Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1911 conjugacy classes of subgroups.
The subgroup lattice has 665 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#3340 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #59821 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #36187 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #30568 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #28687 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #28511 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #26231 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #26139 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | #25701 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #25385 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #25340 | C2 | 1 | 2 | yes | maximal, complemented |
| 12 | #25011 | C2 | 1 | 2 | yes | maximal, complemented |
| 13 | #24121 | C2 | 1 | 2 | yes | maximal, complemented |
| 14 | #23173 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 15 | #22658 | C2 | 1 | 2 | yes | maximal, complemented |
| 16 | C4×#474 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 17 | C4×#461 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 18 | C2×#556 | C22 | 1 | 2 | yes | — |
| 19 | C2×#555 | C22 | 1 | 2 | yes | — |
| 20 | #3340 | — | 2 | 16 | no | — |
| 21 | C2×#498 | C22 | 1 | 2 | yes | — |
| 22 | C2×#485 | C22 | 1 | 4 | yes | — |
| 23 | C2×#482 | C22 | 1 | 4 | yes | — |
| 24 | C2×#477 | C22 | 1 | 2 | yes | — |
| 25 | C2×#477 | C22 | 1 | 2 | yes | — |
| 26 | C2×#474 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×#465 | C22 | 1 | 4 | yes | — |
| 28 | C2×#461 | C22 | 1 | 1 | yes | characteristic |
| 29 | C2×C4×SD16 | C22 | 1 | 2 | yes | — |
| 30 | C2×C8×D4 | C22 | 1 | 2 | yes | — |
| 31 | C2×C8×D4 | C22 | 1 | 2 | yes | — |
| 32 | C2×#427 | C22 | 1 | 4 | yes | — |
| 33 | #2868 | C22 | 1 | 1 | yes | characteristic |
| 34 | #2856 | — | 2 | 1 | no | — |
| 35 | #2727 | — | 2 | 2 | no | — |
| 36 | #2563 | C22 | 1 | 2 | yes | — |
| 37 | #2548 | — | 2 | 1 | no | — |
| 38 | #2542 | C22 | 1 | 2 | yes | — |
| 39 | #2538 | C22 | 1 | 2 | yes | — |
| 40 | #2537 | C22 | 1 | 2 | yes | — |
| 41 | #2537 | C22 | 1 | 2 | yes | — |
| 42 | #2512 | — | 2 | 2 | no | — |
| 43 | #2500 | — | 2 | 2 | no | — |
| 44 | #2491 | — | 2 | 2 | no | — |
| 45 | #2491 | — | 2 | 2 | no | — |
| 46 | #2448 | — | 2 | 2 | no | — |
| 47 | #2415 | C22 | 1 | 4 | yes | — |
| 48 | #2354 | C22 | 1 | 2 | yes | — |
| 49 | #2354 | C22 | 1 | 2 | yes | — |
| 50 | #2354 | C22 | 1 | 2 | yes | — |
| 51 | Q8×C4⋊C4 | — | 2 | 1 | no | — |
| 52 | C4×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 53 | C4×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 54 | C4×#127 | C22 | 1 | 1 | yes | characteristic |
| 55 | #2285 | — | 2 | 1 | no | — |
| 56 | C42×D4 | C22 | 1 | 1 | yes | characteristic |
| 57 | #2095 | C22 | 1 | 4 | yes | — |
| 58 | #2084 | C22 | 1 | 4 | yes | — |
| 59 | #2076 | C22 | 1 | 4 | yes | — |
| 60 | #2036 | C22 | 1 | 4 | yes | — |
| 61 | #2035 | C22 | 1 | 4 | yes | — |
| 62 | #2035 | C22 | 1 | 4 | yes | — |
| 63 | #2033 | C22 | 1 | 4 | yes | — |
| 64 | #2000 | C22 | 1 | 4 | yes | — |
| 65 | #1990 | C22 | 1 | 2 | yes | — |
| 66 | #1988 | C22 | 1 | 1 | yes | characteristic |
| 67 | #1981 | C22 | 1 | 1 | yes | characteristic |
| 68 | #1956 | C22 | 1 | 2 | yes | — |
| 69 | #1954 | C22 | 1 | 2 | yes | — |
| 70 | #1954 | C22 | 1 | 2 | yes | — |
| 71 | #1948 | C22 | 1 | 2 | yes | — |
| 72 | #1948 | C22 | 1 | 2 | yes | — |
| 73 | #1948 | C22 | 1 | 1 | yes | characteristic |
| 74 | #1947 | C22 | 1 | 1 | yes | characteristic |
| 75 | #1927 | C22 | 1 | 2 | yes | — |
| 76 | #1918 | C22 | 1 | 2 | yes | — |
| 77 | #1892 | C22 | 1 | 4 | yes | — |
| 78 | #1877 | C22 | 1 | 2 | yes | — |
| 79 | #1873 | C22 | 1 | 4 | yes | — |
| 80 | #1869 | C22 | 1 | 4 | yes | — |
| 81 | #1862 | C22 | 1 | 2 | yes | — |
| 82 | #1854 | C22 | 1 | 4 | yes | — |
| 83 | #1854 | C22 | 1 | 4 | yes | — |
| 84 | #1854 | C22 | 1 | 4 | yes | — |
| 85 | #1845 | C22 | 1 | 2 | yes | — |
| 86 | #1808 | C22 | 1 | 4 | yes | — |
| 87 | #1802 | C22 | 1 | 1 | yes | characteristic |
| 88 | #1802 | C22 | 1 | 1 | yes | characteristic |
| 89 | C4×#107 | C22 | 1 | 2 | yes | — |
| 90 | C4×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 91 | C4×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 92 | C4×#103 | C22 | 1 | 2 | yes | — |
| 93 | C4×#102 | C22 | 1 | 2 | yes | — |
| 94 | C22×SD16 | C23 | 1 | 2 | yes | — |
| 95 | C23×C8 | C23 | 1 | 4 | yes | — |
| 96 | #556 | — | 2 | 8 | no | — |
| 97 | #556 | — | 4 | 4 | no | — |
| 98 | #555 | — | 2 | 8 | no | — |
| 99 | #555 | — | 4 | 4 | no | — |
| 100 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 101 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 102 | C2×C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 103 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 104 | C2×C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 105 | C2×#127 | D4 | 1 | 1 | yes | characteristic |
| 106 | C2×#126 | — | 2 | 4 | no | — |
| 107 | C2×#124 | — | 2 | 2 | no | — |
| 108 | C2×#124 | — | 2 | 4 | no | — |
| 109 | C2×C22⋊Q8 | C23 | 1 | 4 | yes | — |
| 110 | C2×C22⋊Q8 | C23 | 1 | 4 | yes | — |
| 111 | C2×#121 | — | 2 | 2 | no | — |
| 112 | C2×#121 | — | 2 | 2 | no | — |
| 113 | C2×#121 | — | 2 | 2 | no | — |
| 114 | C2×#121 | C23 | 1 | 4 | yes | — |
| 115 | C2×C4×Q8 | — | 2 | 2 | no | — |
| 116 | C2×C4×Q8 | — | 2 | 1 | no | — |
| 117 | C2×C4×Q8 | C23 | 1 | 2 | yes | — |
| 118 | C2×C4×D4 | C23 | 1 | 1 | yes | characteristic |
| 119 | C2×C4×D4 | C23 | 1 | 1 | yes | characteristic |
| 120 | C2×C4×D4 | C23 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×D4 | C23 | 1 | 2 | yes | — |
| 122 | C2×C4×D4 | C23 | 1 | 2 | yes | — |
| 123 | C2×C4×D4 | C23 | 1 | 2 | yes | — |
| 124 | C2×#117 | — | 2 | 2 | no | — |
| 125 | C2×#117 | C23 | 1 | 4 | yes | — |
| 126 | C2×#117 | C23 | 1 | 4 | yes | — |
| 127 | C22×C42 | — | 2 | 1 | no | — |
| 128 | #498 | — | 2 | 8 | no | — |
| 129 | #485 | — | 2 | 16 | no | — |
| 130 | #482 | — | 2 | 16 | no | — |
| 131 | #477 | — | 2 | 8 | no | — |
| 132 | #477 | — | 4 | 4 | no | — |
| 133 | #474 | D4 | 1 | 8 | yes | complemented |
| 134 | #465 | — | 2 | 16 | no | — |
| 135 | #461 | — | 2 | 4 | no | — |
| 136 | C4×SD16 | — | 2 | 8 | no | — |
| 137 | C8×D4 | — | 2 | 8 | no | — |
| 138 | C8×D4 | — | 2 | 8 | no | — |
| 139 | #427 | — | 2 | 16 | no | — |
| 140 | C2×#107 | C23 | 1 | 4 | yes | — |
| 141 | C2×#107 | C23 | 1 | 4 | yes | — |
| 142 | C2×#107 | C23 | 1 | 2 | yes | — |
| 143 | C2×#107 | C23 | 1 | 2 | yes | — |
| 144 | C2×#106 | C23 | 1 | 4 | yes | — |
| 145 | C2×#106 | C23 | 1 | 4 | yes | — |
| 146 | C2×#106 | C23 | 1 | 2 | yes | — |
| 147 | C2×C4⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 148 | C2×C4⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 149 | C2×C4⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 150 | C2×C4⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 151 | C2×#103 | C23 | 1 | 4 | yes | — |
| 152 | C2×#103 | C23 | 1 | 2 | yes | — |
| 153 | C2×#103 | C23 | 1 | 2 | yes | — |
| 154 | C2×#103 | C23 | 1 | 4 | yes | — |
| 155 | C2×#103 | C23 | 1 | 2 | yes | — |
| 156 | C2×#102 | C23 | 1 | 2 | yes | — |
| 157 | C2×#102 | C23 | 1 | 2 | yes | — |
| 158 | C2×#102 | C23 | 1 | 2 | yes | — |
| 159 | C2×C22⋊C8 | C23 | 1 | 4 | yes | — |
| 160 | C2×C22⋊C8 | C23 | 1 | 4 | yes | — |
| 161 | C2×C22⋊C8 | C23 | 1 | 4 | yes | — |
| 162 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 163 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 164 | #399 | — | 2 | 4 | no | — |
| 165 | #397 | — | 2 | 4 | no | — |
| 166 | #396 | — | 2 | 4 | no | — |
| 167 | #394 | — | 2 | 4 | no | — |
| 168 | #394 | — | 2 | 4 | no | — |
| 169 | #394 | — | 2 | 4 | no | — |
| 170 | #393 | — | 2 | 4 | no | — |
| 171 | #391 | — | 2 | 1 | no | — |
| 172 | #391 | C23 | 1 | 2 | yes | — |
| 173 | #390 | — | 2 | 2 | no | — |
| 174 | #390 | — | 2 | 2 | no | — |
| 175 | #390 | — | 2 | 2 | no | — |
| 176 | #390 | — | 2 | 2 | no | — |
| 177 | #390 | — | 2 | 2 | no | — |
| 178 | #390 | — | 2 | 2 | no | — |
| 179 | #390 | D4 | 1 | 2 | yes | — |
| 180 | #390 | D4 | 1 | 2 | yes | — |
| 181 | #390 | D4 | 1 | 2 | yes | — |
| 182 | #389 | — | 2 | 4 | no | — |
| 183 | #389 | — | 2 | 2 | no | — |
| 184 | #389 | — | 2 | 1 | no | — |
| 185 | #389 | — | 2 | 1 | no | — |
| 186 | #389 | — | 2 | 1 | no | — |
| 187 | #389 | C23 | 1 | 2 | yes | — |
| 188 | #388 | — | 2 | 4 | no | — |
| 189 | #388 | — | 2 | 4 | no | — |
| 190 | #388 | — | 2 | 4 | no | — |
| 191 | #388 | — | 2 | 4 | no | — |
| 192 | #384 | — | 2 | 2 | no | — |
| 193 | #384 | — | 2 | 1 | no | — |
| 194 | #384 | — | 2 | 1 | no | — |
| 195 | #384 | D4 | 1 | 1 | yes | characteristic |
| 196 | #384 | C23 | 1 | 1 | yes | characteristic |
| 197 | #384 | C23 | 1 | 1 | yes | characteristic |
| 198 | #384 | C23 | 1 | 2 | yes | — |
| 199 | #384 | C23 | 1 | 2 | yes | — |
| 200 | #382 | — | 2 | 2 | no | — |
| 201 | #380 | — | 2 | 2 | no | — |
| 202 | #380 | — | 2 | 2 | no | — |
| 203 | #380 | C23 | 1 | 4 | yes | — |
| 204 | #380 | C23 | 1 | 4 | yes | — |
| 205 | #380 | C23 | 1 | 4 | yes | — |
| 206 | C4×C4⋊C4 | — | 2 | 2 | no | — |
| 207 | C4×C4⋊C4 | — | 2 | 2 | no | — |
| 208 | C4×C4⋊C4 | — | 2 | 2 | no | — |
| 209 | C4×C4⋊C4 | — | 2 | 1 | no | — |
| 210 | C4×C4⋊C4 | — | 2 | 1 | no | — |
| 211 | C4×C4⋊C4 | — | 2 | 1 | no | — |
| 212 | C4×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 213 | C4×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 214 | C4×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 215 | C4×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 216 | C4×C22⋊C4 | — | 2 | 1 | no | — |
| 217 | C4×C22⋊C4 | — | 2 | 1 | no | — |
| 218 | C4×C22⋊C4 | — | 2 | 1 | no | — |
| 219 | C43 | C23 | 1 | 1 | yes | characteristic |
| 220 | #340 | C23 | 1 | 2 | yes | — |
| 221 | #340 | C23 | 1 | 2 | yes | — |
| 222 | #340 | C23 | 1 | 2 | yes | — |
| 223 | #340 | C23 | 1 | 4 | yes | — |
| 224 | #340 | C23 | 1 | 2 | yes | — |
| 225 | #340 | C23 | 1 | 4 | yes | — |
| 226 | #340 | C23 | 1 | 2 | yes | — |
| 227 | #340 | C23 | 1 | 2 | yes | — |
| 228 | #336 | C23 | 1 | 2 | yes | — |
| 229 | #336 | C23 | 1 | 2 | yes | — |
| 230 | C22×Q8 | — | 2 | 1 | no | — |
| 231 | C22×Q8 | — | 2 | 2 | no | — |
| 232 | C22×Q8 | C24 | 1 | 2 | yes | — |
| 233 | C22×D4 | — | 2 | 4 | no | — |
| 234 | C22×D4 | C2×D4 | 1 | 2 | yes | — |
| 235 | C22×D4 | C2×D4 | 1 | 2 | yes | — |
| 236 | C22×D4 | C2×D4 | 1 | 2 | yes | — |
| 237 | C22×D4 | C24 | 1 | 1 | yes | characteristic |
| 238 | C23×C4 | — | 2 | 1 | no | — |
| 239 | C23×C4 | — | 2 | 1 | no | — |
| 240 | C23×C4 | — | 2 | 1 | no | — |
| 241 | C23×C4 | C24 | 1 | 4 | yes | — |
| 242 | C2×SD16 | — | 2 | 4 | no | — |
| 243 | C2×SD16 | — | 2 | 4 | no | — |
| 244 | C2×SD16 | — | 4 | 2 | no | — |
| 245 | C22×C8 | — | 2 | 8 | no | — |
| 246 | C22×C8 | — | 2 | 8 | no | — |
| 247 | C22×C8 | — | 4 | 4 | no | — |
| 248 | C22×C8 | C24 | 1 | 2 | yes | — |
| 249 | C22×C8 | C24 | 1 | 2 | yes | — |
| 250 | C22×C8 | C24 | 1 | 2 | yes | — |
| 251 | C22×C8 | C24 | 1 | 2 | yes | — |
| 252 | C4⋊Q8 | — | 2 | 4 | no | — |
| 253 | C4⋊Q8 | — | 2 | 2 | no | — |
| 254 | C4⋊Q8 | — | 2 | 2 | no | — |
| 255 | C4⋊Q8 | — | 2 | 4 | no | — |
| 256 | C4⋊Q8 | — | 2 | 2 | no | — |
| 257 | C4⋊Q8 | — | 8 | 2 | no | — |
| 258 | C4⋊Q8 | — | 8 | 2 | no | — |
| 259 | C4⋊Q8 | C2×D4 | 1 | 4 | yes | — |
| 260 | #127 | — | 2 | 4 | no | — |
| 261 | #126 | — | 4 | 16 | no | — |
| 262 | #124 | — | 4 | 16 | no | — |
| 263 | #124 | — | 4 | 8 | no | — |
| 264 | C22⋊Q8 | — | 2 | 8 | no | — |
| 265 | C22⋊Q8 | — | 4 | 4 | no | — |
| 266 | C22⋊Q8 | — | 4 | 4 | no | — |
| 267 | C22⋊Q8 | — | 4 | 4 | no | — |
| 268 | #121 | — | 2 | 8 | no | — |
| 269 | #121 | — | 4 | 4 | no | — |
| 270 | #121 | — | 4 | 8 | no | — |
| 271 | #121 | — | 8 | 4 | no | — |
| 272 | #121 | — | 8 | 4 | no | — |
| 273 | C4×Q8 | — | 2 | 4 | no | — |
| 274 | C4×Q8 | — | 2 | 4 | no | — |
| 275 | C4×Q8 | — | 4 | 8 | no | — |
| 276 | C4×Q8 | — | 4 | 4 | no | — |
| 277 | C4×D4 | — | 2 | 4 | no | — |
| 278 | C4×D4 | — | 2 | 4 | no | — |
| 279 | C4×D4 | — | 2 | 2 | no | — |
| 280 | C4×D4 | — | 2 | 2 | no | — |
| 281 | C4×D4 | — | 2 | 4 | no | — |
| 282 | C4×D4 | — | 2 | 4 | no | — |
| 283 | C4×D4 | — | 2 | 2 | no | — |
| 284 | C4×D4 | — | 4 | 1 | no | — |
| 285 | C4×D4 | — | 4 | 1 | no | — |
| 286 | C4×D4 | — | 4 | 2 | no | — |
| 287 | C4×D4 | — | 4 | 2 | no | — |
| 288 | C4×D4 | C2×D4 | 1 | 4 | yes | — |
| 289 | #117 | — | 2 | 8 | no | — |
| 290 | #117 | — | 2 | 8 | no | — |
| 291 | #117 | — | 4 | 4 | no | — |
| 292 | #117 | — | 4 | 4 | no | — |
| 293 | #117 | — | 4 | 8 | no | — |
| 294 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 295 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 296 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 297 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 298 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 299 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 300 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 301 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 302 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 303 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 304 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 305 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 306 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 307 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 308 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 309 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 310 | C2×C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 311 | C2×C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 312 | C2×C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 313 | C2×C4⋊C4 | C24 | 1 | 2 | yes | — |
| 314 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 315 | C2×C4⋊C4 | C24 | 1 | 1 | yes | characteristic |
| 316 | C2×C4⋊C4 | C24 | 1 | 1 | yes | characteristic |
| 317 | C2×C4⋊C4 | C24 | 1 | 2 | yes | — |
| 318 | C2×C4⋊C4 | C24 | 1 | 1 | yes | characteristic |
| 319 | C2×C4⋊C4 | C24 | 1 | 2 | yes | — |
| 320 | C2×C22⋊C4 | — | 2 | 1 | no | — |
| 321 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 322 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 323 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 324 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 325 | C2×C22⋊C4 | — | 2 | 1 | no | — |
| 326 | C2×C22⋊C4 | — | 2 | 1 | no | — |
| 327 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 328 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 329 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 330 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 331 | C2×C22⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 332 | C2×C22⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 333 | C2×C22⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 334 | C2×C42 | — | 2 | 4 | no | — |
| 335 | C2×C42 | — | 2 | 4 | no | — |
| 336 | C2×C42 | — | 2 | 2 | no | — |
| 337 | C2×C42 | — | 2 | 2 | no | — |
| 338 | C2×C42 | — | 2 | 1 | no | — |
| 339 | C2×C42 | — | 2 | 1 | no | — |
| 340 | C2×C42 | — | 2 | 1 | no | — |
| 341 | C2×C42 | — | 4 | 4 | no | — |
| 342 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 343 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 344 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 345 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 346 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 347 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 348 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 349 | #107 | — | 2 | 8 | no | — |
| 350 | #107 | — | 4 | 2 | no | — |
| 351 | #107 | — | 4 | 4 | no | — |
| 352 | #107 | C2×D4 | 1 | 8 | yes | — |
| 353 | #106 | — | 2 | 4 | no | — |
| 354 | #106 | — | 2 | 8 | no | — |
| 355 | #106 | — | 2 | 8 | no | — |
| 356 | C4⋊C8 | — | 2 | 2 | no | — |
| 357 | C4⋊C8 | — | 2 | 2 | no | — |
| 358 | C4⋊C8 | C2×D4 | 1 | 4 | yes | — |
| 359 | C4⋊C8 | #41 | 1 | 4 | yes | — |
| 360 | #103 | — | 2 | 8 | no | — |
| 361 | #103 | — | 2 | 4 | no | — |
| 362 | #103 | — | 2 | 4 | no | — |
| 363 | #103 | — | 4 | 2 | no | — |
| 364 | #103 | — | 4 | 4 | no | — |
| 365 | #102 | — | 2 | 4 | no | — |
| 366 | #102 | — | 4 | 2 | no | — |
| 367 | #102 | C2×D4 | 1 | 8 | yes | — |
| 368 | C22⋊C8 | — | 2 | 8 | no | — |
| 369 | C22⋊C8 | — | 2 | 8 | no | — |
| 370 | C22⋊C8 | — | 4 | 4 | no | — |
| 371 | C4×C8 | — | 2 | 4 | no | — |
| 372 | C4×C8 | — | 2 | 4 | no | — |
| 373 | #95 | — | 2 | 1 | no | — |
| 374 | #95 | — | 2 | 1 | no | — |
| 375 | #95 | — | 2 | 1 | no | — |
| 376 | #95 | — | 2 | 4 | no | — |
| 377 | #95 | — | 2 | 4 | no | — |
| 378 | #95 | — | 2 | 4 | no | — |
| 379 | #95 | — | 2 | 2 | no | — |
| 380 | #95 | — | 2 | 2 | no | — |
| 381 | #95 | — | 2 | 2 | no | — |
| 382 | #95 | — | 2 | 2 | no | — |
| 383 | C24 | — | 2 | 1 | no | — |
| 384 | C24 | C22×D4 | 1 | 4 | yes | — |
| 385 | C2×Q8 | — | 2 | 4 | no | — |
| 386 | C2×Q8 | — | 2 | 2 | no | — |
| 387 | C2×Q8 | — | 2 | 2 | no | — |
| 388 | C2×Q8 | — | 2 | 2 | no | — |
| 389 | C2×Q8 | — | 2 | 2 | no | — |
| 390 | C2×Q8 | — | 2 | 2 | no | — |
| 391 | C2×Q8 | — | 2 | 2 | no | — |
| 392 | C2×Q8 | — | 4 | 4 | no | — |
| 393 | C2×Q8 | — | 4 | 2 | no | — |
| 394 | C2×Q8 | — | 4 | 2 | no | — |
| 395 | C2×Q8 | — | 8 | 2 | no | — |
| 396 | C2×Q8 | — | 8 | 2 | no | — |
| 397 | C2×D4 | — | 2 | 4 | no | — |
| 398 | C2×D4 | — | 2 | 4 | no | — |
| 399 | C2×D4 | — | 2 | 4 | no | — |
| 400 | C2×D4 | — | 2 | 4 | no | — |
| 401 | C2×D4 | — | 2 | 4 | no | — |
| 402 | C2×D4 | — | 2 | 4 | no | — |
| 403 | C2×D4 | — | 2 | 4 | 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 | — | 2 | 1 | no | — |
| 409 | C2×D4 | — | 4 | 2 | no | — |
| 410 | C2×D4 | — | 4 | 2 | no | — |
| 411 | C2×D4 | — | 4 | 8 | no | — |
| 412 | C2×D4 | — | 8 | 4 | no | — |
| 413 | C2×D4 | — | 8 | 4 | no | — |
| 414 | C2×D4 | C22×D4 | 1 | 2 | yes | — |
| 415 | C22×C4 | — | 2 | 1 | no | — |
| 416 | C22×C4 | — | 2 | 2 | no | — |
| 417 | C22×C4 | — | 2 | 4 | no | — |
| 418 | C22×C4 | — | 2 | 4 | no | — |
| 419 | C22×C4 | — | 2 | 4 | no | — |
| 420 | C22×C4 | — | 2 | 4 | no | — |
| 421 | C22×C4 | — | 2 | 4 | no | — |
| 422 | C22×C4 | — | 2 | 4 | no | — |
| 423 | C22×C4 | — | 2 | 4 | no | — |
| 424 | C22×C4 | — | 2 | 1 | no | — |
| 425 | C22×C4 | — | 2 | 1 | no | — |
| 426 | C22×C4 | — | 2 | 2 | no | — |
| 427 | C22×C4 | — | 2 | 2 | no | — |
| 428 | C22×C4 | — | 2 | 2 | no | — |
| 429 | C22×C4 | — | 2 | 4 | no | — |
| 430 | C22×C4 | — | 4 | 2 | no | — |
| 431 | C22×C4 | — | 4 | 2 | no | — |
| 432 | C22×C4 | — | 4 | 2 | no | — |
| 433 | C22×C4 | — | 4 | 2 | no | — |
| 434 | C22×C4 | — | 4 | 2 | no | — |
| 435 | C22×C4 | — | 4 | 2 | no | — |
| 436 | C22×C4 | — | 4 | 2 | no | — |
| 437 | C22×C4 | — | 4 | 2 | no | — |
| 438 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 439 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 440 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 441 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 442 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 443 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 444 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 445 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 446 | SD16 | — | 4 | 8 | no | — |
| 447 | C2×C8 | — | 2 | 2 | no | — |
| 448 | C2×C8 | — | 2 | 2 | no | — |
| 449 | C2×C8 | — | 2 | 2 | no | — |
| 450 | C2×C8 | — | 2 | 2 | no | — |
| 451 | C2×C8 | — | 2 | 2 | no | — |
| 452 | C2×C8 | — | 2 | 2 | no | — |
| 453 | C2×C8 | — | 2 | 2 | no | — |
| 454 | C2×C8 | — | 2 | 2 | no | — |
| 455 | C2×C8 | — | 2 | 2 | no | — |
| 456 | C2×C8 | — | 2 | 2 | no | — |
| 457 | C2×C8 | — | 4 | 16 | no | — |
| 458 | C2×C8 | C22×D4 | 1 | 4 | yes | — |
| 459 | C2×C8 | C2×#41 | 1 | 4 | yes | — |
| 460 | C4⋊C4 | — | 2 | 4 | no | — |
| 461 | C4⋊C4 | — | 2 | 4 | no | — |
| 462 | C4⋊C4 | — | 2 | 4 | no | — |
| 463 | C4⋊C4 | — | 2 | 1 | no | — |
| 464 | C4⋊C4 | — | 2 | 2 | no | — |
| 465 | C4⋊C4 | — | 2 | 2 | no | — |
| 466 | C4⋊C4 | — | 2 | 4 | no | — |
| 467 | C4⋊C4 | — | 2 | 1 | no | — |
| 468 | C4⋊C4 | — | 2 | 4 | no | — |
| 469 | C4⋊C4 | — | 2 | 2 | no | — |
| 470 | C4⋊C4 | — | 2 | 1 | no | — |
| 471 | C4⋊C4 | — | 2 | 2 | no | — |
| 472 | C4⋊C4 | — | 2 | 1 | no | — |
| 473 | C4⋊C4 | — | 2 | 1 | no | — |
| 474 | C4⋊C4 | — | 2 | 2 | no | — |
| 475 | C4⋊C4 | — | 4 | 2 | no | — |
| 476 | C4⋊C4 | — | 4 | 8 | no | — |
| 477 | C4⋊C4 | — | 4 | 4 | no | — |
| 478 | C4⋊C4 | — | 4 | 4 | no | — |
| 479 | C4⋊C4 | — | 4 | 4 | no | — |
| 480 | C4⋊C4 | — | 4 | 2 | no | — |
| 481 | C4⋊C4 | — | 4 | 2 | no | — |
| 482 | C4⋊C4 | — | 4 | 4 | no | — |
| 483 | C4⋊C4 | — | 4 | 2 | no | — |
| 484 | C4⋊C4 | — | 4 | 2 | no | — |
| 485 | C4⋊C4 | — | 4 | 4 | no | — |
| 486 | C4⋊C4 | — | 8 | 1 | no | — |
| 487 | C4⋊C4 | — | 8 | 1 | no | — |
| 488 | C4⋊C4 | — | 8 | 4 | no | — |
| 489 | C4⋊C4 | — | 8 | 2 | no | — |
| 490 | C4⋊C4 | — | 8 | 2 | no | — |
| 491 | C4⋊C4 | C22×D4 | 1 | 4 | yes | — |
| 492 | C4⋊C4 | C22×D4 | 1 | 2 | yes | — |
| 493 | C22⋊C4 | — | 2 | 4 | no | — |
| 494 | C22⋊C4 | — | 2 | 8 | no | — |
| 495 | C22⋊C4 | — | 2 | 8 | no | — |
| 496 | C22⋊C4 | — | 4 | 4 | no | — |
| 497 | C22⋊C4 | — | 4 | 8 | no | — |
| 498 | C22⋊C4 | — | 4 | 8 | no | — |
| 499 | C22⋊C4 | — | 4 | 8 | no | — |
| 500 | C22⋊C4 | — | 4 | 8 | no | — |
| 501 | C22⋊C4 | — | 4 | 2 | no | — |
| 502 | C22⋊C4 | — | 4 | 2 | no | — |
| 503 | C22⋊C4 | — | 8 | 4 | no | — |
| 504 | C22⋊C4 | — | 8 | 4 | no | — |
| 505 | C22⋊C4 | — | 8 | 4 | no | — |
| 506 | C22⋊C4 | — | 8 | 4 | no | — |
| 507 | C42 | — | 2 | 2 | no | — |
| 508 | C42 | — | 2 | 2 | no | — |
| 509 | C42 | — | 2 | 1 | no | — |
| 510 | C42 | — | 2 | 1 | no | — |
| 511 | C42 | — | 4 | 1 | no | — |
| 512 | C42 | — | 4 | 8 | no | — |
| 513 | C42 | — | 4 | 8 | no | — |
| 514 | C42 | — | 4 | 4 | no | — |
| 515 | C42 | — | 4 | 2 | no | — |
| 516 | C42 | — | 4 | 2 | no | — |
| 517 | C42 | — | 4 | 2 | no | — |
| 518 | C42 | — | 4 | 4 | no | — |
| 519 | C42 | #135 | 1 | 4 | yes | — |
| 520 | C42 | C22×D4 | 1 | 2 | yes | — |
| 521 | C42 | C2×#41 | 1 | 2 | yes | — |
| 522 | C42 | H(5,𝔽2) | 1 | 2 | yes | — |
| 523 | C42 | 2−1+4 | 1 | 2 | yes | — |
| 524 | C23 | — | 2 | 4 | no | — |
| 525 | C23 | — | 2 | 4 | no | — |
| 526 | C23 | — | 2 | 2 | no | — |
| 527 | C23 | — | 2 | 4 | no | — |
| 528 | C23 | — | 2 | 4 | no | — |
| 529 | C23 | — | 2 | 4 | no | — |
| 530 | C23 | — | 2 | 4 | no | — |
| 531 | C23 | — | 2 | 4 | no | — |
| 532 | C23 | — | 4 | 1 | no | — |
| 533 | C23 | — | 4 | 1 | no | — |
| 534 | C23 | — | 4 | 1 | no | — |
| 535 | C23 | — | 4 | 1 | no | — |
| 536 | C23 | — | 4 | 1 | no | — |
| 537 | C23 | — | 4 | 1 | no | — |
| 538 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 539 | Q8 | — | 4 | 8 | no | — |
| 540 | Q8 | — | 4 | 2 | no | — |
| 541 | Q8 | — | 4 | 2 | no | — |
| 542 | Q8 | — | 4 | 2 | no | — |
| 543 | Q8 | — | 4 | 2 | no | — |
| 544 | Q8 | — | 16 | 4 | no | — |
| 545 | D4 | — | 2 | 2 | no | — |
| 546 | D4 | — | 2 | 2 | no | — |
| 547 | D4 | — | 4 | 8 | no | — |
| 548 | D4 | — | 4 | 1 | no | — |
| 549 | D4 | — | 4 | 8 | no | — |
| 550 | D4 | — | 4 | 1 | no | — |
| 551 | D4 | — | 8 | 4 | no | — |
| 552 | D4 | — | 16 | 8 | no | — |
| 553 | C2×C4 | — | 2 | 1 | no | — |
| 554 | C2×C4 | — | 2 | 1 | no | — |
| 555 | C2×C4 | — | 2 | 1 | no | — |
| 556 | C2×C4 | — | 2 | 1 | no | — |
| 557 | C2×C4 | — | 2 | 1 | no | — |
| 558 | C2×C4 | — | 2 | 1 | no | — |
| 559 | C2×C4 | — | 2 | 2 | no | — |
| 560 | C2×C4 | — | 2 | 4 | no | — |
| 561 | C2×C4 | — | 2 | 4 | no | — |
| 562 | C2×C4 | — | 2 | 4 | no | — |
| 563 | C2×C4 | — | 2 | 1 | no | — |
| 564 | C2×C4 | — | 2 | 4 | no | — |
| 565 | C2×C4 | — | 2 | 1 | no | — |
| 566 | C2×C4 | — | 2 | 1 | no | — |
| 567 | C2×C4 | — | 4 | 1 | no | — |
| 568 | C2×C4 | — | 4 | 2 | no | — |
| 569 | C2×C4 | — | 4 | 1 | no | — |
| 570 | C2×C4 | — | 4 | 2 | no | — |
| 571 | C2×C4 | — | 4 | 4 | no | — |
| 572 | C2×C4 | — | 4 | 4 | no | — |
| 573 | C2×C4 | — | 4 | 4 | no | — |
| 574 | C2×C4 | — | 4 | 8 | no | — |
| 575 | C2×C4 | — | 4 | 4 | no | — |
| 576 | C2×C4 | — | 4 | 1 | no | — |
| 577 | C2×C4 | — | 4 | 2 | no | — |
| 578 | C2×C4 | — | 4 | 2 | no | — |
| 579 | C2×C4 | — | 4 | 4 | no | — |
| 580 | C2×C4 | — | 4 | 2 | no | — |
| 581 | C2×C4 | — | 4 | 4 | no | — |
| 582 | C2×C4 | — | 4 | 2 | no | — |
| 583 | C2×C4 | — | 4 | 1 | no | — |
| 584 | C2×C4 | — | 4 | 2 | no | — |
| 585 | C2×C4 | — | 4 | 1 | no | — |
| 586 | C2×C4 | — | 4 | 2 | no | — |
| 587 | C2×C4 | — | 4 | 2 | no | — |
| 588 | C2×C4 | — | 4 | 1 | no | — |
| 589 | C2×C4 | — | 4 | 1 | no | — |
| 590 | C2×C4 | — | 4 | 2 | no | — |
| 591 | C2×C4 | — | 4 | 1 | no | — |
| 592 | C2×C4 | — | 4 | 4 | no | — |
| 593 | C2×C4 | — | 4 | 2 | no | — |
| 594 | C2×C4 | — | 8 | 4 | no | — |
| 595 | C2×C4 | — | 8 | 4 | no | — |
| 596 | C2×C4 | D42 | 1 | 2 | yes | — |
| 597 | C2×C4 | D42 | 1 | 2 | yes | — |
| 598 | C2×C4 | #547 | 1 | 2 | yes | — |
| 599 | C2×C4 | #547 | 1 | 2 | yes | — |
| 600 | C2×C4 | C2×#135 | 1 | 2 | yes | — |
| 601 | C2×C4 | C2×#135 | 1 | 2 | yes | — |
| 602 | C2×C4 | C2×#135 | 1 | 2 | yes | — |
| 603 | C2×C4 | #577 | 1 | 2 | yes | — |
| 604 | C2×C4 | #578 | 1 | 2 | yes | — |
| 605 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 606 | C2×C4 | C22×#41 | 1 | 1 | yes | characteristic |
| 607 | C2×C4 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 608 | C2×C4 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 609 | C2×C4 | #585 | 1 | 1 | yes | characteristic |
| 610 | C2×C4 | #585 | 1 | 1 | yes | characteristic |
| 611 | C8 | — | 2 | 4 | no | — |
| 612 | C8 | — | 2 | 4 | no | — |
| 613 | C8 | — | 4 | 2 | no | — |
| 614 | C8 | — | 4 | 2 | no | — |
| 615 | C22 | — | 4 | 4 | no | — |
| 616 | C22 | — | 4 | 4 | no | — |
| 617 | C22 | — | 4 | 4 | no | — |
| 618 | C22 | — | 4 | 4 | no | — |
| 619 | C22 | — | 4 | 2 | no | — |
| 620 | C22 | — | 4 | 4 | no | — |
| 621 | C22 | — | 4 | 2 | no | — |
| 622 | C22 | — | 4 | 4 | no | — |
| 623 | C22 | — | 4 | 4 | no | — |
| 624 | C22 | — | 4 | 2 | no | — |
| 625 | C22 | — | 8 | 1 | no | — |
| 626 | C22 | — | 8 | 1 | no | — |
| 627 | C22 | — | 8 | 1 | no | — |
| 628 | C22 | — | 8 | 1 | no | — |
| 629 | C22 | C2×D42 | 1 | 1 | yes | characteristic |
| 630 | C22 | C2×#547 | 1 | 1 | yes | characteristic |
| 631 | C22 | #3485 | 1 | 1 | yes | characteristic |
| 632 | C22 | C22×#135 | 1 | 1 | yes | characteristic |
| 633 | C22 | C2×#577 | 1 | 1 | yes | characteristic |
| 634 | C22 | C2×#578 | 1 | 1 | yes | characteristic |
| 635 | C22 | #3585 | 1 | 1 | yes | characteristic |
| 636 | C4 | — | 2 | 2 | no | — |
| 637 | C4 | — | 2 | 2 | no | — |
| 638 | C4 | — | 2 | 2 | no | — |
| 639 | C4 | — | 4 | 1 | no | — |
| 640 | C4 | — | 4 | 2 | no | — |
| 641 | C4 | — | 4 | 4 | no | — |
| 642 | C4 | — | 4 | 4 | no | — |
| 643 | C4 | — | 8 | 2 | no | — |
| 644 | C4 | — | 8 | 1 | no | — |
| 645 | C4 | — | 8 | 1 | no | — |
| 646 | C4 | — | 8 | 2 | no | — |
| 647 | C4 | — | 8 | 4 | no | — |
| 648 | C4 | — | 8 | 2 | no | — |
| 649 | C4 | #3283 | 1 | 4 | yes | — |
| 650 | C4 | #3340 | 1 | 4 | yes | complemented |
| 651 | C4 | D4×#41 | 1 | 1 | yes | characteristic |
| 652 | C4 | #3497 | 1 | 1 | yes | characteristic |
| 653 | C4 | #3517 | 1 | 1 | yes | characteristic |
| 654 | C4 | #3554 | 1 | 1 | yes | characteristic |
| 655 | C2 | — | 8 | 4 | no | — |
| 656 | C2 | — | 8 | 2 | no | — |
| 657 | C2 | #59066 | 1 | 1 | yes | characteristic |
| 658 | C2 | C2×#3283 | 1 | 1 | yes | characteristic |
| 659 | C2 | #61095 | 1 | 1 | yes | characteristic |
| 660 | C2 | C2×#3340 | 1 | 1 | yes | characteristic |
| 661 | C2 | #61282 | 1 | 1 | yes | characteristic |
| 662 | C2 | #61700 | 1 | 1 | yes | characteristic |
| 663 | C2 | #61789 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36187 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#3340 | C2 | semidirect |
| 2. #59821 | C2 | semidirect |
| 4. #30568 | C2 | semidirect |
| 5. #28687 | C2 | semidirect |
| 6. #28511 | C2 | semidirect |
| 7. #26231 | C2 | semidirect |
| 8. #26139 | C2 | semidirect |
| 9. #25701 | C2 | semidirect |
| 10. #25385 | C2 | semidirect |
| 11. #25340 | C2 | semidirect |
| 12. #25011 | C2 | semidirect |
| 13. #24121 | C2 | semidirect |
| 14. #23173 | C2 | semidirect |
| 15. #22658 | C2 | semidirect |
| 16. C4×#474 | C2 | semidirect |
| 17. C4×#461 | C2 | semidirect |
| 133. #474 | D4 | semidirect |
| 650. C4 | #3340 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3340 | 2 |
| #59821 | 2 |
| #36187 | 2 |
| #30568 | 2 |
| #28687 | 2 |
| #28511 | 2 |
| #26231 | 2 |
| #26139 | 2 |
| #25701 | 2 |
| #25385 | 2 |
| #25340 | 2 |
| #25011 | 2 |
| #24121 | 2 |
| #23173 | 2 |
| #22658 | 2 |
| C4×#474 | 2 |
| C4×#461 | 2 |