Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
418 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #31077 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #13568 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #13564 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #13053 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #13052 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #8918 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #7503 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #3018 | — | 2 | 1 | no | — |
| 9 | #3018 | — | 2 | 1 | no | — |
| 10 | #3018 | C22 | 1 | 1 | yes | characteristic |
| 11 | #3018 | C22 | 1 | 1 | yes | characteristic |
| 12 | #2980 | C22 | 1 | 1 | yes | characteristic |
| 13 | #2132 | C22 | 1 | 1 | yes | characteristic |
| 14 | #2126 | C22 | 1 | 1 | yes | characteristic |
| 15 | #2050 | — | 2 | 1 | no | — |
| 16 | #2050 | — | 2 | 1 | no | — |
| 17 | #2050 | — | 2 | 1 | no | — |
| 18 | #1963 | — | 2 | 1 | no | — |
| 19 | #1963 | C22 | 1 | 1 | yes | characteristic |
| 20 | #1794 | — | 2 | 1 | no | — |
| 21 | #1791 | — | 2 | 1 | no | — |
| 22 | #1569 | — | 2 | 1 | no | — |
| 23 | #1525 | C22 | 1 | 1 | yes | characteristic |
| 24 | #1404 | — | 2 | 1 | no | — |
| 25 | #1404 | — | 2 | 1 | no | — |
| 26 | #1403 | — | 2 | 1 | no | — |
| 27 | #1403 | — | 2 | 1 | no | — |
| 28 | #1395 | — | 2 | 1 | no | — |
| 29 | #1392 | — | 2 | 1 | no | — |
| 30 | #1300 | — | 2 | 1 | no | — |
| 31 | #585 | D4 | 1 | 1 | yes | characteristic |
| 32 | #585 | D4 | 1 | 1 | yes | characteristic |
| 33 | #575 | — | 2 | 1 | no | — |
| 34 | #575 | — | 2 | 1 | no | — |
| 35 | #575 | — | 2 | 1 | no | — |
| 36 | #532 | — | 2 | 1 | no | — |
| 37 | #532 | C23 | 1 | 1 | yes | characteristic |
| 38 | #456 | — | 2 | 1 | no | — |
| 39 | #456 | — | 2 | 1 | no | — |
| 40 | #456 | — | 4 | 1 | no | — |
| 41 | #456 | — | 4 | 1 | no | — |
| 42 | #455 | — | 2 | 1 | no | — |
| 43 | #455 | — | 2 | 1 | no | — |
| 44 | #455 | — | 4 | 1 | no | — |
| 45 | #455 | — | 4 | 1 | no | — |
| 46 | #454 | — | 2 | 1 | no | — |
| 47 | #454 | — | 2 | 1 | no | — |
| 48 | #454 | — | 4 | 1 | no | — |
| 49 | #454 | — | 4 | 1 | no | — |
| 50 | #453 | — | 2 | 1 | no | — |
| 51 | #453 | — | 2 | 1 | no | — |
| 52 | #453 | — | 4 | 1 | no | — |
| 53 | #453 | — | 4 | 1 | no | — |
| 54 | #435 | — | 4 | 1 | no | — |
| 55 | #435 | — | 4 | 1 | no | — |
| 56 | #432 | — | 2 | 1 | no | — |
| 57 | #423 | — | 2 | 1 | no | — |
| 58 | #423 | D4 | 1 | 1 | yes | characteristic |
| 59 | #421 | — | 4 | 1 | no | — |
| 60 | C2×#104 | — | 2 | 1 | no | — |
| 61 | C2×#104 | — | 2 | 1 | no | — |
| 62 | C2×#104 | — | 2 | 1 | no | — |
| 63 | C2×#104 | — | 2 | 1 | no | — |
| 64 | C2×#104 | — | 2 | 1 | no | — |
| 65 | C2×#104 | — | 2 | 1 | no | — |
| 66 | #416 | — | 4 | 1 | no | — |
| 67 | #413 | — | 2 | 1 | no | — |
| 68 | #413 | — | 2 | 1 | no | — |
| 69 | #413 | — | 2 | 1 | no | — |
| 70 | #413 | — | 4 | 1 | no | — |
| 71 | #413 | D4 | 1 | 1 | yes | characteristic |
| 72 | #413 | D4 | 1 | 1 | yes | characteristic |
| 73 | #410 | — | 2 | 1 | no | — |
| 74 | #408 | — | 2 | 1 | no | — |
| 75 | #408 | — | 2 | 1 | no | — |
| 76 | #408 | — | 4 | 1 | no | — |
| 77 | C4×M16 | — | 2 | 1 | no | — |
| 78 | #376 | D4 | 1 | 1 | yes | characteristic |
| 79 | #356 | — | 2 | 1 | no | — |
| 80 | #355 | — | 2 | 1 | no | — |
| 81 | #354 | — | 2 | 1 | no | — |
| 82 | #353 | — | 2 | 1 | no | — |
| 83 | #338 | — | 2 | 1 | no | — |
| 84 | #335 | — | 4 | 1 | no | — |
| 85 | #334 | — | 4 | 1 | no | — |
| 86 | #329 | — | 2 | 1 | no | — |
| 87 | #329 | D4 | 1 | 1 | yes | characteristic, complemented |
| 88 | #329 | D4 | 1 | 1 | yes | characteristic, complemented |
| 89 | 2−1+4 | — | 2 | 1 | no | — |
| 90 | 2−1+4 | — | 2 | 1 | no | — |
| 91 | 2−1+4 | — | 4 | 1 | no | — |
| 92 | 2−1+4 | — | 4 | 1 | no | — |
| 93 | H(5,𝔽2) | — | 2 | 1 | no | — |
| 94 | H(5,𝔽2) | — | 2 | 1 | no | — |
| 95 | H(5,𝔽2) | — | 4 | 1 | no | — |
| 96 | H(5,𝔽2) | — | 4 | 1 | no | — |
| 97 | H(5,𝔽2) | — | 4 | 1 | no | — |
| 98 | H(5,𝔽2) | — | 4 | 1 | no | — |
| 99 | C2×#41 | — | 2 | 1 | no | — |
| 100 | C2×#41 | — | 2 | 1 | no | — |
| 101 | C2×#41 | — | 4 | 1 | no | — |
| 102 | C2×#41 | — | 4 | 1 | no | — |
| 103 | C2×#41 | — | 8 | 1 | no | — |
| 104 | C2×#41 | — | 8 | 1 | no | — |
| 105 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 106 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 107 | #137 | — | 4 | 1 | no | — |
| 108 | #137 | — | 4 | 1 | no | — |
| 109 | #137 | — | 4 | 1 | no | — |
| 110 | #137 | — | 4 | 1 | no | — |
| 111 | #137 | — | 4 | 1 | no | — |
| 112 | #137 | — | 4 | 1 | no | — |
| 113 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 114 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 115 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 116 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 117 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 118 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 119 | #135 | — | 8 | 1 | no | — |
| 120 | #135 | — | 8 | 1 | no | — |
| 121 | #135 | — | 8 | 1 | no | — |
| 122 | C2×M16 | — | 2 | 1 | no | — |
| 123 | C2×M16 | — | 2 | 1 | no | — |
| 124 | C2×M16 | — | 2 | 1 | no | — |
| 125 | C2×M16 | — | 2 | 1 | no | — |
| 126 | C2×M16 | — | 2 | 1 | no | — |
| 127 | C2×M16 | — | 4 | 1 | no | — |
| 128 | C2×M16 | — | 4 | 1 | no | — |
| 129 | C22×C8 | — | 2 | 1 | no | — |
| 130 | C22×C8 | — | 2 | 1 | no | — |
| 131 | C4⋊Q8 | — | 2 | 1 | no | — |
| 132 | C4⋊Q8 | #41 | 1 | 1 | yes | characteristic |
| 133 | #127 | — | 2 | 1 | no | — |
| 134 | #127 | #41 | 1 | 1 | yes | characteristic |
| 135 | #124 | — | 2 | 1 | no | — |
| 136 | #124 | — | 2 | 1 | no | — |
| 137 | #124 | C2×D4 | 1 | 1 | yes | characteristic |
| 138 | #124 | #41 | 1 | 1 | yes | characteristic |
| 139 | #121 | — | 4 | 1 | no | — |
| 140 | #121 | — | 8 | 1 | no | — |
| 141 | C4×D4 | — | 4 | 1 | no | — |
| 142 | C4×D4 | — | 8 | 1 | no | — |
| 143 | #117 | — | 4 | 1 | no | — |
| 144 | C2×C42 | — | 2 | 1 | no | — |
| 145 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 146 | C4⋊C8 | — | 2 | 1 | no | — |
| 147 | C4⋊C8 | — | 2 | 1 | no | — |
| 148 | C4⋊C8 | — | 4 | 1 | no | — |
| 149 | C4⋊C8 | — | 8 | 1 | no | — |
| 150 | #104 | — | 4 | 1 | no | — |
| 151 | #104 | — | 4 | 1 | no | — |
| 152 | #104 | — | 4 | 1 | no | — |
| 153 | #104 | — | 4 | 1 | no | — |
| 154 | #104 | — | 4 | 1 | no | — |
| 155 | #104 | — | 4 | 1 | no | — |
| 156 | #104 | — | 4 | 1 | no | — |
| 157 | #104 | — | 4 | 1 | no | — |
| 158 | #104 | — | 4 | 1 | no | — |
| 159 | #104 | — | 4 | 1 | no | — |
| 160 | #104 | — | 4 | 1 | no | — |
| 161 | #104 | — | 4 | 1 | no | — |
| 162 | #104 | — | 8 | 2 | no | — |
| 163 | #103 | — | 8 | 1 | no | — |
| 164 | #102 | — | 8 | 1 | no | — |
| 165 | #101 | — | 2 | 1 | no | — |
| 166 | #101 | — | 2 | 1 | no | — |
| 167 | #101 | — | 4 | 1 | no | — |
| 168 | #101 | — | 4 | 1 | no | — |
| 169 | #101 | — | 4 | 1 | no | — |
| 170 | #101 | — | 8 | 1 | no | — |
| 171 | #100 | — | 2 | 1 | no | — |
| 172 | #100 | — | 2 | 1 | no | — |
| 173 | #100 | — | 4 | 1 | no | — |
| 174 | #100 | — | 4 | 1 | no | — |
| 175 | #100 | — | 4 | 1 | no | — |
| 176 | #100 | — | 8 | 1 | no | — |
| 177 | #99 | — | 4 | 1 | no | — |
| 178 | #99 | — | 4 | 1 | no | — |
| 179 | C22⋊C8 | — | 4 | 1 | no | — |
| 180 | C22⋊C8 | — | 4 | 1 | no | — |
| 181 | C22⋊C8 | — | 4 | 1 | no | — |
| 182 | C22⋊C8 | — | 8 | 2 | no | — |
| 183 | #97 | — | 2 | 1 | no | — |
| 184 | #97 | — | 2 | 1 | no | — |
| 185 | #97 | — | 4 | 1 | no | — |
| 186 | C4×C8 | — | 4 | 1 | no | — |
| 187 | #95 | — | 4 | 1 | no | — |
| 188 | #41 | — | 4 | 1 | no | — |
| 189 | #41 | — | 4 | 1 | no | — |
| 190 | #41 | — | 4 | 1 | no | — |
| 191 | #41 | — | 4 | 1 | no | — |
| 192 | #41 | — | 4 | 1 | no | — |
| 193 | #41 | — | 4 | 1 | no | — |
| 194 | #41 | — | 4 | 1 | no | — |
| 195 | #41 | — | 4 | 1 | no | — |
| 196 | #41 | — | 4 | 2 | no | — |
| 197 | #41 | — | 4 | 2 | no | — |
| 198 | #41 | — | 4 | 1 | no | — |
| 199 | #41 | — | 4 | 1 | no | — |
| 200 | #41 | — | 4 | 1 | no | — |
| 201 | #41 | — | 4 | 1 | no | — |
| 202 | #41 | — | 8 | 1 | no | — |
| 203 | #41 | — | 8 | 1 | no | — |
| 204 | #41 | — | 8 | 2 | no | — |
| 205 | #41 | — | 8 | 1 | no | — |
| 206 | #41 | — | 8 | 1 | no | — |
| 207 | #41 | — | 8 | 2 | no | — |
| 208 | #41 | — | 8 | 1 | no | — |
| 209 | #41 | — | 8 | 1 | no | — |
| 210 | #41 | — | 8 | 1 | no | — |
| 211 | #41 | — | 8 | 1 | no | — |
| 212 | C2×Q8 | — | 2 | 1 | no | — |
| 213 | C2×Q8 | — | 2 | 1 | no | — |
| 214 | C2×Q8 | — | 4 | 1 | no | — |
| 215 | C2×Q8 | — | 4 | 1 | no | — |
| 216 | C2×Q8 | — | 8 | 1 | no | — |
| 217 | C2×Q8 | — | 8 | 1 | no | — |
| 218 | C2×Q8 | #121 | 1 | 1 | yes | characteristic |
| 219 | C2×Q8 | #123 | 1 | 1 | yes | characteristic |
| 220 | C2×D4 | — | 2 | 1 | no | — |
| 221 | C2×D4 | — | 2 | 1 | no | — |
| 222 | C2×D4 | — | 2 | 1 | no | — |
| 223 | C2×D4 | — | 2 | 1 | no | — |
| 224 | C2×D4 | — | 4 | 1 | no | — |
| 225 | C2×D4 | — | 4 | 1 | no | — |
| 226 | C2×D4 | — | 4 | 1 | no | — |
| 227 | C2×D4 | — | 4 | 2 | no | — |
| 228 | C2×D4 | — | 4 | 1 | no | — |
| 229 | C2×D4 | — | 4 | 1 | no | — |
| 230 | C2×D4 | — | 8 | 1 | no | — |
| 231 | C2×D4 | — | 8 | 1 | no | — |
| 232 | C2×D4 | — | 8 | 1 | no | — |
| 233 | C2×D4 | — | 8 | 2 | no | — |
| 234 | C2×D4 | — | 8 | 2 | no | — |
| 235 | C2×D4 | — | 8 | 1 | no | — |
| 236 | C2×D4 | #121 | 1 | 1 | yes | characteristic |
| 237 | C2×D4 | #123 | 1 | 1 | yes | characteristic |
| 238 | C22×C4 | — | 2 | 1 | no | — |
| 239 | C22×C4 | — | 2 | 1 | no | — |
| 240 | C22×C4 | — | 2 | 1 | no | — |
| 241 | C22×C4 | — | 4 | 1 | no | — |
| 242 | C22×C4 | — | 4 | 1 | no | — |
| 243 | C22×C4 | — | 4 | 2 | no | — |
| 244 | C22×C4 | — | 4 | 1 | no | — |
| 245 | C22×C4 | — | 8 | 1 | no | — |
| 246 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 247 | Q16 | — | 8 | 1 | no | — |
| 248 | Q16 | — | 8 | 1 | no | — |
| 249 | Q16 | — | 8 | 1 | no | — |
| 250 | SD16 | — | 8 | 1 | no | — |
| 251 | SD16 | — | 8 | 1 | no | — |
| 252 | SD16 | — | 8 | 1 | no | — |
| 253 | SD16 | — | 8 | 1 | no | — |
| 254 | SD16 | — | 8 | 1 | no | — |
| 255 | SD16 | — | 8 | 1 | no | — |
| 256 | D8 | — | 8 | 1 | no | — |
| 257 | D8 | — | 8 | 1 | no | — |
| 258 | D8 | — | 8 | 1 | no | — |
| 259 | M16 | — | 4 | 1 | no | — |
| 260 | M16 | — | 4 | 1 | no | — |
| 261 | M16 | — | 4 | 1 | no | — |
| 262 | M16 | — | 4 | 1 | no | — |
| 263 | M16 | — | 4 | 1 | no | — |
| 264 | M16 | — | 4 | 1 | no | — |
| 265 | M16 | — | 8 | 2 | no | — |
| 266 | M16 | — | 8 | 1 | no | — |
| 267 | M16 | — | 8 | 1 | no | — |
| 268 | M16 | — | 8 | 1 | no | — |
| 269 | M16 | — | 8 | 1 | no | — |
| 270 | C2×C8 | — | 4 | 1 | no | — |
| 271 | C2×C8 | — | 4 | 1 | no | — |
| 272 | C2×C8 | — | 4 | 1 | no | — |
| 273 | C2×C8 | — | 4 | 1 | no | — |
| 274 | C2×C8 | — | 4 | 1 | no | — |
| 275 | C2×C8 | — | 4 | 1 | no | — |
| 276 | C2×C8 | — | 4 | 1 | no | — |
| 277 | C2×C8 | — | 8 | 1 | no | — |
| 278 | C2×C8 | — | 8 | 1 | no | — |
| 279 | C2×C8 | — | 8 | 1 | no | — |
| 280 | C4⋊C4 | — | 4 | 1 | no | — |
| 281 | C4⋊C4 | — | 8 | 1 | no | — |
| 282 | C4⋊C4 | — | 8 | 1 | no | — |
| 283 | C22⋊C4 | — | 4 | 1 | no | — |
| 284 | C22⋊C4 | — | 4 | 1 | no | — |
| 285 | C22⋊C4 | — | 8 | 1 | no | — |
| 286 | C22⋊C4 | — | 8 | 1 | no | — |
| 287 | C22⋊C4 | — | 8 | 1 | no | — |
| 288 | C42 | — | 2 | 1 | no | — |
| 289 | C42 | — | 2 | 1 | no | — |
| 290 | C42 | — | 2 | 1 | no | — |
| 291 | C42 | — | 4 | 1 | no | — |
| 292 | C42 | — | 8 | 1 | no | — |
| 293 | C42 | #121 | 1 | 1 | yes | characteristic |
| 294 | C42 | #124 | 1 | 1 | yes | characteristic |
| 295 | C23 | — | 2 | 1 | no | — |
| 296 | C23 | — | 2 | 1 | no | — |
| 297 | C23 | — | 4 | 1 | no | — |
| 298 | C23 | — | 4 | 1 | no | — |
| 299 | C23 | — | 4 | 2 | no | — |
| 300 | C23 | — | 8 | 1 | no | — |
| 301 | C23 | #458 | 1 | 1 | yes | characteristic |
| 302 | Q8 | — | 4 | 1 | no | — |
| 303 | Q8 | — | 4 | 1 | no | — |
| 304 | Q8 | — | 4 | 1 | no | — |
| 305 | Q8 | — | 4 | 1 | no | — |
| 306 | Q8 | — | 4 | 2 | no | — |
| 307 | Q8 | — | 4 | 1 | no | — |
| 308 | Q8 | — | 4 | 1 | no | — |
| 309 | Q8 | — | 8 | 1 | no | — |
| 310 | D4 | — | 4 | 1 | no | — |
| 311 | D4 | — | 4 | 1 | no | — |
| 312 | D4 | — | 4 | 1 | no | — |
| 313 | D4 | — | 4 | 1 | no | — |
| 314 | D4 | — | 4 | 2 | no | — |
| 315 | D4 | — | 4 | 1 | no | — |
| 316 | D4 | — | 4 | 1 | no | — |
| 317 | D4 | — | 8 | 1 | no | — |
| 318 | D4 | — | 8 | 1 | no | — |
| 319 | D4 | — | 8 | 1 | no | — |
| 320 | D4 | — | 8 | 1 | no | — |
| 321 | D4 | — | 8 | 1 | no | — |
| 322 | D4 | — | 8 | 2 | no | — |
| 323 | D4 | — | 8 | 2 | no | — |
| 324 | D4 | — | 8 | 1 | no | — |
| 325 | D4 | — | 8 | 1 | no | — |
| 326 | D4 | — | 16 | 1 | no | — |
| 327 | D4 | — | 16 | 2 | no | — |
| 328 | C2×C4 | — | 2 | 1 | no | — |
| 329 | C2×C4 | — | 2 | 1 | no | — |
| 330 | C2×C4 | — | 2 | 1 | no | — |
| 331 | C2×C4 | — | 2 | 1 | no | — |
| 332 | C2×C4 | — | 2 | 1 | no | — |
| 333 | C2×C4 | — | 2 | 1 | no | — |
| 334 | C2×C4 | — | 4 | 1 | no | — |
| 335 | C2×C4 | — | 4 | 1 | no | — |
| 336 | C2×C4 | — | 4 | 1 | no | — |
| 337 | C2×C4 | — | 4 | 1 | no | — |
| 338 | C2×C4 | — | 4 | 1 | no | — |
| 339 | C2×C4 | — | 4 | 1 | no | — |
| 340 | C2×C4 | — | 4 | 1 | no | — |
| 341 | C2×C4 | — | 4 | 1 | no | — |
| 342 | C2×C4 | — | 4 | 2 | no | — |
| 343 | C2×C4 | — | 8 | 1 | no | — |
| 344 | C2×C4 | — | 8 | 1 | no | — |
| 345 | C2×C4 | — | 8 | 1 | no | — |
| 346 | C2×C4 | — | 8 | 2 | no | — |
| 347 | C2×C4 | — | 8 | 2 | no | — |
| 348 | C2×C4 | — | 8 | 1 | no | — |
| 349 | C2×C4 | — | 8 | 1 | no | — |
| 350 | C2×C4 | — | 8 | 1 | no | — |
| 351 | C2×C4 | — | 16 | 1 | no | — |
| 352 | C2×C4 | — | 16 | 1 | no | — |
| 353 | C2×C4 | #394 | 1 | 1 | yes | characteristic |
| 354 | C2×C4 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 355 | C8 | — | 4 | 2 | no | — |
| 356 | C8 | — | 8 | 1 | no | — |
| 357 | C8 | — | 8 | 1 | no | — |
| 358 | C8 | — | 8 | 1 | no | — |
| 359 | C8 | — | 8 | 1 | no | — |
| 360 | C8 | — | 8 | 1 | no | — |
| 361 | C8 | — | 8 | 1 | no | — |
| 362 | C8 | — | 16 | 1 | no | — |
| 363 | C22 | — | 2 | 1 | no | — |
| 364 | C22 | — | 4 | 1 | no | — |
| 365 | C22 | — | 4 | 1 | no | — |
| 366 | C22 | — | 4 | 1 | no | — |
| 367 | C22 | — | 8 | 1 | no | — |
| 368 | C22 | — | 8 | 1 | no | — |
| 369 | C22 | — | 8 | 1 | no | — |
| 370 | C22 | — | 8 | 1 | no | — |
| 371 | C22 | — | 16 | 1 | no | — |
| 372 | C22 | — | 16 | 1 | no | — |
| 373 | C22 | — | 16 | 2 | no | — |
| 374 | C22 | — | 32 | 1 | no | — |
| 375 | C22 | #2044 | 1 | 1 | yes | characteristic |
| 376 | C4 | — | 2 | 1 | no | — |
| 377 | C4 | — | 4 | 1 | no | — |
| 378 | C4 | — | 4 | 1 | no | — |
| 379 | C4 | — | 4 | 1 | no | — |
| 380 | C4 | — | 4 | 1 | no | — |
| 381 | C4 | — | 8 | 1 | no | — |
| 382 | C4 | — | 8 | 1 | no | — |
| 383 | C4 | — | 8 | 1 | no | — |
| 384 | C4 | — | 8 | 1 | no | — |
| 385 | C4 | — | 32 | 1 | no | — |
| 386 | C4 | #2196 | 1 | 1 | yes | characteristic |
| 387 | C4 | #2199 | 1 | 1 | yes | characteristic |
| 388 | C2 | — | 2 | 1 | no | — |
| 389 | C2 | — | 4 | 1 | no | — |
| 390 | C2 | — | 8 | 1 | no | — |
| 391 | C2 | — | 8 | 1 | no | — |
| 392 | C2 | — | 16 | 1 | no | — |
| 393 | C2 | — | 16 | 1 | no | — |
| 394 | C2 | #13419 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2199 |
| commutator, Frattini, cosocle | #532 | C23 |
| socle | C2 | #13419 |
| prime radical | #31077 | C2 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13568 | C2 | semidirect |
| 3. #13564 | C2 | semidirect |
| 4. #13053 | C2 | semidirect |
| 5. #13052 | C2 | semidirect |
| 6. #8918 | C2 | semidirect |
| 7. #7503 | C2 | semidirect |
| 87. #329 | D4 | semidirect |
| 88. #329 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #31077 | 2 |
| #13568 | 2 |
| #13564 | 2 |
| #13053 | 2 |
| #13052 | 2 |
| #8918 | 2 |
| #7503 | 2 |