Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1312 conjugacy classes of subgroups.
The subgroup lattice has 358 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #62416 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | Q16×#41 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #33962 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #33930 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #33866 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | D4×Q32 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #33815 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #33775 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | #33664 | C2 | 1 | 3 | yes | maximal, complemented |
| 10 | C2×#2209 | C2 | 1 | 3 | yes | maximal, complemented |
| 11 | #33502 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #3503 | — | 2 | 1 | no | — |
| 13 | Q8×#41 | — | 2 | 1 | no | — |
| 14 | C22×Q32 | C22 | 1 | 6 | yes | — |
| 15 | #3395 | — | 2 | 1 | no | — |
| 16 | #3395 | C22 | 1 | 2 | yes | — |
| 17 | #3395 | C22 | 1 | 1 | yes | characteristic |
| 18 | Q8×Q16 | C22 | 1 | 1 | yes | characteristic |
| 19 | #3363 | — | 2 | 1 | no | — |
| 20 | #3338 | C22 | 1 | 3 | yes | — |
| 21 | #3299 | — | 2 | 3 | no | — |
| 22 | D4×Q16 | — | 2 | 3 | no | — |
| 23 | D4×Q16 | C22 | 1 | 3 | yes | — |
| 24 | D4×Q16 | C22 | 1 | 6 | yes | — |
| 25 | #3209 | — | 2 | 3 | no | — |
| 26 | #3170 | C22 | 1 | 3 | yes | — |
| 27 | #3170 | C22 | 1 | 3 | yes | — |
| 28 | C2×#494 | C22 | 1 | 3 | yes | — |
| 29 | #3104 | — | 2 | 3 | no | — |
| 30 | C8×#41 | C22 | 1 | 1 | yes | characteristic |
| 31 | C2×C4×Q16 | C22 | 1 | 3 | yes | — |
| 32 | #2247 | C22 | 1 | 6 | yes | — |
| 33 | #2241 | C22 | 1 | 6 | yes | — |
| 34 | #2236 | C22 | 1 | 6 | yes | — |
| 35 | #2227 | C22 | 1 | 3 | yes | — |
| 36 | #2227 | C22 | 1 | 6 | yes | — |
| 37 | #2227 | C22 | 1 | 1 | yes | characteristic |
| 38 | #2216 | C22 | 1 | 12 | yes | — |
| 39 | #2209 | C22 | 1 | 6 | yes | — |
| 40 | #2209 | C22 | 1 | 6 | yes | — |
| 41 | #2209 | C22 | 1 | 3 | yes | — |
| 42 | #2209 | C22 | 1 | 6 | yes | — |
| 43 | #2209 | C22 | 1 | 6 | yes | — |
| 44 | #2209 | C22 | 1 | 3 | yes | — |
| 45 | #2188 | C22 | 1 | 6 | yes | — |
| 46 | #2188 | C22 | 1 | 12 | yes | — |
| 47 | Q8×C16 | C22 | 1 | 2 | yes | — |
| 48 | C4×Q32 | C22 | 1 | 6 | yes | — |
| 49 | C4×Q32 | C22 | 1 | 6 | yes | — |
| 50 | D4×C16 | C22 | 1 | 6 | yes | — |
| 51 | #2162 | C22 | 1 | 3 | yes | — |
| 52 | C2×C4⋊C16 | C22 | 1 | 3 | yes | — |
| 53 | C2×#358 | C22 | 1 | 6 | yes | — |
| 54 | C22×Q16 | — | 2 | 3 | no | — |
| 55 | C22×Q16 | C23 | 1 | 3 | yes | — |
| 56 | C22×Q16 | C23 | 1 | 6 | yes | — |
| 57 | Q82 | — | 2 | 1 | no | — |
| 58 | Q82 | — | 4 | 1 | no | — |
| 59 | #557 | — | 2 | 1 | no | — |
| 60 | #557 | — | 2 | 2 | no | — |
| 61 | #557 | — | 4 | 1 | no | — |
| 62 | #554 | — | 2 | 3 | no | — |
| 63 | #554 | — | 4 | 3 | no | — |
| 64 | #554 | — | 4 | 3 | no | — |
| 65 | D4×Q8 | — | 2 | 3 | no | — |
| 66 | D4×Q8 | — | 2 | 6 | no | — |
| 67 | #544 | — | 4 | 3 | no | — |
| 68 | #533 | — | 2 | 3 | no | — |
| 69 | #533 | — | 2 | 3 | no | — |
| 70 | #533 | — | 4 | 3 | no | — |
| 71 | C2×C4⋊Q8 | — | 2 | 3 | no | — |
| 72 | #520 | — | 4 | 1 | no | — |
| 73 | C4×#41 | — | 4 | 1 | no | — |
| 74 | C4×#41 | D4 | 1 | 1 | yes | characteristic |
| 75 | C2×C4×Q8 | — | 2 | 3 | no | — |
| 76 | C2×Q32 | — | 4 | 12 | no | — |
| 77 | C2×Q32 | C23 | 1 | 6 | yes | — |
| 78 | C2×Q32 | C23 | 1 | 12 | yes | — |
| 79 | C2×Q32 | C23 | 1 | 6 | yes | — |
| 80 | C22×C16 | C23 | 1 | 6 | yes | — |
| 81 | #500 | C23 | 1 | 3 | yes | — |
| 82 | #500 | C23 | 1 | 1 | yes | characteristic |
| 83 | #494 | — | 2 | 3 | no | — |
| 84 | #494 | — | 2 | 3 | no | — |
| 85 | #494 | C23 | 1 | 6 | yes | — |
| 86 | #494 | C23 | 1 | 3 | yes | — |
| 87 | #494 | C23 | 1 | 3 | yes | — |
| 88 | #487 | — | 2 | 3 | no | — |
| 89 | #487 | — | 2 | 3 | no | — |
| 90 | #484 | — | 2 | 6 | no | — |
| 91 | #477 | — | 2 | 3 | no | — |
| 92 | #477 | — | 2 | 3 | no | — |
| 93 | #467 | — | 2 | 3 | no | — |
| 94 | #467 | — | 2 | 3 | no | — |
| 95 | #467 | C23 | 1 | 6 | yes | — |
| 96 | #467 | C23 | 1 | 6 | yes | — |
| 97 | #462 | — | 2 | 3 | no | — |
| 98 | #462 | — | 2 | 3 | no | — |
| 99 | #462 | — | 2 | 6 | no | — |
| 100 | #462 | — | 2 | 6 | no | — |
| 101 | #451 | — | 2 | 6 | no | — |
| 102 | #451 | — | 2 | 12 | no | — |
| 103 | C8×Q8 | D4 | 1 | 1 | yes | characteristic |
| 104 | C8×Q8 | C23 | 1 | 1 | yes | characteristic |
| 105 | C4×Q16 | — | 2 | 3 | no | — |
| 106 | C4×Q16 | — | 2 | 3 | no | — |
| 107 | C4×Q16 | — | 2 | 3 | no | — |
| 108 | C4×Q16 | — | 2 | 3 | no | — |
| 109 | C4×Q16 | C23 | 1 | 3 | yes | — |
| 110 | C4×Q16 | C23 | 1 | 3 | yes | — |
| 111 | C4×Q16 | C23 | 1 | 6 | yes | — |
| 112 | C4×Q16 | C23 | 1 | 3 | yes | — |
| 113 | C4×Q16 | C23 | 1 | 1 | yes | characteristic |
| 114 | C8×D4 | D4 | 1 | 3 | yes | — |
| 115 | C8×D4 | C23 | 1 | 3 | yes | — |
| 116 | #431 | D4 | 1 | 3 | yes | — |
| 117 | C2×#107 | C23 | 1 | 3 | yes | — |
| 118 | C2×#103 | — | 2 | 3 | no | — |
| 119 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 120 | #366 | C23 | 1 | 2 | yes | — |
| 121 | #366 | C23 | 1 | 6 | yes | — |
| 122 | C4⋊C16 | C23 | 1 | 3 | yes | — |
| 123 | C4⋊C16 | C23 | 1 | 6 | yes | — |
| 124 | C4⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 125 | #358 | C23 | 1 | 6 | yes | — |
| 126 | #358 | C23 | 1 | 12 | yes | — |
| 127 | #358 | C23 | 1 | 2 | yes | — |
| 128 | #358 | C23 | 1 | 6 | yes | — |
| 129 | #358 | C23 | 1 | 6 | yes | — |
| 130 | C22⋊C16 | C23 | 1 | 6 | yes | — |
| 131 | C4×C16 | C23 | 1 | 6 | yes | — |
| 132 | C2×#41 | D8 | 1 | 1 | yes | characteristic |
| 133 | C22×Q8 | — | 2 | 3 | no | — |
| 134 | C22×Q8 | — | 2 | 6 | no | — |
| 135 | C2×Q16 | — | 2 | 6 | no | — |
| 136 | C2×Q16 | — | 2 | 3 | no | — |
| 137 | C2×Q16 | — | 2 | 3 | no | — |
| 138 | C2×Q16 | — | 2 | 6 | no | — |
| 139 | C2×Q16 | — | 2 | 6 | no | — |
| 140 | C2×Q16 | — | 2 | 3 | no | — |
| 141 | C2×Q16 | — | 4 | 3 | no | — |
| 142 | C2×Q16 | — | 4 | 6 | no | — |
| 143 | C2×Q16 | — | 4 | 6 | no | — |
| 144 | C2×Q16 | — | 8 | 6 | no | — |
| 145 | C2×Q16 | C24 | 1 | 6 | yes | — |
| 146 | C2×Q16 | C24 | 1 | 3 | yes | — |
| 147 | C2×Q16 | C24 | 1 | 1 | yes | characteristic |
| 148 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 149 | C22×C8 | C24 | 1 | 3 | yes | — |
| 150 | C4⋊Q8 | — | 2 | 1 | no | — |
| 151 | C4⋊Q8 | — | 2 | 3 | no | — |
| 152 | C4⋊Q8 | — | 2 | 3 | no | — |
| 153 | C4⋊Q8 | — | 2 | 6 | no | — |
| 154 | C4⋊Q8 | — | 2 | 3 | no | — |
| 155 | C4⋊Q8 | — | 4 | 3 | no | — |
| 156 | C4⋊Q8 | — | 4 | 3 | no | — |
| 157 | C4⋊Q8 | — | 4 | 6 | no | — |
| 158 | C4⋊Q8 | — | 4 | 3 | no | — |
| 159 | C4⋊Q8 | — | 4 | 3 | no | — |
| 160 | #125 | — | 4 | 3 | no | — |
| 161 | #125 | — | 4 | 3 | no | — |
| 162 | #125 | — | 4 | 6 | no | — |
| 163 | #125 | — | 4 | 3 | no | — |
| 164 | #125 | — | 4 | 3 | no | — |
| 165 | C22⋊Q8 | — | 2 | 6 | no | — |
| 166 | C22⋊Q8 | — | 2 | 6 | no | — |
| 167 | C22⋊Q8 | — | 4 | 3 | no | — |
| 168 | C22⋊Q8 | — | 4 | 3 | no | — |
| 169 | C22⋊Q8 | — | 4 | 6 | no | — |
| 170 | C22⋊Q8 | — | 4 | 6 | no | — |
| 171 | C22⋊Q8 | — | 4 | 12 | no | — |
| 172 | C4×Q8 | — | 2 | 1 | no | — |
| 173 | C4×Q8 | — | 2 | 3 | no | — |
| 174 | C4×Q8 | — | 2 | 3 | no | — |
| 175 | C4×Q8 | — | 2 | 3 | no | — |
| 176 | C4×Q8 | — | 2 | 6 | no | — |
| 177 | C4×Q8 | — | 4 | 1 | no | — |
| 178 | C4×Q8 | — | 4 | 1 | no | — |
| 179 | C4×Q8 | — | 4 | 2 | no | — |
| 180 | C4×Q8 | — | 4 | 3 | no | — |
| 181 | C4×Q8 | — | 4 | 3 | no | — |
| 182 | C4×Q8 | — | 4 | 3 | no | — |
| 183 | C4×Q8 | — | 4 | 3 | no | — |
| 184 | C4×Q8 | D8 | 1 | 1 | yes | characteristic |
| 185 | C4×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 186 | C4×D4 | — | 4 | 3 | no | — |
| 187 | C4×D4 | — | 4 | 3 | no | — |
| 188 | C4×D4 | — | 4 | 6 | no | — |
| 189 | C4×D4 | D8 | 1 | 3 | yes | — |
| 190 | C4×D4 | C2×D4 | 1 | 3 | yes | — |
| 191 | #117 | — | 4 | 3 | no | — |
| 192 | #117 | — | 4 | 3 | no | — |
| 193 | #117 | D8 | 1 | 3 | yes | — |
| 194 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 195 | C2×C4⋊C4 | — | 4 | 3 | no | — |
| 196 | C2×C4⋊C4 | — | 4 | 3 | no | — |
| 197 | C2×C4⋊C4 | — | 4 | 3 | no | — |
| 198 | C2×C42 | — | 4 | 3 | no | — |
| 199 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 200 | Q32 | — | 2 | 12 | no | — |
| 201 | Q32 | — | 4 | 12 | no | — |
| 202 | Q32 | — | 4 | 6 | no | — |
| 203 | C2×C16 | — | 4 | 6 | no | — |
| 204 | C2×C16 | C24 | 1 | 2 | yes | — |
| 205 | C2×C16 | C24 | 1 | 6 | yes | — |
| 206 | #107 | — | 2 | 3 | no | — |
| 207 | #107 | — | 2 | 1 | no | — |
| 208 | #107 | — | 2 | 3 | no | — |
| 209 | #107 | C24 | 1 | 1 | yes | characteristic |
| 210 | #107 | C24 | 1 | 3 | yes | — |
| 211 | #107 | C24 | 1 | 2 | yes | — |
| 212 | C4⋊C8 | C2×D4 | 1 | 3 | yes | — |
| 213 | C4⋊C8 | C2×D4 | 1 | 3 | yes | — |
| 214 | #103 | — | 2 | 1 | no | — |
| 215 | #103 | — | 2 | 3 | no | — |
| 216 | #103 | — | 2 | 3 | no | — |
| 217 | #103 | — | 2 | 3 | no | — |
| 218 | #103 | — | 2 | 6 | no | — |
| 219 | #103 | — | 2 | 6 | no | — |
| 220 | #103 | — | 2 | 6 | no | — |
| 221 | C22⋊C8 | C2×D4 | 1 | 6 | yes | — |
| 222 | C4×C8 | C2×D4 | 1 | 3 | yes | — |
| 223 | C4×C8 | C2×D4 | 1 | 3 | yes | — |
| 224 | C4×C8 | C24 | 1 | 3 | yes | — |
| 225 | C4×C8 | C24 | 1 | 1 | yes | characteristic |
| 226 | #41 | Q32 | 1 | 8 | yes | complemented |
| 227 | C2×Q8 | — | 2 | 1 | no | — |
| 228 | C2×Q8 | — | 2 | 3 | no | — |
| 229 | C2×Q8 | — | 2 | 6 | no | — |
| 230 | C2×Q8 | — | 4 | 6 | no | — |
| 231 | C2×Q8 | — | 4 | 1 | no | — |
| 232 | C2×Q8 | — | 4 | 3 | no | — |
| 233 | C2×Q8 | — | 4 | 3 | no | — |
| 234 | C2×Q8 | — | 4 | 6 | no | — |
| 235 | C2×Q8 | — | 8 | 3 | no | — |
| 236 | C2×Q8 | — | 8 | 6 | no | — |
| 237 | C2×Q8 | — | 8 | 6 | no | — |
| 238 | C2×Q8 | C2×D8 | 1 | 1 | yes | characteristic |
| 239 | C2×D4 | C2×D8 | 1 | 3 | yes | — |
| 240 | C22×C4 | — | 4 | 3 | no | — |
| 241 | C22×C4 | — | 4 | 3 | no | — |
| 242 | C22×C4 | — | 4 | 6 | no | — |
| 243 | C22×C4 | C2×D8 | 1 | 3 | yes | — |
| 244 | C22×C4 | C2×D8 | 1 | 3 | yes | — |
| 245 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 246 | Q16 | — | 2 | 6 | no | — |
| 247 | Q16 | — | 2 | 3 | no | — |
| 248 | Q16 | — | 2 | 1 | no | — |
| 249 | Q16 | — | 2 | 3 | no | — |
| 250 | Q16 | — | 2 | 6 | no | — |
| 251 | Q16 | — | 4 | 6 | no | — |
| 252 | Q16 | — | 4 | 6 | no | — |
| 253 | Q16 | — | 8 | 3 | no | — |
| 254 | Q16 | — | 8 | 6 | no | — |
| 255 | Q16 | — | 8 | 6 | no | — |
| 256 | Q16 | H(5,𝔽2) | 1 | 2 | yes | — |
| 257 | C2×C8 | — | 2 | 3 | no | — |
| 258 | C2×C8 | — | 2 | 3 | no | — |
| 259 | C2×C8 | — | 4 | 3 | no | — |
| 260 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 261 | C2×C8 | C22×D4 | 1 | 3 | yes | — |
| 262 | C2×C8 | C22×D4 | 1 | 3 | yes | — |
| 263 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 264 | C4⋊C4 | — | 2 | 2 | no | — |
| 265 | C4⋊C4 | — | 2 | 1 | no | — |
| 266 | C4⋊C4 | — | 2 | 3 | no | — |
| 267 | C4⋊C4 | — | 4 | 3 | no | — |
| 268 | C4⋊C4 | — | 4 | 1 | no | — |
| 269 | C4⋊C4 | — | 4 | 3 | no | — |
| 270 | C4⋊C4 | — | 4 | 3 | no | — |
| 271 | C4⋊C4 | — | 4 | 6 | no | — |
| 272 | C4⋊C4 | — | 4 | 3 | no | — |
| 273 | C4⋊C4 | — | 4 | 3 | no | — |
| 274 | C4⋊C4 | — | 4 | 3 | no | — |
| 275 | C4⋊C4 | — | 4 | 6 | no | — |
| 276 | C4⋊C4 | — | 4 | 1 | no | — |
| 277 | C4⋊C4 | — | 4 | 3 | no | — |
| 278 | C4⋊C4 | — | 4 | 3 | no | — |
| 279 | C4⋊C4 | — | 4 | 6 | no | — |
| 280 | C4⋊C4 | — | 4 | 6 | no | — |
| 281 | C4⋊C4 | — | 4 | 6 | no | — |
| 282 | C4⋊C4 | — | 4 | 3 | no | — |
| 283 | C4⋊C4 | C2×D8 | 1 | 3 | yes | — |
| 284 | C4⋊C4 | C2×D8 | 1 | 3 | yes | — |
| 285 | C22⋊C4 | — | 4 | 6 | no | — |
| 286 | C22⋊C4 | — | 4 | 6 | no | — |
| 287 | C22⋊C4 | C2×D8 | 1 | 6 | yes | — |
| 288 | C42 | — | 4 | 1 | no | — |
| 289 | C42 | — | 4 | 3 | no | — |
| 290 | C42 | — | 4 | 3 | no | — |
| 291 | C42 | — | 4 | 3 | no | — |
| 292 | C42 | — | 4 | 6 | no | — |
| 293 | C42 | C2×D8 | 1 | 3 | yes | — |
| 294 | C42 | C2×D8 | 1 | 3 | yes | — |
| 295 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 296 | C42 | C22×D4 | 1 | 3 | yes | — |
| 297 | C16 | — | 2 | 6 | no | — |
| 298 | C16 | — | 2 | 2 | no | — |
| 299 | C23 | C22×D8 | 1 | 3 | yes | — |
| 300 | Q8 | — | 2 | 2 | no | — |
| 301 | Q8 | — | 4 | 1 | no | — |
| 302 | Q8 | — | 4 | 6 | no | — |
| 303 | Q8 | — | 4 | 3 | no | — |
| 304 | Q8 | — | 4 | 3 | no | — |
| 305 | Q8 | — | 4 | 6 | no | — |
| 306 | Q8 | — | 8 | 6 | no | — |
| 307 | Q8 | — | 8 | 2 | no | — |
| 308 | Q8 | — | 16 | 3 | no | — |
| 309 | Q8 | — | 16 | 6 | no | — |
| 310 | Q8 | C2×Q32 | 1 | 4 | yes | complemented |
| 311 | D4 | C2×Q32 | 1 | 12 | yes | complemented |
| 312 | C2×C4 | — | 2 | 3 | no | — |
| 313 | C2×C4 | — | 2 | 3 | no | — |
| 314 | C2×C4 | — | 4 | 3 | no | — |
| 315 | C2×C4 | — | 4 | 1 | no | — |
| 316 | C2×C4 | — | 4 | 1 | no | — |
| 317 | C2×C4 | — | 4 | 2 | no | — |
| 318 | C2×C4 | — | 4 | 3 | no | — |
| 319 | C2×C4 | — | 4 | 3 | no | — |
| 320 | C2×C4 | — | 4 | 6 | no | — |
| 321 | C2×C4 | — | 8 | 6 | no | — |
| 322 | C2×C4 | — | 16 | 3 | no | — |
| 323 | C2×C4 | — | 16 | 6 | no | — |
| 324 | C2×C4 | C2×Q32 | 1 | 12 | yes | complemented |
| 325 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 326 | C2×C4 | C22×D8 | 1 | 3 | yes | — |
| 327 | C2×C4 | C22×D8 | 1 | 3 | yes | — |
| 328 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 329 | C2×C4 | C22×D8 | 1 | 3 | yes | — |
| 330 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 331 | C8 | — | 2 | 1 | no | — |
| 332 | C8 | — | 2 | 3 | no | — |
| 333 | C8 | — | 2 | 3 | no | — |
| 334 | C8 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 335 | C8 | #585 | 1 | 1 | yes | characteristic |
| 336 | C22 | — | 2 | 3 | no | — |
| 337 | C22 | — | 2 | 3 | no | — |
| 338 | C22 | C22×Q32 | 1 | 6 | yes | — |
| 339 | C22 | C23×D8 | 1 | 1 | yes | characteristic |
| 340 | C4 | — | 2 | 1 | no | — |
| 341 | C4 | — | 2 | 3 | no | — |
| 342 | C4 | — | 2 | 3 | no | — |
| 343 | C4 | — | 4 | 2 | no | — |
| 344 | C4 | — | 8 | 1 | no | — |
| 345 | C4 | — | 8 | 3 | no | — |
| 346 | C4 | — | 8 | 2 | no | — |
| 347 | C4 | — | 8 | 3 | no | — |
| 348 | C4 | — | 8 | 6 | no | — |
| 349 | C4 | C22×Q32 | 1 | 2 | yes | — |
| 350 | C4 | C22×Q32 | 1 | 6 | yes | — |
| 351 | C4 | #3488 | 1 | 1 | yes | characteristic |
| 352 | C4 | #3585 | 1 | 1 | yes | characteristic |
| 353 | C2 | — | 2 | 6 | no | — |
| 354 | C2 | #61947 | 1 | 1 | yes | characteristic |
| 355 | C2 | C23×Q32 | 1 | 1 | yes | characteristic |
| 356 | C2 | #62637 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #33962 | C2 | semidirect |
| 4. #33930 | C2 | semidirect |
| 5. #33866 | C2 | semidirect |
| 6. D4×Q32 | C2 | semidirect |
| 7. #33815 | C2 | semidirect |
| 8. #33775 | C2 | semidirect |
| 9. #33664 | C2 | semidirect |
| 10. C2×#2209 | C2 | semidirect |
| 226. #41 | Q32 | semidirect |
| 310. Q8 | C2×Q32 | semidirect |
| 311. D4 | C2×Q32 | semidirect |
| 324. C2×C4 | C2×Q32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62416 | 2 |
| Q16×#41 | 2 |
| #33962 | 2 |
| #33930 | 2 |
| #33866 | 2 |
| D4×Q32 | 2 |
| #33815 | 2 |
| #33775 | 2 |
| #33664 | 2 |
| C2×#2209 | 2 |
| #33502 | 2 |