Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
962 conjugacy classes of subgroups.
The subgroup lattice has 507 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#3168 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#2207 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#2206 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #33171 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #13365 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #13326 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #13305 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #13304 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #12812 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #12811 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #12783 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C2×#529 | — | 2 | 1 | no | — |
| 13 | C22×SD32 | C22 | 1 | 2 | yes | — |
| 14 | C22×D16 | C22 | 1 | 2 | yes | — |
| 15 | #3168 | — | 2 | 4 | no | — |
| 16 | #3168 | — | 2 | 4 | no | — |
| 17 | C2×#499 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#495 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#493 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#488 | — | 2 | 1 | no | — |
| 21 | C2×#467 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×#466 | — | 2 | 1 | no | — |
| 23 | C2×#466 | C22 | 1 | 1 | yes | characteristic |
| 24 | C2×#466 | C22 | 1 | 2 | yes | — |
| 25 | C2×#465 | — | 2 | 1 | no | — |
| 26 | C2×C4×Q16 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×C4×SD16 | — | 2 | 1 | no | — |
| 28 | C2×C4×D8 | C22 | 1 | 1 | yes | characteristic |
| 29 | C22×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 30 | #2856 | — | 2 | 1 | no | — |
| 31 | #2207 | — | 2 | 4 | no | — |
| 32 | #2206 | — | 2 | 4 | no | — |
| 33 | C2×C4⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 34 | C2×C4⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 35 | C2×#358 | C22 | 1 | 2 | yes | — |
| 36 | C2×#357 | C22 | 1 | 2 | yes | — |
| 37 | C2×#357 | C22 | 1 | 2 | yes | — |
| 38 | C2×#357 | C22 | 1 | 2 | yes | — |
| 39 | C2×C22⋊C16 | C22 | 1 | 2 | yes | — |
| 40 | #1970 | — | 2 | 1 | no | — |
| 41 | #1970 | C22 | 1 | 2 | yes | — |
| 42 | #1957 | — | 2 | 1 | no | — |
| 43 | #1845 | C22 | 1 | 1 | yes | characteristic |
| 44 | #1380 | C22 | 1 | 2 | yes | — |
| 45 | #1380 | C22 | 1 | 2 | yes | — |
| 46 | #1374 | C22 | 1 | 2 | yes | — |
| 47 | C22×Q16 | C23 | 1 | 1 | yes | characteristic |
| 48 | C22×SD16 | — | 2 | 1 | no | — |
| 49 | C22×D8 | — | 2 | 1 | no | — |
| 50 | C22×D8 | C23 | 1 | 2 | yes | — |
| 51 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 52 | C23×C8 | D4 | 1 | 1 | yes | characteristic |
| 53 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 54 | C2×#127 | — | 2 | 1 | no | — |
| 55 | #529 | — | 2 | 8 | no | — |
| 56 | #529 | — | 4 | 4 | no | — |
| 57 | C2×#126 | — | 4 | 1 | no | — |
| 58 | C2×#125 | — | 2 | 1 | no | — |
| 59 | C2×#124 | — | 2 | 1 | no | — |
| 60 | C2×#123 | — | 4 | 1 | no | — |
| 61 | C2×C22⋊Q8 | — | 2 | 1 | no | — |
| 62 | C2×#121 | — | 2 | 1 | no | — |
| 63 | C2×#121 | — | 2 | 2 | no | — |
| 64 | C2×#121 | — | 4 | 1 | no | — |
| 65 | C2×#121 | — | 4 | 1 | no | — |
| 66 | C2×C4×Q8 | — | 2 | 1 | no | — |
| 67 | C2×C4×D4 | — | 2 | 1 | no | — |
| 68 | C2×C4×D4 | — | 4 | 1 | no | — |
| 69 | C2×#117 | — | 4 | 1 | no | — |
| 70 | C22×C42 | D4 | 1 | 1 | yes | characteristic |
| 71 | C2×SD32 | — | 2 | 4 | no | — |
| 72 | C2×SD32 | — | 4 | 2 | no | — |
| 73 | C2×SD32 | — | 4 | 2 | no | — |
| 74 | C2×D16 | — | 2 | 4 | no | — |
| 75 | C2×D16 | — | 4 | 2 | no | — |
| 76 | C2×D16 | — | 4 | 2 | no | — |
| 77 | C22×C16 | C23 | 1 | 2 | yes | — |
| 78 | C22×C16 | C23 | 1 | 2 | yes | — |
| 79 | #499 | — | 2 | 2 | no | — |
| 80 | #499 | — | 2 | 2 | no | — |
| 81 | #495 | — | 2 | 2 | no | — |
| 82 | #495 | — | 2 | 2 | no | — |
| 83 | #493 | — | 2 | 2 | no | — |
| 84 | #493 | — | 2 | 2 | no | — |
| 85 | #488 | — | 2 | 8 | no | — |
| 86 | #467 | — | 2 | 2 | no | — |
| 87 | #467 | — | 2 | 2 | no | — |
| 88 | #466 | — | 2 | 2 | no | — |
| 89 | #466 | — | 2 | 2 | no | — |
| 90 | #466 | — | 4 | 2 | no | — |
| 91 | #466 | — | 4 | 2 | no | — |
| 92 | #466 | — | 4 | 4 | no | — |
| 93 | #465 | — | 2 | 8 | no | — |
| 94 | C4×Q16 | — | 2 | 2 | no | — |
| 95 | C4×Q16 | D4 | 1 | 4 | yes | complemented |
| 96 | C4×SD16 | — | 2 | 8 | no | — |
| 97 | C4×D8 | — | 2 | 2 | no | — |
| 98 | C4×D8 | D4 | 1 | 4 | yes | complemented |
| 99 | C2×#107 | — | 2 | 1 | no | — |
| 100 | C2×#107 | C23 | 1 | 2 | yes | — |
| 101 | C2×#107 | C23 | 1 | 1 | yes | characteristic |
| 102 | C2×#107 | C23 | 1 | 1 | yes | characteristic |
| 103 | C2×#106 | — | 2 | 1 | no | — |
| 104 | C2×#103 | — | 2 | 1 | no | — |
| 105 | C2×#103 | — | 2 | 1 | no | — |
| 106 | C2×#102 | — | 2 | 1 | no | — |
| 107 | C2×#102 | — | 2 | 1 | no | — |
| 108 | C2×#102 | — | 2 | 2 | no | — |
| 109 | C2×#102 | — | 2 | 2 | no | — |
| 110 | C2×C4×C8 | — | 2 | 2 | no | — |
| 111 | C2×C4×C8 | — | 2 | 2 | no | — |
| 112 | C2×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 113 | C2×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 114 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 115 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 116 | #390 | — | 2 | 2 | no | — |
| 117 | #390 | — | 4 | 1 | no | — |
| 118 | #390 | — | 4 | 1 | no | — |
| 119 | #384 | — | 2 | 1 | no | — |
| 120 | C4⋊C16 | — | 2 | 2 | no | — |
| 121 | C4⋊C16 | — | 2 | 2 | no | — |
| 122 | #358 | — | 4 | 2 | no | — |
| 123 | #357 | — | 2 | 4 | no | — |
| 124 | #357 | — | 2 | 4 | no | — |
| 125 | #357 | — | 4 | 2 | no | — |
| 126 | C22⋊C16 | — | 4 | 2 | no | — |
| 127 | C22×Q8 | — | 2 | 1 | no | — |
| 128 | C22×D4 | — | 2 | 1 | no | — |
| 129 | C22×D4 | — | 2 | 2 | no | — |
| 130 | C22×D4 | — | 4 | 1 | no | — |
| 131 | C22×D4 | — | 4 | 2 | no | — |
| 132 | C22×D4 | — | 4 | 1 | no | — |
| 133 | C22×D4 | — | 4 | 1 | no | — |
| 134 | C23×C4 | — | 4 | 1 | no | — |
| 135 | C23×C4 | D8 | 1 | 1 | yes | characteristic |
| 136 | C23×C4 | D8 | 1 | 1 | yes | characteristic |
| 137 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 138 | C2×Q16 | — | 2 | 1 | no | — |
| 139 | C2×Q16 | — | 2 | 1 | no | — |
| 140 | C2×Q16 | — | 2 | 1 | no | — |
| 141 | C2×Q16 | — | 2 | 1 | no | — |
| 142 | C2×Q16 | — | 2 | 1 | no | — |
| 143 | C2×Q16 | C2×D4 | 1 | 2 | yes | — |
| 144 | C2×SD16 | — | 2 | 4 | no | — |
| 145 | C2×SD16 | — | 4 | 2 | no | — |
| 146 | C2×SD16 | — | 4 | 2 | no | — |
| 147 | C2×D8 | — | 2 | 1 | no | — |
| 148 | C2×D8 | — | 2 | 2 | no | — |
| 149 | C2×D8 | — | 2 | 2 | no | — |
| 150 | C2×D8 | — | 2 | 1 | no | — |
| 151 | C2×D8 | — | 2 | 1 | no | — |
| 152 | C2×D8 | — | 2 | 1 | no | — |
| 153 | C2×D8 | — | 2 | 2 | no | — |
| 154 | C2×D8 | — | 2 | 2 | no | — |
| 155 | C2×D8 | — | 2 | 2 | no | — |
| 156 | C2×D8 | — | 2 | 1 | no | — |
| 157 | C2×D8 | — | 2 | 2 | no | — |
| 158 | C2×D8 | — | 4 | 2 | no | — |
| 159 | C2×D8 | — | 4 | 2 | no | — |
| 160 | C2×D8 | — | 4 | 2 | no | — |
| 161 | C2×D8 | C2×D4 | 1 | 2 | yes | — |
| 162 | C22×C8 | — | 2 | 2 | no | — |
| 163 | C22×C8 | — | 2 | 2 | no | — |
| 164 | C22×C8 | — | 2 | 1 | no | — |
| 165 | C22×C8 | — | 2 | 1 | no | — |
| 166 | C22×C8 | — | 2 | 1 | no | — |
| 167 | C22×C8 | — | 2 | 1 | no | — |
| 168 | C22×C8 | — | 2 | 1 | no | — |
| 169 | C22×C8 | — | 2 | 1 | no | — |
| 170 | C22×C8 | — | 4 | 1 | no | — |
| 171 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 172 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 173 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 174 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 175 | #127 | — | 4 | 2 | no | — |
| 176 | #127 | — | 4 | 2 | no | — |
| 177 | #126 | — | 4 | 8 | no | — |
| 178 | #125 | — | 2 | 4 | no | — |
| 179 | #125 | — | 4 | 2 | no | — |
| 180 | #124 | — | 2 | 4 | no | — |
| 181 | #124 | — | 4 | 2 | no | — |
| 182 | #123 | — | 4 | 8 | no | — |
| 183 | C22⋊Q8 | — | 2 | 4 | no | — |
| 184 | C22⋊Q8 | — | 4 | 2 | no | — |
| 185 | #121 | — | 2 | 4 | no | — |
| 186 | #121 | — | 4 | 2 | no | — |
| 187 | #121 | — | 4 | 4 | no | — |
| 188 | #121 | — | 4 | 4 | no | — |
| 189 | #121 | — | 8 | 4 | no | — |
| 190 | #121 | — | 8 | 4 | no | — |
| 191 | C4×Q8 | — | 2 | 4 | no | — |
| 192 | C4×Q8 | — | 4 | 2 | no | — |
| 193 | C4×D4 | — | 2 | 4 | no | — |
| 194 | C4×D4 | — | 4 | 8 | no | — |
| 195 | C4×D4 | — | 4 | 2 | no | — |
| 196 | #117 | — | 4 | 8 | no | — |
| 197 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 198 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 199 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 200 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 201 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 202 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 203 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 204 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 205 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 206 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 207 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 208 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 209 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 210 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 211 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 212 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 213 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 214 | C2×C42 | — | 2 | 2 | no | — |
| 215 | C2×C42 | — | 2 | 2 | no | — |
| 216 | C2×C42 | — | 4 | 1 | no | — |
| 217 | C2×C42 | D8 | 1 | 1 | yes | characteristic |
| 218 | C2×C42 | D8 | 1 | 1 | yes | characteristic |
| 219 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 220 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 221 | SD32 | — | 4 | 8 | no | — |
| 222 | D16 | — | 4 | 8 | no | — |
| 223 | C2×C16 | — | 2 | 2 | no | — |
| 224 | C2×C16 | — | 2 | 2 | no | — |
| 225 | C2×C16 | — | 2 | 2 | no | — |
| 226 | C2×C16 | — | 2 | 2 | no | — |
| 227 | C2×C16 | — | 2 | 2 | no | — |
| 228 | C2×C16 | — | 2 | 2 | no | — |
| 229 | #107 | — | 2 | 1 | no | — |
| 230 | #107 | — | 2 | 1 | no | — |
| 231 | #107 | — | 2 | 2 | no | — |
| 232 | #107 | — | 2 | 2 | no | — |
| 233 | #107 | — | 4 | 2 | no | — |
| 234 | #107 | C2×D4 | 1 | 2 | yes | — |
| 235 | #107 | C2×D4 | 1 | 2 | yes | — |
| 236 | #106 | — | 2 | 4 | no | — |
| 237 | #103 | — | 2 | 4 | no | — |
| 238 | #103 | — | 2 | 4 | no | — |
| 239 | #102 | — | 2 | 4 | no | — |
| 240 | #102 | — | 2 | 4 | no | — |
| 241 | #102 | — | 4 | 4 | no | — |
| 242 | #102 | — | 4 | 4 | no | — |
| 243 | C4×C8 | — | 2 | 2 | no | — |
| 244 | C4×C8 | — | 2 | 1 | no | — |
| 245 | C4×C8 | — | 2 | 1 | no | — |
| 246 | C4×C8 | — | 2 | 2 | no | — |
| 247 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 248 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 249 | C24 | — | 4 | 1 | no | — |
| 250 | C24 | — | 4 | 2 | no | — |
| 251 | C24 | C2×D8 | 1 | 1 | yes | characteristic |
| 252 | C2×Q8 | — | 2 | 2 | no | — |
| 253 | C2×Q8 | — | 4 | 1 | no | — |
| 254 | C2×Q8 | — | 4 | 1 | no | — |
| 255 | C2×Q8 | — | 4 | 1 | no | — |
| 256 | C2×Q8 | — | 4 | 1 | no | — |
| 257 | C2×Q8 | — | 4 | 1 | no | — |
| 258 | C2×D4 | — | 2 | 2 | no | — |
| 259 | C2×D4 | — | 4 | 4 | no | — |
| 260 | C2×D4 | — | 4 | 2 | no | — |
| 261 | C2×D4 | — | 4 | 1 | no | — |
| 262 | C2×D4 | — | 4 | 2 | no | — |
| 263 | C2×D4 | — | 4 | 1 | no | — |
| 264 | C2×D4 | — | 4 | 2 | no | — |
| 265 | C2×D4 | — | 4 | 1 | no | — |
| 266 | C2×D4 | — | 4 | 1 | no | — |
| 267 | C2×D4 | — | 4 | 2 | no | — |
| 268 | C2×D4 | — | 4 | 2 | no | — |
| 269 | C2×D4 | — | 4 | 1 | no | — |
| 270 | C2×D4 | — | 4 | 2 | no | — |
| 271 | C2×D4 | — | 8 | 2 | no | — |
| 272 | C2×D4 | — | 8 | 2 | no | — |
| 273 | C2×D4 | — | 8 | 2 | no | — |
| 274 | C2×D4 | — | 8 | 2 | no | — |
| 275 | C2×D4 | — | 8 | 4 | no | — |
| 276 | C2×D4 | — | 8 | 4 | no | — |
| 277 | C2×D4 | — | 8 | 4 | no | — |
| 278 | C2×D4 | — | 8 | 2 | no | — |
| 279 | C2×D4 | — | 8 | 2 | no | — |
| 280 | C2×D4 | — | 8 | 2 | no | — |
| 281 | C2×D4 | — | 8 | 2 | no | — |
| 282 | C22×C4 | — | 2 | 2 | no | — |
| 283 | C22×C4 | — | 2 | 1 | no | — |
| 284 | C22×C4 | — | 2 | 1 | no | — |
| 285 | C22×C4 | — | 2 | 1 | no | — |
| 286 | C22×C4 | — | 2 | 1 | no | — |
| 287 | C22×C4 | — | 2 | 1 | no | — |
| 288 | C22×C4 | — | 2 | 1 | no | — |
| 289 | C22×C4 | — | 2 | 2 | no | — |
| 290 | C22×C4 | — | 2 | 2 | no | — |
| 291 | C22×C4 | — | 2 | 2 | no | — |
| 292 | C22×C4 | — | 2 | 2 | no | — |
| 293 | C22×C4 | — | 4 | 1 | no | — |
| 294 | C22×C4 | — | 4 | 1 | no | — |
| 295 | C22×C4 | — | 4 | 1 | no | — |
| 296 | C22×C4 | — | 4 | 1 | no | — |
| 297 | C22×C4 | — | 4 | 4 | no | — |
| 298 | C22×C4 | — | 4 | 2 | no | — |
| 299 | C22×C4 | — | 8 | 2 | no | — |
| 300 | C22×C4 | — | 8 | 2 | no | — |
| 301 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 302 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 303 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 304 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 305 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 306 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 307 | Q16 | — | 2 | 2 | no | — |
| 308 | Q16 | — | 2 | 2 | no | — |
| 309 | Q16 | — | 4 | 1 | no | — |
| 310 | Q16 | — | 4 | 1 | no | — |
| 311 | SD16 | — | 4 | 8 | no | — |
| 312 | D8 | — | 2 | 2 | no | — |
| 313 | D8 | — | 2 | 2 | no | — |
| 314 | D8 | — | 4 | 1 | no | — |
| 315 | D8 | — | 4 | 2 | no | — |
| 316 | D8 | — | 4 | 2 | no | — |
| 317 | D8 | — | 4 | 1 | no | — |
| 318 | D8 | — | 4 | 2 | no | — |
| 319 | D8 | — | 4 | 2 | no | — |
| 320 | D8 | — | 8 | 4 | no | — |
| 321 | C2×C8 | — | 2 | 1 | no | — |
| 322 | C2×C8 | — | 2 | 1 | no | — |
| 323 | C2×C8 | — | 2 | 1 | no | — |
| 324 | C2×C8 | — | 2 | 1 | no | — |
| 325 | C2×C8 | — | 2 | 1 | no | — |
| 326 | C2×C8 | — | 2 | 1 | no | — |
| 327 | C2×C8 | — | 2 | 1 | no | — |
| 328 | C2×C8 | — | 4 | 4 | no | — |
| 329 | C2×C8 | — | 4 | 1 | no | — |
| 330 | C2×C8 | — | 4 | 1 | no | — |
| 331 | C2×C8 | — | 4 | 1 | no | — |
| 332 | C2×C8 | — | 4 | 1 | no | — |
| 333 | C2×C8 | #120 | 1 | 2 | yes | — |
| 334 | C2×C8 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 335 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 336 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 337 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 338 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 339 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 340 | C2×C8 | 2−1+4 | 1 | 1 | yes | characteristic |
| 341 | C4⋊C4 | — | 2 | 2 | no | — |
| 342 | C4⋊C4 | — | 2 | 2 | no | — |
| 343 | C4⋊C4 | — | 4 | 4 | no | — |
| 344 | C4⋊C4 | — | 4 | 4 | no | — |
| 345 | C4⋊C4 | — | 4 | 4 | no | — |
| 346 | C4⋊C4 | — | 4 | 1 | no | — |
| 347 | C4⋊C4 | — | 4 | 1 | no | — |
| 348 | C4⋊C4 | — | 4 | 4 | no | — |
| 349 | C4⋊C4 | — | 4 | 2 | no | — |
| 350 | C4⋊C4 | — | 4 | 2 | no | — |
| 351 | C4⋊C4 | — | 8 | 2 | no | — |
| 352 | C4⋊C4 | — | 8 | 2 | no | — |
| 353 | C22⋊C4 | — | 4 | 4 | no | — |
| 354 | C22⋊C4 | — | 4 | 4 | no | — |
| 355 | C22⋊C4 | — | 4 | 4 | no | — |
| 356 | C22⋊C4 | — | 4 | 4 | no | — |
| 357 | C22⋊C4 | — | 4 | 4 | no | — |
| 358 | C22⋊C4 | — | 8 | 4 | no | — |
| 359 | C22⋊C4 | — | 8 | 4 | no | — |
| 360 | C22⋊C4 | — | 8 | 4 | no | — |
| 361 | C42 | — | 2 | 1 | no | — |
| 362 | C42 | — | 2 | 1 | no | — |
| 363 | C42 | — | 2 | 2 | no | — |
| 364 | C42 | — | 2 | 2 | no | — |
| 365 | C42 | — | 4 | 4 | no | — |
| 366 | C42 | #120 | 1 | 2 | yes | — |
| 367 | C42 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 368 | C16 | — | 4 | 2 | no | — |
| 369 | C16 | — | 4 | 2 | no | — |
| 370 | C23 | — | 2 | 1 | no | — |
| 371 | C23 | — | 2 | 1 | no | — |
| 372 | C23 | — | 2 | 1 | no | — |
| 373 | C23 | — | 2 | 1 | no | — |
| 374 | C23 | — | 2 | 1 | no | — |
| 375 | C23 | — | 2 | 1 | no | — |
| 376 | C23 | — | 4 | 2 | no | — |
| 377 | C23 | — | 8 | 1 | no | — |
| 378 | C23 | — | 8 | 1 | no | — |
| 379 | C23 | — | 8 | 1 | no | — |
| 380 | C23 | — | 8 | 1 | no | — |
| 381 | C23 | — | 8 | 1 | no | — |
| 382 | C23 | — | 8 | 1 | no | — |
| 383 | C23 | — | 8 | 2 | no | — |
| 384 | C23 | — | 8 | 2 | no | — |
| 385 | C23 | — | 8 | 2 | no | — |
| 386 | C23 | — | 8 | 2 | no | — |
| 387 | C23 | — | 8 | 2 | no | — |
| 388 | C23 | — | 8 | 2 | no | — |
| 389 | C23 | — | 8 | 2 | no | — |
| 390 | C23 | #509 | 1 | 2 | yes | — |
| 391 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 392 | Q8 | — | 4 | 2 | no | — |
| 393 | Q8 | — | 4 | 2 | no | — |
| 394 | Q8 | — | 8 | 1 | no | — |
| 395 | Q8 | — | 8 | 1 | no | — |
| 396 | D4 | — | 4 | 2 | no | — |
| 397 | D4 | — | 4 | 2 | no | — |
| 398 | D4 | — | 8 | 8 | no | — |
| 399 | D4 | — | 8 | 2 | no | — |
| 400 | D4 | — | 8 | 1 | no | — |
| 401 | D4 | — | 8 | 2 | no | — |
| 402 | D4 | — | 8 | 1 | no | — |
| 403 | D4 | — | 8 | 2 | no | — |
| 404 | D4 | — | 8 | 2 | no | — |
| 405 | D4 | — | 16 | 4 | no | — |
| 406 | D4 | — | 16 | 8 | no | — |
| 407 | D4 | — | 16 | 4 | no | — |
| 408 | C2×C4 | — | 2 | 1 | no | — |
| 409 | C2×C4 | — | 2 | 1 | no | — |
| 410 | C2×C4 | — | 2 | 1 | no | — |
| 411 | C2×C4 | — | 2 | 1 | no | — |
| 412 | C2×C4 | — | 2 | 1 | no | — |
| 413 | C2×C4 | — | 2 | 1 | no | — |
| 414 | C2×C4 | — | 2 | 1 | no | — |
| 415 | C2×C4 | — | 2 | 1 | no | — |
| 416 | C2×C4 | — | 2 | 1 | no | — |
| 417 | C2×C4 | — | 2 | 1 | no | — |
| 418 | C2×C4 | — | 2 | 1 | no | — |
| 419 | C2×C4 | — | 4 | 4 | no | — |
| 420 | C2×C4 | — | 4 | 1 | no | — |
| 421 | C2×C4 | — | 4 | 4 | no | — |
| 422 | C2×C4 | — | 4 | 1 | no | — |
| 423 | C2×C4 | — | 4 | 1 | no | — |
| 424 | C2×C4 | — | 4 | 1 | no | — |
| 425 | C2×C4 | — | 4 | 2 | no | — |
| 426 | C2×C4 | — | 4 | 2 | no | — |
| 427 | C2×C4 | — | 4 | 2 | no | — |
| 428 | C2×C4 | — | 8 | 1 | no | — |
| 429 | C2×C4 | — | 8 | 1 | no | — |
| 430 | C2×C4 | — | 8 | 1 | no | — |
| 431 | C2×C4 | — | 8 | 1 | no | — |
| 432 | C2×C4 | — | 8 | 1 | no | — |
| 433 | C2×C4 | — | 8 | 1 | no | — |
| 434 | C2×C4 | — | 8 | 8 | no | — |
| 435 | C2×C4 | — | 8 | 2 | no | — |
| 436 | C2×C4 | — | 8 | 2 | no | — |
| 437 | C2×C4 | — | 8 | 2 | no | — |
| 438 | C2×C4 | #447 | 1 | 2 | yes | — |
| 439 | C2×C4 | #447 | 1 | 2 | yes | — |
| 440 | C2×C4 | #508 | 1 | 2 | yes | — |
| 441 | C2×C4 | #509 | 1 | 2 | yes | — |
| 442 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 443 | C2×C4 | #535 | 1 | 1 | yes | characteristic |
| 444 | C2×C4 | #537 | 1 | 1 | yes | characteristic |
| 445 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 446 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 447 | C2×C4 | #575 | 1 | 1 | yes | characteristic |
| 448 | C8 | — | 2 | 2 | no | — |
| 449 | C8 | — | 2 | 2 | no | — |
| 450 | C8 | — | 4 | 1 | no | — |
| 451 | C8 | #535 | 1 | 1 | yes | characteristic |
| 452 | C8 | #535 | 1 | 1 | yes | characteristic |
| 453 | C8 | #537 | 1 | 1 | yes | characteristic |
| 454 | C8 | #537 | 1 | 1 | yes | characteristic |
| 455 | C22 | — | 2 | 2 | no | — |
| 456 | C22 | — | 2 | 2 | no | — |
| 457 | C22 | — | 2 | 2 | no | — |
| 458 | C22 | — | 4 | 1 | no | — |
| 459 | C22 | — | 4 | 1 | no | — |
| 460 | C22 | — | 4 | 1 | no | — |
| 461 | C22 | — | 4 | 1 | no | — |
| 462 | C22 | — | 8 | 2 | no | — |
| 463 | C22 | — | 8 | 2 | no | — |
| 464 | C22 | — | 8 | 2 | no | — |
| 465 | C22 | — | 16 | 1 | no | — |
| 466 | C22 | — | 16 | 1 | no | — |
| 467 | C22 | — | 16 | 1 | no | — |
| 468 | C22 | — | 16 | 1 | no | — |
| 469 | C22 | — | 16 | 2 | no | — |
| 470 | C22 | — | 16 | 2 | no | — |
| 471 | C22 | — | 16 | 2 | no | — |
| 472 | C22 | — | 16 | 2 | no | — |
| 473 | C22 | — | 16 | 2 | no | — |
| 474 | C22 | — | 16 | 2 | no | — |
| 475 | C22 | — | 16 | 2 | no | — |
| 476 | C22 | C2×#447 | 1 | 1 | yes | characteristic |
| 477 | C22 | #3205 | 1 | 1 | yes | characteristic |
| 478 | C22 | #3245 | 1 | 1 | yes | characteristic |
| 479 | C22 | C2×#508 | 1 | 1 | yes | characteristic |
| 480 | C22 | C2×#509 | 1 | 1 | yes | characteristic |
| 481 | C22 | C2×#509 | 1 | 1 | yes | characteristic |
| 482 | C22 | #3414 | 1 | 1 | yes | characteristic |
| 483 | C4 | — | 2 | 2 | no | — |
| 484 | C4 | — | 4 | 1 | no | — |
| 485 | C4 | — | 4 | 1 | no | — |
| 486 | C4 | — | 4 | 1 | no | — |
| 487 | C4 | — | 8 | 2 | no | — |
| 488 | C4 | — | 8 | 2 | no | — |
| 489 | C4 | — | 8 | 2 | no | — |
| 490 | C4 | — | 16 | 2 | no | — |
| 491 | C4 | #2186 | 1 | 4 | yes | — |
| 492 | C4 | #2615 | 1 | 1 | yes | characteristic |
| 493 | C4 | #3006 | 1 | 1 | yes | characteristic |
| 494 | C4 | #3214 | 1 | 1 | yes | characteristic |
| 495 | C4 | #3253 | 1 | 1 | yes | characteristic |
| 496 | C2 | — | 4 | 2 | no | — |
| 497 | C2 | — | 16 | 2 | no | — |
| 498 | C2 | — | 32 | 2 | no | — |
| 499 | C2 | #29660 | 1 | 1 | yes | characteristic |
| 500 | C2 | C2×#2186 | 1 | 1 | yes | characteristic |
| 501 | C2 | #33520 | 1 | 1 | yes | characteristic |
| 502 | C2 | #33778 | 1 | 1 | yes | characteristic |
| 503 | C2 | #33783 | 1 | 1 | yes | characteristic |
| 504 | C2 | #33813 | 1 | 1 | yes | characteristic |
| 505 | C2 | #33817 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#3168 | C2 | semidirect |
| 2. C2×#2207 | C2 | semidirect |
| 3. C2×#2206 | C2 | semidirect |
| 4. #33171 | C2 | semidirect |
| 5. #13365 | C2 | semidirect |
| 6. #13326 | C2 | semidirect |
| 7. #13305 | C2 | semidirect |
| 8. #13304 | C2 | semidirect |
| 9. #12812 | C2 | semidirect |
| 10. #12811 | C2 | semidirect |
| 11. #12783 | C2 | semidirect |
| 95. C4×Q16 | D4 | semidirect |
| 98. C4×D8 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3168 | 2 |
| C2×#2207 | 2 |
| C2×#2206 | 2 |
| #33171 | 2 |
| #13365 | 2 |
| #13326 | 2 |
| #13305 | 2 |
| #13304 | 2 |
| #12812 | 2 |
| #12811 | 2 |
| #12783 | 2 |