Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1814 conjugacy classes of subgroups.
The subgroup lattice has 519 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #36455 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #34193 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #29925 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #27974 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #27646 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #26237 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #26235 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #24968 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | #24966 | C2 | 1 | 3 | yes | maximal, complemented |
| 10 | #23758 | C2 | 1 | 3 | yes | maximal, complemented |
| 11 | #22740 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #22738 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 13 | #22540 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 14 | #22540 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 15 | #22530 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 16 | C2×#550 | — | 2 | 1 | no | — |
| 17 | C2×#550 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×C4×#41 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#482 | C22 | 1 | 6 | yes | — |
| 20 | C2×C8×Q8 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×C8×Q8 | C22 | 1 | 1 | yes | characteristic |
| 22 | #2856 | C22 | 1 | 3 | yes | — |
| 23 | #2541 | C22 | 1 | 3 | yes | — |
| 24 | #2537 | C22 | 1 | 3 | yes | — |
| 25 | #2428 | — | 2 | 3 | no | — |
| 26 | #2414 | — | 2 | 3 | no | — |
| 27 | #2354 | C22 | 1 | 3 | yes | — |
| 28 | #2354 | C22 | 1 | 3 | yes | — |
| 29 | Q8×C4⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 30 | Q8×C4⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 31 | Q8×C4⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 32 | #2347 | — | 2 | 3 | no | — |
| 33 | #2341 | — | 2 | 1 | no | — |
| 34 | #2341 | — | 2 | 1 | no | — |
| 35 | #2319 | — | 2 | 3 | no | — |
| 36 | C4×#121 | C22 | 1 | 3 | yes | — |
| 37 | #2285 | C22 | 1 | 3 | yes | — |
| 38 | #2283 | C22 | 1 | 3 | yes | — |
| 39 | #2276 | — | 2 | 1 | no | — |
| 40 | #2101 | C22 | 1 | 1 | yes | characteristic |
| 41 | #2092 | C22 | 1 | 3 | yes | — |
| 42 | #2085 | C22 | 1 | 3 | yes | — |
| 43 | #2077 | C22 | 1 | 3 | yes | — |
| 44 | #2076 | C22 | 1 | 3 | yes | — |
| 45 | #2073 | C22 | 1 | 3 | yes | — |
| 46 | #2073 | C22 | 1 | 6 | yes | — |
| 47 | #2034 | C22 | 1 | 6 | yes | — |
| 48 | #2034 | C22 | 1 | 3 | yes | — |
| 49 | #2033 | C22 | 1 | 3 | yes | — |
| 50 | #2033 | C22 | 1 | 6 | yes | — |
| 51 | #1982 | C22 | 1 | 3 | yes | — |
| 52 | #1982 | C22 | 1 | 3 | yes | — |
| 53 | #1957 | C22 | 1 | 3 | yes | — |
| 54 | #1957 | C22 | 1 | 3 | yes | — |
| 55 | #1954 | C22 | 1 | 3 | yes | — |
| 56 | #1954 | C22 | 1 | 3 | yes | — |
| 57 | #1948 | C22 | 1 | 3 | yes | — |
| 58 | #1948 | C22 | 1 | 3 | yes | — |
| 59 | #1947 | C22 | 1 | 3 | yes | — |
| 60 | #1947 | C22 | 1 | 3 | yes | — |
| 61 | #1875 | C22 | 1 | 6 | yes | — |
| 62 | #1875 | C22 | 1 | 6 | yes | — |
| 63 | #1834 | C22 | 1 | 3 | yes | — |
| 64 | #1834 | C22 | 1 | 3 | yes | — |
| 65 | #1825 | C22 | 1 | 6 | yes | — |
| 66 | #1802 | C22 | 1 | 3 | yes | — |
| 67 | #1802 | C22 | 1 | 3 | yes | — |
| 68 | #1797 | C22 | 1 | 3 | yes | — |
| 69 | C4×#107 | C22 | 1 | 3 | yes | — |
| 70 | C4×#106 | C22 | 1 | 3 | yes | — |
| 71 | C4×#102 | C22 | 1 | 3 | yes | — |
| 72 | C4×#102 | C22 | 1 | 3 | yes | — |
| 73 | C4×C22⋊C8 | C22 | 1 | 3 | yes | — |
| 74 | C22×#41 | — | 2 | 1 | no | — |
| 75 | C22×#41 | D4 | 1 | 1 | yes | characteristic |
| 76 | #550 | — | 2 | 4 | no | — |
| 77 | #550 | — | 2 | 4 | no | — |
| 78 | #550 | — | 4 | 8 | no | — |
| 79 | C2×#127 | — | 2 | 3 | no | — |
| 80 | C2×#127 | C23 | 1 | 3 | yes | — |
| 81 | C2×#121 | — | 2 | 3 | no | — |
| 82 | C2×#121 | — | 2 | 3 | no | — |
| 83 | C2×#121 | — | 2 | 3 | no | — |
| 84 | C2×#121 | — | 2 | 3 | no | — |
| 85 | C2×#121 | C23 | 1 | 3 | yes | — |
| 86 | C2×#121 | C23 | 1 | 6 | yes | — |
| 87 | C4×#41 | — | 2 | 4 | no | — |
| 88 | C4×#41 | — | 2 | 4 | no | — |
| 89 | C2×C4×Q8 | — | 2 | 1 | no | — |
| 90 | C2×C4×Q8 | — | 2 | 1 | no | — |
| 91 | C2×C4×Q8 | — | 2 | 1 | no | — |
| 92 | C2×C4×Q8 | D4 | 1 | 1 | yes | characteristic |
| 93 | C2×C4×Q8 | C23 | 1 | 1 | yes | characteristic |
| 94 | C2×C4×D4 | — | 2 | 3 | no | — |
| 95 | C2×C4×D4 | D4 | 1 | 3 | yes | — |
| 96 | C2×C4×D4 | C23 | 1 | 3 | yes | — |
| 97 | C2×C4×D4 | C23 | 1 | 3 | yes | — |
| 98 | C2×#117 | D4 | 1 | 3 | yes | — |
| 99 | C2×#117 | C23 | 1 | 6 | yes | — |
| 100 | C22×C42 | C23 | 1 | 3 | yes | — |
| 101 | #482 | — | 4 | 12 | no | — |
| 102 | C8×Q8 | — | 2 | 4 | no | — |
| 103 | C8×Q8 | — | 2 | 4 | no | — |
| 104 | C2×#107 | C23 | 1 | 3 | yes | — |
| 105 | C2×#107 | C23 | 1 | 3 | yes | — |
| 106 | C2×#106 | C23 | 1 | 3 | yes | — |
| 107 | C2×#106 | C23 | 1 | 3 | yes | — |
| 108 | C2×C4⋊C8 | C23 | 1 | 3 | yes | — |
| 109 | C2×C4⋊C8 | C23 | 1 | 3 | yes | — |
| 110 | C2×#102 | C23 | 1 | 6 | yes | — |
| 111 | C2×#102 | C23 | 1 | 6 | yes | — |
| 112 | C2×#102 | C23 | 1 | 3 | yes | — |
| 113 | C2×#102 | C23 | 1 | 3 | yes | — |
| 114 | C2×#102 | C23 | 1 | 3 | yes | — |
| 115 | C2×#102 | C23 | 1 | 3 | yes | — |
| 116 | C2×C22⋊C8 | C23 | 1 | 6 | yes | — |
| 117 | C2×C22⋊C8 | C23 | 1 | 3 | yes | — |
| 118 | C2×C22⋊C8 | C23 | 1 | 3 | yes | — |
| 119 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 120 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 121 | #394 | — | 2 | 6 | no | — |
| 122 | #392 | — | 2 | 6 | no | — |
| 123 | #390 | — | 2 | 3 | no | — |
| 124 | #390 | — | 2 | 3 | no | — |
| 125 | #390 | — | 2 | 3 | no | — |
| 126 | #390 | — | 2 | 3 | no | — |
| 127 | #390 | — | 2 | 3 | no | — |
| 128 | #390 | C23 | 1 | 3 | yes | — |
| 129 | #389 | — | 2 | 3 | no | — |
| 130 | #389 | — | 2 | 3 | no | — |
| 131 | #389 | — | 2 | 3 | no | — |
| 132 | #389 | C23 | 1 | 1 | yes | characteristic |
| 133 | #388 | — | 2 | 6 | no | — |
| 134 | #388 | — | 2 | 6 | no | — |
| 135 | #388 | — | 2 | 3 | no | — |
| 136 | #386 | — | 2 | 3 | no | — |
| 137 | #384 | C23 | 1 | 3 | yes | — |
| 138 | #384 | C23 | 1 | 3 | yes | — |
| 139 | #384 | C23 | 1 | 3 | yes | — |
| 140 | #383 | — | 2 | 3 | no | — |
| 141 | #382 | C23 | 1 | 3 | yes | — |
| 142 | #382 | C23 | 1 | 1 | yes | characteristic |
| 143 | #381 | — | 2 | 3 | no | — |
| 144 | #380 | — | 2 | 3 | no | — |
| 145 | #380 | — | 2 | 3 | no | — |
| 146 | #380 | — | 2 | 3 | no | — |
| 147 | #380 | C23 | 1 | 3 | yes | — |
| 148 | #380 | C23 | 1 | 3 | yes | — |
| 149 | #380 | C23 | 1 | 3 | yes | — |
| 150 | C4×C4⋊C4 | — | 2 | 3 | no | — |
| 151 | C4×C4⋊C4 | — | 2 | 3 | no | — |
| 152 | C4×C4⋊C4 | — | 2 | 3 | no | — |
| 153 | C4×C4⋊C4 | — | 2 | 3 | no | — |
| 154 | C4×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 155 | C4×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 156 | C4×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 157 | C4×C22⋊C4 | — | 2 | 3 | no | — |
| 158 | C4×C22⋊C4 | — | 2 | 3 | no | — |
| 159 | #376 | — | 2 | 3 | no | — |
| 160 | #340 | C23 | 1 | 3 | yes | — |
| 161 | #340 | C23 | 1 | 3 | yes | — |
| 162 | #340 | C23 | 1 | 3 | yes | — |
| 163 | #340 | C23 | 1 | 3 | yes | — |
| 164 | #340 | C23 | 1 | 6 | yes | — |
| 165 | #340 | C23 | 1 | 6 | yes | — |
| 166 | #340 | C23 | 1 | 1 | yes | characteristic |
| 167 | #340 | C23 | 1 | 1 | yes | characteristic |
| 168 | #340 | C23 | 1 | 1 | yes | characteristic |
| 169 | #340 | C23 | 1 | 1 | yes | characteristic |
| 170 | #336 | C23 | 1 | 3 | yes | — |
| 171 | #336 | C23 | 1 | 1 | yes | characteristic |
| 172 | C2×#41 | — | 2 | 4 | no | — |
| 173 | C2×#41 | — | 2 | 4 | no | — |
| 174 | C2×#41 | — | 2 | 4 | no | — |
| 175 | C2×#41 | — | 4 | 4 | no | — |
| 176 | C2×#41 | — | 4 | 4 | no | — |
| 177 | C2×#41 | — | 4 | 4 | no | — |
| 178 | C22×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 179 | C22×D4 | — | 2 | 3 | no | — |
| 180 | C22×D4 | — | 2 | 6 | no | — |
| 181 | C22×D4 | — | 2 | 3 | no | — |
| 182 | C22×D4 | — | 2 | 3 | no | — |
| 183 | C22×D4 | C2×D4 | 1 | 3 | yes | — |
| 184 | C22×D4 | C24 | 1 | 3 | yes | — |
| 185 | C23×C4 | — | 2 | 3 | no | — |
| 186 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 187 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 188 | C23×C4 | C24 | 1 | 3 | yes | — |
| 189 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 190 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 191 | C22×C8 | C24 | 1 | 3 | yes | — |
| 192 | C22×C8 | C24 | 1 | 3 | yes | — |
| 193 | #127 | — | 2 | 6 | no | — |
| 194 | #127 | — | 2 | 6 | no | — |
| 195 | #127 | — | 4 | 12 | no | — |
| 196 | #121 | — | 2 | 6 | no | — |
| 197 | #121 | — | 4 | 3 | no | — |
| 198 | #121 | — | 4 | 12 | no | — |
| 199 | #121 | — | 4 | 12 | no | — |
| 200 | #121 | — | 4 | 12 | no | — |
| 201 | #121 | — | 4 | 6 | no | — |
| 202 | #121 | — | 4 | 6 | no | — |
| 203 | #121 | — | 8 | 6 | no | — |
| 204 | C4×Q8 | — | 2 | 4 | no | — |
| 205 | C4×Q8 | — | 4 | 4 | no | — |
| 206 | C4×Q8 | — | 4 | 4 | no | — |
| 207 | C4×Q8 | — | 4 | 4 | no | — |
| 208 | C4×Q8 | #41 | 1 | 4 | yes | — |
| 209 | C4×Q8 | #41 | 1 | 4 | yes | — |
| 210 | C4×D4 | — | 2 | 12 | no | — |
| 211 | C4×D4 | — | 2 | 6 | no | — |
| 212 | C4×D4 | — | 2 | 6 | no | — |
| 213 | C4×D4 | — | 2 | 6 | no | — |
| 214 | C4×D4 | — | 2 | 6 | no | — |
| 215 | C4×D4 | — | 4 | 12 | no | — |
| 216 | #117 | — | 2 | 12 | no | — |
| 217 | #117 | — | 4 | 6 | no | — |
| 218 | #117 | — | 4 | 6 | no | — |
| 219 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 220 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 221 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 222 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 223 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 224 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 225 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 226 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 227 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 228 | C2×C4⋊C4 | C24 | 1 | 3 | yes | — |
| 229 | C2×C4⋊C4 | C24 | 1 | 3 | yes | — |
| 230 | C2×C4⋊C4 | C24 | 1 | 1 | yes | characteristic |
| 231 | C2×C4⋊C4 | C24 | 1 | 3 | yes | — |
| 232 | C2×C4⋊C4 | C24 | 1 | 1 | yes | characteristic |
| 233 | C2×C4⋊C4 | C24 | 1 | 1 | yes | characteristic |
| 234 | C2×C4⋊C4 | C24 | 1 | 1 | yes | characteristic |
| 235 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 236 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 237 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 238 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 239 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 240 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 241 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 242 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 243 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 244 | C2×C22⋊C4 | C2×D4 | 1 | 6 | yes | — |
| 245 | C2×C42 | — | 2 | 6 | no | — |
| 246 | C2×C42 | — | 2 | 6 | no | — |
| 247 | C2×C42 | — | 2 | 3 | no | — |
| 248 | C2×C42 | — | 2 | 3 | no | — |
| 249 | C2×C42 | — | 2 | 3 | no | — |
| 250 | C2×C42 | — | 2 | 6 | no | — |
| 251 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 252 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 253 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 254 | C2×C42 | C24 | 1 | 3 | yes | — |
| 255 | #107 | — | 2 | 6 | no | — |
| 256 | #107 | — | 4 | 3 | no | — |
| 257 | #106 | — | 2 | 6 | no | — |
| 258 | #106 | — | 4 | 3 | no | — |
| 259 | C4⋊C8 | — | 2 | 6 | no | — |
| 260 | C4⋊C8 | — | 2 | 6 | no | — |
| 261 | #102 | — | 4 | 3 | no | — |
| 262 | #102 | — | 4 | 3 | no | — |
| 263 | #102 | — | 4 | 3 | no | — |
| 264 | #102 | — | 4 | 3 | no | — |
| 265 | #102 | — | 4 | 6 | no | — |
| 266 | #102 | — | 4 | 6 | no | — |
| 267 | C22⋊C8 | — | 4 | 3 | no | — |
| 268 | C22⋊C8 | — | 4 | 3 | no | — |
| 269 | C22⋊C8 | — | 4 | 6 | no | — |
| 270 | C4×C8 | — | 2 | 6 | no | — |
| 271 | C4×C8 | — | 2 | 6 | no | — |
| 272 | #95 | — | 2 | 1 | no | — |
| 273 | #95 | — | 2 | 3 | no | — |
| 274 | #95 | — | 2 | 3 | no | — |
| 275 | #95 | — | 2 | 3 | no | — |
| 276 | #95 | — | 2 | 1 | no | — |
| 277 | #95 | — | 2 | 1 | no | — |
| 278 | #95 | — | 2 | 3 | no | — |
| 279 | #95 | — | 2 | 3 | no | — |
| 280 | #95 | — | 2 | 3 | no | — |
| 281 | #95 | — | 2 | 3 | no | — |
| 282 | #95 | — | 2 | 3 | no | — |
| 283 | C24 | — | 2 | 3 | no | — |
| 284 | C24 | C22×D4 | 1 | 3 | yes | — |
| 285 | #41 | — | 4 | 16 | no | — |
| 286 | #41 | — | 8 | 16 | no | — |
| 287 | C2×Q8 | #123 | 1 | 4 | yes | — |
| 288 | C2×Q8 | #135 | 1 | 4 | yes | — |
| 289 | C2×Q8 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 290 | C2×D4 | — | 2 | 6 | no | — |
| 291 | C2×D4 | — | 2 | 6 | no | — |
| 292 | C2×D4 | — | 2 | 6 | no | — |
| 293 | C2×D4 | — | 2 | 3 | no | — |
| 294 | C2×D4 | — | 2 | 3 | no | — |
| 295 | C2×D4 | — | 2 | 3 | no | — |
| 296 | C2×D4 | — | 2 | 3 | no | — |
| 297 | C2×D4 | — | 2 | 3 | no | — |
| 298 | C2×D4 | — | 2 | 3 | no | — |
| 299 | C2×D4 | — | 4 | 6 | no | — |
| 300 | C2×D4 | — | 4 | 6 | no | — |
| 301 | C2×D4 | — | 4 | 12 | no | — |
| 302 | C2×D4 | — | 4 | 6 | no | — |
| 303 | C2×D4 | — | 4 | 6 | no | — |
| 304 | C2×D4 | — | 4 | 6 | no | — |
| 305 | C2×D4 | — | 4 | 6 | no | — |
| 306 | C2×D4 | — | 8 | 3 | no | — |
| 307 | C2×D4 | — | 8 | 3 | no | — |
| 308 | C2×D4 | — | 8 | 3 | no | — |
| 309 | C2×D4 | — | 8 | 6 | no | — |
| 310 | C2×D4 | — | 8 | 6 | no | — |
| 311 | C22×C4 | — | 2 | 1 | no | — |
| 312 | C22×C4 | — | 2 | 6 | no | — |
| 313 | C22×C4 | — | 2 | 6 | no | — |
| 314 | C22×C4 | — | 2 | 6 | no | — |
| 315 | C22×C4 | — | 2 | 3 | no | — |
| 316 | C22×C4 | — | 2 | 3 | no | — |
| 317 | C22×C4 | — | 2 | 3 | no | — |
| 318 | C22×C4 | — | 2 | 3 | no | — |
| 319 | C22×C4 | — | 2 | 3 | no | — |
| 320 | C22×C4 | — | 2 | 3 | no | — |
| 321 | C22×C4 | — | 2 | 6 | no | — |
| 322 | C22×C4 | — | 2 | 1 | no | — |
| 323 | C22×C4 | — | 2 | 1 | no | — |
| 324 | C22×C4 | — | 2 | 3 | no | — |
| 325 | C22×C4 | — | 2 | 3 | no | — |
| 326 | C22×C4 | — | 2 | 1 | no | — |
| 327 | C22×C4 | — | 2 | 3 | no | — |
| 328 | C22×C4 | — | 4 | 3 | no | — |
| 329 | C22×C4 | — | 4 | 3 | no | — |
| 330 | C22×C4 | — | 4 | 6 | no | — |
| 331 | C22×C4 | — | 4 | 6 | no | — |
| 332 | C22×C4 | — | 4 | 6 | no | — |
| 333 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 334 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 335 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 336 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 337 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 338 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 339 | C2×C8 | — | 2 | 1 | no | — |
| 340 | C2×C8 | — | 2 | 1 | no | — |
| 341 | C2×C8 | — | 2 | 3 | no | — |
| 342 | C2×C8 | — | 2 | 3 | no | — |
| 343 | C2×C8 | — | 2 | 3 | no | — |
| 344 | C2×C8 | — | 2 | 3 | no | — |
| 345 | C2×C8 | — | 2 | 1 | no | — |
| 346 | C2×C8 | — | 2 | 1 | no | — |
| 347 | C2×C8 | — | 2 | 3 | no | — |
| 348 | C2×C8 | — | 2 | 3 | no | — |
| 349 | C2×C8 | — | 2 | 1 | no | — |
| 350 | C2×C8 | — | 2 | 1 | no | — |
| 351 | C4⋊C4 | — | 2 | 6 | no | — |
| 352 | C4⋊C4 | — | 2 | 1 | no | — |
| 353 | C4⋊C4 | — | 2 | 1 | no | — |
| 354 | C4⋊C4 | — | 2 | 1 | no | — |
| 355 | C4⋊C4 | — | 2 | 3 | no | — |
| 356 | C4⋊C4 | — | 2 | 3 | no | — |
| 357 | C4⋊C4 | — | 2 | 1 | no | — |
| 358 | C4⋊C4 | — | 2 | 1 | no | — |
| 359 | C4⋊C4 | — | 2 | 3 | no | — |
| 360 | C4⋊C4 | — | 2 | 3 | no | — |
| 361 | C4⋊C4 | — | 2 | 1 | no | — |
| 362 | C4⋊C4 | — | 2 | 3 | no | — |
| 363 | C4⋊C4 | — | 2 | 1 | no | — |
| 364 | C4⋊C4 | — | 2 | 3 | no | — |
| 365 | C4⋊C4 | — | 4 | 2 | no | — |
| 366 | C4⋊C4 | — | 4 | 6 | no | — |
| 367 | C4⋊C4 | — | 4 | 6 | no | — |
| 368 | C4⋊C4 | — | 4 | 6 | no | — |
| 369 | C4⋊C4 | — | 8 | 3 | no | — |
| 370 | C4⋊C4 | — | 8 | 3 | no | — |
| 371 | C4⋊C4 | — | 8 | 3 | no | — |
| 372 | C4⋊C4 | #123 | 1 | 12 | yes | — |
| 373 | C4⋊C4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 374 | C22⋊C4 | — | 2 | 12 | no | — |
| 375 | C22⋊C4 | — | 4 | 6 | no | — |
| 376 | C22⋊C4 | — | 4 | 6 | no | — |
| 377 | C22⋊C4 | — | 4 | 6 | no | — |
| 378 | C22⋊C4 | — | 4 | 6 | no | — |
| 379 | C22⋊C4 | — | 4 | 6 | no | — |
| 380 | C22⋊C4 | — | 8 | 6 | no | — |
| 381 | C22⋊C4 | — | 8 | 3 | no | — |
| 382 | C22⋊C4 | — | 8 | 3 | no | — |
| 383 | C22⋊C4 | — | 8 | 6 | no | — |
| 384 | C42 | — | 2 | 6 | no | — |
| 385 | C42 | — | 2 | 6 | no | — |
| 386 | C42 | — | 2 | 1 | no | — |
| 387 | C42 | — | 2 | 1 | no | — |
| 388 | C42 | — | 4 | 6 | no | — |
| 389 | C42 | — | 4 | 6 | no | — |
| 390 | C42 | — | 4 | 6 | no | — |
| 391 | C42 | — | 8 | 6 | no | — |
| 392 | C42 | C2×#41 | 1 | 6 | yes | — |
| 393 | C42 | C2×#41 | 1 | 6 | yes | — |
| 394 | C23 | — | 2 | 3 | no | — |
| 395 | C23 | — | 2 | 3 | no | — |
| 396 | C23 | — | 2 | 3 | no | — |
| 397 | C23 | — | 2 | 3 | no | — |
| 398 | C23 | — | 2 | 3 | no | — |
| 399 | C23 | — | 2 | 3 | no | — |
| 400 | C23 | — | 2 | 3 | no | — |
| 401 | C23 | — | 4 | 3 | no | — |
| 402 | C23 | — | 4 | 3 | no | — |
| 403 | C23 | — | 4 | 3 | no | — |
| 404 | C23 | — | 4 | 3 | no | — |
| 405 | C23 | — | 4 | 3 | no | — |
| 406 | C23 | — | 4 | 3 | no | — |
| 407 | C23 | — | 4 | 3 | no | — |
| 408 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 409 | Q8 | #482 | 1 | 16 | yes | complemented |
| 410 | D4 | — | 4 | 12 | no | — |
| 411 | D4 | — | 4 | 3 | no | — |
| 412 | D4 | — | 4 | 3 | no | — |
| 413 | D4 | — | 4 | 3 | no | — |
| 414 | D4 | — | 4 | 3 | no | — |
| 415 | D4 | — | 8 | 12 | no | — |
| 416 | D4 | — | 8 | 12 | no | — |
| 417 | D4 | — | 16 | 6 | no | — |
| 418 | D4 | — | 16 | 12 | no | — |
| 419 | C2×C4 | — | 2 | 3 | no | — |
| 420 | C2×C4 | — | 2 | 3 | no | — |
| 421 | C2×C4 | — | 2 | 1 | no | — |
| 422 | C2×C4 | — | 2 | 1 | no | — |
| 423 | C2×C4 | — | 2 | 1 | no | — |
| 424 | C2×C4 | — | 2 | 1 | no | — |
| 425 | C2×C4 | — | 2 | 3 | no | — |
| 426 | C2×C4 | — | 2 | 3 | no | — |
| 427 | C2×C4 | — | 2 | 2 | no | — |
| 428 | C2×C4 | — | 2 | 3 | no | — |
| 429 | C2×C4 | — | 2 | 1 | no | — |
| 430 | C2×C4 | — | 2 | 1 | no | — |
| 431 | C2×C4 | — | 4 | 1 | no | — |
| 432 | C2×C4 | — | 4 | 1 | no | — |
| 433 | C2×C4 | — | 4 | 12 | no | — |
| 434 | C2×C4 | — | 4 | 3 | no | — |
| 435 | C2×C4 | — | 4 | 3 | no | — |
| 436 | C2×C4 | — | 4 | 3 | no | — |
| 437 | C2×C4 | — | 4 | 3 | no | — |
| 438 | C2×C4 | — | 4 | 1 | no | — |
| 439 | C2×C4 | — | 4 | 3 | no | — |
| 440 | C2×C4 | — | 4 | 3 | no | — |
| 441 | C2×C4 | — | 4 | 1 | no | — |
| 442 | C2×C4 | — | 4 | 3 | no | — |
| 443 | C2×C4 | — | 4 | 1 | no | — |
| 444 | C2×C4 | — | 4 | 3 | no | — |
| 445 | C2×C4 | — | 4 | 3 | no | — |
| 446 | C2×C4 | — | 4 | 3 | no | — |
| 447 | C2×C4 | — | 4 | 1 | no | — |
| 448 | C2×C4 | — | 4 | 1 | no | — |
| 449 | C2×C4 | — | 4 | 1 | no | — |
| 450 | C2×C4 | — | 4 | 3 | no | — |
| 451 | C2×C4 | — | 4 | 1 | no | — |
| 452 | C2×C4 | — | 4 | 1 | no | — |
| 453 | C2×C4 | — | 4 | 3 | no | — |
| 454 | C2×C4 | — | 4 | 3 | no | — |
| 455 | C2×C4 | — | 4 | 1 | no | — |
| 456 | C2×C4 | — | 8 | 6 | no | — |
| 457 | C2×C4 | — | 8 | 12 | no | — |
| 458 | C2×C4 | C2×#123 | 1 | 6 | yes | — |
| 459 | C2×C4 | C2×#123 | 1 | 6 | yes | — |
| 460 | C2×C4 | C2×#135 | 1 | 6 | yes | — |
| 461 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 6 | yes | — |
| 462 | C2×C4 | C22×#41 | 1 | 1 | yes | characteristic |
| 463 | C2×C4 | C22×#41 | 1 | 1 | yes | characteristic |
| 464 | C2×C4 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 465 | C2×C4 | #585 | 1 | 1 | yes | characteristic |
| 466 | C2×C4 | #585 | 1 | 1 | yes | characteristic |
| 467 | C2×C4 | #585 | 1 | 1 | yes | characteristic |
| 468 | C8 | — | 4 | 3 | no | — |
| 469 | C8 | — | 4 | 3 | no | — |
| 470 | C8 | — | 4 | 1 | no | — |
| 471 | C8 | — | 4 | 1 | no | — |
| 472 | C22 | — | 4 | 3 | no | — |
| 473 | C22 | — | 4 | 3 | no | — |
| 474 | C22 | — | 4 | 3 | no | — |
| 475 | C22 | — | 4 | 3 | no | — |
| 476 | C22 | — | 4 | 3 | no | — |
| 477 | C22 | — | 4 | 3 | no | — |
| 478 | C22 | — | 4 | 3 | no | — |
| 479 | C22 | — | 8 | 3 | no | — |
| 480 | C22 | — | 8 | 3 | no | — |
| 481 | C22 | — | 8 | 3 | no | — |
| 482 | C22 | — | 8 | 3 | no | — |
| 483 | C22 | — | 8 | 3 | no | — |
| 484 | C22 | — | 8 | 3 | no | — |
| 485 | C22 | — | 8 | 3 | no | — |
| 486 | C22 | C22×#123 | 1 | 1 | yes | characteristic |
| 487 | C22 | #3488 | 1 | 1 | yes | characteristic |
| 488 | C22 | #3500 | 1 | 1 | yes | characteristic |
| 489 | C22 | C22×#135 | 1 | 1 | yes | characteristic |
| 490 | C22 | C22×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 491 | C22 | #3584 | 1 | 1 | yes | characteristic |
| 492 | C22 | #3585 | 1 | 1 | yes | characteristic |
| 493 | C4 | — | 2 | 6 | no | — |
| 494 | C4 | — | 4 | 2 | no | — |
| 495 | C4 | — | 4 | 1 | no | — |
| 496 | C4 | — | 4 | 1 | no | — |
| 497 | C4 | — | 4 | 3 | no | — |
| 498 | C4 | — | 8 | 3 | no | — |
| 499 | C4 | — | 8 | 3 | no | — |
| 500 | C4 | — | 8 | 1 | no | — |
| 501 | C4 | — | 8 | 1 | no | — |
| 502 | C4 | — | 8 | 1 | no | — |
| 503 | C4 | — | 8 | 3 | no | — |
| 504 | C4 | C2×#482 | 1 | 12 | yes | — |
| 505 | C4 | #3517 | 1 | 1 | yes | characteristic |
| 506 | C4 | #3517 | 1 | 1 | yes | characteristic |
| 507 | C4 | #3524 | 1 | 1 | yes | characteristic |
| 508 | C4 | #3524 | 1 | 1 | yes | characteristic |
| 509 | C2 | — | 8 | 3 | no | — |
| 510 | C2 | — | 16 | 3 | no | — |
| 511 | C2 | #36076 | 1 | 1 | yes | characteristic |
| 512 | C2 | C22×#482 | 1 | 1 | yes | characteristic |
| 513 | C2 | #60523 | 1 | 1 | yes | characteristic |
| 514 | C2 | #61958 | 1 | 1 | yes | characteristic |
| 515 | C2 | #62003 | 1 | 1 | yes | characteristic |
| 516 | C2 | #62534 | 1 | 1 | yes | characteristic |
| 517 | C2 | #62588 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36455 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #34193 | C2 | semidirect |
| 3. #29925 | C2 | semidirect |
| 4. #27974 | C2 | semidirect |
| 5. #27646 | C2 | semidirect |
| 6. #26237 | C2 | semidirect |
| 7. #26235 | C2 | semidirect |
| 8. #24968 | C2 | semidirect |
| 9. #24966 | C2 | semidirect |
| 10. #23758 | C2 | semidirect |
| 11. #22740 | C2 | semidirect |
| 12. #22738 | C2 | semidirect |
| 13. #22540 | C2 | semidirect |
| 14. #22540 | C2 | semidirect |
| 15. #22530 | C2 | semidirect |
| 409. Q8 | #482 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #36455 | 2 |
| #34193 | 2 |
| #29925 | 2 |
| #27974 | 2 |
| #27646 | 2 |
| #26237 | 2 |
| #26235 | 2 |
| #24968 | 2 |
| #24966 | 2 |
| #23758 | 2 |
| #22740 | 2 |
| #22738 | 2 |
| #22540 | 2 |
| #22530 | 2 |