Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1105 conjugacy classes of subgroups.
The subgroup lattice has 265 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#562 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C22×#125 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #2850 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #2820 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #2711 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #2704 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #2694 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #2449 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | #2443 | C2 | 1 | 4 | yes | maximal, complemented |
| 10 | #2416 | C2 | 1 | 4 | yes | maximal, complemented |
| 11 | #2382 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #562 | — | 2 | 8 | no | — |
| 13 | C2×#125 | — | 2 | 8 | no | — |
| 14 | C2×#125 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#125 | C22 | 1 | 2 | yes | — |
| 16 | C2×#125 | C22 | 1 | 4 | yes | — |
| 17 | C2×#125 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#124 | C22 | 1 | 2 | yes | — |
| 19 | C2×#123 | C22 | 1 | 4 | yes | — |
| 20 | C2×C22⋊Q8 | C22 | 1 | 4 | yes | — |
| 21 | C2×C22⋊Q8 | C22 | 1 | 2 | yes | — |
| 22 | C2×#121 | C22 | 1 | 4 | yes | — |
| 23 | C2×#121 | C22 | 1 | 2 | yes | — |
| 24 | C2×#121 | C22 | 1 | 4 | yes | — |
| 25 | C2×C4×D4 | C22 | 1 | 4 | yes | — |
| 26 | C22×C4⋊C4 | C22 | 1 | 4 | yes | — |
| 27 | C22×C4⋊C4 | C22 | 1 | 2 | yes | — |
| 28 | C22×C42 | C22 | 1 | 1 | yes | characteristic |
| 29 | #399 | C22 | 1 | 4 | yes | — |
| 30 | #399 | C22 | 1 | 4 | yes | — |
| 31 | #398 | C22 | 1 | 4 | yes | — |
| 32 | #398 | C22 | 1 | 4 | yes | — |
| 33 | #397 | C22 | 1 | 4 | yes | — |
| 34 | #396 | C22 | 1 | 4 | yes | — |
| 35 | #396 | C22 | 1 | 4 | yes | — |
| 36 | #396 | C22 | 1 | 4 | yes | — |
| 37 | #395 | C22 | 1 | 4 | yes | — |
| 38 | #394 | C22 | 1 | 4 | yes | — |
| 39 | #394 | C22 | 1 | 4 | yes | — |
| 40 | #394 | C22 | 1 | 4 | yes | — |
| 41 | #391 | C22 | 1 | 2 | yes | — |
| 42 | #390 | C22 | 1 | 2 | yes | — |
| 43 | #390 | C22 | 1 | 4 | yes | — |
| 44 | #389 | C22 | 1 | 4 | yes | — |
| 45 | #389 | C22 | 1 | 4 | yes | — |
| 46 | #388 | C22 | 1 | 4 | yes | — |
| 47 | #388 | C22 | 1 | 4 | yes | — |
| 48 | #388 | C22 | 1 | 4 | yes | — |
| 49 | #387 | C22 | 1 | 4 | yes | — |
| 50 | #387 | C22 | 1 | 4 | yes | — |
| 51 | #386 | C22 | 1 | 4 | yes | — |
| 52 | #386 | C22 | 1 | 4 | yes | — |
| 53 | #386 | C22 | 1 | 4 | yes | — |
| 54 | #385 | C22 | 1 | 4 | yes | — |
| 55 | #385 | C22 | 1 | 4 | yes | — |
| 56 | #385 | C22 | 1 | 4 | yes | — |
| 57 | #382 | C22 | 1 | 2 | yes | — |
| 58 | #380 | C22 | 1 | 4 | yes | — |
| 59 | C22×Q8 | C23 | 1 | 2 | yes | — |
| 60 | C22×D4 | C23 | 1 | 4 | yes | — |
| 61 | C22×D4 | C23 | 1 | 4 | yes | — |
| 62 | C22×D4 | C23 | 1 | 2 | yes | — |
| 63 | C23×C4 | C23 | 1 | 1 | yes | characteristic |
| 64 | C23×C4 | C23 | 1 | 4 | yes | — |
| 65 | C23×C4 | C23 | 1 | 2 | yes | — |
| 66 | C23×C4 | C23 | 1 | 4 | yes | — |
| 67 | #125 | — | 2 | 16 | no | — |
| 68 | #125 | — | 2 | 8 | no | — |
| 69 | #125 | — | 2 | 4 | no | — |
| 70 | #125 | D4 | 1 | 8 | yes | complemented |
| 71 | #124 | — | 2 | 8 | no | — |
| 72 | #123 | — | 2 | 16 | no | — |
| 73 | C22⋊Q8 | — | 2 | 16 | no | — |
| 74 | C22⋊Q8 | — | 4 | 4 | no | — |
| 75 | #121 | — | 2 | 16 | no | — |
| 76 | #121 | — | 4 | 8 | no | — |
| 77 | #121 | — | 4 | 4 | no | — |
| 78 | C4×D4 | — | 4 | 8 | no | — |
| 79 | C2×C4⋊C4 | — | 2 | 16 | no | — |
| 80 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 81 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 82 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 83 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 84 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 85 | C2×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 86 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 87 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 88 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 89 | C2×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 90 | C2×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 91 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 92 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 93 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 94 | C2×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 95 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 96 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 97 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 98 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 99 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 100 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 101 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 102 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 103 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 104 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 105 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 106 | C2×C42 | — | 2 | 4 | no | — |
| 107 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 108 | C2×C42 | C23 | 1 | 2 | yes | — |
| 109 | C2×C42 | C23 | 1 | 4 | yes | — |
| 110 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 111 | #95 | C23 | 1 | 4 | yes | — |
| 112 | #95 | C23 | 1 | 4 | yes | — |
| 113 | #95 | C23 | 1 | 4 | yes | — |
| 114 | #95 | C23 | 1 | 4 | yes | — |
| 115 | #95 | C23 | 1 | 4 | yes | — |
| 116 | #95 | C23 | 1 | 4 | yes | — |
| 117 | #95 | C23 | 1 | 4 | yes | — |
| 118 | #95 | C23 | 1 | 4 | yes | — |
| 119 | C24 | C24 | 1 | 4 | yes | — |
| 120 | C24 | C24 | 1 | 1 | yes | characteristic |
| 121 | C2×Q8 | — | 2 | 4 | no | — |
| 122 | C2×Q8 | — | 4 | 2 | no | — |
| 123 | C2×Q8 | — | 4 | 2 | no | — |
| 124 | C2×D4 | — | 2 | 8 | no | — |
| 125 | C2×D4 | — | 2 | 8 | no | — |
| 126 | C2×D4 | — | 2 | 8 | no | — |
| 127 | C2×D4 | — | 2 | 8 | no | — |
| 128 | C2×D4 | — | 2 | 4 | no | — |
| 129 | C2×D4 | — | 4 | 4 | no | — |
| 130 | C2×D4 | — | 4 | 2 | no | — |
| 131 | C2×D4 | — | 4 | 4 | no | — |
| 132 | C2×D4 | — | 4 | 2 | no | — |
| 133 | C22×C4 | — | 2 | 2 | no | — |
| 134 | C22×C4 | — | 2 | 2 | no | — |
| 135 | C22×C4 | — | 2 | 2 | no | — |
| 136 | C22×C4 | — | 2 | 8 | no | — |
| 137 | C22×C4 | — | 2 | 4 | no | — |
| 138 | C22×C4 | — | 2 | 4 | no | — |
| 139 | C22×C4 | — | 2 | 8 | no | — |
| 140 | C22×C4 | — | 2 | 8 | no | — |
| 141 | C22×C4 | — | 4 | 4 | no | — |
| 142 | C22×C4 | — | 4 | 2 | no | — |
| 143 | C22×C4 | — | 4 | 4 | no | — |
| 144 | C22×C4 | — | 4 | 4 | no | — |
| 145 | C22×C4 | C24 | 1 | 4 | yes | — |
| 146 | C22×C4 | C24 | 1 | 4 | yes | — |
| 147 | C22×C4 | C24 | 1 | 4 | yes | — |
| 148 | C22×C4 | C24 | 1 | 2 | yes | — |
| 149 | C22×C4 | C24 | 1 | 4 | yes | — |
| 150 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 151 | C22×C4 | C24 | 1 | 2 | yes | — |
| 152 | C22×C4 | C24 | 1 | 4 | yes | — |
| 153 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 154 | C4⋊C4 | — | 2 | 8 | no | — |
| 155 | C4⋊C4 | — | 2 | 8 | no | — |
| 156 | C4⋊C4 | — | 2 | 8 | no | — |
| 157 | C4⋊C4 | — | 2 | 4 | no | — |
| 158 | C4⋊C4 | — | 2 | 8 | no | — |
| 159 | C4⋊C4 | — | 2 | 8 | no | — |
| 160 | C4⋊C4 | — | 4 | 4 | no | — |
| 161 | C4⋊C4 | — | 4 | 4 | no | — |
| 162 | C4⋊C4 | — | 4 | 4 | no | — |
| 163 | C4⋊C4 | — | 4 | 2 | no | — |
| 164 | C4⋊C4 | — | 4 | 2 | no | — |
| 165 | C4⋊C4 | — | 4 | 4 | no | — |
| 166 | C4⋊C4 | C2×D4 | 1 | 16 | yes | — |
| 167 | C4⋊C4 | C2×D4 | 1 | 8 | yes | — |
| 168 | C22⋊C4 | — | 2 | 8 | no | — |
| 169 | C22⋊C4 | — | 2 | 8 | no | — |
| 170 | C22⋊C4 | — | 2 | 8 | no | — |
| 171 | C22⋊C4 | — | 2 | 8 | no | — |
| 172 | C22⋊C4 | — | 2 | 8 | no | — |
| 173 | C22⋊C4 | — | 4 | 4 | no | — |
| 174 | C22⋊C4 | — | 4 | 4 | no | — |
| 175 | C22⋊C4 | — | 4 | 4 | no | — |
| 176 | C22⋊C4 | — | 4 | 4 | no | — |
| 177 | C22⋊C4 | — | 4 | 4 | no | — |
| 178 | C22⋊C4 | — | 4 | 4 | no | — |
| 179 | C42 | — | 2 | 4 | no | — |
| 180 | C42 | — | 2 | 2 | no | — |
| 181 | C42 | — | 4 | 4 | no | — |
| 182 | C42 | C2×D4 | 1 | 4 | yes | — |
| 183 | C23 | — | 2 | 1 | no | — |
| 184 | C23 | — | 2 | 1 | no | — |
| 185 | C23 | — | 2 | 2 | no | — |
| 186 | C23 | — | 2 | 4 | no | — |
| 187 | C23 | — | 2 | 4 | no | — |
| 188 | C23 | — | 2 | 4 | no | — |
| 189 | C23 | — | 2 | 4 | no | — |
| 190 | C23 | — | 2 | 4 | no | — |
| 191 | C23 | — | 2 | 4 | no | — |
| 192 | C23 | — | 2 | 4 | no | — |
| 193 | C23 | H(5,𝔽2) | 1 | 4 | yes | — |
| 194 | C23 | 2−1+4 | 1 | 2 | yes | — |
| 195 | C23 | C25 | 1 | 1 | yes | characteristic |
| 196 | Q8 | — | 8 | 4 | no | — |
| 197 | D4 | — | 4 | 16 | no | — |
| 198 | D4 | — | 8 | 8 | no | — |
| 199 | D4 | — | 8 | 4 | no | — |
| 200 | C2×C4 | — | 2 | 8 | no | — |
| 201 | C2×C4 | — | 2 | 1 | no | — |
| 202 | C2×C4 | — | 2 | 4 | no | — |
| 203 | C2×C4 | — | 2 | 2 | no | — |
| 204 | C2×C4 | — | 2 | 1 | no | — |
| 205 | C2×C4 | — | 2 | 1 | no | — |
| 206 | C2×C4 | — | 2 | 1 | no | — |
| 207 | C2×C4 | — | 2 | 4 | no | — |
| 208 | C2×C4 | — | 2 | 2 | no | — |
| 209 | C2×C4 | — | 2 | 2 | no | — |
| 210 | C2×C4 | — | 2 | 4 | no | — |
| 211 | C2×C4 | — | 2 | 4 | no | — |
| 212 | C2×C4 | — | 2 | 4 | no | — |
| 213 | C2×C4 | — | 2 | 4 | no | — |
| 214 | C2×C4 | — | 2 | 4 | no | — |
| 215 | C2×C4 | — | 2 | 4 | no | — |
| 216 | C2×C4 | — | 2 | 2 | no | — |
| 217 | C2×C4 | — | 2 | 4 | no | — |
| 218 | C2×C4 | — | 2 | 4 | no | — |
| 219 | C2×C4 | — | 2 | 4 | no | — |
| 220 | C2×C4 | — | 2 | 4 | no | — |
| 221 | C2×C4 | — | 2 | 2 | no | — |
| 222 | C2×C4 | — | 2 | 4 | no | — |
| 223 | C2×C4 | — | 2 | 4 | no | — |
| 224 | C2×C4 | — | 4 | 16 | no | — |
| 225 | C2×C4 | — | 4 | 8 | no | — |
| 226 | C2×C4 | — | 8 | 8 | no | — |
| 227 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 228 | C2×C4 | C22×D4 | 1 | 4 | yes | — |
| 229 | C2×C4 | C22×D4 | 1 | 8 | yes | — |
| 230 | C2×C4 | 2−1+4 | 1 | 2 | yes | — |
| 231 | C22 | — | 2 | 2 | no | — |
| 232 | C22 | — | 2 | 4 | no | — |
| 233 | C22 | — | 2 | 2 | no | — |
| 234 | C22 | — | 2 | 4 | no | — |
| 235 | C22 | — | 4 | 4 | no | — |
| 236 | C22 | — | 4 | 4 | no | — |
| 237 | C22 | — | 4 | 4 | no | — |
| 238 | C22 | — | 4 | 4 | no | — |
| 239 | C22 | — | 4 | 4 | no | — |
| 240 | C22 | — | 4 | 4 | no | — |
| 241 | C22 | — | 4 | 4 | no | — |
| 242 | C22 | #562 | 1 | 4 | yes | complemented |
| 243 | C22 | C23×D4 | 1 | 1 | yes | characteristic |
| 244 | C22 | C2×H(5,𝔽2) | 1 | 2 | yes | — |
| 245 | C22 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 246 | C22 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 247 | C22 | #585 | 1 | 2 | yes | — |
| 248 | C4 | — | 2 | 2 | no | — |
| 249 | C4 | — | 2 | 2 | no | — |
| 250 | C4 | — | 2 | 8 | no | — |
| 251 | C4 | — | 2 | 4 | no | — |
| 252 | C4 | — | 4 | 2 | no | — |
| 253 | C4 | — | 4 | 4 | no | — |
| 254 | C4 | — | 4 | 4 | no | — |
| 255 | C4 | — | 4 | 4 | no | — |
| 256 | C4 | — | 4 | 4 | no | — |
| 257 | C2 | — | 2 | 4 | no | — |
| 258 | C2 | — | 8 | 4 | no | — |
| 259 | C2 | #3486 | 1 | 1 | yes | characteristic |
| 260 | C2 | #3488 | 1 | 2 | yes | — |
| 261 | C2 | C2×#562 | 1 | 1 | yes | characteristic |
| 262 | C2 | #3553 | 1 | 1 | yes | characteristic |
| 263 | C2 | #3559 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#562 | C2 | semidirect |
| 2. C22×#125 | C2 | semidirect |
| 3. #2850 | C2 | semidirect |
| 4. #2820 | C2 | semidirect |
| 5. #2711 | C2 | semidirect |
| 6. #2704 | C2 | semidirect |
| 7. #2694 | C2 | semidirect |
| 8. #2449 | C2 | semidirect |
| 9. #2443 | C2 | semidirect |
| 10. #2416 | C2 | semidirect |
| 11. #2382 | C2 | semidirect |
| 70. #125 | D4 | semidirect |
| 242. C22 | #562 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#562 | 2 |
| C22×#125 | 2 |
| #2850 | 2 |
| #2820 | 2 |
| #2711 | 2 |
| #2704 | 2 |
| #2694 | 2 |
| #2449 | 2 |
| #2443 | 2 |
| #2416 | 2 |
| #2382 | 2 |