Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
677 conjugacy classes of subgroups.
The subgroup lattice has 398 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #62360 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #33812 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #33736 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #19261 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #12406 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #12382 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #12354 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #12330 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #11983 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #11672 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | C2×Q8⋊C16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #3502 | — | 2 | 1 | no | — |
| 13 | #3395 | — | 2 | 1 | no | — |
| 14 | #3395 | C22 | 1 | 2 | yes | — |
| 15 | Q8×Q16 | — | 2 | 1 | no | — |
| 16 | Q8×Q16 | C22 | 1 | 2 | yes | — |
| 17 | #3363 | — | 2 | 2 | no | — |
| 18 | #3363 | — | 2 | 2 | no | — |
| 19 | #3249 | — | 2 | 2 | no | — |
| 20 | #3249 | — | 2 | 1 | no | — |
| 21 | #3170 | — | 2 | 2 | no | — |
| 22 | #3170 | C22 | 1 | 1 | yes | characteristic |
| 23 | #3170 | C22 | 1 | 1 | yes | characteristic |
| 24 | #3169 | C22 | 1 | 1 | yes | characteristic |
| 25 | #3037 | — | 2 | 1 | no | — |
| 26 | C2×C8×Q8 | C22 | 1 | 1 | yes | characteristic |
| 27 | #2254 | C22 | 1 | 1 | yes | characteristic |
| 28 | #2248 | C22 | 1 | 1 | yes | characteristic |
| 29 | #2241 | C22 | 1 | 1 | yes | characteristic |
| 30 | #2240 | C22 | 1 | 1 | yes | characteristic |
| 31 | #2226 | — | 2 | 1 | no | — |
| 32 | #2225 | C22 | 1 | 2 | yes | — |
| 33 | #2217 | — | 2 | 2 | no | — |
| 34 | #2217 | — | 2 | 2 | no | — |
| 35 | #2208 | — | 2 | 1 | no | — |
| 36 | #2207 | C22 | 1 | 2 | yes | — |
| 37 | C4×SD32 | — | 2 | 2 | no | — |
| 38 | C2×C4⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 39 | C2×C4×C16 | C22 | 1 | 1 | yes | characteristic |
| 40 | #1682 | — | 2 | 2 | no | — |
| 41 | #1674 | C22 | 1 | 2 | yes | — |
| 42 | #1668 | C22 | 1 | 2 | yes | — |
| 43 | #1567 | C22 | 1 | 1 | yes | characteristic |
| 44 | #1547 | — | 2 | 1 | no | — |
| 45 | #1365 | C22 | 1 | 2 | yes | — |
| 46 | #1362 | C22 | 1 | 2 | yes | — |
| 47 | Q8⋊C16 | C22 | 1 | 2 | yes | — |
| 48 | Q8⋊C16 | C22 | 1 | 2 | yes | — |
| 49 | #1332 | C22 | 1 | 2 | yes | — |
| 50 | #1332 | C22 | 1 | 2 | yes | — |
| 51 | C2×#137 | — | 4 | 1 | no | — |
| 52 | Q82 | — | 2 | 2 | no | — |
| 53 | Q82 | — | 2 | 2 | no | — |
| 54 | #557 | — | 2 | 2 | no | — |
| 55 | #557 | — | 2 | 2 | no | — |
| 56 | #557 | — | 4 | 2 | no | — |
| 57 | #557 | — | 4 | 2 | no | — |
| 58 | #544 | — | 4 | 2 | no | — |
| 59 | #544 | — | 4 | 1 | no | — |
| 60 | #533 | — | 2 | 4 | no | — |
| 61 | #533 | — | 2 | 1 | no | — |
| 62 | #533 | — | 2 | 1 | no | — |
| 63 | #532 | — | 2 | 1 | no | — |
| 64 | #519 | — | 4 | 1 | no | — |
| 65 | C2×C4×Q8 | D4 | 1 | 1 | yes | characteristic |
| 66 | C2×SD32 | — | 2 | 2 | no | — |
| 67 | C2×SD32 | — | 2 | 2 | no | — |
| 68 | C22×C16 | — | 2 | 1 | no | — |
| 69 | C22×C16 | — | 2 | 1 | no | — |
| 70 | #500 | — | 2 | 2 | no | — |
| 71 | #500 | D4 | 1 | 1 | yes | characteristic |
| 72 | #500 | C23 | 1 | 1 | yes | characteristic |
| 73 | #500 | C23 | 1 | 1 | yes | characteristic |
| 74 | #494 | — | 2 | 2 | no | — |
| 75 | #494 | D4 | 1 | 1 | yes | characteristic |
| 76 | #494 | C23 | 1 | 1 | yes | characteristic |
| 77 | #493 | C23 | 1 | 1 | yes | characteristic |
| 78 | #487 | — | 2 | 4 | no | — |
| 79 | #487 | — | 2 | 1 | no | — |
| 80 | #487 | — | 2 | 1 | no | — |
| 81 | #486 | — | 2 | 1 | no | — |
| 82 | #477 | — | 2 | 2 | no | — |
| 83 | #477 | — | 2 | 4 | no | — |
| 84 | #477 | — | 2 | 4 | no | — |
| 85 | #477 | — | 2 | 2 | no | — |
| 86 | #467 | — | 2 | 4 | no | — |
| 87 | #467 | — | 2 | 1 | no | — |
| 88 | #467 | — | 2 | 1 | no | — |
| 89 | #467 | D4 | 1 | 2 | yes | — |
| 90 | #467 | D4 | 1 | 2 | yes | — |
| 91 | #466 | — | 2 | 1 | no | — |
| 92 | #466 | — | 2 | 1 | no | — |
| 93 | #464 | — | 4 | 2 | no | — |
| 94 | #462 | — | 2 | 2 | no | — |
| 95 | #462 | — | 2 | 4 | no | — |
| 96 | #462 | — | 2 | 4 | no | — |
| 97 | #462 | — | 2 | 2 | no | — |
| 98 | #462 | — | 2 | 2 | no | — |
| 99 | #460 | — | 2 | 2 | no | — |
| 100 | C8×Q8 | — | 2 | 1 | no | — |
| 101 | C8×Q8 | — | 2 | 1 | no | — |
| 102 | C8×Q8 | D4 | 1 | 2 | yes | — |
| 103 | C8×Q8 | C23 | 1 | 2 | yes | — |
| 104 | C4×Q16 | — | 2 | 4 | no | — |
| 105 | C4×Q16 | — | 2 | 2 | no | — |
| 106 | C4×Q16 | — | 2 | 2 | no | — |
| 107 | C4×Q16 | — | 2 | 1 | no | — |
| 108 | C4×Q16 | — | 2 | 1 | no | — |
| 109 | C4×Q16 | D4 | 1 | 2 | yes | — |
| 110 | C4×Q16 | C23 | 1 | 2 | yes | — |
| 111 | C4×D8 | — | 2 | 1 | no | — |
| 112 | C4×D8 | — | 2 | 1 | no | — |
| 113 | #423 | — | 2 | 1 | no | — |
| 114 | C2×C4⋊C8 | — | 2 | 1 | no | — |
| 115 | C2×C4⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 116 | #417 | — | 4 | 1 | no | — |
| 117 | C2×C4×C8 | — | 2 | 1 | no | — |
| 118 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 119 | #367 | — | 2 | 2 | no | — |
| 120 | #367 | — | 2 | 2 | no | — |
| 121 | C4⋊C16 | — | 2 | 1 | no | — |
| 122 | C4⋊C16 | C23 | 1 | 2 | yes | — |
| 123 | #358 | — | 2 | 2 | no | — |
| 124 | #358 | — | 2 | 2 | no | — |
| 125 | #357 | — | 2 | 2 | no | — |
| 126 | #357 | — | 2 | 2 | no | — |
| 127 | C4×C16 | — | 2 | 1 | no | — |
| 128 | C4×C16 | C23 | 1 | 1 | yes | characteristic |
| 129 | C4×C16 | C23 | 1 | 1 | yes | characteristic |
| 130 | #335 | — | 2 | 1 | no | — |
| 131 | #335 | C23 | 1 | 2 | yes | — |
| 132 | #332 | — | 2 | 2 | no | — |
| 133 | #331 | — | 2 | 2 | no | — |
| 134 | C2×#41 | — | 4 | 1 | no | — |
| 135 | C22×Q8 | D8 | 1 | 1 | yes | characteristic |
| 136 | #137 | — | 8 | 4 | no | — |
| 137 | C2×Q16 | — | 2 | 4 | no | — |
| 138 | C2×Q16 | — | 2 | 1 | no | — |
| 139 | C2×Q16 | — | 2 | 1 | no | — |
| 140 | C2×Q16 | — | 4 | 2 | no | — |
| 141 | C2×Q16 | C2×D4 | 1 | 2 | yes | — |
| 142 | C2×Q16 | C2×D4 | 1 | 2 | yes | — |
| 143 | C2×SD16 | — | 4 | 2 | no | — |
| 144 | C2×D8 | — | 2 | 1 | no | — |
| 145 | C2×D8 | — | 2 | 1 | no | — |
| 146 | C2×M16 | — | 4 | 1 | no | — |
| 147 | C22×C8 | — | 2 | 1 | no | — |
| 148 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 149 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 150 | C4⋊Q8 | — | 2 | 4 | no | — |
| 151 | C4⋊Q8 | — | 2 | 4 | no | — |
| 152 | C4⋊Q8 | — | 2 | 1 | no | — |
| 153 | C4⋊Q8 | — | 2 | 1 | no | — |
| 154 | C4⋊Q8 | — | 2 | 1 | no | — |
| 155 | C4⋊Q8 | — | 2 | 1 | no | — |
| 156 | C4⋊Q8 | — | 2 | 1 | no | — |
| 157 | C4⋊Q8 | — | 4 | 4 | no | — |
| 158 | C4⋊Q8 | — | 4 | 4 | no | — |
| 159 | C4⋊Q8 | — | 4 | 2 | no | — |
| 160 | C4⋊Q8 | — | 4 | 2 | no | — |
| 161 | #127 | — | 2 | 1 | no | — |
| 162 | #125 | — | 4 | 4 | no | — |
| 163 | #125 | — | 4 | 4 | no | — |
| 164 | #125 | — | 4 | 4 | no | — |
| 165 | #125 | — | 4 | 2 | no | — |
| 166 | #125 | — | 4 | 2 | no | — |
| 167 | #125 | — | 4 | 1 | no | — |
| 168 | #125 | — | 4 | 1 | no | — |
| 169 | #124 | — | 4 | 1 | no | — |
| 170 | C22⋊Q8 | — | 2 | 2 | no | — |
| 171 | C22⋊Q8 | — | 2 | 2 | no | — |
| 172 | C22⋊Q8 | — | 4 | 4 | no | — |
| 173 | C22⋊Q8 | — | 4 | 1 | no | — |
| 174 | C22⋊Q8 | — | 4 | 1 | no | — |
| 175 | #121 | — | 4 | 1 | no | — |
| 176 | #121 | — | 4 | 1 | no | — |
| 177 | C4×Q8 | — | 2 | 1 | no | — |
| 178 | C4×Q8 | — | 2 | 1 | no | — |
| 179 | C4×Q8 | — | 2 | 4 | no | — |
| 180 | C4×Q8 | — | 2 | 4 | no | — |
| 181 | C4×Q8 | — | 2 | 2 | no | — |
| 182 | C4×Q8 | — | 2 | 2 | no | — |
| 183 | C4×Q8 | — | 4 | 4 | no | — |
| 184 | C4×Q8 | — | 4 | 2 | no | — |
| 185 | C4×Q8 | — | 4 | 2 | no | — |
| 186 | C4×Q8 | — | 4 | 2 | no | — |
| 187 | C4×Q8 | — | 4 | 2 | no | — |
| 188 | C4×Q8 | — | 4 | 1 | no | — |
| 189 | C4×Q8 | — | 4 | 1 | no | — |
| 190 | C4×Q8 | D8 | 1 | 2 | yes | — |
| 191 | C4×Q8 | C2×D4 | 1 | 2 | yes | — |
| 192 | C4×D4 | — | 4 | 1 | no | — |
| 193 | C4×D4 | — | 4 | 1 | no | — |
| 194 | #117 | — | 4 | 4 | no | — |
| 195 | #117 | — | 4 | 1 | no | — |
| 196 | #117 | — | 4 | 1 | no | — |
| 197 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 198 | C2×C4⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 199 | C2×C42 | — | 2 | 1 | no | — |
| 200 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 201 | SD32 | — | 4 | 4 | no | — |
| 202 | SD32 | — | 8 | 2 | no | — |
| 203 | C2×C16 | — | 2 | 1 | no | — |
| 204 | C2×C16 | — | 2 | 1 | no | — |
| 205 | C2×C16 | — | 2 | 2 | no | — |
| 206 | C2×C16 | — | 4 | 2 | no | — |
| 207 | C2×C16 | — | 8 | 1 | no | — |
| 208 | #107 | — | 2 | 4 | no | — |
| 209 | #107 | — | 2 | 1 | no | — |
| 210 | #107 | — | 2 | 1 | no | — |
| 211 | #107 | — | 2 | 1 | no | — |
| 212 | #107 | — | 2 | 1 | no | — |
| 213 | #107 | C2×D4 | 1 | 2 | yes | — |
| 214 | #107 | C2×D4 | 1 | 2 | yes | — |
| 215 | C4⋊C8 | — | 2 | 2 | no | — |
| 216 | C4⋊C8 | — | 2 | 2 | no | — |
| 217 | C4⋊C8 | — | 2 | 1 | no | — |
| 218 | C4⋊C8 | — | 2 | 2 | no | — |
| 219 | C4⋊C8 | — | 4 | 1 | no | — |
| 220 | C4⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 221 | #103 | — | 2 | 4 | no | — |
| 222 | #103 | — | 2 | 4 | no | — |
| 223 | #103 | — | 2 | 2 | no | — |
| 224 | #103 | — | 2 | 2 | no | — |
| 225 | #103 | — | 2 | 2 | no | — |
| 226 | #103 | — | 2 | 2 | no | — |
| 227 | #103 | — | 4 | 2 | no | — |
| 228 | #102 | — | 4 | 1 | no | — |
| 229 | #102 | — | 4 | 1 | no | — |
| 230 | #102 | — | 4 | 2 | no | — |
| 231 | C4×C8 | — | 2 | 2 | no | — |
| 232 | C4×C8 | — | 2 | 2 | no | — |
| 233 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 234 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 235 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 236 | C4×C8 | C24 | 1 | 1 | yes | characteristic |
| 237 | #41 | — | 8 | 2 | no | — |
| 238 | #41 | — | 8 | 2 | no | — |
| 239 | C2×Q8 | — | 2 | 1 | no | — |
| 240 | C2×Q8 | — | 2 | 2 | no | — |
| 241 | C2×Q8 | — | 2 | 2 | no | — |
| 242 | C2×Q8 | — | 2 | 2 | no | — |
| 243 | C2×Q8 | — | 4 | 4 | no | — |
| 244 | C2×Q8 | — | 4 | 1 | no | — |
| 245 | C2×Q8 | — | 4 | 1 | no | — |
| 246 | C2×Q8 | — | 4 | 1 | no | — |
| 247 | C2×Q8 | SD32 | 1 | 4 | yes | complemented |
| 248 | C2×Q8 | C2×D8 | 1 | 2 | yes | — |
| 249 | C2×D4 | — | 4 | 1 | no | — |
| 250 | C2×D4 | — | 4 | 1 | no | — |
| 251 | C2×D4 | — | 4 | 1 | no | — |
| 252 | C2×D4 | — | 8 | 1 | no | — |
| 253 | C22×C4 | — | 2 | 1 | no | — |
| 254 | C22×C4 | — | 2 | 1 | no | — |
| 255 | C22×C4 | — | 8 | 1 | no | — |
| 256 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 257 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 258 | C22×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 259 | Q16 | — | 2 | 2 | no | — |
| 260 | Q16 | — | 2 | 2 | no | — |
| 261 | Q16 | — | 4 | 2 | no | — |
| 262 | Q16 | — | 4 | 1 | no | — |
| 263 | Q16 | — | 4 | 1 | no | — |
| 264 | Q16 | — | 8 | 1 | no | — |
| 265 | Q16 | — | 8 | 4 | no | — |
| 266 | Q16 | — | 8 | 4 | no | — |
| 267 | SD16 | — | 8 | 4 | no | — |
| 268 | D8 | — | 4 | 2 | no | — |
| 269 | D8 | — | 4 | 1 | no | — |
| 270 | D8 | — | 4 | 1 | no | — |
| 271 | M16 | — | 8 | 2 | no | — |
| 272 | C2×C8 | — | 2 | 1 | no | — |
| 273 | C2×C8 | — | 2 | 1 | no | — |
| 274 | C2×C8 | — | 2 | 2 | no | — |
| 275 | C2×C8 | — | 4 | 1 | no | — |
| 276 | C2×C8 | — | 4 | 2 | no | — |
| 277 | C2×C8 | — | 8 | 1 | no | — |
| 278 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 279 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 280 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 281 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 282 | C4⋊C4 | — | 2 | 2 | no | — |
| 283 | C4⋊C4 | — | 2 | 2 | no | — |
| 284 | C4⋊C4 | — | 2 | 1 | no | — |
| 285 | C4⋊C4 | — | 2 | 2 | no | — |
| 286 | C4⋊C4 | — | 2 | 2 | no | — |
| 287 | C4⋊C4 | — | 4 | 4 | no | — |
| 288 | C4⋊C4 | — | 4 | 4 | no | — |
| 289 | C4⋊C4 | — | 4 | 4 | no | — |
| 290 | C4⋊C4 | — | 4 | 4 | no | — |
| 291 | C4⋊C4 | — | 4 | 4 | no | — |
| 292 | C4⋊C4 | — | 4 | 1 | no | — |
| 293 | C4⋊C4 | — | 4 | 1 | no | — |
| 294 | C4⋊C4 | — | 4 | 2 | no | — |
| 295 | C4⋊C4 | — | 4 | 1 | no | — |
| 296 | C4⋊C4 | — | 4 | 1 | no | — |
| 297 | C4⋊C4 | — | 4 | 1 | no | — |
| 298 | C4⋊C4 | — | 4 | 2 | no | — |
| 299 | C4⋊C4 | — | 4 | 2 | no | — |
| 300 | C4⋊C4 | — | 4 | 2 | no | — |
| 301 | C4⋊C4 | — | 4 | 1 | no | — |
| 302 | C4⋊C4 | — | 4 | 1 | no | — |
| 303 | C4⋊C4 | — | 4 | 1 | no | — |
| 304 | C4⋊C4 | C2×D8 | 1 | 2 | yes | — |
| 305 | C22⋊C4 | — | 4 | 2 | no | — |
| 306 | C22⋊C4 | — | 4 | 2 | no | — |
| 307 | C22⋊C4 | — | 8 | 1 | no | — |
| 308 | C22⋊C4 | — | 8 | 1 | no | — |
| 309 | C42 | — | 2 | 2 | no | — |
| 310 | C42 | — | 2 | 2 | no | — |
| 311 | C42 | — | 4 | 4 | no | — |
| 312 | C42 | — | 4 | 4 | no | — |
| 313 | C42 | — | 4 | 2 | no | — |
| 314 | C42 | — | 4 | 2 | no | — |
| 315 | C42 | #120 | 1 | 1 | yes | characteristic |
| 316 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 317 | C42 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 318 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 319 | C16 | — | 2 | 2 | no | — |
| 320 | C16 | — | 4 | 1 | no | — |
| 321 | C16 | — | 4 | 2 | no | — |
| 322 | C23 | — | 8 | 1 | no | — |
| 323 | C23 | #447 | 1 | 1 | yes | characteristic |
| 324 | Q8 | — | 2 | 2 | no | — |
| 325 | Q8 | — | 2 | 4 | no | — |
| 326 | Q8 | — | 4 | 2 | no | — |
| 327 | Q8 | — | 4 | 2 | no | — |
| 328 | Q8 | — | 4 | 2 | no | — |
| 329 | Q8 | — | 4 | 2 | no | — |
| 330 | Q8 | — | 8 | 1 | no | — |
| 331 | Q8 | — | 8 | 1 | no | — |
| 332 | Q8 | — | 16 | 1 | no | — |
| 333 | Q8 | — | 16 | 1 | no | — |
| 334 | Q8 | — | 16 | 4 | no | — |
| 335 | Q8 | C2×SD32 | 1 | 4 | yes | — |
| 336 | D4 | — | 8 | 1 | no | — |
| 337 | D4 | — | 8 | 1 | no | — |
| 338 | D4 | — | 8 | 1 | no | — |
| 339 | D4 | — | 8 | 1 | no | — |
| 340 | D4 | — | 16 | 1 | no | — |
| 341 | D4 | — | 16 | 2 | no | — |
| 342 | C2×C4 | — | 2 | 4 | no | — |
| 343 | C2×C4 | — | 2 | 2 | no | — |
| 344 | C2×C4 | — | 2 | 1 | no | — |
| 345 | C2×C4 | — | 2 | 1 | no | — |
| 346 | C2×C4 | — | 2 | 2 | no | — |
| 347 | C2×C4 | — | 4 | 1 | no | — |
| 348 | C2×C4 | — | 4 | 2 | no | — |
| 349 | C2×C4 | — | 4 | 2 | no | — |
| 350 | C2×C4 | — | 4 | 2 | no | — |
| 351 | C2×C4 | — | 4 | 2 | no | — |
| 352 | C2×C4 | — | 8 | 1 | no | — |
| 353 | C2×C4 | — | 8 | 1 | no | — |
| 354 | C2×C4 | — | 16 | 2 | no | — |
| 355 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 356 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 357 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 358 | C2×C4 | C2×SD32 | 1 | 2 | yes | — |
| 359 | C2×C4 | #510 | 1 | 2 | yes | — |
| 360 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 361 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 362 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 363 | C8 | — | 2 | 1 | no | — |
| 364 | C8 | — | 2 | 1 | no | — |
| 365 | C8 | — | 2 | 1 | no | — |
| 366 | C8 | — | 4 | 2 | no | — |
| 367 | C8 | — | 8 | 2 | no | — |
| 368 | C8 | #577 | 1 | 1 | yes | characteristic |
| 369 | C8 | #578 | 1 | 1 | yes | characteristic |
| 370 | C22 | — | 2 | 1 | no | — |
| 371 | C22 | — | 2 | 1 | no | — |
| 372 | C22 | — | 16 | 1 | no | — |
| 373 | C22 | — | 16 | 1 | no | — |
| 374 | C22 | — | 16 | 1 | no | — |
| 375 | C22 | #2185 | 1 | 2 | yes | — |
| 376 | C22 | C2×#447 | 1 | 1 | yes | characteristic |
| 377 | C4 | — | 2 | 1 | no | — |
| 378 | C4 | — | 2 | 1 | no | — |
| 379 | C4 | — | 2 | 4 | no | — |
| 380 | C4 | — | 2 | 1 | no | — |
| 381 | C4 | — | 4 | 2 | no | — |
| 382 | C4 | — | 8 | 2 | no | — |
| 383 | C4 | — | 8 | 2 | no | — |
| 384 | C4 | — | 8 | 2 | no | — |
| 385 | C4 | — | 8 | 2 | no | — |
| 386 | C4 | — | 16 | 1 | no | — |
| 387 | C4 | #2185 | 1 | 2 | yes | — |
| 388 | C4 | #3013 | 1 | 1 | yes | characteristic |
| 389 | C4 | #3018 | 1 | 1 | yes | characteristic |
| 390 | C4 | C22×SD32 | 1 | 1 | yes | characteristic |
| 391 | C4 | C2×#510 | 1 | 1 | yes | characteristic |
| 392 | C2 | — | 2 | 2 | no | — |
| 393 | C2 | — | 32 | 1 | no | — |
| 394 | C2 | #18674 | 1 | 1 | yes | characteristic |
| 395 | C2 | C2×#2185 | 1 | 1 | yes | characteristic |
| 396 | C2 | #33538 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #62360 | C2 | semidirect |
| 5. #12406 | C2 | semidirect |
| 6. #12382 | C2 | semidirect |
| 7. #12354 | C2 | semidirect |
| 8. #12330 | C2 | semidirect |
| 9. #11983 | C2 | semidirect |
| 10. #11672 | C2 | semidirect |
| 11. C2×Q8⋊C16 | C2 | semidirect |
| 247. C2×Q8 | SD32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62360 | 2 |
| #33812 | 2 |
| #33736 | 2 |
| #19261 | 2 |
| #12406 | 2 |
| #12382 | 2 |
| #12354 | 2 |
| #12330 | 2 |
| #11983 | 2 |
| #11672 | 2 |
| C2×Q8⋊C16 | 2 |