← ΓL(1,𝔽8)×#6894 #19103178
Order 5103.
932 conjugacy classes of subgroups.
The subgroup lattice has 240 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C3×ΓL(1,𝔽8)×3−1+2 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | C32×C9×ΓL(1,𝔽8) | C3 | 1 | 1 | yes | maximal, characteristic |
| 3 | #13133566 | C3 | 1 | 2 | yes | maximal, complemented |
| 4 | #13133565 | C3 | 1 | 2 | yes | maximal, complemented |
| 5 | #13133564 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #13133563 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #13133562 | C3 | 1 | 6 | yes | maximal, complemented |
| 8 | #13133561 | C3 | 1 | 6 | yes | maximal, complemented |
| 9 | #13133560 | C3 | 1 | 2 | yes | maximal, complemented |
| 10 | #13133559 | C3 | 1 | 3 | yes | maximal, complemented |
| 11 | #13133558 | C3 | 1 | 3 | yes | maximal, complemented |
| 12 | ΓL(1,𝔽8)×C9⋊C9 | C3 | 1 | 6 | yes | maximal, complemented |
| 13 | ΓL(1,𝔽8)×C9⋊C9 | C3 | 1 | 3 | yes | maximal, complemented |
| 14 | ΓL(1,𝔽8)×C32⋊C9 | C3 | 1 | 2 | yes | maximal |
| 15 | C7×#6894 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 16 | C3×#6894 | — | 7 | 1 | no | maximal, Sylow |
| 17 | C33×ΓL(1,𝔽8) | C32 | 1 | 1 | yes | characteristic |
| 18 | C3×C7×3−1+2 | C32 | 1 | 1 | yes | characteristic |
| 19 | C32×C7×C9 | C32 | 1 | 1 | yes | characteristic |
| 20 | C3×#4430 | C32 | 1 | 1 | yes | characteristic |
| 21 | C3×#4430 | C32 | 1 | 1 | yes | characteristic |
| 22 | C32×C7⋊C9 | C32 | 1 | 2 | yes | — |
| 23 | ΓL(1,𝔽8)×3−1+2 | — | 3 | 9 | no | — |
| 24 | C3×#4428 | C32 | 1 | 3 | yes | — |
| 25 | C3×#4427 | C32 | 1 | 3 | yes | — |
| 26 | C3×C9×ΓL(1,𝔽8) | — | 3 | 9 | no | — |
| 27 | C3×C9×ΓL(1,𝔽8) | C32 | 1 | 2 | yes | — |
| 28 | C3×C9×ΓL(1,𝔽8) | C32 | 1 | 1 | yes | characteristic |
| 29 | C3×C9×ΓL(1,𝔽8) | C32 | 1 | 1 | yes | characteristic |
| 30 | C3×C9×ΓL(1,𝔽8) | C32 | 1 | 2 | yes | — |
| 31 | C3×C9×ΓL(1,𝔽8) | C32 | 1 | 2 | yes | — |
| 32 | C3×C9×ΓL(1,𝔽8) | C32 | 1 | 3 | yes | — |
| 33 | C3×C9×ΓL(1,𝔽8) | C32 | 1 | 6 | yes | — |
| 34 | C3×C9×ΓL(1,𝔽8) | C32 | 1 | 1 | yes | characteristic |
| 35 | C7×C9⋊C9 | C32 | 1 | 3 | yes | complemented |
| 36 | C7×C9⋊C9 | C32 | 1 | 6 | yes | complemented |
| 37 | C7×C32⋊C9 | C32 | 1 | 2 | yes | — |
| 38 | C32⋊(C7⋊C9) | C32 | 1 | 2 | yes | — |
| 39 | C32⋊(C7⋊C9) | C32 | 1 | 2 | yes | — |
| 40 | (C3×ΓL(1,𝔽8))⋊C9 | C32 | 1 | 6 | yes | — |
| 41 | (C3×ΓL(1,𝔽8))⋊C9 | C32 | 1 | 6 | yes | — |
| 42 | #10588036 | C32 | 1 | 3 | yes | complemented |
| 43 | #10588036 | C32 | 1 | 6 | yes | complemented |
| 44 | #10588035 | C32 | 1 | 3 | yes | complemented |
| 45 | #10588035 | C32 | 1 | 6 | yes | complemented |
| 46 | (C7⋊C9)⋊C9 | C32 | 1 | 6 | yes | complemented |
| 47 | (C7⋊C9)⋊C9 | C32 | 1 | 6 | yes | complemented |
| 48 | (C7⋊C9)⋊C9 | C32 | 1 | 6 | yes | complemented |
| 49 | (C7⋊C9)⋊C9 | C32 | 1 | 6 | yes | complemented |
| 50 | (C7⋊C9)⋊C9 | C32 | 1 | 6 | yes | complemented |
| 51 | (C7⋊C9)⋊C9 | C32 | 1 | 6 | yes | complemented |
| 52 | (C7⋊C9)⋊C9 | C32 | 1 | 6 | yes | complemented |
| 53 | (C7⋊C9)⋊C9 | C32 | 1 | 6 | yes | complemented |
| 54 | (C7⋊C9)⋊C9 | C32 | 1 | 6 | yes | complemented |
| 55 | C32×3−1+2 | — | 7 | 1 | no | — |
| 56 | C33×C9 | — | 7 | 1 | no | — |
| 57 | #6894 | — | 7 | 2 | no | — |
| 58 | #6894 | — | 7 | 1 | no | — |
| 59 | #6894 | — | 7 | 1 | no | — |
| 60 | #6894 | — | 7 | 3 | no | — |
| 61 | #6894 | — | 7 | 3 | no | — |
| 62 | #6894 | — | 7 | 6 | no | — |
| 63 | #6894 | — | 7 | 6 | no | — |
| 64 | #6894 | — | 7 | 2 | no | — |
| 65 | #6894 | — | 7 | 2 | no | — |
| 66 | #6894 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor |
| 67 | C3×C9⋊C9 | — | 7 | 3 | no | — |
| 68 | C3×C9⋊C9 | — | 7 | 6 | no | — |
| 69 | C3×C32⋊C9 | — | 7 | 2 | no | — |
| 70 | C33×C7 | C33 | 1 | 1 | yes | characteristic |
| 71 | C32×ΓL(1,𝔽8) | — | 3 | 3 | no | — |
| 72 | C32×ΓL(1,𝔽8) | — | 3 | 6 | no | — |
| 73 | C32×ΓL(1,𝔽8) | 3−1+2 | 1 | 9 | yes | — |
| 74 | C32×ΓL(1,𝔽8) | C33 | 1 | 1 | yes | characteristic |
| 75 | C32×ΓL(1,𝔽8) | C33 | 1 | 1 | yes | characteristic |
| 76 | C32×ΓL(1,𝔽8) | C33 | 1 | 1 | yes | characteristic |
| 77 | C7×3−1+2 | — | 3 | 3 | no | — |
| 78 | C3×C7×C9 | — | 3 | 3 | no | — |
| 79 | C3×C7×C9 | C33 | 1 | 2 | yes | — |
| 80 | C3×C7×C9 | C33 | 1 | 3 | yes | — |
| 81 | C3×C7×C9 | C33 | 1 | 6 | yes | — |
| 82 | C3×C7×C9 | C33 | 1 | 1 | yes | characteristic |
| 83 | #4430 | — | 3 | 3 | no | — |
| 84 | #4430 | — | 3 | 3 | no | — |
| 85 | C3×C7⋊C9 | — | 3 | 6 | no | — |
| 86 | C3×C7⋊C9 | C33 | 1 | 6 | yes | — |
| 87 | C3×C7⋊C9 | C33 | 1 | 6 | yes | — |
| 88 | C3×C7⋊C9 | C33 | 1 | 2 | yes | — |
| 89 | C3×C7⋊C9 | C33 | 1 | 3 | yes | — |
| 90 | C3×C7⋊C9 | C33 | 1 | 2 | yes | — |
| 91 | C3×C7⋊C9 | C33 | 1 | 3 | yes | — |
| 92 | C3×C7⋊C9 | C33 | 1 | 2 | yes | — |
| 93 | #4428 | — | 3 | 9 | no | — |
| 94 | #4427 | — | 3 | 9 | no | — |
| 95 | C9×ΓL(1,𝔽8) | — | 3 | 6 | no | — |
| 96 | C9×ΓL(1,𝔽8) | — | 3 | 9 | no | — |
| 97 | C9×ΓL(1,𝔽8) | — | 3 | 18 | no | — |
| 98 | C9×ΓL(1,𝔽8) | — | 3 | 3 | no | — |
| 99 | C9×ΓL(1,𝔽8) | — | 3 | 3 | no | — |
| 100 | C9×ΓL(1,𝔽8) | — | 3 | 6 | no | — |
| 101 | C9×ΓL(1,𝔽8) | — | 3 | 6 | no | — |
| 102 | C9×ΓL(1,𝔽8) | 3−1+2 | 1 | 9 | yes | complemented |
| 103 | C34 | — | 7 | 1 | no | — |
| 104 | C3×3−1+2 | — | 7 | 1 | no | — |
| 105 | C3×3−1+2 | — | 7 | 1 | no | — |
| 106 | C3×3−1+2 | — | 7 | 3 | no | — |
| 107 | C3×3−1+2 | — | 7 | 3 | no | — |
| 108 | C3×3−1+2 | — | 21 | 9 | no | — |
| 109 | C3×3−1+2 | C3×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 110 | C32×C9 | — | 7 | 1 | no | — |
| 111 | C32×C9 | — | 7 | 2 | no | — |
| 112 | C32×C9 | — | 7 | 2 | no | — |
| 113 | C32×C9 | — | 7 | 2 | no | — |
| 114 | C32×C9 | — | 7 | 2 | no | — |
| 115 | C32×C9 | — | 7 | 1 | no | — |
| 116 | C32×C9 | — | 7 | 1 | no | — |
| 117 | C32×C9 | — | 7 | 3 | no | — |
| 118 | C32×C9 | — | 7 | 6 | no | — |
| 119 | C32×C9 | — | 21 | 9 | no | — |
| 120 | C32×C9 | C3×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 121 | C9⋊C9 | — | 7 | 6 | no | — |
| 122 | C9⋊C9 | — | 7 | 6 | no | — |
| 123 | C9⋊C9 | — | 7 | 6 | no | — |
| 124 | C9⋊C9 | — | 7 | 6 | no | — |
| 125 | C9⋊C9 | — | 7 | 6 | no | — |
| 126 | C9⋊C9 | — | 7 | 6 | no | — |
| 127 | C9⋊C9 | — | 7 | 6 | no | — |
| 128 | C9⋊C9 | — | 7 | 6 | no | — |
| 129 | C9⋊C9 | — | 7 | 6 | no | — |
| 130 | C9⋊C9 | — | 7 | 3 | no | — |
| 131 | C9⋊C9 | — | 7 | 6 | no | — |
| 132 | C9⋊C9 | — | 7 | 3 | no | — |
| 133 | C9⋊C9 | — | 7 | 6 | no | — |
| 134 | C9⋊C9 | C3×ΓL(1,𝔽8) | 1 | 6 | yes | complemented |
| 135 | C9⋊C9 | C3×ΓL(1,𝔽8) | 1 | 3 | yes | complemented |
| 136 | C32⋊C9 | — | 7 | 2 | no | — |
| 137 | C32⋊C9 | — | 7 | 2 | no | — |
| 138 | C32⋊C9 | — | 7 | 6 | no | — |
| 139 | C32⋊C9 | — | 7 | 6 | no | — |
| 140 | C32⋊C9 | C3×ΓL(1,𝔽8) | 1 | 2 | yes | — |
| 141 | C32×C7 | — | 3 | 1 | no | — |
| 142 | C32×C7 | — | 3 | 2 | no | — |
| 143 | C32×C7 | C3×3−1+2 | 1 | 3 | yes | — |
| 144 | C32×C7 | C34 | 1 | 1 | yes | characteristic |
| 145 | C3×ΓL(1,𝔽8) | — | 3 | 1 | no | — |
| 146 | C3×ΓL(1,𝔽8) | — | 3 | 1 | no | — |
| 147 | C3×ΓL(1,𝔽8) | — | 3 | 2 | no | — |
| 148 | C3×ΓL(1,𝔽8) | — | 3 | 2 | no | — |
| 149 | C3×ΓL(1,𝔽8) | — | 3 | 27 | no | — |
| 150 | C3×ΓL(1,𝔽8) | C3×3−1+2 | 1 | 3 | yes | — |
| 151 | C3×ΓL(1,𝔽8) | C3×3−1+2 | 1 | 3 | yes | — |
| 152 | C3×ΓL(1,𝔽8) | C3×3−1+2 | 1 | 3 | yes | — |
| 153 | C3×ΓL(1,𝔽8) | C3×3−1+2 | 1 | 3 | yes | — |
| 154 | C3×ΓL(1,𝔽8) | #737 | 1 | 6 | yes | — |
| 155 | C7×C9 | — | 3 | 2 | no | — |
| 156 | C7×C9 | — | 3 | 3 | no | — |
| 157 | C7×C9 | — | 3 | 6 | no | — |
| 158 | C7×C9 | C3×3−1+2 | 1 | 3 | yes | complemented |
| 159 | C7⋊C9 | — | 3 | 2 | no | — |
| 160 | C7⋊C9 | — | 3 | 2 | no | — |
| 161 | C7⋊C9 | — | 3 | 3 | no | — |
| 162 | C7⋊C9 | — | 3 | 3 | no | — |
| 163 | C7⋊C9 | — | 3 | 6 | no | — |
| 164 | C7⋊C9 | — | 3 | 6 | no | — |
| 165 | C7⋊C9 | C3×3−1+2 | 1 | 6 | yes | complemented |
| 166 | C33 | — | 7 | 1 | no | — |
| 167 | C33 | — | 7 | 9 | no | — |
| 168 | C33 | — | 7 | 1 | no | — |
| 169 | C33 | — | 7 | 1 | no | — |
| 170 | C33 | — | 21 | 3 | no | — |
| 171 | C33 | — | 21 | 6 | no | — |
| 172 | C33 | C32×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 173 | 3−1+2 | — | 3 | 3 | no | — |
| 174 | 3−1+2 | — | 21 | 9 | no | — |
| 175 | 3−1+2 | — | 21 | 9 | no | — |
| 176 | 3−1+2 | — | 21 | 3 | no | — |
| 177 | 3−1+2 | — | 21 | 3 | no | — |
| 178 | C3×C9 | — | 3 | 3 | no | — |
| 179 | C3×C9 | — | 7 | 2 | no | — |
| 180 | C3×C9 | — | 7 | 9 | no | — |
| 181 | C3×C9 | — | 7 | 6 | no | — |
| 182 | C3×C9 | — | 7 | 6 | no | — |
| 183 | C3×C9 | — | 7 | 2 | no | — |
| 184 | C3×C9 | — | 7 | 2 | no | — |
| 185 | C3×C9 | — | 7 | 3 | no | — |
| 186 | C3×C9 | — | 7 | 3 | no | — |
| 187 | C3×C9 | — | 21 | 6 | no | — |
| 188 | C3×C9 | — | 21 | 6 | no | — |
| 189 | C3×C9 | — | 21 | 18 | no | — |
| 190 | C3×C9 | — | 21 | 6 | no | — |
| 191 | C3×C9 | — | 21 | 9 | no | — |
| 192 | C3×C9 | — | 21 | 6 | no | — |
| 193 | C3×C9 | — | 21 | 3 | no | — |
| 194 | C3×C9 | — | 21 | 3 | no | — |
| 195 | C3×C9 | C32×ΓL(1,𝔽8) | 1 | 6 | yes | — |
| 196 | C3×C9 | C32×ΓL(1,𝔽8) | 1 | 2 | yes | — |
| 197 | C3×C9 | C32×ΓL(1,𝔽8) | 1 | 3 | yes | — |
| 198 | C3×C9 | C32×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 199 | C3×C7 | — | 3 | 3 | no | — |
| 200 | C3×C7 | C32×3−1+2 | 1 | 1 | yes | characteristic |
| 201 | C3×C7 | C32×3−1+2 | 1 | 1 | yes | characteristic |
| 202 | C3×C7 | C3×#737 | 1 | 2 | yes | — |
| 203 | ΓL(1,𝔽8) | — | 3 | 3 | no | — |
| 204 | ΓL(1,𝔽8) | — | 3 | 3 | no | — |
| 205 | ΓL(1,𝔽8) | #6894 | 1 | 9 | yes | direct product factor |
| 206 | C32 | — | 3 | 1 | no | — |
| 207 | C32 | — | 3 | 2 | no | — |
| 208 | C32 | — | 7 | 3 | no | — |
| 209 | C32 | — | 7 | 3 | no | — |
| 210 | C32 | — | 7 | 6 | no | — |
| 211 | C32 | — | 7 | 3 | no | — |
| 212 | C32 | — | 7 | 3 | no | — |
| 213 | C32 | — | 21 | 27 | no | — |
| 214 | C32 | — | 21 | 1 | no | — |
| 215 | C32 | — | 21 | 1 | no | — |
| 216 | C32 | — | 21 | 2 | no | — |
| 217 | C32 | — | 21 | 2 | no | — |
| 218 | C32 | ΓL(1,𝔽8)×3−1+2 | 1 | 3 | yes | — |
| 219 | C32 | C33×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 220 | C9 | — | 3 | 2 | no | — |
| 221 | C9 | — | 3 | 3 | no | — |
| 222 | C9 | — | 3 | 6 | no | — |
| 223 | C9 | — | 7 | 6 | no | — |
| 224 | C9 | — | 21 | 2 | no | — |
| 225 | C9 | — | 21 | 2 | no | — |
| 226 | C9 | — | 21 | 3 | no | — |
| 227 | C9 | — | 21 | 3 | no | — |
| 228 | C9 | — | 21 | 6 | no | — |
| 229 | C9 | — | 21 | 6 | no | — |
| 230 | C9 | ΓL(1,𝔽8)×3−1+2 | 1 | 3 | yes | complemented |
| 231 | C7 | C3×#6894 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 232 | C3 | — | 3 | 3 | no | — |
| 233 | C3 | — | 7 | 9 | no | — |
| 234 | C3 | — | 21 | 3 | no | — |
| 235 | C3 | — | 21 | 3 | no | — |
| 236 | C3 | C3×ΓL(1,𝔽8)×3−1+2 | 1 | 1 | yes | characteristic |
| 237 | C3 | C3×ΓL(1,𝔽8)×3−1+2 | 1 | 1 | yes | characteristic |
| 238 | C3 | ΓL(1,𝔽8)×#737 | 1 | 2 | yes | — |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Finite Group Databasebeta