Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
552 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 97 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×#89485 | C2 | 1 | 2 | yes | maximal |
| 2 | #11731662 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#89260 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #11727307 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C7×#1295 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#6437 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#6433 | C22 | 1 | 2 | yes | — |
| 8 | C2×C7×#97 | C22 | 1 | 2 | yes | — |
| 9 | C7×#382 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#6395 | — | 2 | 2 | no | — |
| 11 | C2×#6395 | — | 2 | 2 | no | — |
| 12 | #89755 | — | 2 | 2 | no | — |
| 13 | C2×#6383 | — | 2 | 2 | no | — |
| 14 | C2×#6377 | C22 | 1 | 1 | yes | characteristic |
| 15 | #89485 | — | 2 | 4 | no | — |
| 16 | #89485 | — | 2 | 4 | no | — |
| 17 | #89433 | — | 2 | 2 | no | — |
| 18 | #89431 | — | 2 | 2 | no | — |
| 19 | #89260 | — | 2 | 4 | no | — |
| 20 | #10080 | — | 7 | 1 | no | maximal, Sylow |
| 21 | C22×D28 | — | 2 | 2 | no | — |
| 22 | C22×D28 | — | 2 | 1 | no | — |
| 23 | C22×C7×C8 | — | 2 | 2 | no | — |
| 24 | C2×C7×C4⋊C4 | D4 | 1 | 2 | yes | — |
| 25 | C2×C42×C7 | C23 | 1 | 1 | yes | characteristic |
| 26 | C2×C4⋊Dic7 | — | 2 | 2 | no | — |
| 27 | C2×C4⋊Dic7 | D4 | 1 | 2 | yes | — |
| 28 | C2×C4⋊Dic7 | D4 | 1 | 2 | yes | — |
| 29 | C2×Dic7⋊C4 | — | 2 | 2 | no | — |
| 30 | C2×Dic7⋊C4 | — | 2 | 2 | no | — |
| 31 | C22×C7⋊C8 | — | 2 | 1 | no | — |
| 32 | #6437 | — | 2 | 2 | no | — |
| 33 | #6437 | — | 2 | 1 | no | — |
| 34 | #6437 | D4 | 1 | 2 | yes | — |
| 35 | #6433 | — | 2 | 2 | no | — |
| 36 | #6433 | — | 2 | 2 | no | — |
| 37 | #6433 | — | 2 | 2 | no | — |
| 38 | #6433 | — | 2 | 2 | no | — |
| 39 | C7×#97 | — | 2 | 2 | no | — |
| 40 | C7×#97 | D4 | 1 | 4 | yes | — |
| 41 | C7×#95 | — | 2 | 2 | no | — |
| 42 | C7⋊#95 | — | 2 | 2 | no | — |
| 43 | C7⋊#95 | — | 2 | 4 | no | — |
| 44 | #6395 | — | 2 | 8 | no | — |
| 45 | #6395 | — | 2 | 8 | no | — |
| 46 | #6383 | — | 4 | 4 | no | — |
| 47 | #6377 | — | 2 | 2 | no | — |
| 48 | C2×#490 | — | 7 | 2 | no | — |
| 49 | #2822 | — | 7 | 1 | no | — |
| 50 | #1961 | — | 7 | 1 | no | — |
| 51 | C2×#330 | — | 7 | 2 | no | — |
| 52 | #1295 | D7 | 1 | 1 | yes | characteristic, complemented |
| 53 | C23×D7 | — | 4 | 1 | no | — |
| 54 | C22×C4×C7 | — | 2 | 2 | no | — |
| 55 | C22×C4×C7 | C2×D4 | 1 | 2 | yes | — |
| 56 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 57 | C22×Dic7 | — | 2 | 2 | no | — |
| 58 | C22×Dic7 | — | 2 | 2 | no | — |
| 59 | C2×D28 | — | 2 | 4 | no | — |
| 60 | C2×D28 | — | 2 | 2 | no | — |
| 61 | C2×D28 | — | 4 | 2 | no | — |
| 62 | C2×D28 | — | 4 | 8 | no | — |
| 63 | C2×D28 | — | 4 | 2 | no | — |
| 64 | C2×D28 | — | 4 | 2 | no | — |
| 65 | C2×D28 | — | 4 | 1 | no | — |
| 66 | C2×C7×C8 | — | 2 | 4 | no | — |
| 67 | C2×C7×C8 | — | 4 | 4 | no | — |
| 68 | C7×C4⋊C4 | — | 4 | 2 | no | — |
| 69 | C42×C7 | C2×D4 | 1 | 2 | yes | — |
| 70 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 71 | C42×C7 | #41 | 1 | 1 | yes | characteristic |
| 72 | C4⋊Dic7 | — | 2 | 2 | no | — |
| 73 | C4⋊Dic7 | — | 2 | 2 | no | — |
| 74 | C4⋊Dic7 | — | 2 | 2 | no | — |
| 75 | C4⋊Dic7 | — | 2 | 2 | no | — |
| 76 | C4⋊Dic7 | — | 8 | 2 | no | — |
| 77 | Dic7⋊C4 | — | 4 | 2 | no | — |
| 78 | Dic7⋊C4 | — | 4 | 2 | no | — |
| 79 | Dic7⋊C4 | — | 4 | 2 | no | — |
| 80 | Dic7⋊C4 | — | 4 | 2 | no | — |
| 81 | C2×C7⋊C8 | — | 2 | 2 | no | — |
| 82 | C2×C7⋊C8 | — | 4 | 1 | no | — |
| 83 | C2×C7⋊C8 | — | 4 | 1 | no | — |
| 84 | C2×#127 | — | 7 | 1 | no | — |
| 85 | C2×#125 | — | 7 | 2 | no | — |
| 86 | #490 | — | 14 | 4 | no | — |
| 87 | #490 | — | 14 | 4 | no | — |
| 88 | C2×#102 | — | 14 | 2 | no | — |
| 89 | C2×#102 | — | 14 | 2 | no | — |
| 90 | C2×#102 | — | 14 | 2 | no | — |
| 91 | C2×#97 | — | 7 | 1 | no | — |
| 92 | C2×#97 | C2×D7 | 1 | 2 | yes | — |
| 93 | #400 | — | 14 | 2 | no | — |
| 94 | #398 | — | 14 | 2 | no | — |
| 95 | #398 | — | 14 | 2 | no | — |
| 96 | #382 | C2×D7 | 1 | 1 | yes | characteristic |
| 97 | #330 | — | 14 | 4 | no | — |
| 98 | C23×C7 | #120 | 1 | 1 | yes | characteristic |
| 99 | C22×D7 | — | 4 | 2 | no | — |
| 100 | C22×D7 | — | 8 | 1 | no | — |
| 101 | C22×D7 | — | 8 | 1 | no | — |
| 102 | C22×D7 | — | 8 | 4 | no | — |
| 103 | C2×C4×C7 | — | 2 | 1 | no | — |
| 104 | C2×C4×C7 | — | 2 | 1 | no | — |
| 105 | C2×C4×C7 | — | 2 | 4 | no | — |
| 106 | C2×C4×C7 | — | 4 | 2 | no | — |
| 107 | C2×C4×C7 | — | 4 | 4 | no | — |
| 108 | C2×C4×C7 | #120 | 1 | 2 | yes | — |
| 109 | C2×C4×C7 | #121 | 1 | 1 | yes | characteristic |
| 110 | C2×C4×C7 | #121 | 1 | 2 | yes | — |
| 111 | C2×C4×C7 | #127 | 1 | 1 | yes | characteristic |
| 112 | C2×Dic7 | — | 4 | 2 | no | — |
| 113 | C2×Dic7 | — | 4 | 2 | no | — |
| 114 | C2×Dic7 | — | 4 | 2 | no | — |
| 115 | C2×Dic7 | — | 4 | 2 | no | — |
| 116 | C2×Dic7 | — | 4 | 4 | no | — |
| 117 | D28 | — | 4 | 8 | no | — |
| 118 | D28 | — | 4 | 4 | no | — |
| 119 | D28 | — | 8 | 4 | no | — |
| 120 | D28 | — | 8 | 2 | no | — |
| 121 | C7×C8 | — | 4 | 4 | no | — |
| 122 | C7⋊C8 | — | 4 | 2 | no | — |
| 123 | C22×D4 | — | 14 | 1 | no | — |
| 124 | C22×D4 | — | 14 | 2 | no | — |
| 125 | C22×C8 | — | 2 | 2 | no | — |
| 126 | C22×C8 | — | 14 | 1 | no | — |
| 127 | #127 | — | 7 | 2 | no | — |
| 128 | #127 | — | 14 | 2 | no | — |
| 129 | #127 | — | 14 | 1 | no | — |
| 130 | #125 | — | 14 | 2 | no | — |
| 131 | #125 | — | 14 | 2 | no | — |
| 132 | #125 | — | 14 | 2 | no | — |
| 133 | #125 | — | 14 | 2 | no | — |
| 134 | C2×C4⋊C4 | — | 7 | 2 | no | — |
| 135 | C2×C4⋊C4 | — | 7 | 2 | no | — |
| 136 | C2×C4⋊C4 | — | 14 | 2 | no | — |
| 137 | C2×C4⋊C4 | — | 14 | 2 | no | — |
| 138 | C2×C4⋊C4 | — | 14 | 2 | no | — |
| 139 | C2×C4⋊C4 | D28 | 1 | 2 | yes | — |
| 140 | C2×C42 | C22×D7 | 1 | 1 | yes | characteristic |
| 141 | #102 | — | 14 | 8 | no | — |
| 142 | #102 | — | 14 | 8 | no | — |
| 143 | #102 | — | 28 | 4 | no | — |
| 144 | #97 | — | 2 | 2 | no | — |
| 145 | #97 | — | 14 | 2 | no | — |
| 146 | #97 | D28 | 1 | 4 | yes | — |
| 147 | #95 | — | 2 | 2 | no | — |
| 148 | #95 | — | 14 | 4 | no | — |
| 149 | #95 | — | 14 | 2 | no | — |
| 150 | C22×C7 | #392 | 1 | 1 | yes | characteristic |
| 151 | C22×C7 | #453 | 1 | 1 | yes | characteristic |
| 152 | C22×C7 | #454 | 1 | 1 | yes | characteristic |
| 153 | C22×C7 | #454 | 1 | 4 | yes | — |
| 154 | C2×D7 | — | 8 | 4 | no | — |
| 155 | C2×D7 | — | 8 | 2 | no | — |
| 156 | C2×D7 | — | 16 | 2 | no | — |
| 157 | C2×D7 | — | 16 | 2 | no | — |
| 158 | C4×C7 | — | 2 | 2 | no | — |
| 159 | C4×C7 | — | 2 | 2 | no | — |
| 160 | C4×C7 | — | 2 | 1 | no | — |
| 161 | C4×C7 | — | 2 | 1 | no | — |
| 162 | C4×C7 | — | 8 | 2 | no | — |
| 163 | Dic7 | — | 8 | 2 | no | — |
| 164 | Dic7 | — | 8 | 2 | no | — |
| 165 | C24 | — | 28 | 1 | no | — |
| 166 | C2×D4 | — | 14 | 4 | no | — |
| 167 | C2×D4 | — | 14 | 2 | no | — |
| 168 | C2×D4 | — | 28 | 2 | no | — |
| 169 | C2×D4 | — | 28 | 2 | no | — |
| 170 | C2×D4 | — | 28 | 8 | no | — |
| 171 | C2×D4 | — | 28 | 1 | no | — |
| 172 | C2×D4 | — | 28 | 2 | no | — |
| 173 | C22×C4 | — | 2 | 2 | no | — |
| 174 | C22×C4 | — | 14 | 2 | no | — |
| 175 | C22×C4 | — | 14 | 2 | no | — |
| 176 | C22×C4 | C2×D28 | 1 | 1 | yes | characteristic |
| 177 | C22×C4 | D4×D7 | 1 | 2 | yes | — |
| 178 | C2×C8 | — | 2 | 4 | no | — |
| 179 | C2×C8 | — | 4 | 4 | no | — |
| 180 | C2×C8 | — | 14 | 2 | no | — |
| 181 | C2×C8 | — | 28 | 1 | no | — |
| 182 | C2×C8 | — | 28 | 1 | no | — |
| 183 | C4⋊C4 | — | 4 | 2 | no | — |
| 184 | C4⋊C4 | — | 14 | 2 | no | — |
| 185 | C4⋊C4 | — | 14 | 2 | no | — |
| 186 | C4⋊C4 | — | 14 | 2 | no | — |
| 187 | C4⋊C4 | — | 14 | 2 | no | — |
| 188 | C4⋊C4 | — | 28 | 2 | no | — |
| 189 | C4⋊C4 | — | 28 | 2 | no | — |
| 190 | C4⋊C4 | — | 28 | 2 | no | — |
| 191 | C4⋊C4 | — | 28 | 2 | no | — |
| 192 | C4⋊C4 | — | 56 | 2 | no | — |
| 193 | C42 | C2×D28 | 1 | 2 | yes | — |
| 194 | C42 | D4×D7 | 1 | 1 | yes | characteristic |
| 195 | C42 | #1161 | 1 | 1 | yes | characteristic |
| 196 | C2×C7 | #1657 | 1 | 2 | yes | — |
| 197 | C2×C7 | #1658 | 1 | 2 | yes | — |
| 198 | C2×C7 | #2003 | 1 | 2 | yes | — |
| 199 | C2×C7 | #2006 | 1 | 1 | yes | characteristic |
| 200 | D7 | — | 16 | 2 | no | — |
| 201 | C23 | — | 28 | 2 | no | — |
| 202 | C23 | — | 56 | 1 | no | — |
| 203 | C23 | — | 56 | 1 | no | — |
| 204 | C23 | — | 56 | 4 | no | — |
| 205 | C23 | #6445 | 1 | 1 | yes | characteristic |
| 206 | D4 | — | 28 | 4 | no | — |
| 207 | D4 | — | 28 | 8 | no | — |
| 208 | D4 | — | 56 | 2 | no | — |
| 209 | D4 | — | 56 | 4 | no | — |
| 210 | C2×C4 | — | 2 | 1 | no | — |
| 211 | C2×C4 | — | 2 | 1 | no | — |
| 212 | C2×C4 | — | 2 | 4 | no | — |
| 213 | C2×C4 | — | 4 | 2 | no | — |
| 214 | C2×C4 | — | 4 | 4 | no | — |
| 215 | C2×C4 | — | 28 | 2 | no | — |
| 216 | C2×C4 | — | 28 | 2 | no | — |
| 217 | C2×C4 | — | 28 | 2 | no | — |
| 218 | C2×C4 | — | 28 | 2 | no | — |
| 219 | C2×C4 | — | 28 | 4 | no | — |
| 220 | C2×C4 | #6437 | 1 | 1 | yes | characteristic |
| 221 | C2×C4 | #6445 | 1 | 2 | yes | — |
| 222 | C2×C4 | #6458 | 1 | 2 | yes | — |
| 223 | C2×C4 | #6458 | 1 | 1 | yes | characteristic |
| 224 | C8 | — | 4 | 4 | no | — |
| 225 | C8 | — | 28 | 2 | no | — |
| 226 | C7 | #10080 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 227 | C22 | — | 56 | 2 | no | — |
| 228 | C22 | — | 56 | 4 | no | — |
| 229 | C22 | — | 112 | 2 | no | — |
| 230 | C22 | — | 112 | 2 | no | — |
| 231 | C22 | #89443 | 1 | 1 | yes | characteristic |
| 232 | C22 | #89519 | 1 | 1 | yes | characteristic |
| 233 | C22 | #89525 | 1 | 1 | yes | characteristic |
| 234 | C22 | #89599 | 1 | 4 | yes | — |
| 235 | C4 | — | 2 | 1 | no | — |
| 236 | C4 | — | 2 | 1 | no | — |
| 237 | C4 | — | 2 | 2 | no | — |
| 238 | C4 | — | 2 | 2 | no | — |
| 239 | C4 | — | 8 | 2 | no | — |
| 240 | C4 | — | 56 | 2 | no | — |
| 241 | C4 | — | 56 | 2 | no | — |
| 242 | C2 | — | 112 | 2 | no | — |
| 243 | C2 | #11726108 | 1 | 2 | yes | — |
| 244 | C2 | #11726129 | 1 | 2 | yes | — |
| 245 | C2 | #11726521 | 1 | 2 | yes | — |
| 246 | C2 | #11726762 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11731662 | C2 | semidirect |
| 3. C2×#89260 | C2 | semidirect |
| 5. C7×#1295 | C2 | semidirect |
| 52. #1295 | D7 | semidirect |
| 226. C7 | #10080 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#89485 | 2 |
| #11731662 | 2 |
| C2×#89260 | 2 |
| #11727307 | 2 |
| C7×#1295 | 2 |
| #10080 | 7 |