Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
326 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 41 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 | A4×Dic14 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C4⋊(C7⋊S4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C4×C7×S4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10621480 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | (C7⋊S4)⋊C4 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #90243 | — | 3 | 1 | no | maximal, Hall |
| 7 | C2×C7⋊S4 | C22 | 1 | 2 | yes | — |
| 8 | C2×C7×S4 | C22 | 1 | 1 | yes | characteristic |
| 9 | C4×C7×A4 | C22 | 1 | 1 | yes | characteristic |
| 10 | #66673 | — | 4 | 1 | no | maximal |
| 11 | A4×Dic7 | C22 | 1 | 2 | yes | — |
| 12 | C7×SL(2,ℤ/4) | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#1157 | — | 3 | 1 | no | — |
| 14 | C22×Dic14 | D3 | 1 | 1 | yes | characteristic, complemented |
| 15 | C4×C7×D4 | — | 3 | 1 | no | — |
| 16 | #6502 | — | 3 | 2 | no | — |
| 17 | #6493 | — | 3 | 1 | no | — |
| 18 | #6487 | — | 3 | 1 | no | — |
| 19 | #6459 | — | 3 | 2 | no | — |
| 20 | #6448 | — | 3 | 2 | no | — |
| 21 | (C22⋊D7)⋊C4 | — | 3 | 2 | no | — |
| 22 | #6438 | — | 3 | 1 | no | — |
| 23 | C4×Dic14 | — | 3 | 1 | no | — |
| 24 | #5919 | — | 7 | 1 | no | maximal, Hall |
| 25 | C2×C7×A4 | C23 | 1 | 1 | yes | characteristic |
| 26 | C7⋊S4 | — | 2 | 2 | no | — |
| 27 | C7×S4 | — | 2 | 1 | no | — |
| 28 | D84 | — | 4 | 1 | no | — |
| 29 | C4×D3×C7 | — | 4 | 1 | no | — |
| 30 | C3×Dic14 | — | 4 | 1 | no | — |
| 31 | #4247 | — | 4 | 2 | no | — |
| 32 | D21⋊C4 | — | 4 | 2 | no | — |
| 33 | C2×C7×D4 | — | 3 | 1 | no | — |
| 34 | C22×C4×C7 | — | 3 | 1 | no | — |
| 35 | C22×C4×C7 | C2×D3 | 1 | 1 | yes | characteristic |
| 36 | C2×C22⋊D7 | — | 3 | 2 | no | — |
| 37 | C2×C22⋊D7 | — | 3 | 2 | no | — |
| 38 | C22×Dic7 | C2×D3 | 1 | 2 | yes | — |
| 39 | #1157 | — | 6 | 1 | no | — |
| 40 | #1157 | — | 6 | 2 | no | — |
| 41 | #1157 | — | 6 | 1 | no | — |
| 42 | C2×D28 | — | 3 | 1 | no | — |
| 43 | C2×C4×D7 | — | 3 | 2 | no | — |
| 44 | C2×Dic14 | — | 3 | 1 | no | — |
| 45 | C2×Dic14 | — | 3 | 1 | no | — |
| 46 | C2×Dic14 | — | 3 | 2 | no | — |
| 47 | C7×C4⋊C4 | — | 3 | 1 | no | — |
| 48 | C7×C22⋊C4 | — | 3 | 1 | no | — |
| 49 | C7×C22⋊C4 | — | 3 | 1 | no | — |
| 50 | C42×C7 | — | 3 | 1 | no | — |
| 51 | C22⋊Dic7 | — | 3 | 2 | no | — |
| 52 | (C2×D7)⋊C4 | — | 3 | 2 | no | — |
| 53 | (C2×D7)⋊C4 | — | 3 | 2 | no | — |
| 54 | (C2×D7)⋊C4 | — | 3 | 2 | no | — |
| 55 | C4⋊Dic7 | — | 3 | 1 | no | — |
| 56 | Dic7⋊C4 | — | 3 | 2 | no | — |
| 57 | Dic7⋊C4 | — | 3 | 2 | no | — |
| 58 | C4×Dic7 | — | 3 | 2 | no | — |
| 59 | Q8×A4 | — | 7 | 1 | no | — |
| 60 | C4⋊S4 | — | 7 | 2 | no | — |
| 61 | C4⋊S4 | — | 7 | 1 | no | — |
| 62 | C4×S4 | — | 7 | 2 | no | — |
| 63 | C4×S4 | D7 | 1 | 1 | yes | characteristic, complemented |
| 64 | C2×D21 | — | 4 | 2 | no | — |
| 65 | C2×D3×C7 | — | 4 | 1 | no | — |
| 66 | C7×A4 | #41 | 1 | 1 | yes | characteristic, complemented |
| 67 | C3×C4×C7 | — | 4 | 1 | no | — |
| 68 | C3×Dic7 | — | 4 | 2 | no | — |
| 69 | C7×Dic3 | — | 4 | 1 | no | — |
| 70 | #548 | — | 21 | 1 | no | Sylow |
| 71 | C23×C7 | — | 3 | 1 | no | — |
| 72 | C23×C7 | C22×D3 | 1 | 1 | yes | characteristic |
| 73 | C22×D7 | — | 3 | 2 | no | — |
| 74 | C7×D4 | — | 6 | 1 | no | — |
| 75 | C7×D4 | — | 6 | 1 | no | — |
| 76 | C2×C4×C7 | — | 3 | 1 | no | — |
| 77 | C2×C4×C7 | — | 3 | 1 | no | — |
| 78 | C2×C4×C7 | — | 3 | 1 | no | — |
| 79 | C2×C4×C7 | — | 3 | 1 | no | — |
| 80 | C2×C4×C7 | — | 3 | 1 | no | — |
| 81 | C2×C4×C7 | — | 6 | 1 | no | — |
| 82 | C2×C4×C7 | — | 6 | 1 | no | — |
| 83 | C22⋊D7 | — | 6 | 2 | no | — |
| 84 | C22⋊D7 | — | 6 | 2 | no | — |
| 85 | C22⋊D7 | — | 6 | 2 | no | — |
| 86 | C22⋊D7 | — | 6 | 2 | no | — |
| 87 | C2×Dic7 | — | 3 | 2 | no | — |
| 88 | C2×Dic7 | — | 3 | 2 | no | — |
| 89 | C2×Dic7 | — | 3 | 2 | no | — |
| 90 | D28 | — | 6 | 1 | no | — |
| 91 | D28 | — | 6 | 1 | no | — |
| 92 | C4×D7 | — | 6 | 2 | no | — |
| 93 | C4×D7 | — | 6 | 2 | no | — |
| 94 | Dic14 | — | 3 | 1 | no | — |
| 95 | Dic14 | — | 3 | 2 | no | — |
| 96 | Dic14 | — | 6 | 1 | no | — |
| 97 | Dic14 | S4 | 1 | 1 | yes | characteristic, complemented |
| 98 | C2×S4 | — | 7 | 2 | no | — |
| 99 | C2×S4 | C2×D7 | 1 | 1 | yes | characteristic |
| 100 | #239 | — | 28 | 1 | no | — |
| 101 | C4×A4 | — | 7 | 2 | no | — |
| 102 | C4×A4 | C2×D7 | 1 | 1 | yes | characteristic |
| 103 | SL(2,ℤ/4) | C2×D7 | 1 | 1 | yes | characteristic |
| 104 | C2×C3×C7 | — | 4 | 1 | no | — |
| 105 | D21 | — | 8 | 2 | no | — |
| 106 | D3×C7 | — | 8 | 1 | no | — |
| 107 | C2×#41 | — | 21 | 1 | no | — |
| 108 | C22×Q8 | — | 7 | 1 | no | — |
| 109 | #124 | — | 21 | 1 | no | — |
| 110 | #124 | — | 21 | 2 | no | — |
| 111 | C22⋊Q8 | — | 21 | 2 | no | — |
| 112 | C22⋊Q8 | — | 21 | 1 | no | — |
| 113 | #121 | — | 21 | 2 | no | — |
| 114 | #121 | — | 21 | 1 | no | — |
| 115 | C4×Q8 | — | 21 | 1 | no | — |
| 116 | C4×D4 | — | 3 | 1 | no | — |
| 117 | C4×D4 | — | 21 | 2 | no | — |
| 118 | C22×C7 | — | 3 | 1 | no | — |
| 119 | C22×C7 | — | 3 | 1 | no | — |
| 120 | C22×C7 | — | 6 | 1 | no | — |
| 121 | C22×C7 | — | 6 | 1 | no | — |
| 122 | C22×C7 | — | 6 | 1 | no | — |
| 123 | C22×C7 | #239 | 1 | 1 | yes | characteristic, complemented |
| 124 | C2×D7 | — | 6 | 2 | no | — |
| 125 | C2×D7 | — | 6 | 2 | no | — |
| 126 | C2×D7 | — | 6 | 2 | no | — |
| 127 | C4×C7 | — | 3 | 1 | no | — |
| 128 | C4×C7 | — | 6 | 1 | no | — |
| 129 | C4×C7 | — | 6 | 1 | no | — |
| 130 | C4×C7 | — | 6 | 1 | no | — |
| 131 | C4×C7 | C2×S4 | 1 | 1 | yes | characteristic |
| 132 | Dic7 | — | 3 | 2 | no | — |
| 133 | Dic7 | — | 6 | 2 | no | — |
| 134 | Dic7 | C2×S4 | 1 | 2 | yes | — |
| 135 | C2×A4 | C22×D7 | 1 | 1 | yes | characteristic |
| 136 | S4 | — | 2 | 1 | no | — |
| 137 | S4 | — | 14 | 2 | no | — |
| 138 | C3×Q8 | — | 28 | 1 | no | — |
| 139 | D12 | — | 28 | 1 | no | — |
| 140 | D12 | — | 28 | 2 | no | — |
| 141 | C4×D3 | — | 4 | 1 | no | — |
| 142 | C4×D3 | — | 28 | 2 | no | — |
| 143 | C3×C7 | — | 4 | 1 | no | Hall |
| 144 | #41 | — | 42 | 1 | no | — |
| 145 | #41 | — | 42 | 2 | no | — |
| 146 | #41 | — | 42 | 1 | no | — |
| 147 | C2×Q8 | — | 21 | 1 | no | — |
| 148 | C2×Q8 | — | 21 | 1 | no | — |
| 149 | C2×Q8 | — | 21 | 2 | no | — |
| 150 | C2×D4 | — | 3 | 1 | no | — |
| 151 | C2×D4 | — | 21 | 1 | no | — |
| 152 | C2×D4 | — | 21 | 2 | no | — |
| 153 | C2×D4 | — | 21 | 2 | no | — |
| 154 | C22×C4 | — | 3 | 1 | no | — |
| 155 | C22×C4 | — | 7 | 2 | no | — |
| 156 | C22×C4 | — | 21 | 2 | no | — |
| 157 | C22×C4 | D3×D7 | 1 | 1 | yes | characteristic |
| 158 | C4⋊C4 | — | 3 | 1 | no | — |
| 159 | C4⋊C4 | — | 21 | 2 | no | — |
| 160 | C4⋊C4 | — | 21 | 2 | no | — |
| 161 | C4⋊C4 | — | 21 | 1 | no | — |
| 162 | C22⋊C4 | — | 3 | 1 | no | — |
| 163 | C22⋊C4 | — | 3 | 1 | no | — |
| 164 | C22⋊C4 | — | 21 | 2 | no | — |
| 165 | C22⋊C4 | — | 21 | 2 | no | — |
| 166 | C22⋊C4 | — | 21 | 2 | no | — |
| 167 | C22⋊C4 | — | 21 | 2 | no | — |
| 168 | C42 | — | 3 | 1 | no | — |
| 169 | C42 | — | 21 | 2 | no | — |
| 170 | C2×C7 | — | 3 | 1 | no | — |
| 171 | C2×C7 | — | 3 | 1 | no | — |
| 172 | C2×C7 | — | 12 | 1 | no | — |
| 173 | C2×C7 | C22×S4 | 1 | 1 | yes | characteristic |
| 174 | D7 | — | 12 | 2 | no | — |
| 175 | C2×D3 | — | 4 | 1 | no | — |
| 176 | C2×D3 | — | 28 | 2 | no | — |
| 177 | A4 | #1157 | 1 | 1 | yes | characteristic, complemented |
| 178 | C3×C4 | — | 4 | 1 | no | — |
| 179 | C3×C4 | — | 28 | 2 | no | — |
| 180 | Dic3 | — | 4 | 1 | no | — |
| 181 | C23 | — | 3 | 1 | no | — |
| 182 | C23 | — | 21 | 2 | no | — |
| 183 | C23 | C2×D3×D7 | 1 | 1 | yes | characteristic |
| 184 | Q8 | — | 7 | 1 | no | — |
| 185 | Q8 | — | 21 | 2 | no | — |
| 186 | Q8 | — | 21 | 1 | no | — |
| 187 | Q8 | — | 42 | 1 | no | — |
| 188 | D4 | — | 6 | 1 | no | — |
| 189 | D4 | — | 6 | 1 | no | — |
| 190 | D4 | — | 42 | 1 | no | — |
| 191 | D4 | — | 42 | 2 | no | — |
| 192 | D4 | — | 42 | 2 | no | — |
| 193 | D4 | — | 42 | 2 | no | — |
| 194 | D4 | — | 42 | 2 | no | — |
| 195 | D4 | — | 42 | 1 | no | — |
| 196 | C2×C4 | — | 3 | 1 | no | — |
| 197 | C2×C4 | — | 3 | 1 | no | — |
| 198 | C2×C4 | — | 3 | 1 | no | — |
| 199 | C2×C4 | — | 3 | 1 | no | — |
| 200 | C2×C4 | — | 3 | 1 | no | — |
| 201 | C2×C4 | — | 6 | 1 | no | — |
| 202 | C2×C4 | — | 6 | 1 | no | — |
| 203 | C2×C4 | — | 21 | 2 | no | — |
| 204 | C2×C4 | — | 21 | 2 | no | — |
| 205 | C2×C4 | — | 21 | 2 | no | — |
| 206 | C2×C4 | — | 42 | 2 | no | — |
| 207 | C2×C4 | — | 42 | 2 | no | — |
| 208 | C7 | #5919 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 209 | C2×C3 | — | 4 | 1 | no | — |
| 210 | D3 | — | 8 | 1 | no | — |
| 211 | D3 | — | 56 | 2 | no | — |
| 212 | C22 | — | 3 | 1 | no | — |
| 213 | C22 | — | 3 | 1 | no | — |
| 214 | C22 | — | 6 | 1 | no | — |
| 215 | C22 | — | 6 | 1 | no | — |
| 216 | C22 | — | 6 | 1 | no | — |
| 217 | C22 | — | 42 | 2 | no | — |
| 218 | C22 | — | 42 | 2 | no | — |
| 219 | C22 | — | 42 | 2 | no | — |
| 220 | C22 | #66673 | 1 | 1 | yes | characteristic, complemented |
| 221 | C4 | — | 3 | 1 | no | — |
| 222 | C4 | — | 6 | 1 | no | — |
| 223 | C4 | — | 6 | 1 | no | — |
| 224 | C4 | — | 6 | 1 | no | — |
| 225 | C4 | — | 7 | 2 | no | — |
| 226 | C4 | — | 21 | 2 | no | — |
| 227 | C4 | — | 42 | 2 | no | — |
| 228 | C4 | D7×S4 | 1 | 1 | yes | characteristic |
| 229 | C3 | — | 4 | 1 | no | Sylow |
| 230 | C2 | — | 3 | 1 | no | — |
| 231 | C2 | — | 3 | 1 | no | — |
| 232 | C2 | — | 12 | 1 | no | — |
| 233 | C2 | — | 84 | 2 | no | — |
| 234 | C2 | C2×D7×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D7×S4 |
| commutator, cosocle | C2×C7×A4 | C23 |
| Fitting | C22×C4×C7 | C2×D3 |
| socle | C23×C7 | C22×D3 |
| hypercenter | C4 | D7×S4 |
| hypocenter | C7×A4 | #41 |
| LCS | G ▷ C2×C7×A4 ▷ C7×A4 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A4×Dic14 | C2 | semidirect |
| 2. C4⋊(C7⋊S4) | C2 | semidirect |
| 3. C4×C7×S4 | C2 | semidirect |
| 4. #10621480 | C2 | semidirect |
| 5. (C7⋊S4)⋊C4 | C2 | semidirect |
| 14. C22×Dic14 | D3 | semidirect |
| 63. C4×S4 | D7 | semidirect |
| 66. C7×A4 | #41 | semidirect |
| 97. Dic14 | S4 | semidirect |
| 123. C22×C7 | #239 | semidirect |
| 177. A4 | #1157 | semidirect |
| 208. C7 | #5919 | semidirect |
| 220. C22 | #66673 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×Dic14 | 2 |
| C4⋊(C7⋊S4) | 2 |
| C4×C7×S4 | 2 |
| #10621480 | 2 |
| (C7⋊S4)⋊C4 | 2 |
| #90243 | 3 |
| #66673 | 4 |
| #5919 | 7 |