C22 | — | 8 | 2 | no | — |
| 567 | C22 | — | 8 | 2 | no | — |
| 568 | C22 | — | 8 | 1 | no | — |
| 569 | C22 | — | 8 | 1 | no | — |
| 570 | C22 | C22×#121 | 1 | 1 | yes | characteristic |
| 571 | C22 | Q8×#41 | 1 | 1 | yes | characteristic |
| 572 | C22 | #3523 | 1 | 1 | yes | characteristic |
| 573 | C4 | — | 2 | 3 | no | — |
| 574 | C4 | — | 2 | 3 | no | — |
| 575 | C4 | — | 2 | 3 | no | — |
| 576 | C4 | — | 2 | 2 | no | — |
| 577 | C4 | — | 4 | 3 | no | — |
| 578 | C4 | — | 4 | 3 | no | — |
| 579 | C4 | — | 4 | 2 | no | — |
| 580 | C4 | — | 8 | 3 | no | — |
| 581 | C4 | — | 8 | 3 | no | — |
| 582 | C4 | C2×#471 | 1 | 6 | yes | — |
| 583 | C4 | C2×D4×Q8 | 1 | 1 | yes | characteristic |
| 584 | C4 | C2×D4×Q8 | 1 | 1 | yes | characteristic |
| 585 | C4 | #3489 | 1 | 1 | yes | characteristic |
| 586 | C4 | #3497 | 1 | 1 | yes | characteristic |
| 587 | C2 | — | 2 | 1 | no | — |
| 588 | C2 | — | 2 | 1 | no | — |
| 589 | C2 | — | 4 | 2 | no | — |
| 590 | C2 | — | 8 | 2 | no | — |
| 591 | C2 | Q8×#121 | 1 | 1 | yes | characteristic |
| 592 | C2 | C22×#471 | 1 | 1 | yes | characteristic |
| 593 | C2 | #60470 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×AGL(1,ℤ/8) | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×#137 | C2 | semidirect |
| 3. C2×Q8×SD16 | C2 | semidirect |
| 4. Q8×#131 | C2 | semidirect |
| 5. #33220 | C2 | semidirect |
| 6. #28700 | C2 | semidirect |
| 7. #28296 | C2 | semidirect |
| 8. #25492 | C2 | semidirect |
| 9. #25490 | C2 | semidirect |
| 10. #24858 | C2 | semidirect |
| 12. Q8×#108 | C2 | semidirect |
| 13. Q8×#101 | C2 | semidirect |
| 14. Q8×#100 | C2 | semidirect |
| 144. #471 | Q8 | direct (N × H) |
| 231. #137 | C2×Q8 | semidirect |
| 240. C2×SD16 | C2×Q8 | semidirect |
| 245. #131 | C2×Q8 | semidirect |
| 322. #108 | C2×Q8 | semidirect |
| 359. #101 | C2×Q8 | semidirect |
| 361. #100 | C2×Q8 | semidirect |
| 512. Q8 | #471 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×#137 | 2 |
| Q8×AGL(1,ℤ/8) | 2 |
| C2×Q8×SD16 | 2 |
| Q8×#131 | 2 |
| #33220 | 2 |
| #28700 | 2 |
| #28296 | 2 |
| #25492 | 2 |
| #25490 | 2 |
| #24858 | 2 |
| #24856 | 2 |
| Q8×#108 | 2 |
| Q8×#101 | 2 |
| Q8×#100 | 2 |
| C4×#471 | 2 |