Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2012 conjugacy classes of subgroups.
The subgroup lattice has 420 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 | #34183 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #29920 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #27973 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #27644 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #26230 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #24965 | C2 | 1 | 6 | yes | maximal, complemented |
| 8 | #23754 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | #22740 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #22540 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #22530 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C2×#550 | — | 2 | 1 | no | — |
| 13 | C2×#550 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#520 | — | 2 | 1 | no | — |
| 15 | C2×C4×#41 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×#480 | C22 | 1 | 6 | yes | — |
| 17 | C2×C8×Q8 | C22 | 1 | 2 | yes | — |
| 18 | #2856 | C22 | 1 | 3 | yes | — |
| 19 | #2541 | C22 | 1 | 3 | yes | — |
| 20 | #2537 | C22 | 1 | 3 | yes | — |
| 21 | #2428 | — | 2 | 3 | no | — |
| 22 | #2414 | — | 2 | 3 | no | — |
| 23 | Q8×C4⋊C4 | — | 2 | 1 | no | — |
| 24 | Q8×C4⋊C4 | C22 | 1 | 2 | yes | — |
| 25 | Q8×C4⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 26 | D4×C4⋊C4 | — | 2 | 3 | no | — |
| 27 | D4×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 28 | #2341 | — | 2 | 1 | no | — |
| 29 | #2317 | — | 2 | 3 | no | — |
| 30 | C4×#121 | C22 | 1 | 3 | yes | — |
| 31 | #2285 | C22 | 1 | 3 | yes | — |
| 32 | C2×#384 | C22 | 1 | 3 | yes | — |
| 33 | #2099 | C22 | 1 | 1 | yes | characteristic |
| 34 | #2093 | C22 | 1 | 3 | yes | — |
| 35 | #2083 | C22 | 1 | 3 | yes | — |
| 36 | #2075 | C22 | 1 | 6 | yes | — |
| 37 | #2071 | C22 | 1 | 3 | yes | — |
| 38 | #2071 | C22 | 1 | 6 | yes | — |
| 39 | #2033 | C22 | 1 | 6 | yes | — |
| 40 | #2033 | C22 | 1 | 12 | yes | — |
| 41 | #1982 | C22 | 1 | 6 | yes | — |
| 42 | #1957 | C22 | 1 | 6 | yes | — |
| 43 | #1954 | C22 | 1 | 6 | yes | — |
| 44 | #1947 | C22 | 1 | 6 | yes | — |
| 45 | #1947 | C22 | 1 | 6 | yes | — |
| 46 | #1874 | C22 | 1 | 12 | yes | — |
| 47 | #1834 | C22 | 1 | 6 | yes | — |
| 48 | #1823 | C22 | 1 | 6 | yes | — |
| 49 | #1802 | C22 | 1 | 6 | yes | — |
| 50 | #1797 | C22 | 1 | 3 | yes | — |
| 51 | C4×#107 | C22 | 1 | 6 | yes | — |
| 52 | C4×#102 | C22 | 1 | 6 | yes | — |
| 53 | C4×C22⋊C8 | C22 | 1 | 3 | yes | — |
| 54 | C22×#41 | — | 2 | 1 | no | — |
| 55 | C22×#41 | D4 | 1 | 1 | yes | characteristic |
| 56 | #550 | — | 2 | 8 | no | — |
| 57 | #550 | — | 4 | 8 | no | — |
| 58 | C2×#127 | — | 2 | 3 | no | — |
| 59 | C2×#127 | C23 | 1 | 3 | yes | — |
| 60 | C2×#121 | — | 2 | 3 | no | — |
| 61 | C2×#121 | — | 2 | 3 | no | — |
| 62 | C2×#121 | — | 2 | 3 | no | — |
| 63 | C2×#121 | — | 2 | 3 | no | — |
| 64 | C2×#121 | C23 | 1 | 3 | yes | — |
| 65 | C2×#121 | C23 | 1 | 6 | yes | — |
| 66 | #520 | — | 4 | 8 | no | — |
| 67 | C4×#41 | — | 2 | 8 | no | — |
| 68 | C2×C4×Q8 | — | 2 | 1 | no | — |
| 69 | C2×C4×Q8 | — | 2 | 2 | no | — |
| 70 | C2×C4×Q8 | D4 | 1 | 1 | yes | characteristic |
| 71 | C2×C4×Q8 | C23 | 1 | 1 | yes | characteristic |
| 72 | C2×C4×D4 | — | 2 | 6 | no | — |
| 73 | C2×C4×D4 | — | 2 | 3 | no | — |
| 74 | C2×C4×D4 | D4 | 1 | 3 | yes | — |
| 75 | C2×C4×D4 | C23 | 1 | 3 | yes | — |
| 76 | C2×C4×D4 | C23 | 1 | 3 | yes | — |
| 77 | C2×#117 | — | 2 | 3 | no | — |
| 78 | C2×#117 | D4 | 1 | 3 | yes | — |
| 79 | C22×C4⋊C4 | — | 2 | 3 | no | — |
| 80 | C22×C4⋊C4 | — | 2 | 3 | no | — |
| 81 | C22×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 82 | C22×C42 | C23 | 1 | 3 | yes | — |
| 83 | #480 | — | 4 | 12 | no | — |
| 84 | C8×Q8 | — | 2 | 8 | no | — |
| 85 | C2×#107 | C23 | 1 | 6 | yes | — |
| 86 | C2×#107 | C23 | 1 | 6 | yes | — |
| 87 | C2×C4⋊C8 | C23 | 1 | 6 | yes | — |
| 88 | C2×#102 | C23 | 1 | 12 | yes | — |
| 89 | C2×#102 | C23 | 1 | 6 | yes | — |
| 90 | C2×#102 | C23 | 1 | 6 | yes | — |
| 91 | C2×C22⋊C8 | C23 | 1 | 6 | yes | — |
| 92 | C2×C22⋊C8 | C23 | 1 | 6 | yes | — |
| 93 | C2×C4×C8 | C23 | 1 | 6 | yes | — |
| 94 | #394 | — | 2 | 6 | no | — |
| 95 | #392 | — | 2 | 6 | no | — |
| 96 | #390 | — | 2 | 3 | no | — |
| 97 | #390 | — | 2 | 3 | no | — |
| 98 | #390 | — | 2 | 3 | no | — |
| 99 | #390 | C23 | 1 | 3 | yes | — |
| 100 | #389 | — | 2 | 6 | no | — |
| 101 | #389 | — | 2 | 6 | no | — |
| 102 | #389 | — | 2 | 3 | no | — |
| 103 | #389 | C23 | 1 | 1 | yes | characteristic |
| 104 | #388 | — | 2 | 3 | no | — |
| 105 | #386 | — | 2 | 3 | no | — |
| 106 | #385 | — | 2 | 12 | no | — |
| 107 | #384 | — | 2 | 3 | no | — |
| 108 | #384 | — | 2 | 3 | no | — |
| 109 | #384 | C23 | 1 | 6 | yes | — |
| 110 | #384 | C23 | 1 | 3 | yes | — |
| 111 | #384 | C23 | 1 | 3 | yes | — |
| 112 | #384 | C23 | 1 | 1 | yes | characteristic |
| 113 | #382 | — | 2 | 3 | no | — |
| 114 | #380 | — | 2 | 3 | no | — |
| 115 | #380 | — | 2 | 3 | no | — |
| 116 | #380 | C23 | 1 | 3 | yes | — |
| 117 | #380 | C23 | 1 | 6 | yes | — |
| 118 | C4×C4⋊C4 | — | 2 | 3 | no | — |
| 119 | C4×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 120 | C4×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 121 | C4×C22⋊C4 | — | 2 | 3 | no | — |
| 122 | #340 | C23 | 1 | 6 | yes | — |
| 123 | #340 | C23 | 1 | 6 | yes | — |
| 124 | #340 | C23 | 1 | 12 | yes | — |
| 125 | #340 | C23 | 1 | 2 | yes | — |
| 126 | #340 | C23 | 1 | 2 | yes | — |
| 127 | #336 | C23 | 1 | 3 | yes | — |
| 128 | #336 | C23 | 1 | 1 | yes | characteristic |
| 129 | C2×#41 | — | 2 | 4 | no | — |
| 130 | C2×#41 | — | 2 | 4 | no | — |
| 131 | C2×#41 | — | 4 | 4 | no | — |
| 132 | C2×#41 | — | 4 | 4 | no | — |
| 133 | C2×#41 | — | 4 | 4 | no | — |
| 134 | C2×#41 | D8 | 1 | 8 | yes | — |
| 135 | C22×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 136 | C22×D4 | — | 2 | 3 | no | — |
| 137 | C22×D4 | — | 2 | 6 | no | — |
| 138 | C22×D4 | — | 2 | 3 | no | — |
| 139 | C22×D4 | — | 2 | 3 | no | — |
| 140 | C22×D4 | C2×D4 | 1 | 3 | yes | — |
| 141 | C22×D4 | C24 | 1 | 3 | yes | — |
| 142 | C23×C4 | — | 2 | 6 | no | — |
| 143 | C23×C4 | — | 2 | 3 | no | — |
| 144 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 145 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 146 | C23×C4 | C24 | 1 | 3 | yes | — |
| 147 | C22×C8 | C24 | 1 | 2 | yes | — |
| 148 | C22×C8 | C24 | 1 | 6 | yes | — |
| 149 | #127 | — | 2 | 12 | no | — |
| 150 | #127 | — | 4 | 12 | no | — |
| 151 | #121 | — | 2 | 6 | no | — |
| 152 | #121 | — | 4 | 3 | no | — |
| 153 | #121 | — | 4 | 12 | no | — |
| 154 | #121 | — | 4 | 12 | no | — |
| 155 | #121 | — | 4 | 12 | no | — |
| 156 | #121 | — | 4 | 12 | no | — |
| 157 | #121 | — | 8 | 6 | no | — |
| 158 | C4×Q8 | — | 2 | 4 | no | — |
| 159 | C4×Q8 | — | 4 | 8 | no | — |
| 160 | C4×Q8 | — | 4 | 4 | no | — |
| 161 | C4×Q8 | #41 | 1 | 8 | yes | — |
| 162 | C4×D4 | — | 2 | 12 | no | — |
| 163 | C4×D4 | — | 2 | 12 | no | — |
| 164 | C4×D4 | — | 2 | 12 | no | — |
| 165 | C4×D4 | — | 4 | 24 | no | — |
| 166 | C4×D4 | — | 4 | 12 | no | — |
| 167 | #117 | — | 2 | 12 | no | — |
| 168 | #117 | — | 4 | 12 | no | — |
| 169 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 170 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 171 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 172 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 173 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 174 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 175 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 176 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 177 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 178 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 179 | C2×C4⋊C4 | — | 4 | 6 | no | — |
| 180 | C2×C4⋊C4 | — | 4 | 6 | no | — |
| 181 | C2×C4⋊C4 | — | 4 | 12 | no | — |
| 182 | C2×C4⋊C4 | — | 8 | 6 | no | — |
| 183 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 184 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 185 | C2×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 186 | C2×C4⋊C4 | C24 | 1 | 1 | yes | characteristic |
| 187 | C2×C4⋊C4 | C24 | 1 | 3 | yes | — |
| 188 | C2×C4⋊C4 | C24 | 1 | 2 | yes | — |
| 189 | C2×C4⋊C4 | C24 | 1 | 1 | yes | characteristic |
| 190 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 191 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 192 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 193 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 194 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 195 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 196 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 197 | C2×C22⋊C4 | C2×D4 | 1 | 6 | yes | — |
| 198 | C2×C42 | — | 2 | 12 | no | — |
| 199 | C2×C42 | — | 2 | 6 | no | — |
| 200 | C2×C42 | — | 2 | 3 | no | — |
| 201 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 202 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 203 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 204 | C2×C42 | C24 | 1 | 3 | yes | — |
| 205 | #107 | — | 2 | 12 | no | — |
| 206 | #107 | — | 4 | 6 | no | — |
| 207 | C4⋊C8 | — | 2 | 12 | no | — |
| 208 | #102 | — | 4 | 6 | no | — |
| 209 | #102 | — | 4 | 6 | no | — |
| 210 | #102 | — | 4 | 12 | no | — |
| 211 | C22⋊C8 | — | 4 | 6 | no | — |
| 212 | C22⋊C8 | — | 4 | 6 | no | — |
| 213 | C4×C8 | — | 2 | 12 | no | — |
| 214 | #95 | — | 2 | 1 | no | — |
| 215 | #95 | — | 2 | 3 | no | — |
| 216 | #95 | — | 2 | 6 | no | — |
| 217 | #95 | — | 2 | 6 | no | — |
| 218 | #95 | — | 2 | 3 | no | — |
| 219 | C24 | — | 2 | 3 | no | — |
| 220 | C24 | C22×D4 | 1 | 3 | yes | — |
| 221 | #41 | — | 2 | 32 | no | — |
| 222 | #41 | — | 8 | 16 | no | — |
| 223 | C2×Q8 | #123 | 1 | 4 | yes | — |
| 224 | C2×Q8 | C2×D8 | 1 | 4 | yes | — |
| 225 | C2×Q8 | #137 | 1 | 4 | yes | — |
| 226 | C2×D4 | — | 2 | 6 | no | — |
| 227 | C2×D4 | — | 2 | 6 | no | — |
| 228 | C2×D4 | — | 2 | 6 | no | — |
| 229 | C2×D4 | — | 2 | 3 | no | — |
| 230 | C2×D4 | — | 2 | 3 | no | — |
| 231 | C2×D4 | — | 2 | 6 | no | — |
| 232 | C2×D4 | — | 4 | 6 | no | — |
| 233 | C2×D4 | — | 4 | 6 | no | — |
| 234 | C2×D4 | — | 4 | 12 | no | — |
| 235 | C2×D4 | — | 4 | 6 | no | — |
| 236 | C2×D4 | — | 4 | 6 | no | — |
| 237 | C2×D4 | — | 4 | 6 | no | — |
| 238 | C2×D4 | — | 4 | 6 | no | — |
| 239 | C2×D4 | — | 8 | 3 | no | — |
| 240 | C2×D4 | — | 8 | 3 | no | — |
| 241 | C2×D4 | — | 8 | 3 | no | — |
| 242 | C2×D4 | — | 8 | 6 | no | — |
| 243 | C2×D4 | — | 8 | 6 | no | — |
| 244 | C2×D4 | C2×D8 | 1 | 12 | yes | — |
| 245 | C22×C4 | — | 2 | 6 | no | — |
| 246 | C22×C4 | — | 2 | 6 | no | — |
| 247 | C22×C4 | — | 2 | 1 | no | — |
| 248 | C22×C4 | — | 2 | 6 | no | — |
| 249 | C22×C4 | — | 2 | 6 | no | — |
| 250 | C22×C4 | — | 2 | 6 | no | — |
| 251 | C22×C4 | — | 2 | 3 | no | — |
| 252 | C22×C4 | — | 2 | 6 | no | — |
| 253 | C22×C4 | — | 2 | 3 | no | — |
| 254 | C22×C4 | — | 2 | 2 | no | — |
| 255 | C22×C4 | — | 2 | 1 | no | — |
| 256 | C22×C4 | — | 2 | 3 | no | — |
| 257 | C22×C4 | — | 2 | 6 | no | — |
| 258 | C22×C4 | — | 4 | 3 | no | — |
| 259 | C22×C4 | — | 4 | 12 | no | — |
| 260 | C22×C4 | — | 4 | 6 | no | — |
| 261 | C22×C4 | — | 4 | 6 | no | — |
| 262 | C22×C4 | — | 4 | 6 | no | — |
| 263 | C22×C4 | — | 8 | 6 | no | — |
| 264 | C22×C4 | — | 8 | 6 | no | — |
| 265 | C22×C4 | C2×D8 | 1 | 12 | yes | — |
| 266 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 267 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 268 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 269 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 270 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 271 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 272 | C2×C8 | — | 2 | 2 | no | — |
| 273 | C2×C8 | — | 2 | 6 | no | — |
| 274 | C2×C8 | — | 2 | 6 | no | — |
| 275 | C2×C8 | — | 2 | 2 | no | — |
| 276 | C2×C8 | — | 2 | 6 | no | — |
| 277 | C2×C8 | — | 2 | 2 | no | — |
| 278 | C4⋊C4 | — | 2 | 6 | no | — |
| 279 | C4⋊C4 | — | 2 | 1 | no | — |
| 280 | C4⋊C4 | — | 2 | 2 | no | — |
| 281 | C4⋊C4 | — | 2 | 6 | no | — |
| 282 | C4⋊C4 | — | 2 | 2 | no | — |
| 283 | C4⋊C4 | — | 2 | 6 | no | — |
| 284 | C4⋊C4 | — | 2 | 2 | no | — |
| 285 | C4⋊C4 | — | 2 | 6 | no | — |
| 286 | C4⋊C4 | — | 4 | 6 | no | — |
| 287 | C4⋊C4 | — | 4 | 6 | no | — |
| 288 | C4⋊C4 | — | 4 | 12 | no | — |
| 289 | C4⋊C4 | — | 4 | 2 | no | — |
| 290 | C4⋊C4 | — | 4 | 2 | no | — |
| 291 | C4⋊C4 | — | 4 | 6 | no | — |
| 292 | C4⋊C4 | — | 4 | 2 | no | — |
| 293 | C4⋊C4 | — | 8 | 6 | no | — |
| 294 | C4⋊C4 | — | 8 | 6 | no | — |
| 295 | C4⋊C4 | — | 8 | 3 | no | — |
| 296 | C4⋊C4 | #123 | 1 | 12 | yes | — |
| 297 | C4⋊C4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 298 | C22⋊C4 | — | 2 | 12 | no | — |
| 299 | C22⋊C4 | — | 4 | 6 | no | — |
| 300 | C22⋊C4 | — | 4 | 12 | no | — |
| 301 | C22⋊C4 | — | 4 | 6 | no | — |
| 302 | C22⋊C4 | — | 4 | 6 | no | — |
| 303 | C22⋊C4 | — | 4 | 6 | no | — |
| 304 | C22⋊C4 | — | 4 | 6 | no | — |
| 305 | C22⋊C4 | — | 8 | 6 | no | — |
| 306 | C42 | — | 2 | 6 | no | — |
| 307 | C42 | — | 2 | 6 | no | — |
| 308 | C42 | — | 2 | 2 | no | — |
| 309 | C42 | — | 4 | 12 | no | — |
| 310 | C42 | — | 4 | 6 | no | — |
| 311 | C42 | C2×#41 | 1 | 12 | yes | — |
| 312 | C23 | — | 2 | 3 | no | — |
| 313 | C23 | — | 2 | 3 | no | — |
| 314 | C23 | — | 2 | 3 | no | — |
| 315 | C23 | — | 2 | 3 | no | — |
| 316 | C23 | — | 2 | 3 | no | — |
| 317 | C23 | — | 2 | 3 | no | — |
| 318 | C23 | — | 4 | 3 | no | — |
| 319 | C23 | — | 4 | 3 | no | — |
| 320 | C23 | — | 4 | 3 | no | — |
| 321 | C23 | — | 4 | 3 | no | — |
| 322 | C23 | — | 4 | 3 | no | — |
| 323 | C23 | — | 4 | 3 | no | — |
| 324 | C23 | — | 4 | 3 | no | — |
| 325 | C23 | C22×D8 | 1 | 6 | yes | — |
| 326 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 327 | Q8 | #480 | 1 | 16 | yes | complemented |
| 328 | D4 | — | 2 | 24 | no | — |
| 329 | D4 | — | 4 | 6 | no | — |
| 330 | D4 | — | 4 | 6 | no | — |
| 331 | D4 | — | 8 | 12 | no | — |
| 332 | D4 | — | 8 | 12 | no | — |
| 333 | D4 | — | 16 | 6 | no | — |
| 334 | D4 | — | 16 | 12 | no | — |
| 335 | C2×C4 | — | 2 | 1 | no | — |
| 336 | C2×C4 | — | 2 | 3 | no | — |
| 337 | C2×C4 | — | 2 | 1 | no | — |
| 338 | C2×C4 | — | 2 | 1 | no | — |
| 339 | C2×C4 | — | 2 | 1 | no | — |
| 340 | C2×C4 | — | 2 | 3 | no | — |
| 341 | C2×C4 | — | 2 | 24 | no | — |
| 342 | C2×C4 | — | 2 | 3 | no | — |
| 343 | C2×C4 | — | 2 | 2 | no | — |
| 344 | C2×C4 | — | 2 | 3 | no | — |
| 345 | C2×C4 | — | 2 | 3 | no | — |
| 346 | C2×C4 | — | 4 | 12 | no | — |
| 347 | C2×C4 | — | 4 | 1 | no | — |
| 348 | C2×C4 | — | 4 | 1 | no | — |
| 349 | C2×C4 | — | 4 | 6 | no | — |
| 350 | C2×C4 | — | 4 | 6 | no | — |
| 351 | C2×C4 | — | 4 | 6 | no | — |
| 352 | C2×C4 | — | 4 | 1 | no | — |
| 353 | C2×C4 | — | 4 | 6 | no | — |
| 354 | C2×C4 | — | 4 | 3 | no | — |
| 355 | C2×C4 | — | 4 | 2 | no | — |
| 356 | C2×C4 | — | 4 | 2 | no | — |
| 357 | C2×C4 | — | 4 | 3 | no | — |
| 358 | C2×C4 | — | 4 | 6 | no | — |
| 359 | C2×C4 | — | 4 | 1 | no | — |
| 360 | C2×C4 | — | 4 | 1 | no | — |
| 361 | C2×C4 | — | 4 | 3 | no | — |
| 362 | C2×C4 | — | 4 | 2 | no | — |
| 363 | C2×C4 | — | 8 | 12 | no | — |
| 364 | C2×C4 | — | 16 | 12 | no | — |
| 365 | C2×C4 | C2×#123 | 1 | 6 | yes | — |
| 366 | C2×C4 | C2×#123 | 1 | 6 | yes | — |
| 367 | C2×C4 | C22×D8 | 1 | 2 | yes | — |
| 368 | C2×C4 | C22×D8 | 1 | 6 | yes | — |
| 369 | C2×C4 | C2×#137 | 1 | 6 | yes | — |
| 370 | C2×C4 | #577 | 1 | 2 | yes | — |
| 371 | C2×C4 | C22×#41 | 1 | 2 | yes | — |
| 372 | C2×C4 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 373 | C2×C4 | #585 | 1 | 2 | yes | — |
| 374 | C2×C4 | #585 | 1 | 1 | yes | characteristic |
| 375 | C8 | — | 4 | 6 | no | — |
| 376 | C8 | — | 4 | 2 | no | — |
| 377 | C22 | — | 2 | 6 | no | — |
| 378 | C22 | — | 2 | 6 | no | — |
| 379 | C22 | — | 2 | 6 | no | — |
| 380 | C22 | — | 4 | 3 | no | — |
| 381 | C22 | — | 4 | 3 | no | — |
| 382 | C22 | — | 4 | 3 | no | — |
| 383 | C22 | — | 4 | 3 | no | — |
| 384 | C22 | — | 8 | 3 | no | — |
| 385 | C22 | — | 8 | 3 | no | — |
| 386 | C22 | — | 8 | 3 | no | — |
| 387 | C22 | — | 8 | 3 | no | — |
| 388 | C22 | — | 8 | 3 | no | — |
| 389 | C22 | — | 8 | 3 | no | — |
| 390 | C22 | — | 8 | 3 | no | — |
| 391 | C22 | C22×#123 | 1 | 1 | yes | characteristic |
| 392 | C22 | #3488 | 1 | 1 | yes | characteristic |
| 393 | C22 | #3500 | 1 | 1 | yes | characteristic |
| 394 | C22 | C23×D8 | 1 | 1 | yes | characteristic |
| 395 | C22 | C22×#137 | 1 | 1 | yes | characteristic |
| 396 | C22 | C2×#577 | 1 | 1 | yes | characteristic |
| 397 | C22 | #3585 | 1 | 1 | yes | characteristic |
| 398 | C4 | — | 2 | 2 | no | — |
| 399 | C4 | — | 2 | 2 | no | — |
| 400 | C4 | — | 2 | 6 | no | — |
| 401 | C4 | — | 4 | 3 | no | — |
| 402 | C4 | — | 4 | 2 | no | — |
| 403 | C4 | — | 8 | 2 | no | — |
| 404 | C4 | — | 8 | 3 | no | — |
| 405 | C4 | — | 8 | 1 | no | — |
| 406 | C4 | — | 8 | 6 | no | — |
| 407 | C4 | C2×#480 | 1 | 12 | yes | — |
| 408 | C4 | #3517 | 1 | 2 | yes | — |
| 409 | C4 | #3524 | 1 | 2 | yes | — |
| 410 | C2 | — | 4 | 6 | no | — |
| 411 | C2 | — | 16 | 3 | no | — |
| 412 | C2 | #36076 | 1 | 1 | yes | characteristic |
| 413 | C2 | C22×#480 | 1 | 1 | yes | characteristic |
| 414 | C2 | #60525 | 1 | 1 | yes | characteristic |
| 415 | C2 | #61947 | 1 | 1 | yes | characteristic |
| 416 | C2 | #62002 | 1 | 1 | yes | characteristic |
| 417 | C2 | #62529 | 1 | 1 | yes | characteristic |
| 418 | C2 | #62594 | 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. #34183 | C2 | semidirect |
| 3. #29920 | C2 | semidirect |
| 4. #27973 | C2 | semidirect |
| 5. #27644 | C2 | semidirect |
| 6. #26230 | C2 | semidirect |
| 7. #24965 | C2 | semidirect |
| 8. #23754 | C2 | semidirect |
| 9. #22740 | C2 | semidirect |
| 10. #22540 | C2 | semidirect |
| 11. #22530 | C2 | semidirect |
| 327. Q8 | #480 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #36455 | 2 |
| #34183 | 2 |
| #29920 | 2 |
| #27973 | 2 |
| #27644 | 2 |
| #26230 | 2 |
| #24965 | 2 |
| #23754 | 2 |
| #22740 | 2 |
| #22540 | 2 |
| #22530 | 2 |