← #90210⋊C3 #13057256
Subgroups — #90210⋊C3 #13057256
Order 1344.
412 conjugacy classes of subgroups.
The subgroup lattice has 208 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | ΓL(1,𝔽8)×#117 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #6489⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×C4⋊(ΓL(1,𝔽8)⋊C4) | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #6453⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #6451⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #6441⋊C3 | C2 | 1 | 4 | yes | maximal |
| 7 | #6433⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #6432⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #90210 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 10 | C22×C4×ΓL(1,𝔽8) | C22 | 1 | 1 | yes | characteristic |
| 11 | C22×ΓL(1,𝔽8)⋊C4 | C22 | 1 | 2 | yes | — |
| 12 | C2×ΓL(1,𝔽8)⋊Q8 | C22 | 1 | 4 | yes | — |
| 13 | C4⋊C4×ΓL(1,𝔽8) | C22 | 1 | 1 | yes | characteristic |
| 14 | C4⋊C4×ΓL(1,𝔽8) | C22 | 1 | 1 | yes | characteristic |
| 15 | C22⋊C4×ΓL(1,𝔽8) | C22 | 1 | 2 | yes | — |
| 16 | C42×ΓL(1,𝔽8) | C22 | 1 | 2 | yes | — |
| 17 | (C22⋊Dic7)⋊C3 | C22 | 1 | 2 | yes | — |
| 18 | C4⋊(ΓL(1,𝔽8)⋊C4) | C22 | 1 | 1 | yes | characteristic |
| 19 | C4⋊(ΓL(1,𝔽8)⋊C4) | C22 | 1 | 2 | yes | — |
| 20 | C4⋊(ΓL(1,𝔽8)⋊C4) | C22 | 1 | 4 | yes | — |
| 21 | C4⋊(ΓL(1,𝔽8)⋊C4) | C22 | 1 | 1 | yes | characteristic |
| 22 | C4⋊(ΓL(1,𝔽8)⋊C4) | C22 | 1 | 2 | yes | — |
| 23 | (ΓL(1,𝔽8)⋊C4)⋊C4 | C22 | 1 | 4 | yes | — |
| 24 | (ΓL(1,𝔽8)⋊C4)⋊C4 | C22 | 1 | 4 | yes | — |
| 25 | C4×ΓL(1,𝔽8)⋊C4 | C22 | 1 | 2 | yes | — |
| 26 | C7×#117 | C2×C3 | 1 | 1 | yes | characteristic |
| 27 | #6489 | C2×C3 | 1 | 1 | yes | characteristic |
| 28 | C2×C4⋊Dic7 | C2×C3 | 1 | 1 | yes | characteristic |
| 29 | #6453 | C2×C3 | 1 | 2 | yes | complemented |
| 30 | #6451 | C2×C3 | 1 | 2 | yes | complemented |
| 31 | #6441 | C2×C3 | 1 | 4 | yes | — |
| 32 | #6433 | C2×C3 | 1 | 2 | yes | complemented |
| 33 | #6432 | C2×C3 | 1 | 2 | yes | complemented |
| 34 | C3×#544 | — | 7 | 1 | no | maximal, Hall |
| 35 | C23×ΓL(1,𝔽8) | C23 | 1 | 1 | yes | characteristic |
| 36 | C2×C4×ΓL(1,𝔽8) | Q8 | 1 | 4 | yes | complemented |
| 37 | C2×C4×ΓL(1,𝔽8) | C23 | 1 | 2 | yes | — |
| 38 | C2×C4×ΓL(1,𝔽8) | C23 | 1 | 2 | yes | — |
| 39 | C2×C4×ΓL(1,𝔽8) | C23 | 1 | 1 | yes | characteristic |
| 40 | C2×C4×ΓL(1,𝔽8) | C23 | 1 | 1 | yes | characteristic |
| 41 | C2×ΓL(1,𝔽8)⋊C4 | — | 4 | 2 | no | — |
| 42 | C2×ΓL(1,𝔽8)⋊C4 | C23 | 1 | 2 | yes | — |
| 43 | C2×ΓL(1,𝔽8)⋊C4 | C23 | 1 | 4 | yes | — |
| 44 | C2×ΓL(1,𝔽8)⋊C4 | C23 | 1 | 2 | yes | — |
| 45 | ΓL(1,𝔽8)⋊Q8 | — | 4 | 4 | no | — |
| 46 | C22×C4×C7 | C22×C3 | 1 | 1 | yes | characteristic |
| 47 | C22×Dic7 | C22×C3 | 1 | 2 | yes | — |
| 48 | C2×Dic14 | C22×C3 | 1 | 4 | yes | — |
| 49 | C7×C4⋊C4 | C22×C3 | 1 | 1 | yes | characteristic |
| 50 | C7×C4⋊C4 | C22×C3 | 1 | 1 | yes | characteristic |
| 51 | C7×C22⋊C4 | C22×C3 | 1 | 2 | yes | — |
| 52 | C42×C7 | C22×C3 | 1 | 2 | yes | — |
| 53 | C22⋊Dic7 | C22×C3 | 1 | 2 | yes | — |
| 54 | C4⋊Dic7 | C22×C3 | 1 | 1 | yes | characteristic |
| 55 | C4⋊Dic7 | C22×C3 | 1 | 2 | yes | — |
| 56 | C4⋊Dic7 | C22×C3 | 1 | 4 | yes | — |
| 57 | C4⋊Dic7 | C22×C3 | 1 | 1 | yes | characteristic |
| 58 | C4⋊Dic7 | C22×C3 | 1 | 2 | yes | — |
| 59 | Dic7⋊C4 | C22×C3 | 1 | 4 | yes | — |
| 60 | Dic7⋊C4 | C22×C3 | 1 | 4 | yes | — |
| 61 | C4×Dic7 | C22×C3 | 1 | 2 | yes | — |
| 62 | C3×C4⋊Q8 | — | 7 | 2 | no | — |
| 63 | C3×C4⋊Q8 | — | 7 | 2 | no | — |
| 64 | C3×#125 | — | 7 | 2 | no | — |
| 65 | C3×#125 | — | 7 | 2 | no | — |
| 66 | C3×C22⋊Q8 | — | 7 | 4 | no | — |
| 67 | C3×#117 | — | 7 | 1 | no | — |
| 68 | C3×#117 | — | 7 | 1 | no | — |
| 69 | C2×C3×C4⋊C4 | — | 7 | 1 | no | — |
| 70 | C22×ΓL(1,𝔽8) | — | 2 | 1 | no | — |
| 71 | C22×ΓL(1,𝔽8) | — | 2 | 1 | no | — |
| 72 | C22×ΓL(1,𝔽8) | C2×Q8 | 1 | 2 | yes | — |
| 73 | C22×ΓL(1,𝔽8) | C24 | 1 | 1 | yes | characteristic |
| 74 | C4×ΓL(1,𝔽8) | — | 2 | 2 | no | — |
| 75 | C4×ΓL(1,𝔽8) | — | 2 | 2 | no | — |
| 76 | C4×ΓL(1,𝔽8) | C2×Q8 | 1 | 2 | yes | — |
| 77 | C4×ΓL(1,𝔽8) | C2×Q8 | 1 | 2 | yes | — |
| 78 | ΓL(1,𝔽8)⋊C4 | — | 2 | 2 | no | — |
| 79 | ΓL(1,𝔽8)⋊C4 | — | 2 | 4 | no | — |
| 80 | ΓL(1,𝔽8)⋊C4 | — | 2 | 2 | no | — |
| 81 | #544 | — | 7 | 1 | no | Sylow |
| 82 | C23×C7 | C23×C3 | 1 | 1 | yes | characteristic |
| 83 | C2×C4×C7 | C3×Q8 | 1 | 4 | yes | complemented |
| 84 | C2×C4×C7 | C23×C3 | 1 | 2 | yes | — |
| 85 | C2×C4×C7 | C23×C3 | 1 | 2 | yes | — |
| 86 | C2×C4×C7 | C23×C3 | 1 | 1 | yes | characteristic |
| 87 | C2×C4×C7 | C23×C3 | 1 | 1 | yes | characteristic |
| 88 | C2×Dic7 | — | 4 | 2 | no | — |
| 89 | C2×Dic7 | C23×C3 | 1 | 2 | yes | — |
| 90 | C2×Dic7 | C23×C3 | 1 | 2 | yes | — |
| 91 | C2×Dic7 | C23×C3 | 1 | 4 | yes | — |
| 92 | Dic14 | — | 4 | 4 | no | — |
| 93 | C2×C3×Q8 | — | 7 | 4 | no | — |
| 94 | C22×C3×C4 | — | 7 | 2 | no | — |
| 95 | C22×C3×C4 | — | 7 | 1 | no | — |
| 96 | C3×C4⋊C4 | — | 7 | 2 | no | — |
| 97 | C3×C4⋊C4 | — | 7 | 2 | no | — |
| 98 | C3×C4⋊C4 | — | 7 | 1 | no | — |
| 99 | C3×C4⋊C4 | — | 7 | 1 | no | — |
| 100 | C3×C4⋊C4 | — | 7 | 4 | no | — |
| 101 | C3×C4⋊C4 | — | 7 | 4 | no | — |
| 102 | C3×C4⋊C4 | — | 7 | 4 | no | — |
| 103 | C3×C4⋊C4 | — | 7 | 1 | no | — |
| 104 | C3×C4⋊C4 | — | 7 | 1 | no | — |
| 105 | C3×C22⋊C4 | — | 7 | 2 | no | — |
| 106 | C3×C22⋊C4 | — | 7 | 2 | no | — |
| 107 | C3×C42 | — | 7 | 2 | no | — |
| 108 | C3×C42 | — | 7 | 2 | no | — |
| 109 | C2×ΓL(1,𝔽8) | — | 2 | 2 | no | — |
| 110 | C2×ΓL(1,𝔽8) | C22×Q8 | 1 | 1 | yes | characteristic |
| 111 | C2×ΓL(1,𝔽8) | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 112 | C2×ΓL(1,𝔽8) | 2−1+4 | 1 | 1 | yes | characteristic |
| 113 | C4⋊Q8 | — | 7 | 2 | no | — |
| 114 | C4⋊Q8 | — | 7 | 2 | no | — |
| 115 | #125 | — | 7 | 2 | no | — |
| 116 | #125 | — | 7 | 2 | no | — |
| 117 | C22⋊Q8 | — | 7 | 4 | no | — |
| 118 | #117 | — | 7 | 1 | no | — |
| 119 | #117 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 120 | C2×C4⋊C4 | — | 7 | 1 | no | — |
| 121 | C22×C7 | — | 2 | 1 | no | — |
| 122 | C22×C7 | — | 2 | 1 | no | — |
| 123 | C22×C7 | C2×C3×Q8 | 1 | 2 | yes | — |
| 124 | C22×C7 | C24×C3 | 1 | 1 | yes | characteristic |
| 125 | C4×C7 | — | 2 | 2 | no | — |
| 126 | C4×C7 | — | 2 | 2 | no | — |
| 127 | C4×C7 | C2×C3×Q8 | 1 | 2 | yes | — |
| 128 | C4×C7 | C2×C3×Q8 | 1 | 2 | yes | — |
| 129 | Dic7 | — | 2 | 2 | no | — |
| 130 | Dic7 | — | 2 | 4 | no | — |
| 131 | Dic7 | — | 2 | 2 | no | — |
| 132 | C23×C3 | — | 7 | 1 | no | — |
| 133 | C3×Q8 | — | 28 | 4 | no | — |
| 134 | C2×C3×C4 | — | 7 | 2 | no | — |
| 135 | C2×C3×C4 | — | 7 | 2 | no | — |
| 136 | C2×C3×C4 | — | 7 | 4 | no | — |
| 137 | C2×C3×C4 | — | 7 | 4 | no | — |
| 138 | C2×C3×C4 | — | 7 | 1 | no | — |
| 139 | C2×C3×C4 | — | 7 | 1 | no | — |
| 140 | C2×C3×C4 | — | 7 | 2 | no | — |
| 141 | C2×C3×C4 | — | 7 | 2 | no | — |
| 142 | C2×C3×C4 | — | 28 | 2 | no | — |
| 143 | ΓL(1,𝔽8) | #544 | 1 | 1 | yes | characteristic, complemented, Hall |
| 144 | C2×Q8 | — | 7 | 4 | no | — |
| 145 | C22×C4 | — | 7 | 2 | no | — |
| 146 | C22×C4 | C2×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 147 | C4⋊C4 | — | 7 | 4 | no | — |
| 148 | C4⋊C4 | — | 7 | 4 | no | — |
| 149 | C4⋊C4 | — | 7 | 2 | no | — |
| 150 | C4⋊C4 | — | 7 | 2 | no | — |
| 151 | C4⋊C4 | — | 7 | 1 | no | — |
| 152 | C4⋊C4 | — | 7 | 1 | no | — |
| 153 | C4⋊C4 | — | 7 | 4 | no | — |
| 154 | C4⋊C4 | C2×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 155 | C4⋊C4 | C2×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 156 | C22⋊C4 | — | 7 | 2 | no | — |
| 157 | C22⋊C4 | C2×AGL(1,𝔽7) | 1 | 2 | yes | — |
| 158 | C42 | — | 7 | 2 | no | — |
| 159 | C42 | C2×AGL(1,𝔽7) | 1 | 2 | yes | — |
| 160 | C2×C7 | — | 2 | 2 | no | — |
| 161 | C2×C7 | C22×C3×Q8 | 1 | 1 | yes | characteristic |
| 162 | C2×C7 | C3×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 163 | C2×C7 | C3×2−1+4 | 1 | 1 | yes | characteristic |
| 164 | C22×C3 | — | 7 | 2 | no | — |
| 165 | C22×C3 | — | 7 | 1 | no | — |
| 166 | C22×C3 | — | 14 | 1 | no | — |
| 167 | C22×C3 | — | 14 | 1 | no | — |
| 168 | C3×C4 | — | 7 | 2 | no | — |
| 169 | C3×C4 | — | 7 | 2 | no | — |
| 170 | C3×C4 | — | 14 | 2 | no | — |
| 171 | C3×C4 | — | 14 | 2 | no | — |
| 172 | C3×C4 | — | 14 | 4 | no | — |
| 173 | C3×C4 | — | 14 | 2 | no | — |
| 174 | C3×C4 | — | 14 | 2 | no | — |
| 175 | C23 | C22×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 176 | Q8 | — | 28 | 4 | no | — |
| 177 | C2×C4 | — | 7 | 2 | no | — |
| 178 | C2×C4 | — | 7 | 2 | no | — |
| 179 | C2×C4 | — | 7 | 4 | no | — |
| 180 | C2×C4 | — | 28 | 2 | no | — |
| 181 | C2×C4 | ΓL(1,𝔽8)⋊Q8 | 1 | 4 | yes | complemented |
| 182 | C2×C4 | C22×AGL(1,𝔽7) | 1 | 2 | yes | — |
| 183 | C2×C4 | C22×AGL(1,𝔽7) | 1 | 2 | yes | — |
| 184 | C2×C4 | C22×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 185 | C2×C4 | C22×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 186 | C7 | C3×#544 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 187 | C2×C3 | — | 7 | 1 | no | — |
| 188 | C2×C3 | — | 7 | 1 | no | — |
| 189 | C2×C3 | — | 7 | 1 | no | — |
| 190 | C2×C3 | — | 14 | 2 | no | — |
| 191 | C22 | — | 2 | 1 | no | — |
| 192 | C22 | — | 2 | 1 | no | — |
| 193 | C22 | C2×ΓL(1,𝔽8)⋊Q8 | 1 | 2 | yes | — |
| 194 | C22 | C23×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 195 | C4 | — | 2 | 2 | no | — |
| 196 | C4 | — | 2 | 2 | no | — |
| 197 | C4 | — | 14 | 4 | no | — |
| 198 | C4 | — | 14 | 2 | no | — |
| 199 | C4 | — | 14 | 2 | no | — |
| 200 | C4 | C2×ΓL(1,𝔽8)⋊Q8 | 1 | 2 | yes | — |
| 201 | C4 | C2×ΓL(1,𝔽8)⋊Q8 | 1 | 2 | yes | — |
| 202 | C3 | — | 7 | 1 | no | Sylow |
| 203 | C2 | — | 2 | 2 | no | — |
| 204 | C2 | C22×ΓL(1,𝔽8)⋊Q8 | 1 | 1 | yes | characteristic |
| 205 | C2 | #10621518 | 1 | 1 | yes | characteristic |
| 206 | C2 | #10621519 | 1 | 1 | yes | characteristic |
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