Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
772 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 772 conjugacy classes of subgroups — too many to draw. Only its 133 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 | #23782500 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #23782230 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #23781853 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #23781852 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #23781851 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #23781637 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | B(4,𝔽2)×C163 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #19104782 | — | 2 | 3 | no | — |
| 9 | #19104780 | — | 2 | 3 | no | — |
| 10 | #19104780 | C22 | 1 | 3 | yes | — |
| 11 | C2×#14579698 | — | 2 | 3 | no | — |
| 12 | C2×#14579698 | C22 | 1 | 3 | yes | — |
| 13 | #41×D163 | — | 2 | 3 | no | — |
| 14 | C22×C22⋊D163 | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | #19104731 | — | 2 | 3 | no | — |
| 16 | #19104721 | — | 2 | 3 | no | — |
| 17 | #19104708 | — | 2 | 3 | no | — |
| 18 | #19104707 | — | 2 | 3 | no | — |
| 19 | #19104705 | C22 | 1 | 3 | yes | complemented |
| 20 | #19104704 | C22 | 1 | 3 | yes | complemented |
| 21 | #19104703 | C22 | 1 | 3 | yes | complemented |
| 22 | #19104691 | C22 | 1 | 3 | yes | complemented |
| 23 | #19104650 | C22 | 1 | 3 | yes | complemented |
| 24 | #19104649 | C22 | 1 | 6 | yes | complemented |
| 25 | H(5,𝔽2)×C163 | C22 | 1 | 1 | yes | characteristic |
| 26 | #120×C163 | C22 | 1 | 3 | yes | complemented |
| 27 | #99×C163 | C22 | 1 | 3 | yes | complemented |
| 28 | C23×D163 | C23 | 1 | 1 | yes | characteristic |
| 29 | #14579700 | — | 2 | 1 | no | — |
| 30 | #14579699 | — | 2 | 6 | no | — |
| 31 | #14579699 | — | 2 | 3 | no | — |
| 32 | #14579698 | — | 2 | 3 | no | — |
| 33 | #14579698 | — | 2 | 6 | no | — |
| 34 | #14579698 | — | 2 | 6 | no | — |
| 35 | #14579698 | — | 2 | 6 | no | — |
| 36 | #14579698 | — | 2 | 6 | no | — |
| 37 | C2×Dic326 | — | 2 | 3 | no | — |
| 38 | C2×Dic326 | D4 | 1 | 3 | yes | complemented |
| 39 | Q8×D163 | — | 2 | 3 | no | — |
| 40 | C2×D652 | — | 2 | 3 | no | — |
| 41 | C2×C22⋊D163 | — | 2 | 3 | no | — |
| 42 | C2×C22⋊D163 | — | 2 | 3 | no | — |
| 43 | C2×C22⋊D163 | — | 2 | 3 | no | — |
| 44 | C2×C22⋊D163 | — | 2 | 3 | no | — |
| 45 | C2×C22⋊D163 | — | 2 | 3 | no | — |
| 46 | C2×C22⋊D163 | — | 2 | 3 | no | — |
| 47 | C2×C22⋊D163 | D4 | 1 | 3 | yes | complemented |
| 48 | C2×C22⋊D163 | C23 | 1 | 3 | yes | — |
| 49 | D4×D163 | — | 2 | 3 | no | — |
| 50 | D4×D163 | — | 2 | 6 | no | — |
| 51 | C22×Dic163 | — | 2 | 1 | no | — |
| 52 | C22×Dic163 | D4 | 1 | 3 | yes | complemented |
| 53 | C22×Dic163 | C23 | 1 | 1 | yes | characteristic |
| 54 | C2×C4×D163 | — | 2 | 3 | no | — |
| 55 | C2×C4×D163 | D4 | 1 | 3 | yes | complemented |
| 56 | Dic163⋊C4 | — | 2 | 3 | no | — |
| 57 | C22⋊Dic163 | — | 2 | 3 | no | — |
| 58 | C22⋊Dic163 | C23 | 1 | 3 | yes | — |
| 59 | (C2×D163)⋊C4 | — | 2 | 3 | no | — |
| 60 | (C2×D163)⋊C4 | — | 2 | 3 | no | — |
| 61 | C4×Dic163 | — | 2 | 3 | no | — |
| 62 | C24×C163 | C23 | 1 | 1 | yes | characteristic |
| 63 | #41×C163 | — | 2 | 3 | no | — |
| 64 | C2×D4×C163 | — | 2 | 3 | no | — |
| 65 | C2×D4×C163 | — | 2 | 3 | no | — |
| 66 | C2×D4×C163 | C23 | 1 | 3 | yes | — |
| 67 | C22⋊C4×C163 | — | 2 | 3 | no | — |
| 68 | C22⋊C4×C163 | C23 | 1 | 3 | yes | — |
| 69 | C22×D163 | — | 2 | 3 | no | — |
| 70 | C22×D163 | — | 2 | 3 | no | — |
| 71 | C22×D163 | — | 8 | 1 | no | — |
| 72 | C22×D163 | C2×D4 | 1 | 3 | yes | — |
| 73 | Dic326 | — | 2 | 3 | no | — |
| 74 | Dic326 | — | 2 | 6 | no | — |
| 75 | D652 | — | 2 | 3 | no | — |
| 76 | D652 | — | 8 | 3 | no | — |
| 77 | C22⋊D163 | — | 2 | 6 | no | — |
| 78 | C22⋊D163 | — | 2 | 6 | no | — |
| 79 | C22⋊D163 | — | 2 | 6 | no | — |
| 80 | C22⋊D163 | — | 4 | 1 | no | — |
| 81 | C22⋊D163 | — | 4 | 3 | no | — |
| 82 | C22⋊D163 | — | 8 | 3 | no | — |
| 83 | C22⋊D163 | — | 8 | 3 | no | — |
| 84 | C2×Dic163 | — | 2 | 3 | no | — |
| 85 | C2×Dic163 | — | 2 | 3 | no | — |
| 86 | C2×Dic163 | — | 2 | 3 | no | — |
| 87 | C2×Dic163 | — | 2 | 3 | no | — |
| 88 | C2×Dic163 | C2×D4 | 1 | 3 | yes | — |
| 89 | C2×Dic163 | C2×D4 | 1 | 3 | yes | — |
| 90 | C2×Dic163 | C2×D4 | 1 | 3 | yes | — |
| 91 | C4×D163 | — | 2 | 6 | no | — |
| 92 | C4×D163 | — | 2 | 3 | no | — |
| 93 | C4×D163 | — | 2 | 3 | no | — |
| 94 | C23×C163 | — | 2 | 1 | no | — |
| 95 | C23×C163 | — | 2 | 3 | no | — |
| 96 | C23×C163 | — | 8 | 1 | no | — |
| 97 | C23×C163 | C2×D4 | 1 | 3 | yes | — |
| 98 | C23×C163 | C24 | 1 | 1 | yes | characteristic |
| 99 | Q8×C163 | — | 2 | 1 | no | — |
| 100 | D4×C163 | — | 2 | 3 | no | — |
| 101 | D4×C163 | — | 2 | 6 | no | — |
| 102 | D4×C163 | — | 8 | 3 | no | — |
| 103 | C2×C4×C163 | — | 2 | 3 | no | — |
| 104 | C2×C4×C163 | — | 2 | 3 | no | — |
| 105 | C2×C4×C163 | C2×D4 | 1 | 3 | yes | — |
| 106 | C22×C163 | — | 2 | 3 | no | — |
| 107 | C22×C163 | — | 4 | 1 | no | — |
| 108 | C22×C163 | — | 4 | 3 | no | — |
| 109 | C22×C163 | — | 4 | 3 | no | — |
| 110 | C22×C163 | — | 4 | 3 | no | — |
| 111 | C22×C163 | — | 4 | 1 | no | — |
| 112 | C22×C163 | — | 8 | 1 | no | — |
| 113 | C22×C163 | C22×D4 | 1 | 3 | yes | — |
| 114 | C2×D163 | — | 2 | 3 | no | — |
| 115 | C2×D163 | — | 4 | 3 | no | — |
| 116 | C2×D163 | — | 4 | 3 | no | — |
| 117 | C2×D163 | — | 4 | 3 | no | — |
| 118 | C2×D163 | — | 4 | 1 | no | — |
| 119 | C2×D163 | — | 8 | 3 | no | — |
| 120 | C4×C163 | — | 2 | 3 | no | — |
| 121 | C4×C163 | — | 4 | 3 | no | — |
| 122 | Dic163 | — | 2 | 3 | no | — |
| 123 | Dic163 | — | 4 | 3 | no | — |
| 124 | Dic163 | #120 | 1 | 1 | yes | characteristic |
| 125 | Dic163 | #120 | 1 | 3 | yes | — |
| 126 | C2×C163 | — | 2 | 3 | no | — |
| 127 | C2×C163 | — | 4 | 3 | no | — |
| 128 | C2×C163 | — | 8 | 1 | no | — |
| 129 | C2×C163 | C2×#120 | 1 | 1 | yes | characteristic |
| 130 | D163 | — | 4 | 3 | no | — |
| 131 | D163 | — | 8 | 1 | no | — |
| 132 | C163 | #3027 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 133 | #3027 | — | 163 | 1 | no | maximal, Sylow |
| 134 | #585 | — | 163 | 1 | no | — |
| 135 | #535 | — | 163 | 3 | no | — |
| 136 | B(4,𝔽2) | — | 163 | 1 | no | — |
| 137 | B(4,𝔽2) | — | 163 | 3 | no | — |
| 138 | B(4,𝔽2) | — | 163 | 3 | no | — |
| 139 | B(4,𝔽2) | D163 | 1 | 1 | yes | characteristic, complemented |
| 140 | #410 | — | 163 | 3 | no | — |
| 141 | 2−1+4 | — | 326 | 3 | no | — |
| 142 | H(5,𝔽2) | — | 326 | 3 | no | — |
| 143 | H(5,𝔽2) | — | 163 | 3 | no | — |
| 144 | H(5,𝔽2) | C2×D163 | 1 | 1 | yes | characteristic |
| 145 | C2×#41 | — | 326 | 3 | no | — |
| 146 | C2×#41 | — | 326 | 3 | no | — |
| 147 | C2×#41 | — | 163 | 3 | no | — |
| 148 | C22×D4 | — | 163 | 1 | no | — |
| 149 | #127 | — | 326 | 3 | no | — |
| 150 | #124 | — | 326 | 3 | no | — |
| 151 | #121 | — | 326 | 3 | no | — |
| 152 | #121 | — | 326 | 3 | no | — |
| 153 | #120 | — | 163 | 3 | no | — |
| 154 | #120 | — | 163 | 3 | no | — |
| 155 | #120 | — | 163 | 3 | no | — |
| 156 | #120 | C2×D163 | 1 | 3 | yes | complemented |
| 157 | #117 | — | 163 | 3 | no | — |
| 158 | #99 | — | 163 | 3 | no | — |
| 159 | #99 | — | 163 | 6 | no | — |
| 160 | #99 | C2×D163 | 1 | 3 | yes | complemented |
| 161 | C24 | — | 163 | 1 | no | — |
| 162 | C24 | C22×D163 | 1 | 1 | yes | characteristic |
| 163 | #41 | — | 2 | 3 | no | — |
| 164 | #41 | — | 326 | 6 | no | — |
| 165 | #41 | — | 326 | 3 | no | — |
| 166 | #41 | — | 326 | 6 | no | — |
| 167 | #41 | — | 326 | 1 | no | — |
| 168 | #41 | — | 326 | 6 | no | — |
| 169 | #41 | — | 326 | 6 | no | — |
| 170 | #41 | — | 326 | 6 | no | — |
| 171 | #41 | — | 326 | 3 | no | — |
| 172 | C2×Q8 | — | 326 | 3 | no | — |
| 173 | C2×Q8 | — | 326 | 3 | no | — |
| 174 | C2×Q8 | — | 163 | 3 | no | — |
| 175 | C2×D4 | — | 2 | 3 | no | — |
| 176 | C2×D4 | — | 2 | 3 | no | — |
| 177 | C2×D4 | — | 326 | 3 | no | — |
| 178 | C2×D4 | — | 326 | 3 | no | — |
| 179 | C2×D4 | — | 326 | 6 | no | — |
| 180 | C2×D4 | — | 326 | 3 | no | — |
| 181 | C2×D4 | — | 326 | 3 | no | — |
| 182 | C2×D4 | — | 326 | 3 | no | — |
| 183 | C2×D4 | — | 326 | 3 | no | — |
| 184 | C2×D4 | — | 326 | 3 | no | — |
| 185 | C2×D4 | — | 326 | 3 | no | — |
| 186 | C2×D4 | — | 163 | 3 | no | — |
| 187 | C2×D4 | — | 163 | 3 | no | — |
| 188 | C2×D4 | C22×D163 | 1 | 3 | yes | — |
| 189 | C22×C4 | — | 326 | 3 | no | — |
| 190 | C22×C4 | — | 326 | 1 | no | — |
| 191 | C22×C4 | — | 163 | 3 | no | — |
| 192 | C22×C4 | — | 163 | 3 | no | — |
| 193 | C22×C4 | — | 163 | 1 | no | — |
| 194 | C4⋊C4 | — | 326 | 3 | no | — |
| 195 | C22⋊C4 | — | 2 | 3 | no | — |
| 196 | C22⋊C4 | — | 326 | 3 | no | — |
| 197 | C22⋊C4 | — | 326 | 3 | no | — |
| 198 | C22⋊C4 | — | 326 | 3 | no | — |
| 199 | C22⋊C4 | — | 163 | 3 | no | — |
| 200 | C22⋊C4 | C22×D163 | 1 | 3 | yes | — |
| 201 | C42 | — | 326 | 3 | no | — |
| 202 | C23 | — | 2 | 1 | no | — |
| 203 | C23 | — | 2 | 3 | no | — |
| 204 | C23 | — | 8 | 1 | no | — |
| 205 | C23 | — | 1304 | 1 | no | — |
| 206 | C23 | — | 326 | 3 | no | — |
| 207 | C23 | — | 326 | 3 | no | — |
| 208 | C23 | — | 163 | 3 | no | — |
| 209 | C23 | D4×D163 | 1 | 3 | yes | — |
| 210 | C23 | C23×D163 | 1 | 1 | yes | characteristic |
| 211 | Q8 | — | 2 | 1 | no | — |
| 212 | Q8 | — | 326 | 3 | no | — |
| 213 | Q8 | — | 326 | 6 | no | — |
| 214 | D4 | — | 2 | 3 | no | — |
| 215 | D4 | — | 2 | 6 | no | — |
| 216 | D4 | — | 8 | 3 | no | — |
| 217 | D4 | — | 1304 | 3 | no | — |
| 218 | D4 | — | 1304 | 3 | no | — |
| 219 | D4 | — | 1304 | 3 | no | — |
| 220 | D4 | — | 326 | 3 | no | — |
| 221 | D4 | — | 326 | 6 | no | — |
| 222 | D4 | — | 326 | 6 | no | — |
| 223 | D4 | — | 326 | 6 | no | — |
| 224 | D4 | — | 652 | 1 | no | — |
| 225 | D4 | — | 652 | 3 | no | — |
| 226 | C2×C4 | — | 2 | 3 | no | — |
| 227 | C2×C4 | — | 2 | 3 | no | — |
| 228 | C2×C4 | — | 326 | 6 | no | — |
| 229 | C2×C4 | — | 326 | 3 | no | — |
| 230 | C2×C4 | — | 326 | 3 | no | — |
| 231 | C2×C4 | — | 326 | 3 | no | — |
| 232 | C2×C4 | — | 326 | 3 | no | — |
| 233 | C2×C4 | — | 326 | 3 | no | — |
| 234 | C2×C4 | — | 326 | 3 | no | — |
| 235 | C2×C4 | — | 163 | 3 | no | — |
| 236 | C2×C4 | — | 163 | 3 | no | — |
| 237 | C2×C4 | — | 163 | 3 | no | — |
| 238 | C2×C4 | D4×D163 | 1 | 3 | yes | — |
| 239 | C22 | — | 2 | 3 | no | — |
| 240 | C22 | — | 4 | 1 | no | — |
| 241 | C22 | — | 4 | 3 | no | — |
| 242 | C22 | — | 4 | 3 | no | — |
| 243 | C22 | — | 4 | 3 | no | — |
| 244 | C22 | — | 4 | 1 | no | — |
| 245 | C22 | — | 8 | 1 | no | — |
| 246 | C22 | — | 1304 | 3 | no | — |
| 247 | C22 | — | 326 | 3 | no | — |
| 248 | C22 | — | 652 | 3 | no | — |
| 249 | C22 | — | 652 | 3 | no | — |
| 250 | C22 | — | 652 | 3 | no | — |
| 251 | C22 | — | 652 | 1 | no | — |
| 252 | C22 | C2×D4×D163 | 1 | 3 | yes | — |
| 253 | C4 | — | 2 | 3 | no | — |
| 254 | C4 | — | 4 | 3 | no | — |
| 255 | C4 | — | 326 | 3 | no | — |
| 256 | C4 | — | 652 | 3 | no | — |
| 257 | C4 | — | 163 | 1 | no | — |
| 258 | C4 | — | 163 | 3 | no | — |
| 259 | C2 | — | 2 | 3 | no | — |
| 260 | C2 | — | 4 | 3 | no | — |
| 261 | C2 | — | 8 | 1 | no | — |
| 262 | C2 | — | 1304 | 1 | no | — |
| 263 | C2 | — | 652 | 3 | no | — |
| 264 | C2 | #120×D163 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23782500 | C2 | semidirect |
| 2. #23782230 | C2 | semidirect |
| 3. #23781853 | C2 | semidirect |
| 4. #23781852 | C2 | semidirect |
| 5. #23781851 | C2 | semidirect |
| 6. #23781637 | C2 | semidirect |
| 7. B(4,𝔽2)×C163 | C2 | semidirect |
| 14. C22×C22⋊D163 | C22 | semidirect |
| 19. #19104705 | C22 | semidirect |
| 20. #19104704 | C22 | semidirect |
| 21. #19104703 | C22 | semidirect |
| 22. #19104691 | C22 | semidirect |
| 23. #19104650 | C22 | semidirect |
| 24. #19104649 | C22 | semidirect |
| 26. #120×C163 | C22 | semidirect |
| 27. #99×C163 | C22 | semidirect |
| 38. C2×Dic326 | D4 | semidirect |
| 47. C2×C22⋊D163 | D4 | semidirect |
| 52. C22×Dic163 | D4 | semidirect |
| 55. C2×C4×D163 | D4 | semidirect |
| 132. C163 | #3027 | semidirect |
| 139. B(4,𝔽2) | D163 | semidirect |
| 156. #120 | C2×D163 | semidirect |
| 160. #99 | C2×D163 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23782500 | 2 |
| #23782230 | 2 |
| #23781853 | 2 |
| #23781852 | 2 |
| #23781851 | 2 |
| #23781637 | 2 |
| B(4,𝔽2)×C163 | 2 |
| #3027 | 163 |