Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
3706 conjugacy classes of subgroups; only the 315 normal subgroups are shown.
This group has 3706 conjugacy classes of subgroups — too many to draw. Only its 315 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×C7×#585 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#90015 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #11741535 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #11729896 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #11729893 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C7×#585 | — | 2 | 16 | no | — |
| 7 | C2×C7×2−1+4 | — | 2 | 3 | no | — |
| 8 | C2×C7×H(5,𝔽2) | — | 2 | 3 | no | — |
| 9 | C2×C7×H(5,𝔽2) | C22 | 1 | 4 | yes | — |
| 10 | C22×C7×#41 | — | 2 | 6 | no | — |
| 11 | C22×C7×#41 | C22 | 1 | 3 | yes | — |
| 12 | C2×#6498 | C22 | 1 | 6 | yes | complemented |
| 13 | C22×Dic7⋊C4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | #90015 | — | 2 | 16 | no | — |
| 15 | #90001 | — | 2 | 3 | no | — |
| 16 | #89994 | — | 2 | 3 | no | — |
| 17 | #89992 | C22 | 1 | 6 | yes | complemented |
| 18 | C2×#6408 | C22 | 1 | 12 | yes | complemented |
| 19 | #89983 | C22 | 1 | 3 | yes | complemented |
| 20 | #89982 | — | 2 | 6 | no | — |
| 21 | #32901 | — | 7 | 1 | no | maximal, Sylow |
| 22 | C7×2−1+4 | — | 2 | 48 | no | — |
| 23 | C7×H(5,𝔽2) | — | 2 | 48 | no | — |
| 24 | C7×H(5,𝔽2) | — | 2 | 24 | no | — |
| 25 | C7×H(5,𝔽2) | D4 | 1 | 16 | yes | — |
| 26 | C2×C7×#41 | — | 2 | 4 | no | — |
| 27 | C2×C7×#41 | — | 2 | 12 | no | — |
| 28 | C2×C7×#41 | — | 2 | 48 | no | — |
| 29 | C2×C7×#41 | — | 2 | 48 | no | — |
| 30 | C2×C7×#41 | — | 2 | 24 | no | — |
| 31 | C2×C7×#41 | — | 2 | 6 | no | — |
| 32 | C2×C7×#41 | — | 2 | 12 | no | — |
| 33 | C2×C7×#41 | — | 2 | 6 | no | — |
| 34 | C22×C7×Q8 | — | 2 | 6 | no | — |
| 35 | C22×C7×Q8 | D4 | 1 | 3 | yes | — |
| 36 | C22×C7×D4 | — | 2 | 6 | no | — |
| 37 | C22×C7×D4 | — | 2 | 12 | no | — |
| 38 | C22×C7×D4 | D4 | 1 | 3 | yes | — |
| 39 | C22×C7×D4 | C23 | 1 | 6 | yes | — |
| 40 | C23×C4×C7 | — | 2 | 4 | no | — |
| 41 | C23×C4×C7 | D4 | 1 | 6 | yes | — |
| 42 | C23×C4×C7 | C23 | 1 | 1 | yes | characteristic |
| 43 | C23×Dic7 | C23 | 1 | 1 | yes | characteristic |
| 44 | C23×Dic7 | C23 | 1 | 1 | yes | characteristic |
| 45 | C2×C22⋊Dic7 | — | 2 | 6 | no | — |
| 46 | C2×C22⋊Dic7 | — | 2 | 6 | no | — |
| 47 | C2×C22⋊Dic7 | C23 | 1 | 6 | yes | — |
| 48 | #6498 | — | 2 | 24 | no | — |
| 49 | C2×Dic7⋊C4 | — | 2 | 3 | no | — |
| 50 | C2×Dic7⋊C4 | — | 2 | 3 | no | — |
| 51 | C2×Dic7⋊C4 | — | 2 | 6 | no | — |
| 52 | #6408 | — | 2 | 24 | no | — |
| 53 | C7⋊#95 | — | 2 | 3 | no | — |
| 54 | C7⋊#95 | — | 2 | 6 | no | — |
| 55 | C7⋊#95 | — | 2 | 3 | no | — |
| 56 | C2×#585 | D7 | 1 | 1 | yes | characteristic |
| 57 | C2×#458 | — | 7 | 4 | no | — |
| 58 | #2406 | — | 7 | 3 | no | — |
| 59 | #1790 | — | 7 | 4 | no | — |
| 60 | #1782 | — | 7 | 3 | no | — |
| 61 | C24×C7 | — | 2 | 4 | no | — |
| 62 | C24×C7 | C2×D4 | 1 | 6 | yes | — |
| 63 | C24×C7 | C24 | 1 | 1 | yes | characteristic |
| 64 | C7×#41 | — | 2 | 32 | no | — |
| 65 | C7×#41 | — | 2 | 48 | no | — |
| 66 | C7×#41 | — | 2 | 96 | no | — |
| 67 | C7×#41 | — | 2 | 48 | no | — |
| 68 | C7×#41 | — | 2 | 48 | no | — |
| 69 | C7×#41 | — | 2 | 48 | no | — |
| 70 | C2×C7×Q8 | — | 2 | 4 | no | — |
| 71 | C2×C7×Q8 | — | 2 | 48 | no | — |
| 72 | C2×C7×Q8 | — | 2 | 6 | no | — |
| 73 | C2×C7×Q8 | — | 2 | 12 | no | — |
| 74 | C2×C7×D4 | — | 2 | 12 | no | — |
| 75 | C2×C7×D4 | — | 2 | 48 | no | — |
| 76 | C2×C7×D4 | — | 2 | 48 | no | — |
| 77 | C2×C7×D4 | — | 2 | 48 | no | — |
| 78 | C2×C7×D4 | — | 2 | 12 | no | — |
| 79 | C2×C7×D4 | — | 2 | 12 | no | — |
| 80 | C2×C7×D4 | — | 2 | 12 | no | — |
| 81 | C2×C7×D4 | — | 2 | 6 | no | — |
| 82 | C2×C7×D4 | C2×D4 | 1 | 24 | yes | — |
| 83 | C22×C4×C7 | — | 2 | 12 | no | — |
| 84 | C22×C4×C7 | — | 2 | 32 | no | — |
| 85 | C22×C4×C7 | — | 2 | 12 | no | — |
| 86 | C22×C4×C7 | — | 2 | 4 | no | — |
| 87 | C22×C4×C7 | — | 2 | 6 | no | — |
| 88 | C22×C4×C7 | — | 2 | 6 | no | — |
| 89 | C22×C4×C7 | — | 2 | 12 | no | — |
| 90 | C22×C4×C7 | C2×D4 | 1 | 3 | yes | — |
| 91 | C22×C4×C7 | C2×D4 | 1 | 6 | yes | — |
| 92 | C22×C4×C7 | C2×D4 | 1 | 3 | yes | — |
| 93 | C22×Dic7 | — | 2 | 3 | no | — |
| 94 | C22×Dic7 | — | 2 | 6 | no | — |
| 95 | C22×Dic7 | — | 2 | 3 | no | — |
| 96 | C22×Dic7 | — | 2 | 4 | no | — |
| 97 | C22×Dic7 | — | 4 | 2 | no | — |
| 98 | C22⋊Dic7 | — | 2 | 12 | no | — |
| 99 | C22⋊Dic7 | — | 4 | 12 | no | — |
| 100 | C22⋊Dic7 | — | 4 | 12 | no | — |
| 101 | Dic7⋊C4 | — | 4 | 1 | no | — |
| 102 | Dic7⋊C4 | — | 4 | 3 | no | — |
| 103 | Dic7⋊C4 | — | 4 | 12 | no | — |
| 104 | #585 | — | 2 | 16 | no | — |
| 105 | C2×2−1+4 | — | 2 | 3 | no | — |
| 106 | C2×H(5,𝔽2) | — | 2 | 3 | no | — |
| 107 | C2×H(5,𝔽2) | C2×D7 | 1 | 4 | yes | — |
| 108 | C22×#41 | — | 2 | 6 | no | — |
| 109 | C22×#41 | C2×D7 | 1 | 3 | yes | — |
| 110 | C2×#123 | — | 7 | 6 | no | — |
| 111 | C22×C4⋊C4 | — | 7 | 1 | no | — |
| 112 | #458 | — | 14 | 16 | no | — |
| 113 | C2×#99 | — | 7 | 12 | no | — |
| 114 | #395 | — | 14 | 3 | no | — |
| 115 | #394 | — | 14 | 3 | no | — |
| 116 | #386 | — | 7 | 6 | no | — |
| 117 | #385 | — | 14 | 6 | no | — |
| 118 | #381 | — | 7 | 3 | no | — |
| 119 | C23×C7 | — | 2 | 16 | no | — |
| 120 | C23×C7 | — | 2 | 6 | no | — |
| 121 | C23×C7 | — | 2 | 6 | no | — |
| 122 | C23×C7 | — | 4 | 1 | no | — |
| 123 | C23×C7 | — | 4 | 1 | no | — |
| 124 | C23×C7 | — | 4 | 6 | no | — |
| 125 | C23×C7 | — | 4 | 6 | no | — |
| 126 | C23×C7 | — | 8 | 4 | no | — |
| 127 | C23×C7 | — | 8 | 4 | no | — |
| 128 | C23×C7 | #120 | 1 | 12 | yes | — |
| 129 | C23×C7 | C22×D4 | 1 | 3 | yes | — |
| 130 | C23×C7 | C22×D4 | 1 | 4 | yes | — |
| 131 | C7×Q8 | — | 2 | 16 | no | — |
| 132 | C7×Q8 | — | 2 | 24 | no | — |
| 133 | C7×D4 | — | 2 | 48 | no | — |
| 134 | C7×D4 | — | 2 | 24 | no | — |
| 135 | C7×D4 | — | 2 | 48 | no | — |
| 136 | C2×C4×C7 | — | 2 | 24 | no | — |
| 137 | C2×C4×C7 | — | 2 | 48 | no | — |
| 138 | C2×C4×C7 | — | 2 | 24 | no | — |
| 139 | C2×C4×C7 | — | 2 | 6 | no | — |
| 140 | C2×C4×C7 | — | 2 | 6 | no | — |
| 141 | C2×C4×C7 | — | 2 | 6 | no | — |
| 142 | C2×C4×C7 | — | 2 | 6 | no | — |
| 143 | C2×C4×C7 | #120 | 1 | 1 | yes | characteristic |
| 144 | C2×C4×C7 | #120 | 1 | 3 | yes | — |
| 145 | C2×C4×C7 | #120 | 1 | 12 | yes | — |
| 146 | C2×Dic7 | — | 2 | 6 | no | — |
| 147 | C2×Dic7 | — | 4 | 3 | no | — |
| 148 | C2×Dic7 | — | 4 | 1 | no | — |
| 149 | C2×Dic7 | — | 4 | 6 | no | — |
| 150 | C2×Dic7 | — | 4 | 6 | no | — |
| 151 | C2×Dic7 | — | 4 | 1 | no | — |
| 152 | C2×Dic7 | — | 4 | 6 | no | — |
| 153 | C2×Dic7 | — | 4 | 6 | no | — |
| 154 | 2−1+4 | — | 2 | 48 | no | — |
| 155 | H(5,𝔽2) | — | 2 | 48 | no | — |
| 156 | H(5,𝔽2) | — | 2 | 24 | no | — |
| 157 | H(5,𝔽2) | C22⋊D7 | 1 | 16 | yes | — |
| 158 | C2×#41 | — | 2 | 4 | no | — |
| 159 | C2×#41 | — | 2 | 12 | no | — |
| 160 | C2×#41 | — | 2 | 48 | no | — |
| 161 | C2×#41 | — | 2 | 48 | no | — |
| 162 | C2×#41 | — | 2 | 24 | no | — |
| 163 | C2×#41 | — | 2 | 6 | no | — |
| 164 | C2×#41 | — | 2 | 12 | no | — |
| 165 | C2×#41 | — | 2 | 6 | no | — |
| 166 | C22×Q8 | — | 2 | 6 | no | — |
| 167 | C22×Q8 | C22⋊D7 | 1 | 3 | yes | — |
| 168 | C22×D4 | — | 2 | 6 | no | — |
| 169 | C22×D4 | — | 2 | 12 | no | — |
| 170 | C22×D4 | C22⋊D7 | 1 | 3 | yes | — |
| 171 | C22×D4 | C22×D7 | 1 | 6 | yes | — |
| 172 | C23×C4 | — | 2 | 4 | no | — |
| 173 | C23×C4 | — | 7 | 1 | no | — |
| 174 | C23×C4 | — | 7 | 1 | no | — |
| 175 | C23×C4 | C22⋊D7 | 1 | 6 | yes | — |
| 176 | C23×C4 | C22×D7 | 1 | 1 | yes | characteristic |
| 177 | #123 | — | 14 | 24 | no | — |
| 178 | C2×C4⋊C4 | — | 14 | 3 | no | — |
| 179 | C2×C4⋊C4 | — | 14 | 3 | no | — |
| 180 | C2×C4⋊C4 | — | 14 | 6 | no | — |
| 181 | C2×C22⋊C4 | — | 7 | 6 | no | — |
| 182 | C2×C22⋊C4 | — | 14 | 6 | no | — |
| 183 | C2×C22⋊C4 | — | 14 | 6 | no | — |
| 184 | #99 | — | 14 | 24 | no | — |
| 185 | #95 | — | 14 | 3 | no | — |
| 186 | #95 | — | 14 | 3 | no | — |
| 187 | #95 | — | 14 | 6 | no | — |
| 188 | C22×C7 | — | 2 | 3 | no | — |
| 189 | C22×C7 | — | 2 | 3 | no | — |
| 190 | C22×C7 | — | 2 | 12 | no | — |
| 191 | C22×C7 | — | 4 | 4 | no | — |
| 192 | C22×C7 | — | 4 | 12 | no | — |
| 193 | C22×C7 | — | 4 | 6 | no | — |
| 194 | C22×C7 | — | 4 | 6 | no | — |
| 195 | C22×C7 | — | 8 | 6 | no | — |
| 196 | C22×C7 | — | 8 | 16 | no | — |
| 197 | C22×C7 | C2×#120 | 1 | 6 | yes | — |
| 198 | C22×C7 | C2×#120 | 1 | 1 | yes | characteristic |
| 199 | C4×C7 | — | 2 | 1 | no | — |
| 200 | C4×C7 | — | 2 | 12 | no | — |
| 201 | C4×C7 | — | 2 | 3 | no | — |
| 202 | Dic7 | — | 4 | 2 | no | — |
| 203 | Dic7 | — | 8 | 6 | no | — |
| 204 | Dic7 | — | 8 | 1 | no | — |
| 205 | C24 | — | 2 | 4 | no | — |
| 206 | C24 | C2×C22⋊D7 | 1 | 6 | yes | — |
| 207 | C24 | C23×D7 | 1 | 1 | yes | characteristic |
| 208 | #41 | — | 2 | 32 | no | — |
| 209 | #41 | — | 2 | 48 | no | — |
| 210 | #41 | — | 2 | 96 | no | — |
| 211 | #41 | — | 2 | 48 | no | — |
| 212 | #41 | — | 2 | 48 | no | — |
| 213 | #41 | — | 2 | 48 | no | — |
| 214 | C2×Q8 | — | 2 | 4 | no | — |
| 215 | C2×Q8 | — | 2 | 48 | no | — |
| 216 | C2×Q8 | — | 2 | 6 | no | — |
| 217 | C2×Q8 | — | 2 | 12 | no | — |
| 218 | C2×D4 | — | 2 | 12 | no | — |
| 219 | C2×D4 | — | 2 | 48 | no | — |
| 220 | C2×D4 | — | 2 | 48 | no | — |
| 221 | C2×D4 | — | 2 | 48 | no | — |
| 222 | C2×D4 | — | 2 | 12 | no | — |
| 223 | C2×D4 | — | 2 | 12 | no | — |
| 224 | C2×D4 | — | 2 | 12 | no | — |
| 225 | C2×D4 | — | 2 | 6 | no | — |
| 226 | C2×D4 | C2×C22⋊D7 | 1 | 24 | yes | — |
| 227 | C22×C4 | — | 2 | 12 | no | — |
| 228 | C22×C4 | — | 2 | 32 | no | — |
| 229 | C22×C4 | — | 2 | 12 | no | — |
| 230 | C22×C4 | — | 2 | 4 | no | — |
| 231 | C22×C4 | — | 2 | 6 | no | — |
| 232 | C22×C4 | — | 2 | 6 | no | — |
| 233 | C22×C4 | — | 2 | 12 | no | — |
| 234 | C22×C4 | — | 14 | 3 | no | — |
| 235 | C22×C4 | — | 14 | 6 | no | — |
| 236 | C22×C4 | — | 14 | 3 | no | — |
| 237 | C22×C4 | — | 14 | 4 | no | — |
| 238 | C22×C4 | — | 28 | 2 | no | — |
| 239 | C22×C4 | C2×C22⋊D7 | 1 | 3 | yes | — |
| 240 | C22×C4 | C2×C22⋊D7 | 1 | 6 | yes | — |
| 241 | C22×C4 | C2×C22⋊D7 | 1 | 3 | yes | — |
| 242 | C4⋊C4 | — | 28 | 1 | no | — |
| 243 | C4⋊C4 | — | 28 | 3 | no | — |
| 244 | C4⋊C4 | — | 28 | 12 | no | — |
| 245 | C22⋊C4 | — | 14 | 12 | no | — |
| 246 | C22⋊C4 | — | 28 | 12 | no | — |
| 247 | C22⋊C4 | — | 28 | 12 | no | — |
| 248 | C2×C7 | — | 2 | 6 | no | — |
| 249 | C2×C7 | — | 4 | 12 | no | — |
| 250 | C2×C7 | #2846 | 1 | 1 | yes | characteristic |
| 251 | C2×C7 | #3025 | 1 | 1 | yes | characteristic |
| 252 | C2×C7 | #3028 | 1 | 1 | yes | characteristic |
| 253 | C23 | — | 2 | 16 | no | — |
| 254 | C23 | — | 2 | 6 | no | — |
| 255 | C23 | — | 2 | 6 | no | — |
| 256 | C23 | — | 4 | 1 | no | — |
| 257 | C23 | — | 4 | 1 | no | — |
| 258 | C23 | — | 4 | 6 | no | — |
| 259 | C23 | — | 4 | 6 | no | — |
| 260 | C23 | — | 8 | 4 | no | — |
| 261 | C23 | — | 8 | 4 | no | — |
| 262 | C23 | #6516 | 1 | 12 | yes | — |
| 263 | C23 | C22×C22⋊D7 | 1 | 3 | yes | — |
| 264 | C23 | C22×C22⋊D7 | 1 | 4 | yes | — |
| 265 | Q8 | — | 2 | 16 | no | — |
| 266 | Q8 | — | 2 | 24 | no | — |
| 267 | D4 | — | 2 | 48 | no | — |
| 268 | D4 | — | 2 | 24 | no | — |
| 269 | D4 | — | 2 | 48 | no | — |
| 270 | C2×C4 | — | 2 | 24 | no | — |
| 271 | C2×C4 | — | 2 | 48 | no | — |
| 272 | C2×C4 | — | 2 | 24 | no | — |
| 273 | C2×C4 | — | 2 | 6 | no | — |
| 274 | C2×C4 | — | 2 | 6 | no | — |
| 275 | C2×C4 | — | 2 | 6 | no | — |
| 276 | C2×C4 | — | 2 | 6 | no | — |
| 277 | C2×C4 | — | 14 | 6 | no | — |
| 278 | C2×C4 | — | 28 | 3 | no | — |
| 279 | C2×C4 | — | 28 | 1 | no | — |
| 280 | C2×C4 | — | 28 | 6 | no | — |
| 281 | C2×C4 | — | 28 | 6 | no | — |
| 282 | C2×C4 | — | 28 | 6 | no | — |
| 283 | C2×C4 | — | 28 | 6 | no | — |
| 284 | C2×C4 | — | 28 | 1 | no | — |
| 285 | C2×C4 | #6516 | 1 | 12 | yes | — |
| 286 | C2×C4 | #6516 | 1 | 1 | yes | characteristic |
| 287 | C2×C4 | #6516 | 1 | 3 | yes | — |
| 288 | C7 | #32901 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 289 | C22 | — | 2 | 3 | no | — |
| 290 | C22 | — | 2 | 3 | no | — |
| 291 | C22 | — | 2 | 12 | no | — |
| 292 | C22 | — | 4 | 4 | no | — |
| 293 | C22 | — | 4 | 12 | no | — |
| 294 | C22 | — | 4 | 6 | no | — |
| 295 | C22 | — | 4 | 6 | no | — |
| 296 | C22 | — | 8 | 6 | no | — |
| 297 | C22 | — | 8 | 16 | no | — |
| 298 | C22 | C2×#6516 | 1 | 6 | yes | — |
| 299 | C22 | C2×#6516 | 1 | 1 | yes | characteristic |
| 300 | C4 | — | 2 | 1 | no | — |
| 301 | C4 | — | 2 | 12 | no | — |
| 302 | C4 | — | 2 | 3 | no | — |
| 303 | C4 | — | 28 | 2 | no | — |
| 304 | C4 | — | 56 | 6 | no | — |
| 305 | C4 | — | 56 | 1 | no | — |
| 306 | C2 | — | 2 | 6 | no | — |
| 307 | C2 | — | 4 | 12 | no | — |
| 308 | C2 | #11741569 | 1 | 1 | yes | characteristic |
| 309 | C2 | #11741570 | 1 | 1 | yes | characteristic |
| 310 | C2 | #11741573 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#90015 | C2 | semidirect |
| 3. #11741535 | C2 | semidirect |
| 4. #11729896 | C2 | semidirect |
| 5. #11729893 | C2 | semidirect |
| 12. C2×#6498 | C22 | semidirect |
| 13. C22×Dic7⋊C4 | C22 | semidirect |
| 17. #89992 | C22 | semidirect |
| 18. C2×#6408 | C22 | semidirect |
| 19. #89983 | C22 | semidirect |
| 288. C7 | #32901 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#585 | 2 |
| C2×#90015 | 2 |
| #11741535 | 2 |
| #11729896 | 2 |
| #11729893 | 2 |
| #32901 | 7 |