Finite Group DatabasebetaFinite Group DatabasebetaOrder 2688.
878 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 878 conjugacy classes of subgroups — too many to draw. Only its 36 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×#997 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C7×#5934 | C2 | 1 | 2 | yes | maximal, direct product factor |
| 3 | C2×C7×B(4,𝔽2) | — | 3 | 1 | no | maximal, Hall |
| 4 | C22×C7×S4 | — | 4 | 3 | no | maximal |
| 5 | C7×#997 | C22 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×C7×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 7 | C2×C7×#120 | — | 3 | 3 | no | — |
| 8 | C7×B(4,𝔽2) | — | 3 | 6 | no | — |
| 9 | C7×B(4,𝔽2) | — | 3 | 2 | no | — |
| 10 | C2×C7×#99 | — | 3 | 3 | no | — |
| 11 | C2×#5934 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 12 | C22×C7×A4 | — | 4 | 3 | no | — |
| 13 | C2×C7×S4 | — | 4 | 6 | no | — |
| 14 | C2×C7×S4 | — | 4 | 12 | no | — |
| 15 | C2×C7×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 16 | C25×C7 | — | 3 | 1 | no | — |
| 17 | C7×H(5,𝔽2) | — | 3 | 3 | no | — |
| 18 | C7×H(5,𝔽2) | — | 6 | 1 | no | — |
| 19 | C7×H(5,𝔽2) | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 20 | C2×C7×#41 | — | 6 | 1 | no | — |
| 21 | C22×C7×D4 | — | 3 | 3 | no | — |
| 22 | C22×C7×D4 | — | 6 | 3 | no | — |
| 23 | C7×#120 | — | 3 | 6 | no | — |
| 24 | C7×#120 | — | 3 | 6 | no | — |
| 25 | C7×#120 | — | 6 | 3 | no | — |
| 26 | C7×#120 | — | 6 | 3 | no | — |
| 27 | C2×C7×C22⋊C4 | — | 3 | 3 | no | — |
| 28 | C2×C7×C22⋊C4 | — | 6 | 3 | no | — |
| 29 | C7×#99 | — | 3 | 6 | no | — |
| 30 | C7×#99 | — | 3 | 6 | no | — |
| 31 | C2×#997 | C2×C7 | 1 | 1 | yes | characteristic, complemented |
| 32 | #5934 | C2×C7 | 1 | 2 | yes | direct product factor |
| 33 | C22×D3×C7 | — | 16 | 1 | no | — |
| 34 | C2×C7×A4 | — | 4 | 6 | no | — |
| 35 | C2×C7×A4 | — | 4 | 3 | no | — |
| 36 | C7×S4 | — | 4 | 12 | no | — |
| 37 | C7×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 38 | C2×B(4,𝔽2) | — | 3 | 1 | no | Sylow |
| 39 | C24×C7 | — | 3 | 1 | no | — |
| 40 | C24×C7 | — | 3 | 2 | no | — |
| 41 | C24×C7 | — | 6 | 3 | no | — |
| 42 | C24×C7 | — | 6 | 3 | no | — |
| 43 | C24×C7 | — | 12 | 4 | no | — |
| 44 | C24×C7 | S4 | 1 | 3 | yes | complemented |
| 45 | C7×#41 | — | 6 | 1 | no | — |
| 46 | C7×#41 | — | 6 | 3 | no | — |
| 47 | C7×#41 | — | 6 | 3 | no | — |
| 48 | C7×#41 | — | 6 | 1 | no | — |
| 49 | C2×C7×Q8 | — | 2 | 1 | no | — |
| 50 | C2×C7×D4 | — | 3 | 3 | no | — |
| 51 | C2×C7×D4 | — | 3 | 3 | no | — |
| 52 | C2×C7×D4 | — | 3 | 3 | no | — |
| 53 | C2×C7×D4 | — | 3 | 3 | no | — |
| 54 | C2×C7×D4 | — | 6 | 3 | no | — |
| 55 | C2×C7×D4 | — | 6 | 6 | no | — |
| 56 | C2×C7×D4 | — | 6 | 6 | no | — |
| 57 | C2×C7×D4 | — | 6 | 6 | no | — |
| 58 | C2×C7×D4 | — | 12 | 12 | no | — |
| 59 | C22×C4×C7 | — | 3 | 3 | no | — |
| 60 | C22×C4×C7 | — | 6 | 3 | no | — |
| 61 | C7×C22⋊C4 | — | 3 | 6 | no | — |
| 62 | C7×C22⋊C4 | — | 6 | 6 | no | — |
| 63 | C7×C22⋊C4 | — | 6 | 6 | no | — |
| 64 | C7×C22⋊C4 | — | 6 | 3 | no | — |
| 65 | C22×S4 | — | 4 | 3 | no | — |
| 66 | #997 | C22×C7 | 1 | 1 | yes | characteristic, complemented |
| 67 | C22×C3×C7 | — | 16 | 1 | no | — |
| 68 | C2×D3×C7 | — | 16 | 4 | no | — |
| 69 | C2×D3×C7 | — | 16 | 2 | no | — |
| 70 | C7×A4 | — | 4 | 3 | no | — |
| 71 | C2×H(5,𝔽2) | D3×C7 | 1 | 1 | yes | characteristic, complemented |
| 72 | C2×#120 | — | 3 | 3 | no | — |
| 73 | B(4,𝔽2) | — | 3 | 6 | no | — |
| 74 | B(4,𝔽2) | — | 3 | 2 | no | — |
| 75 | C2×#99 | — | 3 | 3 | no | — |
| 76 | C23×C7 | — | 3 | 1 | no | — |
| 77 | C23×C7 | — | 3 | 3 | no | — |
| 78 | C23×C7 | — | 3 | 3 | no | — |
| 79 | C23×C7 | — | 3 | 3 | no | — |
| 80 | C23×C7 | — | 4 | 6 | no | — |
| 81 | C23×C7 | — | 6 | 6 | no | — |
| 82 | C23×C7 | — | 6 | 12 | no | — |
| 83 | C23×C7 | — | 12 | 3 | no | — |
| 84 | C23×C7 | — | 12 | 2 | no | — |
| 85 | C23×C7 | — | 12 | 4 | no | — |
| 86 | C23×C7 | — | 12 | 6 | no | — |
| 87 | C23×C7 | — | 12 | 12 | no | — |
| 88 | C23×C7 | — | 12 | 1 | no | — |
| 89 | C23×C7 | C2×S4 | 1 | 3 | yes | complemented |
| 90 | C7×Q8 | — | 2 | 1 | no | — |
| 91 | C7×Q8 | — | 6 | 1 | no | — |
| 92 | C7×D4 | — | 6 | 3 | no | — |
| 93 | C7×D4 | — | 6 | 3 | no | — |
| 94 | C7×D4 | — | 6 | 6 | no | — |
| 95 | C7×D4 | — | 12 | 12 | no | — |
| 96 | C7×D4 | — | 12 | 12 | no | — |
| 97 | C2×C4×C7 | — | 3 | 3 | no | — |
| 98 | C2×C4×C7 | — | 3 | 3 | no | — |
| 99 | C2×C4×C7 | — | 6 | 3 | no | — |
| 100 | C2×C4×C7 | — | 6 | 1 | no | — |
| 101 | C2×C4×C7 | — | 6 | 6 | no | — |
| 102 | C2×C4×C7 | — | 12 | 6 | no | — |
| 103 | C22×A4 | — | 4 | 3 | no | — |
| 104 | C2×S4 | — | 4 | 6 | no | — |
| 105 | C2×S4 | — | 4 | 12 | no | — |
| 106 | C2×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 107 | C2×C3×C7 | — | 16 | 2 | no | — |
| 108 | C2×C3×C7 | — | 16 | 1 | no | — |
| 109 | D3×C7 | — | 16 | 4 | no | — |
| 110 | C25 | — | 3 | 1 | no | — |
| 111 | H(5,𝔽2) | — | 3 | 3 | no | — |
| 112 | H(5,𝔽2) | — | 6 | 1 | no | — |
| 113 | H(5,𝔽2) | C2×D3×C7 | 1 | 1 | yes | characteristic, complemented |
| 114 | C2×#41 | — | 6 | 1 | no | — |
| 115 | C22×D4 | — | 3 | 3 | no | — |
| 116 | C22×D4 | — | 6 | 3 | no | — |
| 117 | #120 | — | 3 | 6 | no | — |
| 118 | #120 | — | 3 | 6 | no | — |
| 119 | #120 | — | 6 | 3 | no | — |
| 120 | #120 | — | 6 | 3 | no | — |
| 121 | C2×C22⋊C4 | — | 3 | 3 | no | — |
| 122 | C2×C22⋊C4 | — | 6 | 3 | no | — |
| 123 | #99 | — | 3 | 6 | no | — |
| 124 | #99 | — | 3 | 6 | no | — |
| 125 | C22×C7 | — | 3 | 3 | no | — |
| 126 | C22×C7 | — | 3 | 3 | no | — |
| 127 | C22×C7 | — | 4 | 3 | no | — |
| 128 | C22×C7 | — | 6 | 6 | no | — |
| 129 | C22×C7 | — | 6 | 12 | no | — |
| 130 | C22×C7 | — | 12 | 3 | no | — |
| 131 | C22×C7 | — | 12 | 1 | no | — |
| 132 | C22×C7 | — | 12 | 3 | no | — |
| 133 | C22×C7 | — | 12 | 6 | no | — |
| 134 | C22×C7 | — | 12 | 12 | no | — |
| 135 | C22×C7 | — | 12 | 4 | no | — |
| 136 | C22×C7 | — | 12 | 2 | no | — |
| 137 | C22×C7 | #1020 | 1 | 1 | yes | characteristic |
| 138 | C4×C7 | — | 6 | 1 | no | — |
| 139 | C4×C7 | — | 6 | 1 | no | — |
| 140 | C4×C7 | — | 12 | 6 | no | — |
| 141 | C22×D3 | — | 16 | 1 | no | — |
| 142 | C2×A4 | — | 4 | 6 | no | — |
| 143 | C2×A4 | — | 4 | 3 | no | — |
| 144 | S4 | — | 4 | 12 | no | — |
| 145 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 146 | C3×C7 | — | 16 | 1 | no | Hall |
| 147 | C24 | — | 3 | 1 | no | — |
| 148 | C24 | — | 3 | 2 | no | — |
| 149 | C24 | — | 6 | 3 | no | — |
| 150 | C24 | — | 6 | 3 | no | — |
| 151 | C24 | — | 12 | 4 | no | — |
| 152 | C24 | C7×S4 | 1 | 3 | yes | complemented |
| 153 | #41 | — | 6 | 1 | no | — |
| 154 | #41 | — | 6 | 3 | no | — |
| 155 | #41 | — | 6 | 3 | no | — |
| 156 | #41 | — | 6 | 1 | no | — |
| 157 | C2×Q8 | — | 2 | 1 | no | — |
| 158 | C2×D4 | — | 3 | 3 | no | — |
| 159 | C2×D4 | — | 3 | 3 | no | — |
| 160 | C2×D4 | — | 3 | 3 | no | — |
| 161 | C2×D4 | — | 3 | 3 | no | — |
| 162 | C2×D4 | — | 6 | 3 | no | — |
| 163 | C2×D4 | — | 6 | 6 | no | — |
| 164 | C2×D4 | — | 6 | 6 | no | — |
| 165 | C2×D4 | — | 6 | 6 | no | — |
| 166 | C2×D4 | — | 12 | 12 | no | — |
| 167 | C22×C4 | — | 3 | 3 | no | — |
| 168 | C22×C4 | — | 6 | 3 | no | — |
| 169 | C22⋊C4 | — | 3 | 6 | no | — |
| 170 | C22⋊C4 | — | 6 | 6 | no | — |
| 171 | C22⋊C4 | — | 6 | 6 | no | — |
| 172 | C22⋊C4 | — | 6 | 3 | no | — |
| 173 | C2×C7 | — | 6 | 3 | no | — |
| 174 | C2×C7 | — | 6 | 3 | no | — |
| 175 | C2×C7 | — | 12 | 4 | no | — |
| 176 | C2×C7 | #5934 | 1 | 2 | yes | direct product factor |
| 177 | C2×C7 | C2×#1020 | 1 | 1 | yes | characteristic |
| 178 | C22×C3 | — | 16 | 1 | no | — |
| 179 | C2×D3 | — | 16 | 2 | no | — |
| 180 | C2×D3 | — | 16 | 4 | no | — |
| 181 | A4 | — | 4 | 3 | no | — |
| 182 | C23 | — | 3 | 1 | no | — |
| 183 | C23 | — | 3 | 3 | no | — |
| 184 | C23 | — | 3 | 3 | no | — |
| 185 | C23 | — | 3 | 3 | no | — |
| 186 | C23 | — | 4 | 6 | no | — |
| 187 | C23 | — | 6 | 6 | no | — |
| 188 | C23 | — | 6 | 12 | no | — |
| 189 | C23 | — | 12 | 3 | no | — |
| 190 | C23 | — | 12 | 2 | no | — |
| 191 | C23 | — | 12 | 4 | no | — |
| 192 | C23 | — | 12 | 6 | no | — |
| 193 | C23 | — | 12 | 12 | no | — |
| 194 | C23 | — | 12 | 1 | no | — |
| 195 | C23 | C2×C7×S4 | 1 | 3 | yes | complemented |
| 196 | Q8 | — | 2 | 1 | no | — |
| 197 | Q8 | — | 6 | 1 | no | — |
| 198 | D4 | — | 6 | 3 | no | — |
| 199 | D4 | — | 6 | 3 | no | — |
| 200 | D4 | — | 6 | 6 | no | — |
| 201 | D4 | — | 12 | 12 | no | — |
| 202 | D4 | — | 12 | 12 | no | — |
| 203 | C2×C4 | — | 3 | 3 | no | — |
| 204 | C2×C4 | — | 3 | 3 | no | — |
| 205 | C2×C4 | — | 6 | 3 | no | — |
| 206 | C2×C4 | — | 6 | 1 | no | — |
| 207 | C2×C4 | — | 6 | 6 | no | — |
| 208 | C2×C4 | — | 12 | 6 | no | — |
| 209 | C7 | C2×#5934 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 210 | C2×C3 | — | 16 | 1 | no | — |
| 211 | C2×C3 | — | 16 | 2 | no | — |
| 212 | D3 | — | 16 | 4 | no | — |
| 213 | C22 | — | 3 | 3 | no | — |
| 214 | C22 | — | 3 | 3 | no | — |
| 215 | C22 | — | 4 | 3 | no | — |
| 216 | C22 | — | 6 | 6 | no | — |
| 217 | C22 | — | 6 | 12 | no | — |
| 218 | C22 | — | 12 | 3 | no | — |
| 219 | C22 | — | 12 | 1 | no | — |
| 220 | C22 | — | 12 | 3 | no | — |
| 221 | C22 | — | 12 | 6 | no | — |
| 222 | C22 | — | 12 | 12 | no | — |
| 223 | C22 | — | 12 | 2 | no | — |
| 224 | C22 | — | 12 | 4 | no | — |
| 225 | C22 | C7×#1020 | 1 | 1 | yes | characteristic |
| 226 | C4 | — | 6 | 1 | no | — |
| 227 | C4 | — | 6 | 1 | no | — |
| 228 | C4 | — | 12 | 6 | no | — |
| 229 | C3 | — | 16 | 1 | no | Sylow |
| 230 | C2 | — | 6 | 3 | no | — |
| 231 | C2 | — | 6 | 3 | no | — |
| 232 | C2 | — | 12 | 4 | no | — |
| 233 | C2 | C7×#5934 | 1 | 2 | yes | direct product factor |
| 234 | C2 | C2×C7×#1020 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22×C7 | #1020 |
| commutator, cosocle, hypocenter | #997 | C22×C7 |
| Frattini | C2 | C2×C7×#1020 |
| Fitting | C2×C7×H(5,𝔽2) | D3 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22×C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#997 | C2 | semidirect |
| 2. C7×#5934 | C2 | direct (N × H) |
| 5. C7×#997 | C22 | semidirect |
| 6. C2×C7×H(5,𝔽2) | D3 | semidirect |
| 11. C2×#5934 | C7 | direct (N × H) |
| 19. C7×H(5,𝔽2) | C2×D3 | semidirect |
| 31. C2×#997 | C2×C7 | semidirect |
| 32. #5934 | C2×C7 | direct (N × H) |
| 44. C24×C7 | S4 | semidirect |
| 66. #997 | C22×C7 | semidirect |
| 71. C2×H(5,𝔽2) | D3×C7 | semidirect |
| 89. C23×C7 | C2×S4 | semidirect |
| 113. H(5,𝔽2) | C2×D3×C7 | semidirect |
| 152. C24 | C7×S4 | semidirect |
| 176. C2×C7 | #5934 | direct (N × H) |
| 195. C23 | C2×C7×S4 | semidirect |
| 209. C7 | C2×#5934 | direct (N × H) |
| 233. C2 | C7×#5934 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#997 | 2 |
| C7×#5934 | 2 |
| C2×C7×B(4,𝔽2) | 3 |
| C22×C7×S4 | 4 |
| C2×#5934 | 7 |