← #90342⋊C3 #13057389
Subgroups — #90342⋊C3 #13057389
Order 1344.
392 conjugacy classes of subgroups.
The subgroup lattice has 248 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | ΓL(1,𝔽8)×#123 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #6499⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #6498⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #6487⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #6452⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #6451⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #6442⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #6442⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #6441⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #90342 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 11 | C2×D4×ΓL(1,𝔽8) | C22 | 1 | 1 | yes | characteristic |
| 12 | C22×C4×ΓL(1,𝔽8) | C22 | 1 | 1 | yes | characteristic |
| 13 | C22×ΓL(1,𝔽8)⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×ΓL(1,𝔽8)⋊Q8 | C22 | 1 | 2 | yes | — |
| 15 | C2×ΓL(1,𝔽8)⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 16 | C4⋊C4×ΓL(1,𝔽8) | C22 | 1 | 2 | yes | — |
| 17 | C22⋊C4×ΓL(1,𝔽8) | C22 | 1 | 2 | yes | — |
| 18 | C22⋊C4×ΓL(1,𝔽8) | C22 | 1 | 1 | yes | characteristic |
| 19 | (C22⋊Dic7)⋊C3 | C22 | 1 | 2 | yes | — |
| 20 | (C22⋊Dic7)⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 21 | (C22⋊Dic7)⋊C3 | C22 | 1 | 2 | yes | — |
| 22 | (C22⋊Dic7)⋊C3 | C22 | 1 | 2 | yes | — |
| 23 | C4⋊(ΓL(1,𝔽8)⋊C4) | C22 | 1 | 2 | yes | — |
| 24 | C4⋊(ΓL(1,𝔽8)⋊C4) | C22 | 1 | 2 | yes | — |
| 25 | (ΓL(1,𝔽8)⋊C4)⋊C4 | C22 | 1 | 2 | yes | — |
| 26 | (ΓL(1,𝔽8)⋊C4)⋊C4 | C22 | 1 | 2 | yes | — |
| 27 | (ΓL(1,𝔽8)⋊C4)⋊C4 | C22 | 1 | 2 | yes | — |
| 28 | (ΓL(1,𝔽8)⋊C4)⋊C4 | C22 | 1 | 2 | yes | — |
| 29 | (ΓL(1,𝔽8)⋊C4)⋊C4 | C22 | 1 | 2 | yes | — |
| 30 | C4×ΓL(1,𝔽8)⋊C4 | C22 | 1 | 2 | yes | — |
| 31 | C4×ΓL(1,𝔽8)⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 32 | C7×#123 | C2×C3 | 1 | 1 | yes | characteristic |
| 33 | #6499 | C2×C3 | 1 | 1 | yes | characteristic |
| 34 | #6498 | C2×C3 | 1 | 1 | yes | characteristic |
| 35 | #6487 | C2×C3 | 1 | 2 | yes | complemented |
| 36 | #6452 | C2×C3 | 1 | 2 | yes | complemented |
| 37 | #6451 | C2×C3 | 1 | 2 | yes | complemented |
| 38 | #6442 | C2×C3 | 1 | 2 | yes | complemented |
| 39 | #6442 | C2×C3 | 1 | 2 | yes | complemented |
| 40 | #6441 | C2×C3 | 1 | 2 | yes | complemented |
| 41 | C3×#563 | — | 7 | 1 | no | maximal, Hall |
| 42 | C23×ΓL(1,𝔽8) | C23 | 1 | 1 | yes | characteristic |
| 43 | C23×ΓL(1,𝔽8) | C23 | 1 | 1 | yes | characteristic |
| 44 | D4×ΓL(1,𝔽8) | — | 4 | 1 | no | — |
| 45 | C2×C4×ΓL(1,𝔽8) | — | 4 | 1 | no | — |
| 46 | C2×C4×ΓL(1,𝔽8) | C23 | 1 | 2 | yes | — |
| 47 | C2×C4×ΓL(1,𝔽8) | C23 | 1 | 2 | yes | — |
| 48 | C2×C4×ΓL(1,𝔽8) | C23 | 1 | 1 | yes | characteristic |
| 49 | C2×ΓL(1,𝔽8)⋊C4 | — | 4 | 1 | no | — |
| 50 | C2×ΓL(1,𝔽8)⋊C4 | C23 | 1 | 2 | yes | — |
| 51 | C2×ΓL(1,𝔽8)⋊C4 | C23 | 1 | 2 | yes | — |
| 52 | C2×ΓL(1,𝔽8)⋊C4 | C23 | 1 | 2 | yes | — |
| 53 | C2×ΓL(1,𝔽8)⋊C4 | C23 | 1 | 2 | yes | — |
| 54 | ΓL(1,𝔽8)⋊Q8 | — | 4 | 1 | no | — |
| 55 | ΓL(1,𝔽8)⋊Q8 | — | 4 | 2 | no | — |
| 56 | C2×C7×D4 | C22×C3 | 1 | 1 | yes | characteristic |
| 57 | C22×C4×C7 | C22×C3 | 1 | 1 | yes | characteristic |
| 58 | C22×Dic7 | C22×C3 | 1 | 1 | yes | characteristic |
| 59 | C2×Dic14 | C22×C3 | 1 | 2 | yes | — |
| 60 | C2×Dic14 | C22×C3 | 1 | 1 | yes | characteristic |
| 61 | C7×C4⋊C4 | C22×C3 | 1 | 2 | yes | — |
| 62 | C7×C22⋊C4 | C22×C3 | 1 | 2 | yes | — |
| 63 | C7×C22⋊C4 | C22×C3 | 1 | 1 | yes | characteristic |
| 64 | C22⋊Dic7 | C22×C3 | 1 | 2 | yes | — |
| 65 | C22⋊Dic7 | C22×C3 | 1 | 1 | yes | characteristic |
| 66 | C22⋊Dic7 | C22×C3 | 1 | 2 | yes | — |
| 67 | C22⋊Dic7 | C22×C3 | 1 | 2 | yes | — |
| 68 | C4⋊Dic7 | C22×C3 | 1 | 2 | yes | — |
| 69 | C4⋊Dic7 | C22×C3 | 1 | 2 | yes | — |
| 70 | Dic7⋊C4 | C22×C3 | 1 | 2 | yes | — |
| 71 | Dic7⋊C4 | C22×C3 | 1 | 2 | yes | — |
| 72 | Dic7⋊C4 | C22×C3 | 1 | 2 | yes | — |
| 73 | Dic7⋊C4 | C22×C3 | 1 | 2 | yes | — |
| 74 | Dic7⋊C4 | C22×C3 | 1 | 2 | yes | — |
| 75 | C4×Dic7 | C22×C3 | 1 | 2 | yes | — |
| 76 | C4×Dic7 | C22×C3 | 1 | 1 | yes | characteristic |
| 77 | C3×C4⋊Q8 | — | 7 | 2 | no | — |
| 78 | C3×#126 | — | 7 | 2 | no | — |
| 79 | C3×#126 | — | 7 | 2 | no | — |
| 80 | C3×#125 | — | 7 | 2 | no | — |
| 81 | C3×#124 | — | 7 | 1 | no | — |
| 82 | C3×#123 | — | 7 | 1 | no | — |
| 83 | C3×#123 | — | 7 | 1 | no | — |
| 84 | C3×C22⋊Q8 | — | 7 | 2 | no | — |
| 85 | C3×C22⋊Q8 | — | 7 | 2 | no | — |
| 86 | C22×ΓL(1,𝔽8) | — | 2 | 1 | no | — |
| 87 | C22×ΓL(1,𝔽8) | — | 2 | 1 | no | — |
| 88 | C22×ΓL(1,𝔽8) | — | 2 | 2 | no | — |
| 89 | C22×ΓL(1,𝔽8) | — | 2 | 2 | no | — |
| 90 | C22×ΓL(1,𝔽8) | C24 | 1 | 1 | yes | characteristic |
| 91 | C4×ΓL(1,𝔽8) | — | 2 | 2 | no | — |
| 92 | C4×ΓL(1,𝔽8) | — | 2 | 1 | no | — |
| 93 | C4×ΓL(1,𝔽8) | — | 2 | 2 | no | — |
| 94 | ΓL(1,𝔽8)⋊C4 | — | 2 | 2 | no | — |
| 95 | ΓL(1,𝔽8)⋊C4 | — | 2 | 2 | no | — |
| 96 | ΓL(1,𝔽8)⋊C4 | — | 2 | 2 | no | — |
| 97 | ΓL(1,𝔽8)⋊C4 | — | 2 | 2 | no | — |
| 98 | #563 | — | 7 | 1 | no | Sylow |
| 99 | C23×C7 | C23×C3 | 1 | 1 | yes | characteristic |
| 100 | C23×C7 | C23×C3 | 1 | 1 | yes | characteristic |
| 101 | C7×D4 | — | 4 | 1 | no | — |
| 102 | C2×C4×C7 | — | 4 | 1 | no | — |
| 103 | C2×C4×C7 | C23×C3 | 1 | 2 | yes | — |
| 104 | C2×C4×C7 | C23×C3 | 1 | 1 | yes | characteristic |
| 105 | C2×C4×C7 | C23×C3 | 1 | 2 | yes | — |
| 106 | C2×Dic7 | — | 4 | 1 | no | — |
| 107 | C2×Dic7 | C23×C3 | 1 | 2 | yes | — |
| 108 | C2×Dic7 | C23×C3 | 1 | 2 | yes | — |
| 109 | C2×Dic7 | C23×C3 | 1 | 2 | yes | — |
| 110 | C2×Dic7 | C23×C3 | 1 | 2 | yes | — |
| 111 | Dic14 | — | 4 | 1 | no | — |
| 112 | Dic14 | — | 4 | 2 | no | — |
| 113 | C2×C3×Q8 | — | 7 | 2 | no | — |
| 114 | C2×C3×Q8 | — | 7 | 1 | no | — |
| 115 | C2×C3×D4 | — | 7 | 1 | no | — |
| 116 | C22×C3×C4 | — | 7 | 1 | no | — |
| 117 | C22×C3×C4 | — | 7 | 1 | no | — |
| 118 | C3×C4⋊C4 | — | 7 | 2 | no | — |
| 119 | C3×C4⋊C4 | — | 7 | 2 | no | — |
| 120 | C3×C4⋊C4 | — | 7 | 2 | no | — |
| 121 | C3×C4⋊C4 | — | 7 | 2 | no | — |
| 122 | C3×C4⋊C4 | — | 7 | 2 | no | — |
| 123 | C3×C4⋊C4 | — | 7 | 2 | no | — |
| 124 | C3×C4⋊C4 | — | 7 | 2 | no | — |
| 125 | C3×C4⋊C4 | — | 7 | 2 | no | — |
| 126 | C3×C22⋊C4 | — | 7 | 2 | no | — |
| 127 | C3×C22⋊C4 | — | 7 | 2 | no | — |
| 128 | C3×C22⋊C4 | — | 7 | 2 | no | — |
| 129 | C3×C22⋊C4 | — | 7 | 1 | no | — |
| 130 | C3×C22⋊C4 | — | 7 | 2 | no | — |
| 131 | C3×C22⋊C4 | — | 7 | 1 | no | — |
| 132 | C3×C42 | — | 7 | 1 | no | — |
| 133 | C3×C42 | — | 7 | 2 | no | — |
| 134 | C2×ΓL(1,𝔽8) | — | 4 | 1 | no | — |
| 135 | C2×ΓL(1,𝔽8) | — | 4 | 1 | no | — |
| 136 | C2×ΓL(1,𝔽8) | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 137 | C2×ΓL(1,𝔽8) | 2−1+4 | 1 | 2 | yes | — |
| 138 | C4⋊Q8 | — | 7 | 2 | no | — |
| 139 | #126 | — | 7 | 2 | no | — |
| 140 | #126 | — | 7 | 2 | no | — |
| 141 | #125 | — | 7 | 2 | no | — |
| 142 | #124 | — | 7 | 1 | no | — |
| 143 | #123 | — | 7 | 1 | no | — |
| 144 | #123 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 145 | C22⋊Q8 | — | 7 | 2 | no | — |
| 146 | C22⋊Q8 | — | 7 | 2 | no | — |
| 147 | C22×C7 | — | 2 | 1 | no | — |
| 148 | C22×C7 | — | 2 | 1 | no | — |
| 149 | C22×C7 | — | 2 | 2 | no | — |
| 150 | C22×C7 | — | 2 | 2 | no | — |
| 151 | C22×C7 | C24×C3 | 1 | 1 | yes | characteristic |
| 152 | C4×C7 | — | 2 | 2 | no | — |
| 153 | C4×C7 | — | 2 | 1 | no | — |
| 154 | C4×C7 | — | 2 | 2 | no | — |
| 155 | Dic7 | — | 2 | 2 | no | — |
| 156 | Dic7 | — | 2 | 2 | no | — |
| 157 | Dic7 | — | 2 | 2 | no | — |
| 158 | Dic7 | — | 2 | 2 | no | — |
| 159 | C23×C3 | — | 7 | 1 | no | — |
| 160 | C23×C3 | — | 7 | 1 | no | — |
| 161 | C3×Q8 | — | 28 | 2 | no | — |
| 162 | C3×Q8 | — | 28 | 1 | no | — |
| 163 | C3×D4 | — | 28 | 1 | no | — |
| 164 | C2×C3×C4 | — | 7 | 2 | no | — |
| 165 | C2×C3×C4 | — | 7 | 2 | no | — |
| 166 | C2×C3×C4 | — | 7 | 2 | no | — |
| 167 | C2×C3×C4 | — | 7 | 2 | no | — |
| 168 | C2×C3×C4 | — | 7 | 2 | no | — |
| 169 | C2×C3×C4 | — | 7 | 1 | no | — |
| 170 | C2×C3×C4 | — | 7 | 2 | no | — |
| 171 | C2×C3×C4 | — | 28 | 1 | no | — |
| 172 | C2×C3×C4 | — | 28 | 1 | no | — |
| 173 | ΓL(1,𝔽8) | #563 | 1 | 1 | yes | characteristic, complemented, Hall |
| 174 | C2×Q8 | — | 7 | 2 | no | — |
| 175 | C2×Q8 | — | 7 | 1 | no | — |
| 176 | C2×D4 | C2×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 177 | C22×C4 | — | 7 | 1 | no | — |
| 178 | C22×C4 | C2×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 179 | C4⋊C4 | — | 7 | 2 | no | — |
| 180 | C4⋊C4 | — | 7 | 2 | no | — |
| 181 | C4⋊C4 | — | 7 | 2 | no | — |
| 182 | C4⋊C4 | — | 7 | 2 | no | — |
| 183 | C4⋊C4 | — | 7 | 2 | no | — |
| 184 | C4⋊C4 | — | 7 | 2 | no | — |
| 185 | C4⋊C4 | — | 7 | 2 | no | — |
| 186 | C4⋊C4 | C2×AGL(1,𝔽7) | 1 | 2 | yes | — |
| 187 | C22⋊C4 | — | 7 | 2 | no | — |
| 188 | C22⋊C4 | — | 7 | 2 | no | — |
| 189 | C22⋊C4 | — | 7 | 2 | no | — |
| 190 | C22⋊C4 | — | 7 | 1 | no | — |
| 191 | C22⋊C4 | C2×AGL(1,𝔽7) | 1 | 2 | yes | — |
| 192 | C22⋊C4 | C2×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 193 | C42 | — | 7 | 1 | no | — |
| 194 | C42 | — | 7 | 2 | no | — |
| 195 | C2×C7 | — | 4 | 1 | no | — |
| 196 | C2×C7 | — | 4 | 1 | no | — |
| 197 | C2×C7 | C3×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 198 | C2×C7 | C3×2−1+4 | 1 | 2 | yes | — |
| 199 | C22×C3 | — | 7 | 1 | no | — |
| 200 | C22×C3 | — | 14 | 1 | no | — |
| 201 | C22×C3 | — | 14 | 1 | no | — |
| 202 | C22×C3 | — | 14 | 2 | no | — |
| 203 | C22×C3 | — | 14 | 2 | no | — |
| 204 | C3×C4 | — | 14 | 2 | no | — |
| 205 | C3×C4 | — | 14 | 1 | no | — |
| 206 | C3×C4 | — | 14 | 2 | no | — |
| 207 | C3×C4 | — | 14 | 2 | no | — |
| 208 | C3×C4 | — | 14 | 2 | no | — |
| 209 | C3×C4 | — | 14 | 2 | no | — |
| 210 | C3×C4 | — | 14 | 2 | no | — |
| 211 | C23 | C22×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 212 | C23 | C22×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 213 | Q8 | — | 28 | 2 | no | — |
| 214 | Q8 | — | 28 | 1 | no | — |
| 215 | D4 | — | 4 | 1 | no | — |
| 216 | C2×C4 | — | 4 | 1 | no | — |
| 217 | C2×C4 | — | 7 | 2 | no | — |
| 218 | C2×C4 | — | 7 | 2 | no | — |
| 219 | C2×C4 | — | 7 | 2 | no | — |
| 220 | C2×C4 | — | 7 | 2 | no | — |
| 221 | C2×C4 | — | 28 | 1 | no | — |
| 222 | C2×C4 | C22×AGL(1,𝔽7) | 1 | 2 | yes | — |
| 223 | C2×C4 | C22×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 224 | C2×C4 | C22×AGL(1,𝔽7) | 1 | 2 | yes | — |
| 225 | C7 | C3×#563 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 226 | C2×C3 | — | 7 | 1 | no | — |
| 227 | C2×C3 | — | 7 | 2 | no | — |
| 228 | C2×C3 | — | 28 | 1 | no | — |
| 229 | C2×C3 | — | 28 | 1 | no | — |
| 230 | C22 | — | 2 | 1 | no | — |
| 231 | C22 | — | 2 | 1 | no | — |
| 232 | C22 | — | 2 | 2 | no | — |
| 233 | C22 | — | 2 | 2 | no | — |
| 234 | C22 | C23×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 235 | C4 | — | 2 | 2 | no | — |
| 236 | C4 | — | 2 | 1 | no | — |
| 237 | C4 | — | 2 | 2 | no | — |
| 238 | C4 | — | 14 | 2 | no | — |
| 239 | C4 | — | 14 | 2 | no | — |
| 240 | C4 | — | 14 | 2 | no | — |
| 241 | C4 | — | 14 | 2 | no | — |
| 242 | C3 | — | 7 | 1 | no | Sylow |
| 243 | C2 | — | 4 | 1 | no | — |
| 244 | C2 | — | 4 | 1 | no | — |
| 245 | C2 | #6548⋊C3 | 1 | 1 | yes | characteristic |
| 246 | C2 | #10621519 | 1 | 2 | yes | — |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta