Finite Group DatabasebetaFinite Group DatabasebetaOrder 8576.
260 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 41 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 | #16864135 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #16863587 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #16863582 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #16863566 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #16863188 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #16863187 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | AΣL(1,𝔽16)×C67 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #14524088 | C22 | 1 | 1 | yes | characteristic |
| 9 | #14524071 | — | 2 | 1 | no | — |
| 10 | #14524039 | C22 | 1 | 1 | yes | characteristic |
| 11 | #14524036 | — | 2 | 1 | no | — |
| 12 | #14524027 | — | 2 | 1 | no | — |
| 13 | #14524025 | — | 2 | 1 | no | — |
| 14 | #14524024 | C22 | 1 | 1 | yes | characteristic |
| 15 | #14524016 | — | 2 | 1 | no | — |
| 16 | #14524015 | — | 2 | 1 | no | — |
| 17 | #14524014 | — | 2 | 1 | no | — |
| 18 | #14524012 | — | 2 | 1 | no | — |
| 19 | #14524012 | — | 2 | 1 | no | — |
| 20 | #14523969 | — | 2 | 1 | no | — |
| 21 | #14523958 | C22 | 1 | 1 | yes | characteristic |
| 22 | #14523957 | — | 2 | 1 | no | — |
| 23 | #14523957 | — | 2 | 1 | no | — |
| 24 | #120×C67 | C22 | 1 | 1 | yes | characteristic |
| 25 | #100×C67 | C22 | 1 | 1 | yes | characteristic |
| 26 | #99×C67 | C22 | 1 | 1 | yes | characteristic |
| 27 | #11761628 | — | 2 | 1 | no | — |
| 28 | #11761627 | — | 4 | 1 | no | — |
| 29 | C2×D268 | D4 | 1 | 1 | yes | characteristic, complemented |
| 30 | C2×C22⋊D67 | — | 2 | 1 | no | — |
| 31 | C2×C22⋊D67 | — | 2 | 1 | no | — |
| 32 | C2×C22⋊D67 | — | 2 | 1 | no | — |
| 33 | C2×C22⋊D67 | — | 2 | 1 | no | — |
| 34 | C2×C22⋊D67 | — | 2 | 1 | no | — |
| 35 | C2×C22⋊D67 | — | 2 | 1 | no | — |
| 36 | C2×C22⋊D67 | — | 4 | 1 | no | — |
| 37 | D4×D67 | — | 4 | 1 | no | — |
| 38 | C22×Dic67 | D4 | 1 | 1 | yes | characteristic, complemented |
| 39 | C2×C4×D67 | — | 2 | 1 | no | — |
| 40 | #11761617 | — | 4 | 1 | no | — |
| 41 | D536 | — | 4 | 1 | no | — |
| 42 | C4⋊Dic67 | — | 2 | 1 | no | — |
| 43 | C4⋊Dic67 | — | 4 | 1 | no | — |
| 44 | Dic67⋊C4 | — | 2 | 1 | no | — |
| 45 | Dic67⋊C4 | — | 2 | 1 | no | — |
| 46 | C22⋊Dic67 | — | 2 | 1 | no | — |
| 47 | C22⋊Dic67 | — | 2 | 1 | no | — |
| 48 | C22⋊Dic67 | — | 2 | 1 | no | — |
| 49 | C22⋊Dic67 | — | 2 | 1 | no | — |
| 50 | C22⋊Dic67 | — | 4 | 1 | no | — |
| 51 | C22⋊Dic67 | D4 | 1 | 1 | yes | characteristic, complemented |
| 52 | (C2×D67)⋊C4 | — | 2 | 1 | no | — |
| 53 | (C2×D67)⋊C4 | — | 4 | 1 | no | — |
| 54 | C4×Dic67 | D4 | 1 | 1 | yes | characteristic, complemented |
| 55 | C24×C67 | D4 | 1 | 1 | yes | characteristic |
| 56 | C2×D4×C67 | — | 2 | 1 | no | — |
| 57 | C2×D4×C67 | C23 | 1 | 1 | yes | characteristic |
| 58 | M16×C67 | — | 2 | 1 | no | — |
| 59 | C22⋊C4×C67 | — | 2 | 1 | no | — |
| 60 | C22⋊C4×C67 | — | 2 | 1 | no | — |
| 61 | C22⋊C4×C67 | D4 | 1 | 1 | yes | characteristic |
| 62 | C23×C67 | — | 2 | 1 | no | — |
| 63 | C23×C67 | — | 4 | 1 | no | — |
| 64 | C23×C67 | — | 8 | 1 | no | — |
| 65 | C23×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 66 | C23×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 67 | C22×D67 | — | 2 | 1 | no | — |
| 68 | C22×D67 | — | 2 | 1 | no | — |
| 69 | C22×D67 | — | 2 | 1 | no | — |
| 70 | D4×C67 | — | 4 | 1 | no | — |
| 71 | D4×C67 | — | 8 | 1 | no | — |
| 72 | C2×C4×C67 | — | 2 | 1 | no | — |
| 73 | C2×C4×C67 | — | 4 | 1 | no | — |
| 74 | C2×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C22⋊D67 | — | 4 | 1 | no | — |
| 76 | C22⋊D67 | — | 4 | 1 | no | — |
| 77 | C22⋊D67 | — | 4 | 1 | no | — |
| 78 | C22⋊D67 | — | 8 | 1 | no | — |
| 79 | C22⋊D67 | — | 8 | 1 | no | — |
| 80 | C22⋊D67 | — | 8 | 1 | no | — |
| 81 | C22⋊D67 | — | 8 | 1 | no | — |
| 82 | C22⋊D67 | — | 8 | 2 | no | — |
| 83 | C2×Dic67 | — | 2 | 1 | no | — |
| 84 | C2×Dic67 | — | 2 | 1 | no | — |
| 85 | C2×Dic67 | — | 2 | 1 | no | — |
| 86 | C2×Dic67 | — | 2 | 1 | no | — |
| 87 | C2×Dic67 | — | 2 | 1 | no | — |
| 88 | C2×Dic67 | — | 4 | 1 | no | — |
| 89 | C2×Dic67 | — | 4 | 1 | no | — |
| 90 | D268 | — | 2 | 1 | no | — |
| 91 | D268 | — | 4 | 1 | no | — |
| 92 | C4×D67 | — | 4 | 1 | no | — |
| 93 | C4×D67 | — | 8 | 1 | no | — |
| 94 | Dic134 | — | 2 | 1 | no | — |
| 95 | C8×C67 | — | 4 | 1 | no | — |
| 96 | C22×C67 | — | 2 | 1 | no | — |
| 97 | C22×C67 | — | 2 | 1 | no | — |
| 98 | C22×C67 | — | 4 | 1 | no | — |
| 99 | C22×C67 | — | 4 | 1 | no | — |
| 100 | C22×C67 | — | 4 | 1 | no | — |
| 101 | C22×C67 | — | 4 | 1 | no | — |
| 102 | C22×C67 | — | 4 | 2 | no | — |
| 103 | C22×C67 | — | 4 | 1 | no | — |
| 104 | C22×C67 | — | 8 | 1 | no | — |
| 105 | C22×C67 | #120 | 1 | 1 | yes | characteristic |
| 106 | C2×D67 | — | 4 | 1 | no | — |
| 107 | C2×D67 | — | 4 | 1 | no | — |
| 108 | C2×D67 | — | 4 | 1 | no | — |
| 109 | C2×D67 | — | 4 | 1 | no | — |
| 110 | C2×D67 | — | 8 | 1 | no | — |
| 111 | C2×D67 | — | 8 | 1 | no | — |
| 112 | C4×C67 | — | 2 | 1 | no | — |
| 113 | C4×C67 | — | 4 | 1 | no | — |
| 114 | C4×C67 | — | 8 | 1 | no | — |
| 115 | Dic67 | — | 4 | 1 | no | — |
| 116 | Dic67 | — | 4 | 1 | no | — |
| 117 | Dic67 | — | 4 | 1 | no | — |
| 118 | Dic67 | — | 4 | 1 | no | — |
| 119 | Dic67 | — | 8 | 1 | no | — |
| 120 | C2×C67 | — | 2 | 1 | no | — |
| 121 | C2×C67 | — | 4 | 1 | no | — |
| 122 | C2×C67 | — | 4 | 1 | no | — |
| 123 | C2×C67 | — | 8 | 1 | no | — |
| 124 | C2×C67 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 125 | D67 | — | 8 | 1 | no | — |
| 126 | D67 | — | 8 | 1 | no | — |
| 127 | #2200 | — | 67 | 1 | no | maximal, Sylow |
| 128 | C67 | #2200 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 129 | #560 | — | 67 | 1 | no | — |
| 130 | #458 | — | 67 | 1 | no | — |
| 131 | B(4,𝔽2) | — | 67 | 1 | no | — |
| 132 | #453 | — | 67 | 1 | no | — |
| 133 | #353 | — | 67 | 1 | no | — |
| 134 | #352 | — | 67 | 1 | no | — |
| 135 | AΣL(1,𝔽16) | D67 | 1 | 1 | yes | characteristic, complemented |
| 136 | H(5,𝔽2) | — | 67 | 1 | no | — |
| 137 | AGL(1,ℤ/8) | — | 134 | 1 | no | — |
| 138 | #127 | — | 67 | 1 | no | — |
| 139 | #126 | — | 134 | 1 | no | — |
| 140 | #123 | — | 67 | 1 | no | — |
| 141 | #123 | — | 134 | 1 | no | — |
| 142 | #123 | — | 134 | 1 | no | — |
| 143 | #121 | — | 134 | 1 | no | — |
| 144 | #121 | — | 134 | 1 | no | — |
| 145 | #121 | — | 134 | 1 | no | — |
| 146 | #120 | — | 134 | 1 | no | — |
| 147 | #120 | — | 134 | 1 | no | — |
| 148 | #120 | C2×D67 | 1 | 1 | yes | characteristic |
| 149 | #104 | — | 134 | 1 | no | — |
| 150 | #100 | C2×D67 | 1 | 1 | yes | characteristic |
| 151 | #99 | — | 67 | 1 | no | — |
| 152 | #99 | — | 134 | 1 | no | — |
| 153 | #99 | — | 134 | 1 | no | — |
| 154 | #99 | C2×D67 | 1 | 1 | yes | characteristic |
| 155 | C24 | D268 | 1 | 1 | yes | characteristic |
| 156 | #41 | — | 268 | 1 | no | — |
| 157 | #41 | — | 134 | 1 | no | — |
| 158 | C2×D4 | — | 2 | 1 | no | — |
| 159 | C2×D4 | — | 268 | 1 | no | — |
| 160 | C2×D4 | — | 268 | 1 | no | — |
| 161 | C2×D4 | — | 67 | 1 | no | — |
| 162 | C2×D4 | — | 134 | 1 | no | — |
| 163 | C2×D4 | — | 134 | 1 | no | — |
| 164 | C2×D4 | — | 134 | 1 | no | — |
| 165 | C2×D4 | — | 134 | 1 | no | — |
| 166 | C2×D4 | — | 134 | 1 | no | — |
| 167 | C2×D4 | — | 134 | 1 | no | — |
| 168 | C2×D4 | C22×D67 | 1 | 1 | yes | characteristic |
| 169 | C22×C4 | — | 67 | 1 | no | — |
| 170 | C22×C4 | — | 134 | 1 | no | — |
| 171 | SD16 | — | 268 | 1 | no | — |
| 172 | D8 | — | 268 | 1 | no | — |
| 173 | M16 | — | 2 | 1 | no | — |
| 174 | C4⋊C4 | — | 268 | 1 | no | — |
| 175 | C4⋊C4 | — | 134 | 1 | no | — |
| 176 | C4⋊C4 | — | 134 | 1 | no | — |
| 177 | C4⋊C4 | — | 134 | 1 | no | — |
| 178 | C22⋊C4 | — | 2 | 1 | no | — |
| 179 | C22⋊C4 | — | 2 | 1 | no | — |
| 180 | C22⋊C4 | — | 268 | 1 | no | — |
| 181 | C22⋊C4 | — | 268 | 1 | no | — |
| 182 | C22⋊C4 | — | 67 | 1 | no | — |
| 183 | C22⋊C4 | — | 134 | 1 | no | — |
| 184 | C22⋊C4 | — | 134 | 1 | no | — |
| 185 | C22⋊C4 | — | 134 | 1 | no | — |
| 186 | C22⋊C4 | — | 134 | 1 | no | — |
| 187 | C22⋊C4 | — | 134 | 1 | no | — |
| 188 | C22⋊C4 | D268 | 1 | 1 | yes | characteristic |
| 189 | C42 | — | 67 | 1 | no | — |
| 190 | C23 | — | 2 | 1 | no | — |
| 191 | C23 | — | 4 | 1 | no | — |
| 192 | C23 | — | 8 | 1 | no | — |
| 193 | C23 | — | 134 | 1 | no | — |
| 194 | C23 | — | 134 | 1 | no | — |
| 195 | C23 | — | 134 | 1 | no | — |
| 196 | C23 | D4×D67 | 1 | 1 | yes | characteristic |
| 197 | C23 | C2×D268 | 1 | 1 | yes | characteristic |
| 198 | Q8 | — | 134 | 1 | no | — |
| 199 | D4 | — | 4 | 1 | no | — |
| 200 | D4 | — | 8 | 1 | no | — |
| 201 | D4 | — | 268 | 1 | no | — |
| 202 | D4 | — | 268 | 1 | no | — |
| 203 | D4 | — | 268 | 1 | no | — |
| 204 | D4 | — | 268 | 1 | no | — |
| 205 | D4 | — | 536 | 1 | no | — |
| 206 | D4 | — | 536 | 1 | no | — |
| 207 | D4 | — | 536 | 1 | no | — |
| 208 | D4 | — | 536 | 1 | no | — |
| 209 | D4 | — | 536 | 2 | no | — |
| 210 | D4 | — | 134 | 1 | no | — |
| 211 | C2×C4 | — | 2 | 1 | no | — |
| 212 | C2×C4 | — | 4 | 1 | no | — |
| 213 | C2×C4 | — | 268 | 1 | no | — |
| 214 | C2×C4 | — | 268 | 1 | no | — |
| 215 | C2×C4 | — | 268 | 1 | no | — |
| 216 | C2×C4 | — | 536 | 1 | no | — |
| 217 | C2×C4 | — | 134 | 1 | no | — |
| 218 | C2×C4 | — | 134 | 1 | no | — |
| 219 | C2×C4 | — | 134 | 1 | no | — |
| 220 | C2×C4 | — | 134 | 1 | no | — |
| 221 | C2×C4 | — | 134 | 1 | no | — |
| 222 | C2×C4 | D4×D67 | 1 | 1 | yes | characteristic |
| 223 | C8 | — | 4 | 1 | no | — |
| 224 | C22 | — | 2 | 1 | no | — |
| 225 | C22 | — | 2 | 1 | no | — |
| 226 | C22 | — | 4 | 1 | no | — |
| 227 | C22 | — | 4 | 1 | no | — |
| 228 | C22 | — | 4 | 1 | no | — |
| 229 | C22 | — | 4 | 1 | no | — |
| 230 | C22 | — | 4 | 2 | no | — |
| 231 | C22 | — | 4 | 1 | no | — |
| 232 | C22 | — | 8 | 1 | no | — |
| 233 | C22 | — | 268 | 1 | no | — |
| 234 | C22 | — | 268 | 1 | no | — |
| 235 | C22 | — | 268 | 1 | no | — |
| 236 | C22 | — | 268 | 1 | no | — |
| 237 | C22 | — | 536 | 1 | no | — |
| 238 | C22 | — | 536 | 1 | no | — |
| 239 | C22 | #14524013 | 1 | 1 | yes | characteristic |
| 240 | C4 | — | 2 | 1 | no | — |
| 241 | C4 | — | 4 | 1 | no | — |
| 242 | C4 | — | 8 | 1 | no | — |
| 243 | C4 | — | 268 | 1 | no | — |
| 244 | C4 | — | 268 | 1 | no | — |
| 245 | C4 | — | 268 | 1 | no | — |
| 246 | C4 | — | 268 | 1 | no | — |
| 247 | C4 | — | 536 | 1 | no | — |
| 248 | C2 | — | 2 | 1 | no | — |
| 249 | C2 | — | 4 | 1 | no | — |
| 250 | C2 | — | 4 | 1 | no | — |
| 251 | C2 | — | 8 | 1 | no | — |
| 252 | C2 | — | 536 | 1 | no | — |
| 253 | C2 | — | 536 | 1 | no | — |
| 254 | C2 | #16863584 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16864135 | C2 | semidirect |
| 2. #16863587 | C2 | semidirect |
| 3. #16863582 | C2 | semidirect |
| 4. #16863566 | C2 | semidirect |
| 5. #16863188 | C2 | semidirect |
| 6. #16863187 | C2 | semidirect |
| 7. AΣL(1,𝔽16)×C67 | C2 | semidirect |
| 29. C2×D268 | D4 | semidirect |
| 38. C22×Dic67 | D4 | semidirect |
| 51. C22⋊Dic67 | D4 | semidirect |
| 54. C4×Dic67 | D4 | semidirect |
| 128. C67 | #2200 | semidirect |
| 135. AΣL(1,𝔽16) | D67 | semidirect |