Finite Group DatabasebetaFinite Group DatabasebetaOrder 15616.
1188 conjugacy classes of subgroups; only the 220 normal subgroups are shown.
This group has 1188 conjugacy classes of subgroups — too many to draw. Only its 220 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 | #21462421 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #21462103 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #21462092 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #21459604 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #16840830 | — | 2 | 2 | no | — |
| 6 | #16840742 | — | 2 | 6 | no | — |
| 7 | #16840741 | C22 | 1 | 12 | yes | — |
| 8 | #16840669 | C22 | 1 | 12 | yes | — |
| 9 | #16840667 | C22 | 1 | 4 | yes | — |
| 10 | C2×#14519824 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×#14519824 | C22 | 1 | 1 | yes | characteristic |
| 12 | Q8×D244 | — | 2 | 2 | no | — |
| 13 | #16840416 | — | 2 | 6 | no | — |
| 14 | #16840338 | C4 | 1 | 3 | yes | complemented |
| 15 | #16840338 | C22 | 1 | 3 | yes | — |
| 16 | #16840312 | C22 | 1 | 3 | yes | — |
| 17 | C2×#14519777 | C4 | 1 | 3 | yes | complemented |
| 18 | C4×#41×C61 | C4 | 1 | 1 | yes | characteristic, complemented |
| 19 | Q8×#6923 | — | 4 | 8 | no | — |
| 20 | #14519856 | — | 2 | 12 | no | — |
| 21 | #14519836 | C23 | 1 | 8 | yes | — |
| 22 | #14519835 | — | 2 | 12 | no | — |
| 23 | #14519824 | — | 4 | 4 | no | — |
| 24 | #14519824 | — | 4 | 4 | no | — |
| 25 | C2×#11758222 | C2×C4 | 1 | 3 | yes | — |
| 26 | C2×#11758222 | C23 | 1 | 3 | yes | — |
| 27 | C2×#11758224 | — | 2 | 3 | no | — |
| 28 | C2×#11758224 | D4 | 1 | 6 | yes | — |
| 29 | C22×Dic122 | C2×C4 | 1 | 3 | yes | — |
| 30 | C22×Dic122 | C2×C4 | 1 | 3 | yes | — |
| 31 | C2×Q8×D61 | — | 2 | 1 | no | — |
| 32 | C2×Q8×D61 | D4 | 1 | 2 | yes | — |
| 33 | #14519777 | — | 2 | 6 | no | — |
| 34 | #14519777 | C2×C4 | 1 | 3 | yes | — |
| 35 | #14519777 | C2×C4 | 1 | 6 | yes | — |
| 36 | #14519774 | C23 | 1 | 1 | yes | characteristic |
| 37 | #14519765 | C2×C4 | 1 | 3 | yes | — |
| 38 | #14519765 | C23 | 1 | 3 | yes | — |
| 39 | #14519761 | — | 2 | 6 | no | — |
| 40 | #14519761 | — | 2 | 6 | no | — |
| 41 | #14519757 | — | 2 | 6 | no | — |
| 42 | #14519757 | — | 2 | 6 | no | — |
| 43 | #14519751 | — | 2 | 6 | no | — |
| 44 | C4×D244 | — | 2 | 6 | no | — |
| 45 | C2×C4⋊Dic61 | — | 2 | 6 | no | — |
| 46 | C2×#41×C61 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | C4×Q8×C61 | — | 2 | 1 | no | — |
| 48 | C4×D4×C61 | — | 2 | 3 | no | — |
| 49 | #117×C61 | C2×C4 | 1 | 3 | yes | — |
| 50 | C2×C42×C61 | C2×C4 | 1 | 3 | yes | — |
| 51 | #41×C61 | — | 2 | 4 | no | — |
| 52 | C2×Q8×C61 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 53 | C2×D4×C61 | C22⋊C4 | 1 | 3 | yes | — |
| 54 | C22×C4×C61 | — | 2 | 3 | no | — |
| 55 | C22×C4×C61 | C22×C4 | 1 | 3 | yes | — |
| 56 | C2×C22⋊D61 | — | 2 | 3 | no | — |
| 57 | C2×C22⋊D61 | C2×D4 | 1 | 6 | yes | — |
| 58 | C22×Dic61 | — | 2 | 3 | no | — |
| 59 | C22×Dic61 | C22⋊C4 | 1 | 6 | yes | — |
| 60 | Q8×D61 | — | 2 | 8 | no | — |
| 61 | Q8×D61 | — | 4 | 4 | no | — |
| 62 | #11758224 | — | 4 | 12 | no | — |
| 63 | #11758224 | — | 4 | 12 | no | — |
| 64 | #11758222 | — | 4 | 6 | no | — |
| 65 | #11758222 | — | 4 | 6 | no | — |
| 66 | C2×D244 | — | 2 | 2 | no | — |
| 67 | C2×D244 | C22×C4 | 1 | 1 | yes | characteristic |
| 68 | C2×D244 | C24 | 1 | 1 | yes | characteristic |
| 69 | C2×C4×D61 | — | 2 | 3 | no | — |
| 70 | C2×C4×D61 | C2×D4 | 1 | 6 | yes | — |
| 71 | C2×Dic122 | — | 2 | 3 | no | — |
| 72 | C2×Dic122 | — | 4 | 6 | no | — |
| 73 | C2×Dic122 | — | 4 | 6 | no | — |
| 74 | C2×Dic122 | C22⋊C4 | 1 | 6 | yes | — |
| 75 | C2×Dic122 | C22×C4 | 1 | 3 | yes | — |
| 76 | C2×Dic122 | C22×C4 | 1 | 3 | yes | — |
| 77 | C22⋊#6923 | — | 2 | 8 | no | — |
| 78 | C4⋊#6923 | — | 4 | 12 | no | — |
| 79 | C4×#6923 | — | 4 | 12 | no | — |
| 80 | C4⋊C4×C61 | — | 2 | 3 | no | — |
| 81 | C22⋊C4×C61 | — | 2 | 3 | no | — |
| 82 | C42×C61 | — | 2 | 3 | no | — |
| 83 | C42×C61 | C22×C4 | 1 | 1 | yes | characteristic |
| 84 | C42×C61 | C22×C4 | 1 | 3 | yes | — |
| 85 | #11758198 | — | 2 | 6 | no | — |
| 86 | #11758198 | — | 2 | 6 | no | — |
| 87 | C4⋊Dic61 | — | 2 | 6 | no | — |
| 88 | C4⋊Dic61 | — | 2 | 12 | no | — |
| 89 | C23×C61 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 90 | C22×D61 | — | 2 | 1 | no | — |
| 91 | C22×D61 | C22×D4 | 1 | 2 | yes | — |
| 92 | C2×#6923 | — | 4 | 8 | no | — |
| 93 | Q8×C61 | #99 | 1 | 4 | yes | — |
| 94 | D4×C61 | — | 2 | 6 | no | — |
| 95 | C2×C4×C61 | — | 2 | 1 | no | — |
| 96 | C2×C4×C61 | — | 2 | 6 | no | — |
| 97 | C2×C4×C61 | — | 2 | 3 | no | — |
| 98 | C2×C4×C61 | — | 4 | 6 | no | — |
| 99 | C2×C4×C61 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 100 | C2×C4×C61 | C23×C4 | 1 | 1 | yes | characteristic |
| 101 | C22⋊D61 | — | 4 | 6 | no | — |
| 102 | C22⋊D61 | — | 4 | 6 | no | — |
| 103 | C2×Dic61 | — | 2 | 3 | no | — |
| 104 | C2×Dic61 | — | 4 | 6 | no | — |
| 105 | C2×Dic61 | — | 4 | 6 | no | — |
| 106 | C2×Dic61 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 107 | D244 | — | 4 | 1 | no | — |
| 108 | D244 | — | 4 | 1 | no | — |
| 109 | D244 | — | 4 | 4 | no | — |
| 110 | C4×D61 | — | 2 | 12 | no | — |
| 111 | C4×D61 | — | 4 | 6 | no | — |
| 112 | Dic122 | — | 2 | 12 | no | — |
| 113 | Dic122 | — | 4 | 3 | no | — |
| 114 | Dic122 | — | 4 | 6 | no | — |
| 115 | Dic122 | — | 4 | 3 | no | — |
| 116 | #33360 | — | 61 | 1 | no | maximal, Sylow |
| 117 | C22×C61 | — | 2 | 3 | no | — |
| 118 | C22×C61 | — | 4 | 3 | no | — |
| 119 | C22×C61 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 120 | C2×D61 | — | 2 | 2 | no | — |
| 121 | C2×D61 | — | 4 | 2 | no | — |
| 122 | C2×D61 | — | 4 | 2 | no | — |
| 123 | C2×D61 | — | 4 | 1 | no | — |
| 124 | #6923 | — | 8 | 8 | no | — |
| 125 | C4×C61 | — | 2 | 2 | no | — |
| 126 | C4×C61 | — | 2 | 1 | no | — |
| 127 | C4×C61 | — | 4 | 3 | no | — |
| 128 | C4×C61 | C2×#99 | 1 | 6 | yes | — |
| 129 | Dic61 | — | 2 | 6 | no | — |
| 130 | Dic61 | — | 4 | 3 | no | — |
| 131 | #3499 | — | 61 | 1 | no | — |
| 132 | #2884 | — | 61 | 2 | no | — |
| 133 | #2129 | — | 61 | 6 | no | — |
| 134 | #2126 | — | 61 | 6 | no | — |
| 135 | C2×C61 | — | 2 | 1 | no | — |
| 136 | C2×C61 | — | 4 | 3 | no | — |
| 137 | C2×C61 | C22×#99 | 1 | 1 | yes | characteristic |
| 138 | D61 | — | 4 | 2 | no | — |
| 139 | D61 | — | 8 | 1 | no | — |
| 140 | C2×2−1+4 | — | 61 | 1 | no | — |
| 141 | C2×2−1+4 | — | 61 | 1 | no | — |
| 142 | D4×Q8 | — | 122 | 2 | no | — |
| 143 | #547 | — | 122 | 6 | no | — |
| 144 | #536 | — | 61 | 3 | no | — |
| 145 | #536 | — | 61 | 3 | no | — |
| 146 | #532 | — | 61 | 3 | no | — |
| 147 | C2×C4⋊Q8 | — | 61 | 3 | no | — |
| 148 | #519 | — | 122 | 2 | no | — |
| 149 | C4×#41 | #6923 | 1 | 1 | yes | characteristic, complemented |
| 150 | #410 | — | 61 | 12 | no | — |
| 151 | #410 | — | 122 | 6 | no | — |
| 152 | #354 | — | 61 | 12 | no | — |
| 153 | #353 | — | 61 | 4 | no | — |
| 154 | C61 | #33360 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 155 | 2−1+4 | — | 244 | 4 | no | — |
| 156 | 2−1+4 | — | 244 | 4 | no | — |
| 157 | C2×#41 | — | 61 | 6 | no | — |
| 158 | C2×#41 | — | 61 | 3 | no | — |
| 159 | C2×#41 | — | 61 | 3 | no | — |
| 160 | C2×#41 | — | 122 | 3 | no | — |
| 161 | C2×#41 | C2×#6923 | 1 | 1 | yes | characteristic |
| 162 | C22×Q8 | — | 61 | 2 | no | — |
| 163 | C22×Q8 | — | 61 | 3 | no | — |
| 164 | C22×Q8 | — | 61 | 3 | no | — |
| 165 | C22×Q8 | — | 122 | 1 | no | — |
| 166 | C4⋊Q8 | — | 61 | 3 | no | — |
| 167 | C4⋊Q8 | — | 61 | 6 | no | — |
| 168 | C4⋊Q8 | — | 122 | 6 | no | — |
| 169 | #127 | — | 61 | 1 | no | — |
| 170 | #124 | — | 61 | 3 | no | — |
| 171 | #124 | — | 61 | 3 | no | — |
| 172 | #123 | — | 122 | 6 | no | — |
| 173 | #123 | — | 122 | 6 | no | — |
| 174 | C22⋊Q8 | — | 122 | 6 | no | — |
| 175 | C22⋊Q8 | — | 122 | 6 | no | — |
| 176 | #121 | — | 122 | 6 | no | — |
| 177 | C4×Q8 | — | 2 | 1 | no | — |
| 178 | C4×Q8 | — | 244 | 8 | no | — |
| 179 | C4×D4 | — | 2 | 3 | no | — |
| 180 | C4×D4 | — | 122 | 6 | no | — |
| 181 | #117 | — | 122 | 12 | no | — |
| 182 | #117 | C2×#6923 | 1 | 3 | yes | — |
| 183 | C2×C4⋊C4 | — | 122 | 6 | no | — |
| 184 | C2×C42 | C2×#6923 | 1 | 3 | yes | — |
| 185 | #99 | — | 61 | 8 | no | — |
| 186 | #99 | — | 122 | 12 | no | — |
| 187 | #41 | — | 2 | 4 | no | — |
| 188 | #41 | — | 244 | 6 | no | — |
| 189 | #41 | — | 244 | 6 | no | — |
| 190 | #41 | — | 244 | 12 | no | — |
| 191 | #41 | — | 244 | 12 | no | — |
| 192 | C2×Q8 | — | 61 | 6 | no | — |
| 193 | C2×Q8 | — | 61 | 3 | no | — |
| 194 | C2×Q8 | — | 61 | 3 | no | — |
| 195 | C2×Q8 | — | 122 | 8 | no | — |
| 196 | C2×Q8 | — | 122 | 3 | no | — |
| 197 | C2×Q8 | — | 244 | 6 | no | — |
| 198 | C2×Q8 | — | 244 | 6 | no | — |
| 199 | C2×Q8 | — | 244 | 4 | no | — |
| 200 | C2×Q8 | C22⋊#6923 | 1 | 1 | yes | characteristic |
| 201 | C2×D4 | — | 61 | 6 | no | — |
| 202 | C2×D4 | — | 61 | 1 | no | — |
| 203 | C2×D4 | — | 61 | 1 | no | — |
| 204 | C2×D4 | — | 122 | 2 | no | — |
| 205 | C2×D4 | — | 122 | 3 | no | — |
| 206 | C2×D4 | C22⋊#6923 | 1 | 3 | yes | — |
| 207 | C22×C4 | — | 2 | 3 | no | — |
| 208 | C22×C4 | — | 61 | 6 | no | — |
| 209 | C22×C4 | — | 61 | 6 | no | — |
| 210 | C22×C4 | — | 122 | 3 | no | — |
| 211 | C22×C4 | — | 122 | 3 | no | — |
| 212 | C22×C4 | C22×#6923 | 1 | 3 | yes | — |
| 213 | C4⋊C4 | — | 2 | 3 | no | — |
| 214 | C4⋊C4 | — | 122 | 6 | no | — |
| 215 | C4⋊C4 | — | 122 | 12 | no | — |
| 216 | C4⋊C4 | — | 244 | 12 | no | — |
| 217 | C22⋊C4 | — | 2 | 3 | no | — |
| 218 | C22⋊C4 | — | 122 | 6 | no | — |
| 219 | C22⋊C4 | — | 122 | 6 | no | — |
| 220 | C22⋊C4 | — | 122 | 8 | no | — |
| 221 | C42 | — | 2 | 3 | no | — |
| 222 | C42 | — | 244 | 12 | no | — |
| 223 | C42 | C22×#6923 | 1 | 1 | yes | characteristic |
| 224 | C42 | C22×#6923 | 1 | 3 | yes | — |
| 225 | C23 | — | 61 | 2 | no | — |
| 226 | C23 | — | 122 | 1 | no | — |
| 227 | C23 | C2×C22⋊#6923 | 1 | 3 | yes | — |
| 228 | Q8 | — | 122 | 12 | no | — |
| 229 | Q8 | — | 244 | 3 | no | — |
| 230 | Q8 | — | 244 | 3 | no | — |
| 231 | Q8 | — | 244 | 6 | no | — |
| 232 | Q8 | #14519836 | 1 | 4 | yes | — |
| 233 | D4 | — | 2 | 6 | no | — |
| 234 | D4 | — | 244 | 1 | no | — |
| 235 | D4 | — | 244 | 1 | no | — |
| 236 | D4 | — | 244 | 4 | no | — |
| 237 | D4 | — | 244 | 6 | no | — |
| 238 | D4 | — | 244 | 6 | no | — |
| 239 | C2×C4 | — | 2 | 1 | no | — |
| 240 | C2×C4 | — | 2 | 6 | no | — |
| 241 | C2×C4 | — | 2 | 3 | no | — |
| 242 | C2×C4 | — | 4 | 6 | no | — |
| 243 | C2×C4 | — | 61 | 6 | no | — |
| 244 | C2×C4 | — | 122 | 12 | no | — |
| 245 | C2×C4 | — | 122 | 3 | no | — |
| 246 | C2×C4 | — | 244 | 6 | no | — |
| 247 | C2×C4 | — | 244 | 8 | no | — |
| 248 | C2×C4 | — | 244 | 6 | no | — |
| 249 | C2×C4 | — | 244 | 6 | no | — |
| 250 | C2×C4 | C2×C22⋊#6923 | 1 | 3 | yes | — |
| 251 | C2×C4 | C23×#6923 | 1 | 1 | yes | characteristic |
| 252 | C22 | — | 2 | 3 | no | — |
| 253 | C22 | — | 4 | 3 | no | — |
| 254 | C22 | — | 122 | 2 | no | — |
| 255 | C22 | — | 244 | 2 | no | — |
| 256 | C22 | — | 244 | 2 | no | — |
| 257 | C22 | — | 244 | 1 | no | — |
| 258 | C22 | C22×C22⋊#6923 | 1 | 1 | yes | characteristic |
| 259 | C4 | — | 2 | 2 | no | — |
| 260 | C4 | — | 2 | 1 | no | — |
| 261 | C4 | — | 4 | 3 | no | — |
| 262 | C4 | — | 122 | 6 | no | — |
| 263 | C4 | — | 488 | 8 | no | — |
| 264 | C4 | — | 244 | 3 | no | — |
| 265 | C4 | C2×#14519836 | 1 | 6 | yes | — |
| 266 | C2 | — | 2 | 1 | no | — |
| 267 | C2 | — | 4 | 3 | no | — |
| 268 | C2 | — | 488 | 1 | no | — |
| 269 | C2 | — | 244 | 2 | no | — |
| 270 | C2 | C22×#14519836 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21462421 | C2 | semidirect |
| 2. #21462103 | C2 | semidirect |
| 3. #21462092 | C2 | semidirect |
| 10. C2×#14519824 | C4 | semidirect |
| 14. #16840338 | C4 | semidirect |
| 17. C2×#14519777 | C4 | semidirect |
| 18. C4×#41×C61 | C4 | semidirect |
| 149. C4×#41 | #6923 | semidirect |
| 154. C61 | #33360 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21462421 | 2 |
| #21462103 | 2 |
| #21462092 | 2 |
| #21459604 | 2 |
| #33360 | 61 |