Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
23392 conjugacy classes of subgroups.
The subgroup lattice has 434 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×H(7,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#3484 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #60399 | C2 | 1 | 32 | yes | maximal, direct product factor |
| 4 | C2×#3027 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C2×#3026 | C2 | 1 | 8 | yes | maximal, complemented |
| 6 | C2×#3025 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | C22×B(4,𝔽2) | C2 | 1 | 8 | yes | maximal, complemented |
| 8 | C2×#2883 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | H(7,𝔽2) | — | 2 | 24 | no | — |
| 10 | H(7,𝔽2) | C22 | 1 | 16 | yes | complemented |
| 11 | C2×#585 | — | 2 | 12 | no | — |
| 12 | C2×#585 | C22 | 1 | 4 | yes | complemented |
| 13 | C22×H(5,𝔽2) | — | 2 | 12 | no | — |
| 14 | C22×H(5,𝔽2) | C22 | 1 | 3 | yes | — |
| 15 | C22×H(5,𝔽2) | C22 | 1 | 8 | yes | complemented |
| 16 | C24×D4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | #3484 | — | 2 | 24 | no | — |
| 18 | #3484 | C22 | 1 | 48 | yes | complemented |
| 19 | C2×#546 | — | 2 | 12 | no | — |
| 20 | C2×D42 | — | 2 | 12 | no | — |
| 21 | C2×#535 | C22 | 1 | 6 | yes | complemented |
| 22 | C2×#534 | C22 | 1 | 12 | yes | complemented |
| 23 | C2×#525 | C22 | 1 | 6 | yes | complemented |
| 24 | C22×#120 | C22 | 1 | 12 | yes | complemented |
| 25 | C2×#518 | C22 | 1 | 3 | yes | complemented |
| 26 | #3027 | C22 | 1 | 64 | yes | complemented |
| 27 | #3026 | C22 | 1 | 128 | yes | complemented |
| 28 | #3025 | C22 | 1 | 64 | yes | complemented |
| 29 | C2×#458 | C22 | 1 | 16 | yes | complemented |
| 30 | C2×B(4,𝔽2) | C22 | 1 | 16 | yes | complemented |
| 31 | C2×B(4,𝔽2) | C22 | 1 | 32 | yes | complemented |
| 32 | C2×B(4,𝔽2) | C22 | 1 | 128 | yes | complemented |
| 33 | #2883 | C22 | 1 | 48 | yes | complemented |
| 34 | C2×#410 | C22 | 1 | 12 | yes | complemented |
| 35 | C22×#99 | C22 | 1 | 24 | yes | complemented |
| 36 | C26 | C23 | 1 | 2 | yes | complemented |
| 37 | #585 | — | 2 | 192 | no | — |
| 38 | #585 | — | 2 | 192 | no | — |
| 39 | #585 | — | 2 | 48 | no | — |
| 40 | #585 | C23 | 1 | 32 | yes | complemented |
| 41 | C2×2−1+4 | — | 2 | 16 | no | — |
| 42 | C2×2−1+4 | — | 2 | 12 | no | — |
| 43 | C2×H(5,𝔽2) | — | 2 | 48 | no | — |
| 44 | C2×H(5,𝔽2) | — | 2 | 192 | no | — |
| 45 | C2×H(5,𝔽2) | — | 2 | 24 | no | — |
| 46 | C2×H(5,𝔽2) | — | 2 | 24 | no | — |
| 47 | C2×H(5,𝔽2) | — | 2 | 12 | no | — |
| 48 | C2×H(5,𝔽2) | — | 2 | 48 | no | — |
| 49 | C2×H(5,𝔽2) | — | 2 | 192 | no | — |
| 50 | C2×H(5,𝔽2) | — | 2 | 96 | no | — |
| 51 | C2×H(5,𝔽2) | D4 | 1 | 24 | yes | complemented |
| 52 | C2×H(5,𝔽2) | C23 | 1 | 16 | yes | — |
| 53 | C2×H(5,𝔽2) | C23 | 1 | 24 | yes | — |
| 54 | C2×H(5,𝔽2) | C23 | 1 | 64 | yes | complemented |
| 55 | C22×#41 | — | 2 | 48 | no | — |
| 56 | C22×#41 | — | 2 | 24 | no | — |
| 57 | C22×#41 | — | 2 | 24 | no | — |
| 58 | C22×#41 | D4 | 1 | 12 | yes | complemented |
| 59 | C22×#41 | C23 | 1 | 6 | yes | — |
| 60 | C23×D4 | — | 2 | 16 | no | — |
| 61 | C23×D4 | — | 2 | 24 | no | — |
| 62 | C23×D4 | — | 2 | 12 | no | — |
| 63 | C23×D4 | — | 2 | 12 | no | — |
| 64 | C23×D4 | — | 2 | 12 | no | — |
| 65 | C23×D4 | D4 | 1 | 12 | yes | complemented |
| 66 | C23×D4 | C23 | 1 | 12 | yes | — |
| 67 | C23×D4 | C23 | 1 | 1 | yes | characteristic |
| 68 | C23×D4 | C23 | 1 | 2 | yes | complemented |
| 69 | C23×D4 | C23 | 1 | 1 | yes | characteristic, complemented |
| 70 | C23×D4 | C23 | 1 | 8 | yes | complemented |
| 71 | C24×C4 | C23 | 1 | 1 | yes | characteristic, complemented |
| 72 | #546 | — | 2 | 192 | no | — |
| 73 | D42 | — | 2 | 192 | no | — |
| 74 | #535 | — | 2 | 24 | no | — |
| 75 | #535 | C23 | 1 | 48 | yes | complemented |
| 76 | #534 | — | 2 | 48 | no | — |
| 77 | #534 | C23 | 1 | 96 | yes | complemented |
| 78 | C2×#127 | — | 2 | 6 | no | — |
| 79 | C2×#124 | — | 2 | 6 | no | — |
| 80 | #525 | — | 2 | 24 | no | — |
| 81 | #525 | C23 | 1 | 48 | yes | complemented |
| 82 | C2×#123 | — | 2 | 12 | no | — |
| 83 | C2×#123 | C23 | 1 | 12 | yes | complemented |
| 84 | C2×C22⋊Q8 | — | 2 | 6 | no | — |
| 85 | C2×#121 | — | 2 | 6 | no | — |
| 86 | C2×#121 | — | 2 | 24 | no | — |
| 87 | C2×#121 | — | 2 | 12 | no | — |
| 88 | C2×#120 | — | 2 | 12 | no | — |
| 89 | C2×#120 | — | 2 | 24 | no | — |
| 90 | C2×#120 | — | 2 | 48 | no | — |
| 91 | C2×#120 | C23 | 1 | 12 | yes | complemented |
| 92 | C2×#120 | C23 | 1 | 24 | yes | complemented |
| 93 | C2×#120 | C23 | 1 | 96 | yes | complemented |
| 94 | #518 | — | 2 | 12 | no | — |
| 95 | #518 | C23 | 1 | 24 | yes | complemented |
| 96 | C2×C4×D4 | — | 2 | 12 | no | — |
| 97 | C2×C4×D4 | — | 2 | 12 | no | — |
| 98 | C2×#117 | C23 | 1 | 6 | yes | complemented |
| 99 | C22×C22⋊C4 | — | 2 | 12 | no | — |
| 100 | C22×C22⋊C4 | C23 | 1 | 12 | yes | complemented |
| 101 | #458 | C23 | 1 | 128 | yes | complemented |
| 102 | B(4,𝔽2) | C23 | 1 | 128 | yes | complemented |
| 103 | B(4,𝔽2) | C23 | 1 | 256 | yes | complemented |
| 104 | #410 | C23 | 1 | 96 | yes | complemented |
| 105 | C2×#99 | C23 | 1 | 48 | yes | complemented |
| 106 | C2×#99 | C23 | 1 | 192 | yes | complemented |
| 107 | C25 | — | 2 | 8 | no | — |
| 108 | C25 | — | 2 | 12 | no | — |
| 109 | C25 | — | 2 | 12 | no | — |
| 110 | C25 | — | 8 | 8 | no | — |
| 111 | C25 | C2×D4 | 1 | 12 | yes | complemented |
| 112 | C25 | C24 | 1 | 1 | yes | characteristic |
| 113 | C25 | C24 | 1 | 2 | yes | — |
| 114 | C25 | C24 | 1 | 2 | yes | — |
| 115 | C25 | C24 | 1 | 8 | yes | complemented |
| 116 | 2−1+4 | — | 2 | 256 | no | — |
| 117 | 2−1+4 | — | 2 | 96 | no | — |
| 118 | 2−1+4 | — | 2 | 96 | no | — |
| 119 | H(5,𝔽2) | — | 2 | 96 | no | — |
| 120 | H(5,𝔽2) | — | 2 | 768 | no | — |
| 121 | H(5,𝔽2) | — | 2 | 384 | no | — |
| 122 | H(5,𝔽2) | — | 2 | 192 | no | — |
| 123 | H(5,𝔽2) | — | 2 | 96 | no | — |
| 124 | H(5,𝔽2) | — | 2 | 192 | no | — |
| 125 | H(5,𝔽2) | — | 2 | 96 | no | — |
| 126 | H(5,𝔽2) | — | 2 | 384 | no | — |
| 127 | H(5,𝔽2) | C24 | 1 | 64 | yes | — |
| 128 | C2×#41 | — | 2 | 96 | no | — |
| 129 | C2×#41 | — | 2 | 48 | no | — |
| 130 | C2×#41 | — | 2 | 16 | no | — |
| 131 | C2×#41 | — | 2 | 768 | no | — |
| 132 | C2×#41 | — | 2 | 192 | no | — |
| 133 | C2×#41 | — | 2 | 192 | no | — |
| 134 | C2×#41 | — | 2 | 48 | no | — |
| 135 | C2×#41 | — | 2 | 48 | no | — |
| 136 | C2×#41 | — | 2 | 48 | no | — |
| 137 | C2×#41 | — | 2 | 24 | no | — |
| 138 | C2×#41 | — | 2 | 24 | no | — |
| 139 | C2×#41 | — | 2 | 192 | no | — |
| 140 | C2×#41 | — | 2 | 192 | no | — |
| 141 | C2×#41 | C2×D4 | 1 | 24 | yes | complemented |
| 142 | C2×#41 | C2×D4 | 1 | 96 | yes | complemented |
| 143 | C2×#41 | C24 | 1 | 24 | yes | — |
| 144 | C22×Q8 | — | 2 | 24 | no | — |
| 145 | C22×Q8 | — | 2 | 8 | no | — |
| 146 | C22×Q8 | C2×D4 | 1 | 6 | yes | complemented |
| 147 | C22×D4 | — | 2 | 96 | no | — |
| 148 | C22×D4 | — | 2 | 24 | no | — |
| 149 | C22×D4 | — | 2 | 24 | no | — |
| 150 | C22×D4 | — | 2 | 192 | no | — |
| 151 | C22×D4 | — | 2 | 256 | no | — |
| 152 | C22×D4 | — | 2 | 48 | no | — |
| 153 | C22×D4 | — | 2 | 48 | no | — |
| 154 | C22×D4 | — | 2 | 48 | no | — |
| 155 | C22×D4 | — | 2 | 192 | no | — |
| 156 | C22×D4 | — | 2 | 48 | no | — |
| 157 | C22×D4 | — | 2 | 48 | no | — |
| 158 | C22×D4 | — | 2 | 12 | no | — |
| 159 | C22×D4 | — | 2 | 12 | no | — |
| 160 | C22×D4 | — | 2 | 12 | no | — |
| 161 | C22×D4 | — | 2 | 24 | no | — |
| 162 | C22×D4 | — | 2 | 96 | no | — |
| 163 | C22×D4 | — | 2 | 96 | no | — |
| 164 | C22×D4 | — | 2 | 6 | no | — |
| 165 | C22×D4 | — | 2 | 6 | no | — |
| 166 | C22×D4 | — | 4 | 16 | no | — |
| 167 | C22×D4 | — | 4 | 48 | no | — |
| 168 | C22×D4 | — | 8 | 48 | no | — |
| 169 | C22×D4 | C2×D4 | 1 | 12 | yes | — |
| 170 | C22×D4 | C2×D4 | 1 | 12 | yes | — |
| 171 | C22×D4 | C2×D4 | 1 | 24 | yes | complemented |
| 172 | C22×D4 | C2×D4 | 1 | 6 | yes | complemented |
| 173 | C22×D4 | C2×D4 | 1 | 48 | yes | complemented |
| 174 | C22×D4 | C2×D4 | 1 | 96 | yes | complemented |
| 175 | C22×D4 | C24 | 1 | 12 | yes | — |
| 176 | C22×D4 | C24 | 1 | 48 | yes | — |
| 177 | C22×D4 | C24 | 1 | 4 | yes | — |
| 178 | C22×D4 | C24 | 1 | 8 | yes | complemented |
| 179 | C22×D4 | C24 | 1 | 4 | yes | complemented |
| 180 | C23×C4 | — | 2 | 24 | no | — |
| 181 | C23×C4 | — | 2 | 16 | no | — |
| 182 | C23×C4 | — | 2 | 12 | no | — |
| 183 | C23×C4 | — | 2 | 6 | no | — |
| 184 | C23×C4 | — | 2 | 6 | no | — |
| 185 | C23×C4 | C2×D4 | 1 | 12 | yes | complemented |
| 186 | C23×C4 | C2×D4 | 1 | 12 | yes | complemented |
| 187 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 188 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 189 | C23×C4 | C24 | 1 | 4 | yes | complemented |
| 190 | #127 | — | 2 | 48 | no | — |
| 191 | #124 | — | 2 | 48 | no | — |
| 192 | #123 | — | 2 | 24 | no | — |
| 193 | #123 | — | 2 | 96 | no | — |
| 194 | #123 | C24 | 1 | 48 | yes | complemented |
| 195 | C22⋊Q8 | — | 2 | 48 | no | — |
| 196 | #121 | — | 2 | 48 | no | — |
| 197 | #121 | — | 2 | 192 | no | — |
| 198 | #121 | — | 2 | 96 | no | — |
| 199 | #120 | — | 2 | 96 | no | — |
| 200 | #120 | — | 2 | 24 | no | — |
| 201 | #120 | — | 2 | 48 | no | — |
| 202 | #120 | — | 2 | 192 | no | — |
| 203 | #120 | C24 | 1 | 48 | yes | complemented |
| 204 | #120 | C24 | 1 | 96 | yes | complemented |
| 205 | C4×D4 | — | 2 | 96 | no | — |
| 206 | C4×D4 | — | 2 | 96 | no | — |
| 207 | #117 | — | 2 | 12 | no | — |
| 208 | #117 | C24 | 1 | 24 | yes | complemented |
| 209 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 210 | C2×C4⋊C4 | — | 2 | 12 | no | — |
| 211 | C2×C22⋊C4 | — | 2 | 12 | no | — |
| 212 | C2×C22⋊C4 | — | 2 | 24 | no | — |
| 213 | C2×C22⋊C4 | — | 2 | 12 | no | — |
| 214 | C2×C22⋊C4 | — | 2 | 12 | no | — |
| 215 | C2×C22⋊C4 | — | 2 | 24 | no | — |
| 216 | C2×C22⋊C4 | — | 2 | 96 | no | — |
| 217 | C2×C22⋊C4 | C24 | 1 | 12 | yes | — |
| 218 | C2×C22⋊C4 | C24 | 1 | 48 | yes | complemented |
| 219 | C2×C42 | — | 2 | 6 | no | — |
| 220 | #99 | C24 | 1 | 192 | yes | complemented |
| 221 | C24 | — | 2 | 16 | no | — |
| 222 | C24 | — | 2 | 64 | no | — |
| 223 | C24 | — | 2 | 3 | no | — |
| 224 | C24 | — | 2 | 24 | no | — |
| 225 | C24 | — | 2 | 24 | no | — |
| 226 | C24 | — | 2 | 6 | no | — |
| 227 | C24 | — | 2 | 3 | no | — |
| 228 | C24 | — | 2 | 48 | no | — |
| 229 | C24 | — | 2 | 6 | no | — |
| 230 | C24 | — | 4 | 4 | no | — |
| 231 | C24 | — | 4 | 48 | no | — |
| 232 | C24 | — | 4 | 24 | no | — |
| 233 | C24 | — | 4 | 8 | no | — |
| 234 | C24 | — | 8 | 4 | no | — |
| 235 | C24 | — | 8 | 4 | no | — |
| 236 | C24 | — | 8 | 16 | no | — |
| 237 | C24 | — | 8 | 24 | no | — |
| 238 | C24 | — | 8 | 48 | no | — |
| 239 | C24 | — | 8 | 24 | no | — |
| 240 | C24 | — | 16 | 16 | no | — |
| 241 | C24 | C22×D4 | 1 | 12 | yes | — |
| 242 | C24 | C22×D4 | 1 | 12 | yes | — |
| 243 | C24 | C22×D4 | 1 | 12 | yes | — |
| 244 | C24 | C22×D4 | 1 | 48 | yes | complemented |
| 245 | C24 | C25 | 1 | 2 | yes | — |
| 246 | C24 | C25 | 1 | 4 | yes | — |
| 247 | C24 | C25 | 1 | 1 | yes | characteristic |
| 248 | C24 | C25 | 1 | 4 | yes | — |
| 249 | #41 | — | 2 | 192 | no | — |
| 250 | #41 | — | 2 | 768 | no | — |
| 251 | #41 | — | 2 | 384 | no | — |
| 252 | #41 | — | 2 | 128 | no | — |
| 253 | #41 | — | 2 | 384 | no | — |
| 254 | #41 | — | 2 | 96 | no | — |
| 255 | #41 | — | 2 | 192 | no | — |
| 256 | #41 | — | 2 | 96 | no | — |
| 257 | C2×Q8 | — | 2 | 16 | no | — |
| 258 | C2×Q8 | — | 2 | 192 | no | — |
| 259 | C2×Q8 | — | 2 | 64 | no | — |
| 260 | C2×Q8 | — | 2 | 12 | no | — |
| 261 | C2×Q8 | — | 2 | 12 | no | — |
| 262 | C2×Q8 | C22×D4 | 1 | 24 | yes | complemented |
| 263 | C2×D4 | — | 2 | 48 | no | — |
| 264 | C2×D4 | — | 2 | 192 | no | — |
| 265 | C2×D4 | — | 2 | 768 | no | — |
| 266 | C2×D4 | — | 2 | 192 | no | — |
| 267 | C2×D4 | — | 2 | 192 | no | — |
| 268 | C2×D4 | — | 2 | 48 | no | — |
| 269 | C2×D4 | — | 2 | 48 | no | — |
| 270 | C2×D4 | — | 2 | 192 | no | — |
| 271 | C2×D4 | — | 2 | 24 | no | — |
| 272 | C2×D4 | — | 2 | 192 | no | — |
| 273 | C2×D4 | — | 2 | 24 | no | — |
| 274 | C2×D4 | — | 2 | 192 | no | — |
| 275 | C2×D4 | — | 2 | 384 | no | — |
| 276 | C2×D4 | — | 2 | 48 | no | — |
| 277 | C2×D4 | — | 2 | 48 | no | — |
| 278 | C2×D4 | — | 2 | 96 | no | — |
| 279 | C2×D4 | — | 2 | 12 | no | — |
| 280 | C2×D4 | — | 2 | 12 | no | — |
| 281 | C2×D4 | — | 2 | 24 | no | — |
| 282 | C2×D4 | — | 4 | 48 | no | — |
| 283 | C2×D4 | — | 4 | 24 | no | — |
| 284 | C2×D4 | — | 4 | 8 | no | — |
| 285 | C2×D4 | — | 4 | 128 | no | — |
| 286 | C2×D4 | — | 4 | 12 | no | — |
| 287 | C2×D4 | — | 4 | 4 | no | — |
| 288 | C2×D4 | — | 8 | 48 | no | — |
| 289 | C2×D4 | — | 8 | 192 | no | — |
| 290 | C2×D4 | #120 | 1 | 16 | yes | — |
| 291 | C2×D4 | #120 | 1 | 48 | yes | — |
| 292 | C2×D4 | C22×D4 | 1 | 24 | yes | — |
| 293 | C2×D4 | C22×D4 | 1 | 48 | yes | — |
| 294 | C2×D4 | C22×D4 | 1 | 96 | yes | complemented |
| 295 | C2×D4 | C22×D4 | 1 | 24 | yes | complemented |
| 296 | C2×D4 | C25 | 1 | 24 | yes | — |
| 297 | C22×C4 | — | 2 | 48 | no | — |
| 298 | C22×C4 | — | 2 | 192 | no | — |
| 299 | C22×C4 | — | 2 | 128 | no | — |
| 300 | C22×C4 | — | 2 | 24 | no | — |
| 301 | C22×C4 | — | 2 | 24 | no | — |
| 302 | C22×C4 | — | 2 | 24 | no | — |
| 303 | C22×C4 | — | 2 | 24 | no | — |
| 304 | C22×C4 | — | 2 | 12 | no | — |
| 305 | C22×C4 | — | 2 | 3 | no | — |
| 306 | C22×C4 | — | 2 | 3 | no | — |
| 307 | C22×C4 | — | 2 | 48 | no | — |
| 308 | C22×C4 | — | 2 | 24 | no | — |
| 309 | C22×C4 | — | 4 | 12 | no | — |
| 310 | C22×C4 | — | 4 | 4 | no | — |
| 311 | C22×C4 | — | 8 | 24 | no | — |
| 312 | C22×C4 | C22×D4 | 1 | 12 | yes | — |
| 313 | C22×C4 | C22×D4 | 1 | 6 | yes | — |
| 314 | C22×C4 | C22×D4 | 1 | 6 | yes | — |
| 315 | C22×C4 | C22×D4 | 1 | 48 | yes | complemented |
| 316 | C22×C4 | C22×D4 | 1 | 48 | yes | complemented |
| 317 | C22×C4 | C25 | 1 | 2 | yes | — |
| 318 | C22×C4 | C25 | 1 | 2 | yes | — |
| 319 | C4⋊C4 | — | 2 | 24 | no | — |
| 320 | C4⋊C4 | — | 2 | 48 | no | — |
| 321 | C22⋊C4 | — | 2 | 12 | no | — |
| 322 | C22⋊C4 | — | 2 | 48 | no | — |
| 323 | C22⋊C4 | — | 2 | 48 | no | — |
| 324 | C22⋊C4 | — | 2 | 48 | no | — |
| 325 | C22⋊C4 | — | 2 | 96 | no | — |
| 326 | C22⋊C4 | C25 | 1 | 24 | yes | — |
| 327 | C42 | — | 2 | 24 | no | — |
| 328 | C23 | — | 2 | 12 | no | — |
| 329 | C23 | — | 2 | 64 | no | — |
| 330 | C23 | — | 2 | 6 | no | — |
| 331 | C23 | — | 2 | 12 | no | — |
| 332 | C23 | — | 2 | 12 | no | — |
| 333 | C23 | — | 2 | 96 | no | — |
| 334 | C23 | — | 2 | 12 | no | — |
| 335 | C23 | — | 4 | 24 | no | — |
| 336 | C23 | — | 4 | 2 | no | — |
| 337 | C23 | — | 4 | 4 | no | — |
| 338 | C23 | — | 4 | 6 | no | — |
| 339 | C23 | — | 4 | 12 | no | — |
| 340 | C23 | — | 4 | 24 | no | — |
| 341 | C23 | — | 4 | 24 | no | — |
| 342 | C23 | — | 4 | 96 | no | — |
| 343 | C23 | — | 4 | 3 | no | — |
| 344 | C23 | — | 4 | 1 | no | — |
| 345 | C23 | — | 4 | 32 | no | — |
| 346 | C23 | — | 4 | 6 | no | — |
| 347 | C23 | — | 4 | 2 | no | — |
| 348 | C23 | — | 8 | 48 | no | — |
| 349 | C23 | — | 8 | 32 | no | — |
| 350 | C23 | — | 8 | 12 | no | — |
| 351 | C23 | — | 8 | 4 | no | — |
| 352 | C23 | — | 8 | 4 | no | — |
| 353 | C23 | — | 8 | 48 | no | — |
| 354 | C23 | — | 8 | 48 | no | — |
| 355 | C23 | — | 8 | 12 | no | — |
| 356 | C23 | — | 8 | 32 | no | — |
| 357 | C23 | — | 8 | 96 | no | — |
| 358 | C23 | — | 16 | 24 | no | — |
| 359 | C23 | — | 16 | 64 | no | — |
| 360 | C23 | C2×#120 | 1 | 8 | yes | — |
| 361 | C23 | C2×#120 | 1 | 24 | yes | — |
| 362 | C23 | C23×D4 | 1 | 3 | yes | — |
| 363 | C23 | C23×D4 | 1 | 24 | yes | — |
| 364 | C23 | C23×D4 | 1 | 3 | yes | — |
| 365 | C23 | C23×D4 | 1 | 24 | yes | — |
| 366 | C23 | C26 | 1 | 1 | yes | characteristic |
| 367 | Q8 | — | 2 | 64 | no | — |
| 368 | Q8 | — | 2 | 48 | no | — |
| 369 | D4 | — | 2 | 192 | no | — |
| 370 | D4 | — | 2 | 48 | no | — |
| 371 | D4 | — | 2 | 48 | no | — |
| 372 | D4 | — | 2 | 192 | no | — |
| 373 | D4 | — | 2 | 48 | no | — |
| 374 | D4 | — | 4 | 96 | no | — |
| 375 | D4 | — | 4 | 32 | no | — |
| 376 | D4 | — | 4 | 16 | no | — |
| 377 | D4 | — | 8 | 96 | no | — |
| 378 | D4 | C2×#120 | 1 | 64 | yes | — |
| 379 | C2×C4 | — | 2 | 192 | no | — |
| 380 | C2×C4 | — | 2 | 12 | no | — |
| 381 | C2×C4 | — | 2 | 96 | no | — |
| 382 | C2×C4 | — | 2 | 12 | no | — |
| 383 | C2×C4 | — | 2 | 96 | no | — |
| 384 | C2×C4 | — | 2 | 24 | no | — |
| 385 | C2×C4 | — | 2 | 6 | no | — |
| 386 | C2×C4 | — | 4 | 24 | no | — |
| 387 | C2×C4 | — | 4 | 3 | no | — |
| 388 | C2×C4 | — | 4 | 1 | no | — |
| 389 | C2×C4 | — | 4 | 16 | no | — |
| 390 | C2×C4 | — | 8 | 48 | no | — |
| 391 | C2×C4 | C2×#120 | 1 | 12 | yes | — |
| 392 | C2×C4 | C2×#120 | 1 | 4 | yes | — |
| 393 | C2×C4 | C23×D4 | 1 | 24 | yes | — |
| 394 | C2×C4 | C23×D4 | 1 | 12 | yes | — |
| 395 | C22 | — | 2 | 6 | no | — |
| 396 | C22 | — | 2 | 16 | no | — |
| 397 | C22 | — | 2 | 12 | no | — |
| 398 | C22 | — | 2 | 24 | no | — |
| 399 | C22 | — | 4 | 24 | no | — |
| 400 | C22 | — | 4 | 3 | no | — |
| 401 | C22 | — | 4 | 1 | no | — |
| 402 | C22 | — | 4 | 24 | no | — |
| 403 | C22 | — | 4 | 24 | no | — |
| 404 | C22 | — | 4 | 6 | no | — |
| 405 | C22 | — | 4 | 12 | no | — |
| 406 | C22 | — | 4 | 24 | no | — |
| 407 | C22 | — | 4 | 2 | no | — |
| 408 | C22 | — | 4 | 4 | no | — |
| 409 | C22 | — | 4 | 24 | no | — |
| 410 | C22 | — | 4 | 4 | no | — |
| 411 | C22 | — | 8 | 12 | no | — |
| 412 | C22 | — | 8 | 96 | no | — |
| 413 | C22 | — | 8 | 64 | no | — |
| 414 | C22 | — | 8 | 12 | no | — |
| 415 | C22 | — | 8 | 64 | no | — |
| 416 | C22 | — | 8 | 4 | no | — |
| 417 | C22 | C22×#120 | 1 | 16 | yes | — |
| 418 | C22 | C22×#120 | 1 | 3 | yes | — |
| 419 | C22 | C22×#120 | 1 | 1 | yes | characteristic |
| 420 | C22 | C24×D4 | 1 | 3 | yes | — |
| 421 | C4 | — | 2 | 24 | no | — |
| 422 | C4 | — | 4 | 24 | no | — |
| 423 | C4 | — | 4 | 2 | no | — |
| 424 | C4 | C22×#120 | 1 | 8 | yes | — |
| 425 | C2 | — | 2 | 3 | no | — |
| 426 | C2 | — | 2 | 3 | no | — |
| 427 | C2 | — | 2 | 16 | no | — |
| 428 | C2 | — | 4 | 4 | no | — |
| 429 | C2 | — | 4 | 24 | no | — |
| 430 | C2 | — | 8 | 4 | no | — |
| 431 | C2 | #60399 | 1 | 2 | yes | direct product factor |
| 432 | C2 | C23×#120 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×H(7,𝔽2) | C2 | semidirect |
| 2. C2×#3484 | C2 | semidirect |
| 3. #60399 | C2 | direct (N × H) |
| 4. C2×#3027 | C2 | semidirect |
| 5. C2×#3026 | C2 | semidirect |
| 6. C2×#3025 | C2 | semidirect |
| 7. C22×B(4,𝔽2) | C2 | semidirect |
| 8. C2×#2883 | C2 | semidirect |
| 10. H(7,𝔽2) | C22 | semidirect |
| 12. C2×#585 | C22 | semidirect |
| 15. C22×H(5,𝔽2) | C22 | semidirect |
| 16. C24×D4 | C22 | semidirect |
| 18. #3484 | C22 | semidirect |
| 21. C2×#535 | C22 | semidirect |
| 22. C2×#534 | C22 | semidirect |
| 23. C2×#525 | C22 | semidirect |
| 24. C22×#120 | C22 | semidirect |
| 25. C2×#518 | C22 | semidirect |
| 26. #3027 | C22 | semidirect |
| 27. #3026 | C22 | semidirect |
| 28. #3025 | C22 | semidirect |
| 29. C2×#458 | C22 | semidirect |
| 30. C2×B(4,𝔽2) | C22 | semidirect |
| 31. C2×B(4,𝔽2) | C22 | semidirect |
| 32. C2×B(4,𝔽2) | C22 | semidirect |
| 33. #2883 | C22 | semidirect |
| 34. C2×#410 | C22 | semidirect |
| 35. C22×#99 | C22 | semidirect |
| 36. C26 | C23 | semidirect |
| 40. #585 | C23 | semidirect |
| 51. C2×H(5,𝔽2) | D4 | semidirect |
| 54. C2×H(5,𝔽2) | C23 | semidirect |
| 58. C22×#41 | D4 | semidirect |
| 65. C23×D4 | D4 | semidirect |
| 68. C23×D4 | C23 | semidirect |
| 69. C23×D4 | C23 | semidirect |
| 70. C23×D4 | C23 | semidirect |
| 71. C24×C4 | C23 | semidirect |
| 75. #535 | C23 | semidirect |
| 77. #534 | C23 | semidirect |
| 81. #525 | C23 | semidirect |
| 83. C2×#123 | C23 | semidirect |
| 91. C2×#120 | C23 | semidirect |
| 92. C2×#120 | C23 | semidirect |
| 93. C2×#120 | C23 | semidirect |
| 95. #518 | C23 | semidirect |
| 98. C2×#117 | C23 | semidirect |
| 100. C22×C22⋊C4 | C23 | semidirect |
| 101. #458 | C23 | semidirect |
| 102. B(4,𝔽2) | C23 | semidirect |
| 103. B(4,𝔽2) | C23 | semidirect |
| 104. #410 | C23 | semidirect |
| 105. C2×#99 | C23 | semidirect |
| 106. C2×#99 | C23 | semidirect |
| 111. C25 | C2×D4 | semidirect |
| 115. C25 | C24 | semidirect |
| 141. C2×#41 | C2×D4 | semidirect |
| 142. C2×#41 | C2×D4 | semidirect |
| 146. C22×Q8 | C2×D4 | semidirect |
| 171. C22×D4 | C2×D4 | semidirect |
| 172. C22×D4 | C2×D4 | semidirect |
| 173. C22×D4 | C2×D4 | semidirect |
| 174. C22×D4 | C2×D4 | semidirect |
| 178. C22×D4 | C24 | semidirect |
| 179. C22×D4 | C24 | semidirect |
| 185. C23×C4 | C2×D4 | semidirect |
| 186. C23×C4 | C2×D4 | semidirect |
| 189. C23×C4 | C24 | semidirect |
| 194. #123 | C24 | semidirect |
| 203. #120 | C24 | semidirect |
| 204. #120 | C24 | semidirect |
| 208. #117 | C24 | semidirect |
| 218. C2×C22⋊C4 | C24 | semidirect |
| 220. #99 | C24 | semidirect |
| 244. C24 | C22×D4 | semidirect |
| 262. C2×Q8 | C22×D4 | semidirect |
| 294. C2×D4 | C22×D4 | semidirect |
| 295. C2×D4 | C22×D4 | semidirect |
| 315. C22×C4 | C22×D4 | semidirect |
| 316. C22×C4 | C22×D4 | semidirect |
| 431. C2 | #60399 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(7,𝔽2) | 2 |
| C2×#3484 | 2 |
| #60399 | 2 |
| C2×#3027 | 2 |
| C2×#3026 | 2 |
| C2×#3025 | 2 |
| C22×B(4,𝔽2) | 2 |
| C2×#2883 | 2 |