Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1017 conjugacy classes of subgroups.
The subgroup lattice has 461 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#3293 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#2214 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #33527 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C2×#2179 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #28245 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #13394 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #13308 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #13207 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #13196 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #12820 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #12749 | C2 | 1 | 2 | yes | maximal, complemented |
| 12 | C2×#578 | — | 2 | 1 | no | — |
| 13 | C2×#578 | C22 | 1 | 2 | yes | — |
| 14 | C2×#510 | C22 | 1 | 2 | yes | — |
| 15 | C2×#508 | C22 | 1 | 1 | yes | characteristic |
| 16 | #3293 | — | 2 | 4 | no | — |
| 17 | #3293 | — | 2 | 4 | no | — |
| 18 | C2×#494 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#473 | — | 2 | 1 | no | — |
| 20 | C2×#473 | C22 | 1 | 2 | yes | — |
| 21 | #3061 | C22 | 1 | 2 | yes | — |
| 22 | #3060 | — | 2 | 1 | no | — |
| 23 | C2×#456 | — | 2 | 2 | no | — |
| 24 | C2×#443 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×#443 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×#443 | C22 | 1 | 1 | yes | characteristic |
| 27 | #2214 | — | 2 | 4 | no | — |
| 28 | #2189 | — | 2 | 2 | no | — |
| 29 | #2188 | — | 2 | 2 | no | — |
| 30 | #2186 | — | 2 | 2 | no | — |
| 31 | #2185 | — | 2 | 2 | no | — |
| 32 | #2179 | — | 2 | 4 | no | — |
| 33 | C2×#368 | C22 | 1 | 1 | yes | characteristic |
| 34 | C2×#364 | C22 | 1 | 1 | yes | characteristic |
| 35 | C2×#362 | C22 | 1 | 2 | yes | — |
| 36 | C2×#359 | C22 | 1 | 1 | yes | characteristic |
| 37 | C2×#359 | C22 | 1 | 1 | yes | characteristic |
| 38 | #2139 | C22 | 1 | 2 | yes | — |
| 39 | C2×#358 | C22 | 1 | 2 | yes | — |
| 40 | C2×#358 | C22 | 1 | 2 | yes | — |
| 41 | C2×#350 | C22 | 1 | 2 | yes | — |
| 42 | #2113 | C22 | 1 | 2 | yes | — |
| 43 | C2×#346 | C22 | 1 | 1 | yes | characteristic |
| 44 | #2040 | — | 2 | 2 | no | — |
| 45 | #1846 | C22 | 1 | 1 | yes | characteristic |
| 46 | #1816 | C22 | 1 | 2 | yes | — |
| 47 | #1815 | — | 2 | 1 | no | — |
| 48 | #1387 | C22 | 1 | 2 | yes | — |
| 49 | C2×2−1+4 | — | 2 | 2 | no | — |
| 50 | #578 | — | 2 | 8 | no | — |
| 51 | #578 | — | 2 | 4 | no | — |
| 52 | #578 | — | 2 | 4 | no | — |
| 53 | #578 | — | 2 | 4 | no | — |
| 54 | #578 | — | 4 | 2 | no | — |
| 55 | #578 | — | 4 | 4 | no | — |
| 56 | C2×#137 | — | 2 | 2 | no | — |
| 57 | C2×#137 | — | 2 | 2 | no | — |
| 58 | C2×#137 | — | 4 | 2 | no | — |
| 59 | C2×#137 | — | 4 | 2 | no | — |
| 60 | C2×#135 | — | 2 | 2 | no | — |
| 61 | C2×#135 | — | 4 | 1 | no | — |
| 62 | C2×#135 | D4 | 1 | 1 | yes | characteristic |
| 63 | C2×#135 | C23 | 1 | 1 | yes | characteristic |
| 64 | C2×#135 | C23 | 1 | 1 | yes | characteristic |
| 65 | C22×Q16 | — | 2 | 2 | no | — |
| 66 | C22×Q16 | — | 2 | 1 | no | — |
| 67 | C22×Q16 | — | 4 | 1 | no | — |
| 68 | C22×Q16 | C23 | 1 | 2 | yes | — |
| 69 | C2×#131 | — | 2 | 1 | no | — |
| 70 | C2×#131 | — | 2 | 1 | no | — |
| 71 | C2×#131 | C23 | 1 | 2 | yes | — |
| 72 | #537 | — | 2 | 2 | no | — |
| 73 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 74 | #510 | — | 2 | 4 | no | — |
| 75 | #510 | — | 4 | 2 | no | — |
| 76 | #508 | — | 2 | 2 | no | — |
| 77 | #508 | — | 2 | 2 | no | — |
| 78 | C2×Q32 | — | 2 | 2 | no | — |
| 79 | C2×Q32 | D4 | 1 | 1 | yes | characteristic |
| 80 | C2×SD32 | — | 2 | 2 | no | — |
| 81 | C2×SD32 | D4 | 1 | 1 | yes | characteristic |
| 82 | C2×SD32 | D4 | 1 | 1 | yes | characteristic |
| 83 | C2×D16 | D4 | 1 | 1 | yes | characteristic |
| 84 | C2×M32 | C23 | 1 | 2 | yes | — |
| 85 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 86 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 87 | #494 | — | 2 | 2 | no | — |
| 88 | #494 | D4 | 1 | 4 | yes | complemented |
| 89 | #473 | — | 2 | 8 | no | — |
| 90 | #473 | — | 2 | 4 | no | — |
| 91 | #473 | — | 2 | 4 | no | — |
| 92 | #470 | — | 4 | 1 | no | — |
| 93 | #469 | — | 4 | 1 | no | — |
| 94 | #468 | — | 4 | 2 | no | — |
| 95 | #467 | — | 2 | 2 | no | — |
| 96 | #466 | — | 2 | 2 | no | — |
| 97 | #465 | — | 4 | 2 | no | — |
| 98 | #456 | — | 2 | 16 | no | — |
| 99 | #443 | — | 2 | 2 | no | — |
| 100 | #443 | — | 2 | 2 | no | — |
| 101 | #443 | — | 2 | 2 | no | — |
| 102 | #443 | — | 2 | 2 | no | — |
| 103 | #443 | — | 2 | 2 | no | — |
| 104 | #443 | D4 | 1 | 4 | yes | complemented |
| 105 | C2×#108 | D4 | 1 | 1 | yes | characteristic |
| 106 | C2×#108 | C23 | 1 | 1 | yes | characteristic |
| 107 | C2×#108 | C23 | 1 | 1 | yes | characteristic |
| 108 | C2×#107 | C23 | 1 | 2 | yes | — |
| 109 | C2×#106 | — | 2 | 1 | no | — |
| 110 | C2×#104 | — | 2 | 1 | no | — |
| 111 | C2×#104 | — | 2 | 1 | no | — |
| 112 | C2×#104 | D4 | 1 | 2 | yes | — |
| 113 | #417 | — | 2 | 2 | no | — |
| 114 | #417 | — | 2 | 2 | no | — |
| 115 | C2×#101 | — | 2 | 2 | no | — |
| 116 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 117 | #382 | — | 2 | 1 | no | — |
| 118 | #368 | — | 2 | 2 | no | — |
| 119 | #364 | — | 2 | 2 | no | — |
| 120 | #362 | — | 2 | 4 | no | — |
| 121 | #359 | — | 2 | 2 | no | — |
| 122 | #359 | — | 2 | 2 | no | — |
| 123 | #358 | — | 2 | 2 | no | — |
| 124 | #358 | — | 2 | 2 | no | — |
| 125 | #358 | — | 2 | 2 | no | — |
| 126 | #358 | — | 4 | 2 | no | — |
| 127 | #357 | — | 2 | 2 | no | — |
| 128 | #350 | — | 4 | 2 | no | — |
| 129 | C22⋊C16 | — | 2 | 2 | no | — |
| 130 | C22⋊C16 | — | 2 | 2 | no | — |
| 131 | #346 | — | 2 | 2 | no | — |
| 132 | #343 | — | 2 | 2 | no | — |
| 133 | 2−1+4 | — | 2 | 8 | no | — |
| 134 | 2−1+4 | — | 4 | 4 | no | — |
| 135 | 2−1+4 | — | 4 | 4 | no | — |
| 136 | 2−1+4 | — | 8 | 2 | no | — |
| 137 | C2×#41 | — | 2 | 1 | no | — |
| 138 | C2×#41 | — | 2 | 1 | no | — |
| 139 | C2×#41 | — | 4 | 2 | no | — |
| 140 | C2×#41 | — | 4 | 2 | no | — |
| 141 | C2×#41 | — | 8 | 2 | no | — |
| 142 | C2×#41 | C2×D4 | 1 | 2 | yes | — |
| 143 | C22×Q8 | — | 2 | 2 | no | — |
| 144 | C22×Q8 | — | 4 | 2 | no | — |
| 145 | C22×Q8 | — | 4 | 2 | no | — |
| 146 | #137 | — | 2 | 8 | no | — |
| 147 | #137 | — | 2 | 8 | no | — |
| 148 | #137 | — | 4 | 4 | no | — |
| 149 | #137 | — | 4 | 8 | no | — |
| 150 | #137 | — | 4 | 8 | no | — |
| 151 | #137 | — | 4 | 16 | no | — |
| 152 | #137 | — | 4 | 4 | no | — |
| 153 | #135 | — | 2 | 2 | no | — |
| 154 | #135 | — | 2 | 2 | no | — |
| 155 | #135 | — | 2 | 1 | no | — |
| 156 | #135 | — | 2 | 1 | no | — |
| 157 | #135 | — | 2 | 1 | no | — |
| 158 | #135 | — | 2 | 1 | no | — |
| 159 | #135 | — | 2 | 1 | no | — |
| 160 | #135 | — | 2 | 1 | no | — |
| 161 | #135 | — | 4 | 4 | no | — |
| 162 | #135 | — | 4 | 8 | no | — |
| 163 | #135 | — | 4 | 4 | no | — |
| 164 | #135 | C2×D4 | 1 | 2 | yes | — |
| 165 | #135 | #41 | 1 | 2 | yes | — |
| 166 | C2×Q16 | — | 2 | 4 | no | — |
| 167 | C2×Q16 | — | 2 | 2 | no | — |
| 168 | C2×Q16 | — | 2 | 2 | no | — |
| 169 | C2×Q16 | — | 2 | 2 | no | — |
| 170 | C2×Q16 | — | 2 | 2 | no | — |
| 171 | C2×Q16 | — | 2 | 2 | no | — |
| 172 | C2×Q16 | — | 4 | 2 | no | — |
| 173 | C2×Q16 | — | 4 | 4 | no | — |
| 174 | C2×Q16 | — | 4 | 8 | no | — |
| 175 | C2×Q16 | — | 4 | 2 | no | — |
| 176 | C2×Q16 | — | 4 | 4 | no | — |
| 177 | C2×Q16 | — | 4 | 1 | no | — |
| 178 | C2×Q16 | — | 4 | 1 | no | — |
| 179 | C2×Q16 | — | 4 | 2 | no | — |
| 180 | C2×Q16 | D8 | 1 | 1 | yes | characteristic |
| 181 | C2×Q16 | C2×D4 | 1 | 4 | yes | — |
| 182 | C2×Q16 | C2×D4 | 1 | 1 | yes | characteristic |
| 183 | C2×Q16 | C2×D4 | 1 | 1 | yes | characteristic |
| 184 | C2×SD16 | — | 2 | 1 | no | — |
| 185 | C2×SD16 | — | 2 | 1 | no | — |
| 186 | C2×SD16 | — | 4 | 2 | no | — |
| 187 | C2×SD16 | — | 4 | 2 | no | — |
| 188 | C2×SD16 | D8 | 1 | 2 | yes | — |
| 189 | C2×D8 | D8 | 1 | 1 | yes | characteristic |
| 190 | C2×D8 | C2×D4 | 1 | 1 | yes | characteristic |
| 191 | C2×D8 | C2×D4 | 1 | 1 | yes | characteristic |
| 192 | #131 | — | 2 | 4 | no | — |
| 193 | #131 | — | 2 | 2 | no | — |
| 194 | #131 | — | 2 | 2 | no | — |
| 195 | #131 | — | 2 | 2 | no | — |
| 196 | #131 | — | 2 | 2 | no | — |
| 197 | #131 | — | 4 | 2 | no | — |
| 198 | #131 | — | 4 | 2 | no | — |
| 199 | #131 | — | 4 | 2 | no | — |
| 200 | C2×M16 | — | 2 | 2 | no | — |
| 201 | C2×M16 | — | 2 | 1 | no | — |
| 202 | C2×M16 | — | 2 | 1 | no | — |
| 203 | C2×M16 | — | 4 | 1 | no | — |
| 204 | C2×M16 | — | 4 | 1 | no | — |
| 205 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 206 | C22×C8 | — | 2 | 2 | no | — |
| 207 | C22×C8 | — | 4 | 1 | no | — |
| 208 | C22×C8 | — | 4 | 1 | no | — |
| 209 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 210 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 211 | C4⋊Q8 | — | 2 | 4 | no | — |
| 212 | C4⋊Q8 | — | 8 | 1 | no | — |
| 213 | C22⋊Q8 | — | 4 | 2 | no | — |
| 214 | C22⋊Q8 | — | 4 | 2 | no | — |
| 215 | #121 | — | 4 | 2 | no | — |
| 216 | #121 | — | 4 | 2 | no | — |
| 217 | #121 | — | 8 | 2 | no | — |
| 218 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 219 | C2×C4⋊C4 | — | 8 | 1 | no | — |
| 220 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 221 | Q32 | — | 2 | 2 | no | — |
| 222 | Q32 | — | 4 | 4 | no | — |
| 223 | SD32 | — | 2 | 2 | no | — |
| 224 | SD32 | — | 2 | 2 | no | — |
| 225 | SD32 | — | 4 | 4 | no | — |
| 226 | D16 | — | 2 | 2 | no | — |
| 227 | M32 | — | 2 | 2 | no | — |
| 228 | M32 | — | 2 | 2 | no | — |
| 229 | C2×C16 | — | 2 | 1 | no | — |
| 230 | C2×C16 | — | 2 | 1 | no | — |
| 231 | C2×C16 | — | 2 | 1 | no | — |
| 232 | C2×C16 | — | 2 | 2 | no | — |
| 233 | C2×C16 | — | 2 | 1 | no | — |
| 234 | C2×C16 | — | 2 | 1 | no | — |
| 235 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 236 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 237 | #108 | — | 2 | 2 | no | — |
| 238 | #108 | — | 2 | 1 | no | — |
| 239 | #108 | — | 2 | 1 | no | — |
| 240 | #108 | — | 2 | 1 | no | — |
| 241 | #108 | C2×D4 | 1 | 2 | yes | — |
| 242 | #107 | — | 2 | 2 | no | — |
| 243 | #107 | — | 2 | 2 | no | — |
| 244 | #106 | — | 4 | 1 | no | — |
| 245 | #106 | — | 4 | 1 | no | — |
| 246 | #104 | — | 2 | 4 | no | — |
| 247 | #104 | — | 2 | 4 | no | — |
| 248 | #104 | — | 4 | 2 | no | — |
| 249 | #103 | — | 4 | 2 | no | — |
| 250 | #103 | — | 4 | 2 | no | — |
| 251 | #102 | — | 4 | 2 | no | — |
| 252 | #102 | — | 4 | 2 | no | — |
| 253 | #101 | — | 2 | 8 | no | — |
| 254 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 255 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 256 | #95 | — | 8 | 1 | no | — |
| 257 | #41 | — | 2 | 2 | no | — |
| 258 | #41 | — | 2 | 2 | no | — |
| 259 | #41 | — | 2 | 2 | no | — |
| 260 | #41 | — | 2 | 2 | no | — |
| 261 | #41 | — | 2 | 2 | no | — |
| 262 | #41 | — | 2 | 2 | no | — |
| 263 | #41 | — | 4 | 1 | no | — |
| 264 | #41 | — | 4 | 1 | no | — |
| 265 | #41 | — | 4 | 8 | no | — |
| 266 | #41 | — | 4 | 8 | no | — |
| 267 | #41 | — | 4 | 1 | no | — |
| 268 | #41 | — | 4 | 1 | no | — |
| 269 | #41 | — | 4 | 1 | no | — |
| 270 | #41 | — | 4 | 1 | no | — |
| 271 | #41 | — | 8 | 4 | no | — |
| 272 | #41 | — | 8 | 8 | no | — |
| 273 | #41 | — | 8 | 4 | no | — |
| 274 | #41 | — | 8 | 8 | no | — |
| 275 | C2×Q8 | — | 2 | 4 | no | — |
| 276 | C2×Q8 | — | 2 | 1 | no | — |
| 277 | C2×Q8 | — | 2 | 1 | no | — |
| 278 | C2×Q8 | — | 4 | 2 | no | — |
| 279 | C2×Q8 | — | 4 | 8 | no | — |
| 280 | C2×Q8 | — | 4 | 8 | no | — |
| 281 | C2×Q8 | — | 4 | 2 | no | — |
| 282 | C2×Q8 | — | 4 | 2 | no | — |
| 283 | C2×Q8 | — | 4 | 2 | no | — |
| 284 | C2×Q8 | — | 4 | 2 | no | — |
| 285 | C2×Q8 | — | 8 | 2 | no | — |
| 286 | C2×Q8 | — | 8 | 4 | no | — |
| 287 | C2×Q8 | — | 8 | 4 | no | — |
| 288 | C2×Q8 | C2×D8 | 1 | 2 | yes | — |
| 289 | C2×D4 | — | 2 | 2 | no | — |
| 290 | C2×D4 | — | 2 | 1 | no | — |
| 291 | C2×D4 | — | 2 | 1 | no | — |
| 292 | C2×D4 | — | 4 | 1 | no | — |
| 293 | C2×D4 | — | 4 | 1 | no | — |
| 294 | C2×D4 | — | 8 | 2 | no | — |
| 295 | C2×D4 | — | 8 | 2 | no | — |
| 296 | C2×D4 | C2×D8 | 1 | 2 | yes | — |
| 297 | C22×C4 | — | 2 | 1 | no | — |
| 298 | C22×C4 | — | 2 | 2 | no | — |
| 299 | C22×C4 | — | 4 | 1 | no | — |
| 300 | C22×C4 | — | 4 | 2 | no | — |
| 301 | C22×C4 | — | 4 | 1 | no | — |
| 302 | C22×C4 | — | 4 | 2 | no | — |
| 303 | C22×C4 | — | 8 | 2 | no | — |
| 304 | C22×C4 | — | 8 | 2 | no | — |
| 305 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 306 | Q16 | — | 2 | 1 | no | — |
| 307 | Q16 | — | 2 | 1 | no | — |
| 308 | Q16 | — | 2 | 4 | no | — |
| 309 | Q16 | — | 2 | 1 | no | — |
| 310 | Q16 | — | 2 | 4 | no | — |
| 311 | Q16 | — | 2 | 2 | no | — |
| 312 | Q16 | — | 4 | 8 | no | — |
| 313 | Q16 | — | 4 | 8 | no | — |
| 314 | Q16 | — | 4 | 2 | no | — |
| 315 | Q16 | — | 4 | 2 | no | — |
| 316 | Q16 | — | 4 | 4 | no | — |
| 317 | Q16 | — | 4 | 4 | no | — |
| 318 | Q16 | #121 | 1 | 2 | yes | — |
| 319 | SD16 | — | 2 | 2 | no | — |
| 320 | SD16 | — | 2 | 2 | no | — |
| 321 | SD16 | — | 2 | 4 | no | — |
| 322 | SD16 | — | 4 | 8 | no | — |
| 323 | SD16 | — | 4 | 8 | no | — |
| 324 | SD16 | — | 4 | 1 | no | — |
| 325 | SD16 | — | 4 | 1 | no | — |
| 326 | D8 | — | 2 | 1 | no | — |
| 327 | D8 | — | 2 | 1 | no | — |
| 328 | D8 | — | 2 | 1 | no | — |
| 329 | D8 | — | 2 | 2 | no | — |
| 330 | D8 | #121 | 1 | 2 | yes | — |
| 331 | M16 | — | 2 | 2 | no | — |
| 332 | M16 | — | 2 | 2 | no | — |
| 333 | M16 | — | 2 | 2 | no | — |
| 334 | M16 | — | 4 | 1 | no | — |
| 335 | M16 | — | 4 | 2 | no | — |
| 336 | M16 | — | 4 | 2 | no | — |
| 337 | M16 | — | 4 | 4 | no | — |
| 338 | M16 | — | 4 | 1 | no | — |
| 339 | M16 | — | 4 | 1 | no | — |
| 340 | M16 | — | 8 | 2 | no | — |
| 341 | C2×C8 | — | 2 | 2 | no | — |
| 342 | C2×C8 | — | 4 | 1 | no | — |
| 343 | C2×C8 | — | 4 | 2 | no | — |
| 344 | C2×C8 | — | 4 | 2 | no | — |
| 345 | C2×C8 | — | 4 | 4 | no | — |
| 346 | C2×C8 | — | 4 | 1 | no | — |
| 347 | C2×C8 | — | 8 | 2 | no | — |
| 348 | C2×C8 | #120 | 1 | 2 | yes | — |
| 349 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 350 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 351 | C2×C8 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 352 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 353 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 354 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 355 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 356 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 357 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 358 | C4⋊C4 | — | 4 | 2 | no | — |
| 359 | C4⋊C4 | — | 4 | 2 | no | — |
| 360 | C4⋊C4 | — | 8 | 4 | no | — |
| 361 | C4⋊C4 | — | 8 | 2 | no | — |
| 362 | C22⋊C4 | — | 8 | 2 | no | — |
| 363 | C22⋊C4 | — | 8 | 2 | no | — |
| 364 | C42 | — | 2 | 1 | no | — |
| 365 | C42 | #120 | 1 | 2 | yes | — |
| 366 | C16 | — | 2 | 1 | no | — |
| 367 | C16 | — | 2 | 1 | no | — |
| 368 | C16 | — | 4 | 1 | no | — |
| 369 | C16 | — | 4 | 2 | no | — |
| 370 | C23 | — | 2 | 2 | no | — |
| 371 | C23 | — | 4 | 1 | no | — |
| 372 | C23 | — | 4 | 1 | no | — |
| 373 | C23 | #576 | 1 | 1 | yes | characteristic |
| 374 | Q8 | — | 2 | 2 | no | — |
| 375 | Q8 | — | 2 | 2 | no | — |
| 376 | Q8 | — | 2 | 2 | no | — |
| 377 | Q8 | — | 4 | 1 | no | — |
| 378 | Q8 | — | 4 | 4 | no | — |
| 379 | Q8 | — | 4 | 1 | no | — |
| 380 | Q8 | — | 4 | 4 | no | — |
| 381 | Q8 | — | 4 | 1 | no | — |
| 382 | Q8 | — | 8 | 8 | no | — |
| 383 | Q8 | — | 8 | 2 | no | — |
| 384 | Q8 | — | 8 | 2 | no | — |
| 385 | D4 | — | 2 | 2 | no | — |
| 386 | D4 | — | 2 | 2 | no | — |
| 387 | D4 | — | 2 | 2 | no | — |
| 388 | D4 | — | 4 | 2 | no | — |
| 389 | D4 | — | 4 | 2 | no | — |
| 390 | D4 | — | 4 | 2 | no | — |
| 391 | D4 | — | 4 | 2 | no | — |
| 392 | D4 | — | 4 | 1 | no | — |
| 393 | D4 | — | 4 | 1 | no | — |
| 394 | D4 | — | 4 | 1 | no | — |
| 395 | D4 | — | 8 | 8 | no | — |
| 396 | D4 | — | 8 | 1 | no | — |
| 397 | D4 | — | 8 | 1 | no | — |
| 398 | D4 | — | 32 | 2 | no | — |
| 399 | C2×C4 | — | 2 | 2 | no | — |
| 400 | C2×C4 | — | 2 | 2 | no | — |
| 401 | C2×C4 | — | 4 | 2 | no | — |
| 402 | C2×C4 | — | 4 | 1 | no | — |
| 403 | C2×C4 | — | 4 | 2 | no | — |
| 404 | C2×C4 | — | 4 | 2 | no | — |
| 405 | C2×C4 | — | 4 | 2 | no | — |
| 406 | C2×C4 | — | 4 | 2 | no | — |
| 407 | C2×C4 | — | 4 | 4 | no | — |
| 408 | C2×C4 | — | 4 | 1 | no | — |
| 409 | C2×C4 | — | 8 | 8 | no | — |
| 410 | C2×C4 | — | 8 | 8 | no | — |
| 411 | C2×C4 | — | 8 | 2 | no | — |
| 412 | C2×C4 | — | 8 | 2 | no | — |
| 413 | C2×C4 | — | 8 | 1 | no | — |
| 414 | C2×C4 | — | 8 | 1 | no | — |
| 415 | C2×C4 | — | 8 | 2 | no | — |
| 416 | C2×C4 | — | 16 | 2 | no | — |
| 417 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 418 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 419 | C2×C4 | #546 | 1 | 1 | yes | characteristic |
| 420 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 421 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 422 | C2×C4 | #577 | 1 | 1 | yes | characteristic |
| 423 | C8 | — | 4 | 2 | no | — |
| 424 | C8 | — | 4 | 2 | no | — |
| 425 | C8 | — | 8 | 1 | no | — |
| 426 | C8 | #447 | 1 | 2 | yes | — |
| 427 | C8 | #447 | 1 | 2 | yes | — |
| 428 | C8 | C2×#121 | 1 | 1 | yes | characteristic |
| 429 | C8 | #535 | 1 | 1 | yes | characteristic |
| 430 | C8 | D42 | 1 | 1 | yes | characteristic |
| 431 | C8 | #546 | 1 | 1 | yes | characteristic |
| 432 | C22 | — | 2 | 2 | no | — |
| 433 | C22 | — | 4 | 2 | no | — |
| 434 | C22 | — | 4 | 2 | no | — |
| 435 | C22 | — | 4 | 2 | no | — |
| 436 | C22 | — | 4 | 2 | no | — |
| 437 | C22 | — | 8 | 2 | no | — |
| 438 | C22 | — | 8 | 2 | no | — |
| 439 | C22 | — | 8 | 1 | no | — |
| 440 | C22 | #3012 | 1 | 1 | yes | characteristic |
| 441 | C22 | D4×D8 | 1 | 1 | yes | characteristic |
| 442 | C22 | #3325 | 1 | 1 | yes | characteristic |
| 443 | C4 | — | 4 | 2 | no | — |
| 444 | C4 | — | 4 | 2 | no | — |
| 445 | C4 | — | 4 | 2 | no | — |
| 446 | C4 | — | 8 | 4 | no | — |
| 447 | C4 | — | 8 | 1 | no | — |
| 448 | C4 | — | 16 | 2 | no | — |
| 449 | C4 | #2408 | 1 | 1 | yes | characteristic |
| 450 | C4 | C2×#447 | 1 | 1 | yes | characteristic |
| 451 | C4 | #3272 | 1 | 1 | yes | characteristic |
| 452 | C4 | #3294 | 1 | 1 | yes | characteristic |
| 453 | C2 | — | 2 | 1 | no | — |
| 454 | C2 | — | 2 | 1 | no | — |
| 455 | C2 | — | 8 | 2 | no | — |
| 456 | C2 | — | 8 | 2 | no | — |
| 457 | C2 | — | 16 | 1 | no | — |
| 458 | C2 | #24236 | 1 | 1 | yes | characteristic |
| 459 | C2 | #33855 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#443 ◁ 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×#3293 | C2 | semidirect |
| 2. C2×#2214 | C2 | semidirect |
| 3. #33527 | C2 | semidirect |
| 4. C2×#2179 | C2 | semidirect |
| 5. #28245 | C2 | semidirect |
| 6. #13394 | C2 | semidirect |
| 7. #13308 | C2 | semidirect |
| 8. #13207 | C2 | semidirect |
| 9. #13196 | C2 | semidirect |
| 10. #12820 | C2 | semidirect |
| 11. #12749 | C2 | semidirect |
| 88. #494 | D4 | semidirect |
| 104. #443 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3293 | 2 |
| C2×#2214 | 2 |
| #33527 | 2 |
| C2×#2179 | 2 |
| #28245 | 2 |
| #13394 | 2 |
| #13308 | 2 |
| #13207 | 2 |
| #13196 | 2 |
| #12820 | 2 |
| #12749 | 2 |