Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
270 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 270 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 | #16811366 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #16810811 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #16810809 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #16810803 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #16810417 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #16810411 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C53×AΣL(1,𝔽16) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #13133262 | C22 | 1 | 1 | yes | characteristic |
| 9 | #13133250 | — | 2 | 1 | no | — |
| 10 | #13133204 | — | 2 | 1 | no | — |
| 11 | #13133204 | C22 | 1 | 1 | yes | characteristic |
| 12 | #13133203 | — | 2 | 1 | no | — |
| 13 | #13133203 | — | 2 | 1 | no | — |
| 14 | #13133203 | — | 2 | 1 | no | — |
| 15 | #13133187 | — | 2 | 1 | no | — |
| 16 | #13133186 | — | 2 | 1 | no | — |
| 17 | #13133186 | — | 2 | 1 | no | — |
| 18 | #13133185 | — | 2 | 1 | no | — |
| 19 | #13133185 | C22 | 1 | 1 | yes | characteristic |
| 20 | #13133143 | — | 2 | 1 | no | — |
| 21 | #13133132 | C22 | 1 | 1 | yes | characteristic |
| 22 | #13133131 | — | 2 | 1 | no | — |
| 23 | #13133131 | — | 2 | 1 | no | — |
| 24 | #120×C53 | C22 | 1 | 1 | yes | characteristic |
| 25 | #100×C53 | C22 | 1 | 1 | yes | characteristic |
| 26 | #99×C53 | C22 | 1 | 1 | yes | characteristic |
| 27 | C24×C53 | D4 | 1 | 1 | yes | characteristic, complemented |
| 28 | C23×D53 | D4 | 1 | 1 | yes | characteristic, complemented |
| 29 | C2×D4×C53 | — | 2 | 1 | no | — |
| 30 | C2×D4×C53 | C23 | 1 | 1 | yes | characteristic |
| 31 | C2×C22⋊D53 | — | 2 | 1 | no | — |
| 32 | C2×C22⋊D53 | — | 2 | 1 | no | — |
| 33 | C2×C22⋊D53 | — | 2 | 1 | no | — |
| 34 | C2×C22⋊D53 | — | 2 | 1 | no | — |
| 35 | C2×C22⋊D53 | — | 2 | 1 | no | — |
| 36 | C2×C22⋊D53 | — | 4 | 1 | no | — |
| 37 | #11719947 | — | 4 | 1 | no | — |
| 38 | D4×D53 | — | 4 | 1 | no | — |
| 39 | #11719945 | — | 2 | 1 | no | — |
| 40 | C2×D212 | — | 4 | 1 | no | — |
| 41 | C2×Dic106 | — | 2 | 1 | no | — |
| 42 | C2×Dic106 | D4 | 1 | 1 | yes | characteristic, complemented |
| 43 | M16×C53 | — | 2 | 1 | no | — |
| 44 | C22⋊C4×C53 | — | 2 | 1 | no | — |
| 45 | C22⋊C4×C53 | — | 2 | 1 | no | — |
| 46 | C22⋊C4×C53 | D4 | 1 | 1 | yes | characteristic, complemented |
| 47 | C22⋊Dic53 | — | 2 | 1 | no | — |
| 48 | C22⋊Dic53 | — | 2 | 1 | no | — |
| 49 | C22⋊Dic53 | — | 2 | 1 | no | — |
| 50 | C22⋊Dic53 | — | 2 | 1 | no | — |
| 51 | C22⋊Dic53 | — | 2 | 1 | no | — |
| 52 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 53 | C22⋊Dic53 | D4 | 1 | 1 | yes | characteristic, complemented |
| 54 | #11719921 | — | 2 | 1 | no | — |
| 55 | #11719921 | — | 2 | 1 | no | — |
| 56 | #11719921 | — | 2 | 1 | no | — |
| 57 | #11719921 | — | 4 | 1 | no | — |
| 58 | C4×Dic53 | — | 2 | 1 | no | — |
| 59 | C4×Dic53 | D4 | 1 | 1 | yes | characteristic, complemented |
| 60 | Dic212 | — | 4 | 1 | no | — |
| 61 | #11719913 | — | 4 | 1 | no | — |
| 62 | C23×C53 | — | 2 | 1 | no | — |
| 63 | C23×C53 | — | 4 | 1 | no | — |
| 64 | C23×C53 | — | 8 | 1 | no | — |
| 65 | C23×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 66 | C23×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 67 | C22×D53 | — | 2 | 1 | no | — |
| 68 | C22×D53 | — | 2 | 1 | no | — |
| 69 | C22×D53 | — | 2 | 1 | no | — |
| 70 | C22×D53 | — | 4 | 1 | no | — |
| 71 | C22×D53 | — | 8 | 1 | no | — |
| 72 | D4×C53 | — | 4 | 1 | no | — |
| 73 | D4×C53 | — | 8 | 1 | no | — |
| 74 | C2×C4×C53 | — | 2 | 1 | no | — |
| 75 | C2×C4×C53 | — | 4 | 1 | no | — |
| 76 | C2×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 77 | C22⋊D53 | — | 4 | 1 | no | — |
| 78 | C22⋊D53 | — | 4 | 1 | no | — |
| 79 | C22⋊D53 | — | 4 | 1 | no | — |
| 80 | C22⋊D53 | — | 8 | 1 | no | — |
| 81 | C22⋊D53 | — | 8 | 1 | no | — |
| 82 | C22⋊D53 | — | 8 | 1 | no | — |
| 83 | C22⋊D53 | — | 8 | 2 | no | — |
| 84 | C2×Dic53 | — | 2 | 1 | no | — |
| 85 | C2×Dic53 | — | 2 | 1 | no | — |
| 86 | C2×Dic53 | — | 2 | 1 | no | — |
| 87 | C2×Dic53 | — | 2 | 1 | no | — |
| 88 | C2×Dic53 | — | 2 | 1 | no | — |
| 89 | C2×Dic53 | — | 4 | 1 | no | — |
| 90 | D212 | — | 2 | 1 | no | — |
| 91 | D212 | — | 8 | 2 | no | — |
| 92 | C4×D53 | — | 4 | 1 | no | — |
| 93 | Dic106 | — | 2 | 1 | no | — |
| 94 | Dic106 | — | 4 | 1 | no | — |
| 95 | Dic106 | — | 8 | 1 | no | — |
| 96 | C8×C53 | — | 4 | 1 | no | — |
| 97 | C22×C53 | — | 2 | 1 | no | — |
| 98 | C22×C53 | — | 2 | 1 | no | — |
| 99 | C22×C53 | — | 4 | 1 | no | — |
| 100 | C22×C53 | — | 4 | 1 | no | — |
| 101 | C22×C53 | — | 4 | 1 | no | — |
| 102 | C22×C53 | — | 4 | 1 | no | — |
| 103 | C22×C53 | — | 4 | 2 | no | — |
| 104 | C22×C53 | — | 4 | 1 | no | — |
| 105 | C22×C53 | — | 8 | 1 | no | — |
| 106 | C22×C53 | #120 | 1 | 1 | yes | characteristic |
| 107 | C2×D53 | — | 4 | 1 | no | — |
| 108 | C2×D53 | — | 4 | 1 | no | — |
| 109 | C2×D53 | — | 4 | 2 | no | — |
| 110 | C2×D53 | — | 4 | 1 | no | — |
| 111 | C2×D53 | — | 4 | 1 | no | — |
| 112 | C2×D53 | — | 8 | 1 | no | — |
| 113 | C2×D53 | — | 8 | 1 | no | — |
| 114 | C2×D53 | — | 8 | 1 | no | — |
| 115 | C4×C53 | — | 2 | 1 | no | — |
| 116 | C4×C53 | — | 4 | 1 | no | — |
| 117 | C4×C53 | — | 8 | 1 | no | — |
| 118 | Dic53 | — | 4 | 1 | no | — |
| 119 | Dic53 | — | 4 | 1 | no | — |
| 120 | Dic53 | — | 4 | 1 | no | — |
| 121 | Dic53 | — | 4 | 1 | no | — |
| 122 | Dic53 | — | 8 | 1 | no | — |
| 123 | #2199 | — | 53 | 1 | no | maximal, Sylow |
| 124 | C2×C53 | — | 2 | 1 | no | — |
| 125 | C2×C53 | — | 4 | 1 | no | — |
| 126 | C2×C53 | — | 4 | 1 | no | — |
| 127 | C2×C53 | — | 8 | 1 | no | — |
| 128 | C2×C53 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 129 | D53 | — | 8 | 1 | no | — |
| 130 | D53 | — | 8 | 1 | no | — |
| 131 | H(3,𝔽4) | — | 53 | 1 | no | — |
| 132 | B(4,𝔽2) | — | 53 | 1 | no | — |
| 133 | B(4,𝔽2) | — | 53 | 1 | no | — |
| 134 | #455 | — | 53 | 1 | no | — |
| 135 | #354 | — | 53 | 1 | no | — |
| 136 | AΣL(1,𝔽16) | — | 53 | 1 | no | — |
| 137 | AΣL(1,𝔽16) | D53 | 1 | 1 | yes | characteristic, complemented |
| 138 | C53 | #2199 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 139 | H(5,𝔽2) | — | 53 | 1 | no | — |
| 140 | #137 | — | 106 | 1 | no | — |
| 141 | #124 | — | 53 | 1 | no | — |
| 142 | #124 | — | 106 | 1 | no | — |
| 143 | #124 | — | 106 | 1 | no | — |
| 144 | #124 | — | 106 | 1 | no | — |
| 145 | #124 | — | 106 | 1 | no | — |
| 146 | #120 | — | 53 | 1 | no | — |
| 147 | #120 | — | 106 | 1 | no | — |
| 148 | #120 | — | 106 | 1 | no | — |
| 149 | #120 | — | 106 | 1 | no | — |
| 150 | #120 | — | 106 | 1 | no | — |
| 151 | #120 | C2×D53 | 1 | 1 | yes | characteristic |
| 152 | #104 | — | 106 | 1 | no | — |
| 153 | #100 | C2×D53 | 1 | 1 | yes | characteristic |
| 154 | #99 | — | 53 | 1 | no | — |
| 155 | #99 | — | 106 | 1 | no | — |
| 156 | #99 | — | 106 | 1 | no | — |
| 157 | #99 | C2×D53 | 1 | 1 | yes | characteristic |
| 158 | C24 | — | 53 | 1 | no | — |
| 159 | C24 | D212 | 1 | 1 | yes | characteristic, complemented |
| 160 | #41 | — | 106 | 1 | no | — |
| 161 | #41 | — | 212 | 1 | no | — |
| 162 | C2×Q8 | — | 53 | 1 | no | — |
| 163 | C2×Q8 | — | 106 | 1 | no | — |
| 164 | C2×D4 | — | 2 | 1 | no | — |
| 165 | C2×D4 | — | 106 | 1 | no | — |
| 166 | C2×D4 | — | 106 | 1 | no | — |
| 167 | C2×D4 | — | 106 | 1 | no | — |
| 168 | C2×D4 | — | 106 | 1 | no | — |
| 169 | C2×D4 | — | 106 | 1 | no | — |
| 170 | C2×D4 | — | 212 | 1 | no | — |
| 171 | C2×D4 | — | 212 | 1 | no | — |
| 172 | C2×D4 | — | 212 | 1 | no | — |
| 173 | C2×D4 | C22×D53 | 1 | 1 | yes | characteristic |
| 174 | Q16 | — | 212 | 1 | no | — |
| 175 | SD16 | — | 212 | 1 | no | — |
| 176 | M16 | — | 2 | 1 | no | — |
| 177 | C22⋊C4 | — | 2 | 1 | no | — |
| 178 | C22⋊C4 | — | 2 | 1 | no | — |
| 179 | C22⋊C4 | — | 53 | 1 | no | — |
| 180 | C22⋊C4 | — | 106 | 1 | no | — |
| 181 | C22⋊C4 | — | 106 | 1 | no | — |
| 182 | C22⋊C4 | — | 106 | 1 | no | — |
| 183 | C22⋊C4 | — | 106 | 1 | no | — |
| 184 | C22⋊C4 | — | 106 | 1 | no | — |
| 185 | C22⋊C4 | — | 106 | 1 | no | — |
| 186 | C22⋊C4 | — | 106 | 1 | no | — |
| 187 | C22⋊C4 | — | 106 | 1 | no | — |
| 188 | C22⋊C4 | — | 212 | 1 | no | — |
| 189 | C22⋊C4 | — | 212 | 1 | no | — |
| 190 | C22⋊C4 | D212 | 1 | 1 | yes | characteristic, complemented |
| 191 | C42 | — | 53 | 1 | no | — |
| 192 | C42 | — | 106 | 1 | no | — |
| 193 | C23 | — | 2 | 1 | no | — |
| 194 | C23 | — | 4 | 1 | no | — |
| 195 | C23 | — | 8 | 1 | no | — |
| 196 | C23 | — | 106 | 1 | no | — |
| 197 | C23 | — | 106 | 1 | no | — |
| 198 | C23 | — | 106 | 1 | no | — |
| 199 | C23 | — | 424 | 1 | no | — |
| 200 | C23 | — | 212 | 1 | no | — |
| 201 | C23 | C2×D212 | 1 | 1 | yes | characteristic |
| 202 | C23 | D4×D53 | 1 | 1 | yes | characteristic |
| 203 | Q8 | — | 106 | 1 | no | — |
| 204 | Q8 | — | 424 | 1 | no | — |
| 205 | Q8 | — | 212 | 1 | no | — |
| 206 | D4 | — | 4 | 1 | no | — |
| 207 | D4 | — | 8 | 1 | no | — |
| 208 | D4 | — | 106 | 1 | no | — |
| 209 | D4 | — | 424 | 2 | no | — |
| 210 | D4 | — | 424 | 2 | no | — |
| 211 | D4 | — | 424 | 1 | no | — |
| 212 | D4 | — | 424 | 1 | no | — |
| 213 | D4 | — | 424 | 1 | no | — |
| 214 | D4 | — | 212 | 1 | no | — |
| 215 | D4 | — | 212 | 1 | no | — |
| 216 | D4 | — | 212 | 1 | no | — |
| 217 | C2×C4 | — | 2 | 1 | no | — |
| 218 | C2×C4 | — | 4 | 1 | no | — |
| 219 | C2×C4 | — | 106 | 1 | no | — |
| 220 | C2×C4 | — | 106 | 1 | no | — |
| 221 | C2×C4 | — | 106 | 1 | no | — |
| 222 | C2×C4 | — | 106 | 1 | no | — |
| 223 | C2×C4 | — | 106 | 1 | no | — |
| 224 | C2×C4 | — | 212 | 1 | no | — |
| 225 | C2×C4 | — | 212 | 1 | no | — |
| 226 | C2×C4 | D4×D53 | 1 | 1 | yes | characteristic |
| 227 | C8 | — | 4 | 1 | no | — |
| 228 | C22 | — | 2 | 1 | no | — |
| 229 | C22 | — | 2 | 1 | no | — |
| 230 | C22 | — | 4 | 1 | no | — |
| 231 | C22 | — | 4 | 1 | no | — |
| 232 | C22 | — | 4 | 1 | no | — |
| 233 | C22 | — | 4 | 1 | no | — |
| 234 | C22 | — | 4 | 2 | no | — |
| 235 | C22 | — | 4 | 1 | no | — |
| 236 | C22 | — | 8 | 1 | no | — |
| 237 | C22 | — | 424 | 1 | no | — |
| 238 | C22 | — | 424 | 1 | no | — |
| 239 | C22 | — | 424 | 1 | no | — |
| 240 | C22 | — | 212 | 1 | no | — |
| 241 | C22 | — | 212 | 1 | no | — |
| 242 | C22 | — | 212 | 2 | no | — |
| 243 | C22 | — | 212 | 1 | no | — |
| 244 | C22 | — | 212 | 1 | no | — |
| 245 | C22 | #13133187 | 1 | 1 | yes | characteristic |
| 246 | C4 | — | 2 | 1 | no | — |
| 247 | C4 | — | 4 | 1 | no | — |
| 248 | C4 | — | 8 | 1 | no | — |
| 249 | C4 | — | 424 | 1 | no | — |
| 250 | C4 | — | 212 | 1 | no | — |
| 251 | C4 | — | 212 | 1 | no | — |
| 252 | C4 | — | 212 | 1 | no | — |
| 253 | C4 | — | 212 | 1 | no | — |
| 254 | C2 | — | 2 | 1 | no | — |
| 255 | C2 | — | 4 | 1 | no | — |
| 256 | C2 | — | 4 | 1 | no | — |
| 257 | C2 | — | 8 | 1 | no | — |
| 258 | C2 | — | 424 | 1 | no | — |
| 259 | C2 | — | 424 | 1 | no | — |
| 260 | C2 | #16810811 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811366 | C2 | semidirect |
| 2. #16810811 | C2 | semidirect |
| 3. #16810809 | C2 | semidirect |
| 4. #16810803 | C2 | semidirect |
| 5. #16810417 | C2 | semidirect |
| 6. #16810411 | C2 | semidirect |
| 7. C53×AΣL(1,𝔽16) | C2 | semidirect |
| 27. C24×C53 | D4 | semidirect |
| 28. C23×D53 | D4 | semidirect |
| 42. C2×Dic106 | D4 | semidirect |
| 46. C22⋊C4×C53 | D4 | semidirect |
| 53. C22⋊Dic53 | D4 | semidirect |
| 59. C4×Dic53 | D4 | semidirect |
| 137. AΣL(1,𝔽16) | D53 | semidirect |
| 138. C53 | #2199 | semidirect |
| 159. C24 | D212 | semidirect |
| 190. C22⋊C4 | D212 | semidirect |