Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
1502 conjugacy classes of subgroups; only the 331 normal subgroups are shown.
This group has 1502 conjugacy classes of subgroups — too many to draw. Only its 331 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 | C2×#89487 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | C2×#89484 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×#89468 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C2×#89465 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #11730877 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11730876 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C7×#1726 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×Dic28 | C22 | 1 | 3 | yes | — |
| 9 | C22×D56 | C22 | 1 | 3 | yes | — |
| 10 | C22×#1132 | C22 | 1 | 3 | yes | — |
| 11 | C22×#1132 | C22 | 1 | 6 | yes | — |
| 12 | C22×#1132 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#6438 | C22 | 1 | 3 | yes | — |
| 14 | C2×#6438 | C22 | 1 | 3 | yes | — |
| 15 | C2×#6437 | C22 | 1 | 3 | yes | — |
| 16 | C2×#6432 | C22 | 1 | 3 | yes | — |
| 17 | C2×C7×#97 | C22 | 1 | 3 | yes | — |
| 18 | C2×C4×C7×C8 | C22 | 1 | 3 | yes | — |
| 19 | C7×#376 | C22 | 1 | 1 | yes | characteristic |
| 20 | #89487 | — | 2 | 12 | no | — |
| 21 | #89484 | — | 2 | 12 | no | — |
| 22 | #89468 | — | 2 | 12 | no | — |
| 23 | #89465 | — | 2 | 12 | no | — |
| 24 | #89449 | — | 2 | 4 | no | — |
| 25 | #89443 | — | 2 | 4 | no | — |
| 26 | #33307 | — | 7 | 1 | no | maximal, Sylow |
| 27 | C22×D28 | — | 2 | 6 | no | — |
| 28 | C22×D28 | C23 | 1 | 3 | yes | — |
| 29 | C22×D28 | C23 | 1 | 1 | yes | characteristic |
| 30 | C22×Dic14 | C23 | 1 | 3 | yes | — |
| 31 | C22×Dic14 | C23 | 1 | 1 | yes | characteristic |
| 32 | C22×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 33 | C22×C7×C8 | C23 | 1 | 3 | yes | — |
| 34 | C2×C42×C7 | C23 | 1 | 3 | yes | — |
| 35 | C2×(C2×D7)⋊C4 | — | 2 | 6 | no | — |
| 36 | C2×(C2×D7)⋊C4 | — | 2 | 6 | no | — |
| 37 | C2×C4⋊Dic7 | — | 2 | 6 | no | — |
| 38 | C2×Dic28 | — | 2 | 6 | no | — |
| 39 | C2×Dic28 | — | 2 | 12 | no | — |
| 40 | C2×D56 | — | 2 | 6 | no | — |
| 41 | C2×D56 | — | 2 | 12 | no | — |
| 42 | C2×#1132 | — | 2 | 12 | no | — |
| 43 | C2×#1132 | — | 2 | 12 | no | — |
| 44 | C2×#1132 | — | 2 | 6 | no | — |
| 45 | C2×#1132 | — | 2 | 6 | no | — |
| 46 | C2×#1132 | — | 2 | 12 | no | — |
| 47 | C2×#1132 | — | 2 | 12 | no | — |
| 48 | #6438 | — | 2 | 6 | no | — |
| 49 | #6438 | — | 2 | 6 | no | — |
| 50 | #6438 | — | 2 | 6 | no | — |
| 51 | #6438 | — | 2 | 6 | no | — |
| 52 | #6437 | — | 2 | 6 | no | — |
| 53 | #6437 | — | 2 | 6 | no | — |
| 54 | #6432 | — | 2 | 6 | no | — |
| 55 | #6432 | — | 2 | 6 | no | — |
| 56 | C7×#97 | D4 | 1 | 12 | yes | complemented |
| 57 | C4×C7×C8 | D4 | 1 | 12 | yes | complemented |
| 58 | C7×#95 | D4 | 1 | 4 | yes | — |
| 59 | C2×#497 | — | 7 | 3 | no | — |
| 60 | C2×#496 | — | 7 | 3 | no | — |
| 61 | C2×#495 | — | 7 | 3 | no | — |
| 62 | C2×#492 | — | 7 | 3 | no | — |
| 63 | #2658 | — | 7 | 1 | no | — |
| 64 | #2657 | — | 7 | 1 | no | — |
| 65 | #1726 | D7 | 1 | 1 | yes | characteristic, complemented |
| 66 | C23×D7 | — | 2 | 1 | no | — |
| 67 | C23×D7 | — | 2 | 3 | no | — |
| 68 | C22×C4×C7 | C2×D4 | 1 | 6 | yes | — |
| 69 | C22×C4×C7 | C24 | 1 | 1 | yes | characteristic |
| 70 | C22×Dic7 | — | 2 | 3 | no | — |
| 71 | C22×Dic7 | — | 2 | 1 | no | — |
| 72 | C2×D28 | — | 2 | 6 | no | — |
| 73 | C2×D28 | — | 2 | 6 | no | — |
| 74 | C2×D28 | — | 2 | 3 | no | — |
| 75 | C2×D28 | — | 2 | 3 | no | — |
| 76 | C2×D28 | — | 2 | 3 | no | — |
| 77 | C2×D28 | — | 2 | 3 | no | — |
| 78 | C2×D28 | — | 4 | 12 | no | — |
| 79 | C2×D28 | — | 8 | 12 | no | — |
| 80 | C2×Dic14 | — | 2 | 6 | no | — |
| 81 | C2×Dic14 | — | 2 | 6 | no | — |
| 82 | C2×Dic14 | — | 2 | 3 | no | — |
| 83 | C2×Dic14 | — | 2 | 3 | no | — |
| 84 | C2×Dic14 | — | 2 | 3 | no | — |
| 85 | C2×Dic14 | — | 2 | 3 | no | — |
| 86 | C2×C7×C8 | C2×D4 | 1 | 6 | yes | — |
| 87 | C2×C7×C8 | C2×D4 | 1 | 6 | yes | — |
| 88 | C2×C7×C8 | C2×D4 | 1 | 12 | yes | — |
| 89 | C42×C7 | C2×D4 | 1 | 6 | yes | — |
| 90 | C42×C7 | C2×D4 | 1 | 6 | yes | — |
| 91 | (C2×D7)⋊C4 | — | 4 | 12 | no | — |
| 92 | (C2×D7)⋊C4 | — | 4 | 12 | no | — |
| 93 | C4⋊Dic7 | — | 4 | 12 | no | — |
| 94 | Dic28 | — | 4 | 12 | no | — |
| 95 | D56 | — | 4 | 12 | no | — |
| 96 | #1132 | — | 4 | 12 | no | — |
| 97 | #1132 | — | 4 | 4 | no | — |
| 98 | #1132 | — | 4 | 24 | no | — |
| 99 | C22×Q16 | — | 7 | 3 | no | — |
| 100 | C22×SD16 | — | 7 | 3 | no | — |
| 101 | C22×SD16 | — | 7 | 6 | no | — |
| 102 | C22×SD16 | — | 7 | 1 | no | — |
| 103 | C22×D8 | — | 7 | 3 | no | — |
| 104 | C2×C4⋊Q8 | — | 7 | 3 | no | — |
| 105 | C2×#127 | — | 7 | 3 | no | — |
| 106 | C2×#124 | — | 7 | 3 | no | — |
| 107 | C2×#124 | — | 7 | 3 | no | — |
| 108 | #497 | — | 14 | 12 | no | — |
| 109 | #496 | — | 14 | 12 | no | — |
| 110 | #495 | — | 14 | 12 | no | — |
| 111 | #492 | — | 14 | 12 | no | — |
| 112 | C2×#97 | C2×D7 | 1 | 3 | yes | — |
| 113 | C2×C4×C8 | C2×D7 | 1 | 3 | yes | — |
| 114 | #399 | — | 14 | 4 | no | — |
| 115 | #392 | — | 14 | 4 | no | — |
| 116 | #376 | C2×D7 | 1 | 1 | yes | characteristic |
| 117 | C23×C7 | C22×D4 | 1 | 1 | yes | characteristic |
| 118 | C22×D7 | — | 4 | 3 | no | — |
| 119 | C22×D7 | — | 4 | 3 | no | — |
| 120 | C22×D7 | — | 4 | 6 | no | — |
| 121 | C22×D7 | — | 4 | 12 | no | — |
| 122 | C22×D7 | — | 4 | 4 | no | — |
| 123 | C2×C4×C7 | — | 2 | 12 | no | — |
| 124 | C2×C4×C7 | #127 | 1 | 12 | yes | — |
| 125 | C2×C4×C7 | C22×D4 | 1 | 3 | yes | — |
| 126 | C2×C4×C7 | C22×D4 | 1 | 3 | yes | — |
| 127 | C2×Dic7 | — | 4 | 6 | no | — |
| 128 | C2×Dic7 | — | 4 | 3 | no | — |
| 129 | C2×Dic7 | — | 4 | 3 | no | — |
| 130 | D28 | — | 4 | 3 | no | — |
| 131 | D28 | — | 4 | 6 | no | — |
| 132 | D28 | — | 4 | 1 | no | — |
| 133 | D28 | — | 4 | 3 | no | — |
| 134 | D28 | — | 4 | 3 | no | — |
| 135 | D28 | — | 8 | 24 | no | — |
| 136 | Dic14 | — | 4 | 3 | no | — |
| 137 | Dic14 | — | 4 | 6 | no | — |
| 138 | Dic14 | — | 4 | 1 | no | — |
| 139 | Dic14 | — | 4 | 3 | no | — |
| 140 | Dic14 | — | 4 | 3 | no | — |
| 141 | C7×C8 | #127 | 1 | 4 | yes | — |
| 142 | C7×C8 | #127 | 1 | 12 | yes | — |
| 143 | C22×Q8 | — | 7 | 1 | no | — |
| 144 | C22×Q8 | — | 7 | 3 | no | — |
| 145 | C22×D4 | — | 7 | 1 | no | — |
| 146 | C22×D4 | — | 7 | 3 | no | — |
| 147 | C22×D4 | — | 14 | 6 | no | — |
| 148 | C2×Q16 | — | 14 | 6 | no | — |
| 149 | C2×Q16 | — | 14 | 12 | no | — |
| 150 | C2×SD16 | — | 14 | 12 | no | — |
| 151 | C2×SD16 | — | 14 | 12 | no | — |
| 152 | C2×SD16 | — | 14 | 6 | no | — |
| 153 | C2×SD16 | — | 14 | 6 | no | — |
| 154 | C2×SD16 | — | 14 | 12 | no | — |
| 155 | C2×SD16 | — | 14 | 12 | no | — |
| 156 | C2×D8 | — | 14 | 6 | no | — |
| 157 | C2×D8 | — | 14 | 12 | no | — |
| 158 | C22×C8 | C22×D7 | 1 | 3 | yes | — |
| 159 | C22×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 160 | C4⋊Q8 | — | 14 | 6 | no | — |
| 161 | C4⋊Q8 | — | 14 | 6 | no | — |
| 162 | #127 | — | 14 | 6 | no | — |
| 163 | #127 | — | 14 | 6 | no | — |
| 164 | #124 | — | 14 | 6 | no | — |
| 165 | #124 | — | 14 | 6 | no | — |
| 166 | #124 | — | 14 | 6 | no | — |
| 167 | #124 | — | 14 | 6 | no | — |
| 168 | C2×C4⋊C4 | — | 14 | 6 | no | — |
| 169 | C2×C22⋊C4 | — | 14 | 6 | no | — |
| 170 | C2×C22⋊C4 | — | 14 | 6 | no | — |
| 171 | C2×C42 | C22×D7 | 1 | 3 | yes | — |
| 172 | #97 | D28 | 1 | 12 | yes | complemented |
| 173 | C4×C8 | D28 | 1 | 12 | yes | complemented |
| 174 | #95 | D28 | 1 | 4 | yes | — |
| 175 | C22×C7 | C2×#127 | 1 | 3 | yes | — |
| 176 | C22×C7 | #577 | 1 | 4 | yes | — |
| 177 | C2×D7 | — | 8 | 3 | no | — |
| 178 | C2×D7 | — | 8 | 1 | no | — |
| 179 | C2×D7 | — | 8 | 12 | no | — |
| 180 | C2×D7 | — | 8 | 6 | no | — |
| 181 | C2×D7 | — | 8 | 6 | no | — |
| 182 | C4×C7 | — | 2 | 12 | no | — |
| 183 | C4×C7 | C2×#127 | 1 | 1 | yes | characteristic |
| 184 | C4×C7 | C2×#127 | 1 | 3 | yes | — |
| 185 | Dic7 | — | 8 | 3 | no | — |
| 186 | Dic7 | — | 8 | 1 | no | — |
| 187 | C24 | — | 14 | 1 | no | — |
| 188 | C24 | — | 14 | 3 | no | — |
| 189 | C2×Q8 | — | 14 | 3 | no | — |
| 190 | C2×Q8 | — | 14 | 3 | no | — |
| 191 | C2×Q8 | — | 14 | 3 | no | — |
| 192 | C2×Q8 | — | 14 | 3 | no | — |
| 193 | C2×Q8 | — | 14 | 6 | no | — |
| 194 | C2×Q8 | — | 14 | 6 | no | — |
| 195 | C2×D4 | — | 14 | 3 | no | — |
| 196 | C2×D4 | — | 14 | 3 | no | — |
| 197 | C2×D4 | — | 14 | 3 | no | — |
| 198 | C2×D4 | — | 14 | 3 | no | — |
| 199 | C2×D4 | — | 14 | 6 | no | — |
| 200 | C2×D4 | — | 14 | 6 | no | — |
| 201 | C2×D4 | — | 28 | 12 | no | — |
| 202 | C2×D4 | — | 56 | 12 | no | — |
| 203 | C22×C4 | — | 14 | 1 | no | — |
| 204 | C22×C4 | — | 14 | 3 | no | — |
| 205 | C22×C4 | C2×D28 | 1 | 6 | yes | — |
| 206 | C22×C4 | C23×D7 | 1 | 1 | yes | characteristic |
| 207 | Q16 | — | 28 | 12 | no | — |
| 208 | SD16 | — | 28 | 24 | no | — |
| 209 | SD16 | — | 28 | 12 | no | — |
| 210 | SD16 | — | 28 | 4 | no | — |
| 211 | D8 | — | 28 | 12 | no | — |
| 212 | C2×C8 | C2×D28 | 1 | 6 | yes | — |
| 213 | C2×C8 | C2×D28 | 1 | 6 | yes | — |
| 214 | C2×C8 | C2×D28 | 1 | 12 | yes | — |
| 215 | C4⋊C4 | — | 28 | 12 | no | — |
| 216 | C22⋊C4 | — | 28 | 12 | no | — |
| 217 | C22⋊C4 | — | 28 | 12 | no | — |
| 218 | C42 | C2×D28 | 1 | 6 | yes | — |
| 219 | C42 | C2×D28 | 1 | 6 | yes | — |
| 220 | C2×C7 | #2867 | 1 | 1 | yes | characteristic |
| 221 | C2×C7 | #3154 | 1 | 6 | yes | — |
| 222 | D7 | — | 16 | 3 | no | — |
| 223 | D7 | — | 16 | 1 | no | — |
| 224 | C23 | — | 28 | 3 | no | — |
| 225 | C23 | — | 28 | 3 | no | — |
| 226 | C23 | — | 28 | 6 | no | — |
| 227 | C23 | — | 28 | 12 | no | — |
| 228 | C23 | — | 28 | 4 | no | — |
| 229 | C23 | C22×D28 | 1 | 1 | yes | characteristic |
| 230 | Q8 | — | 28 | 6 | no | — |
| 231 | Q8 | — | 28 | 1 | no | — |
| 232 | Q8 | — | 28 | 3 | no | — |
| 233 | Q8 | — | 28 | 3 | no | — |
| 234 | Q8 | — | 28 | 3 | no | — |
| 235 | D4 | — | 28 | 3 | no | — |
| 236 | D4 | — | 28 | 3 | no | — |
| 237 | D4 | — | 28 | 1 | no | — |
| 238 | D4 | — | 28 | 3 | no | — |
| 239 | D4 | — | 28 | 6 | no | — |
| 240 | D4 | — | 56 | 24 | no | — |
| 241 | C2×C4 | — | 2 | 12 | no | — |
| 242 | C2×C4 | — | 28 | 3 | no | — |
| 243 | C2×C4 | — | 28 | 3 | no | — |
| 244 | C2×C4 | — | 28 | 6 | no | — |
| 245 | C2×C4 | #6437 | 1 | 12 | yes | — |
| 246 | C2×C4 | C22×D28 | 1 | 3 | yes | — |
| 247 | C2×C4 | C22×D28 | 1 | 3 | yes | — |
| 248 | C8 | #6437 | 1 | 12 | yes | — |
| 249 | C8 | #6437 | 1 | 4 | yes | — |
| 250 | C7 | #33307 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 251 | C22 | — | 56 | 1 | no | — |
| 252 | C22 | — | 56 | 3 | no | — |
| 253 | C22 | — | 56 | 12 | no | — |
| 254 | C22 | — | 56 | 6 | no | — |
| 255 | C22 | — | 56 | 6 | no | — |
| 256 | C22 | C2×#6437 | 1 | 3 | yes | — |
| 257 | C22 | #90442 | 1 | 4 | yes | — |
| 258 | C4 | — | 2 | 12 | no | — |
| 259 | C4 | — | 56 | 1 | no | — |
| 260 | C4 | — | 56 | 3 | no | — |
| 261 | C4 | C2×#6437 | 1 | 3 | yes | — |
| 262 | C4 | C2×#6437 | 1 | 1 | yes | characteristic |
| 263 | C2 | — | 112 | 3 | no | — |
| 264 | C2 | — | 112 | 1 | no | — |
| 265 | C2 | #11730763 | 1 | 1 | yes | characteristic |
| 266 | C2 | #11736009 | 1 | 6 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#89487 | C2 | semidirect |
| 2. C2×#89484 | C2 | semidirect |
| 3. C2×#89468 | C2 | semidirect |
| 4. C2×#89465 | C2 | semidirect |
| 5. #11730877 | C2 | semidirect |
| 6. #11730876 | C2 | semidirect |
| 7. C7×#1726 | C2 | semidirect |
| 56. C7×#97 | D4 | semidirect |
| 57. C4×C7×C8 | D4 | semidirect |
| 65. #1726 | D7 | semidirect |
| 172. #97 | D28 | semidirect |
| 173. C4×C8 | D28 | semidirect |
| 250. C7 | #33307 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#89487 | 2 |
| C2×#89484 | 2 |
| C2×#89468 | 2 |
| C2×#89465 | 2 |
| #11730877 | 2 |
| #11730876 | 2 |
| C7×#1726 | 2 |
| #33307 | 7 |