Finite Group DatabasebetaFinite Group DatabasebetaOrder 2688.
19848 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 19848 conjugacy classes of subgroups — too many to draw. Only its 163 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 | C22×C22⋊(C22⋊ΓL(1,𝔽8)) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×D7⋊(C22⋊A4) | C2 | 1 | 6 | yes | maximal, direct product factor |
| 3 | C26×D7 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 4 | C2×C22⋊(C22⋊ΓL(1,𝔽8)) | C22 | 1 | 3 | yes | complemented |
| 5 | D7⋊(C22⋊A4) | C22 | 1 | 4 | yes | direct product factor |
| 6 | C22×D7⋊A4 | — | 4 | 5 | no | maximal |
| 7 | C26×C7 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 8 | C25×D7 | — | 3 | 5 | no | — |
| 9 | C25×D7 | — | 3 | 5 | no | — |
| 10 | C25×D7 | — | 3 | 30 | no | — |
| 11 | C25×D7 | C2×C3 | 1 | 6 | yes | complemented |
| 12 | C23×C22⋊A4 | — | 7 | 1 | no | maximal, Hall |
| 13 | C22⋊(C22⋊ΓL(1,𝔽8)) | C23 | 1 | 1 | yes | characteristic, complemented |
| 14 | C22×C22⋊ΓL(1,𝔽8) | — | 4 | 5 | no | — |
| 15 | C2×D7⋊A4 | — | 4 | 30 | no | — |
| 16 | C25×C7 | — | 3 | 15 | no | — |
| 17 | C25×C7 | — | 3 | 5 | no | — |
| 18 | C25×C7 | C22×C3 | 1 | 3 | yes | complemented |
| 19 | C24×D7 | — | 3 | 180 | no | — |
| 20 | C24×D7 | — | 3 | 240 | no | — |
| 21 | C24×D7 | — | 3 | 180 | no | — |
| 22 | C24×D7 | — | 3 | 30 | no | — |
| 23 | C24×D7 | — | 3 | 30 | no | — |
| 24 | C24×D7 | — | 3 | 40 | no | — |
| 25 | C24×D7 | — | 3 | 10 | no | — |
| 26 | C24×D7 | — | 3 | 5 | no | — |
| 27 | C24×D7 | — | 3 | 30 | no | — |
| 28 | C24×D7 | — | 3 | 30 | no | — |
| 29 | C24×D7 | — | 3 | 30 | no | — |
| 30 | C24×D7 | — | 3 | 60 | no | — |
| 31 | C24×D7 | A4 | 1 | 5 | yes | complemented |
| 32 | C24×D7 | C22×C3 | 1 | 4 | yes | complemented |
| 33 | C22×C22⋊A4 | — | 7 | 6 | no | — |
| 34 | C22×C22⋊A4 | — | 7 | 1 | no | — |
| 35 | C2×C22⋊ΓL(1,𝔽8) | — | 4 | 15 | no | — |
| 36 | D7⋊A4 | — | 4 | 20 | no | — |
| 37 | C22×AGL(1,𝔽7) | — | 16 | 1 | no | — |
| 38 | C27 | — | 7 | 1 | no | Sylow |
| 39 | C24×C7 | — | 3 | 90 | no | — |
| 40 | C24×C7 | — | 3 | 60 | no | — |
| 41 | C24×C7 | — | 3 | 10 | no | — |
| 42 | C24×C7 | — | 3 | 10 | no | — |
| 43 | C24×C7 | — | 3 | 15 | no | — |
| 44 | C24×C7 | — | 3 | 15 | no | — |
| 45 | C24×C7 | — | 3 | 15 | no | — |
| 46 | C24×C7 | C2×A4 | 1 | 5 | yes | complemented |
| 47 | C24×C7 | C23×C3 | 1 | 1 | yes | characteristic, complemented |
| 48 | C23×D7 | — | 3 | 90 | no | — |
| 49 | C23×D7 | — | 3 | 360 | no | — |
| 50 | C23×D7 | — | 3 | 480 | no | — |
| 51 | C23×D7 | — | 3 | 360 | no | — |
| 52 | C23×D7 | — | 3 | 480 | no | — |
| 53 | C23×D7 | — | 3 | 360 | no | — |
| 54 | C23×D7 | — | 3 | 360 | no | — |
| 55 | C23×D7 | — | 3 | 180 | no | — |
| 56 | C23×D7 | — | 3 | 360 | no | — |
| 57 | C23×D7 | — | 3 | 180 | no | — |
| 58 | C23×D7 | — | 3 | 60 | no | — |
| 59 | C23×D7 | — | 3 | 30 | no | — |
| 60 | C23×D7 | — | 3 | 180 | no | — |
| 61 | C23×D7 | — | 3 | 60 | no | — |
| 62 | C23×D7 | — | 3 | 30 | no | — |
| 63 | C23×D7 | — | 3 | 60 | no | — |
| 64 | C23×D7 | — | 3 | 20 | no | — |
| 65 | C23×D7 | — | 3 | 5 | no | — |
| 66 | C23×D7 | — | 3 | 5 | no | — |
| 67 | C23×D7 | — | 3 | 30 | no | — |
| 68 | C23×D7 | — | 3 | 20 | no | — |
| 69 | C23×D7 | C2×A4 | 1 | 30 | yes | complemented |
| 70 | C2×C22⋊A4 | — | 7 | 4 | no | — |
| 71 | C2×C22⋊A4 | — | 7 | 3 | no | — |
| 72 | C23×A4 | — | 28 | 5 | no | — |
| 73 | C22⋊ΓL(1,𝔽8) | — | 4 | 5 | no | — |
| 74 | C22×ΓL(1,𝔽8) | — | 16 | 1 | no | — |
| 75 | C2×AGL(1,𝔽7) | — | 16 | 6 | no | — |
| 76 | C26 | — | 7 | 6 | no | — |
| 77 | C26 | — | 21 | 5 | no | — |
| 78 | C26 | — | 21 | 30 | no | — |
| 79 | C26 | — | 21 | 5 | no | — |
| 80 | C26 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic, complemented |
| 81 | C23×C7 | — | 3 | 15 | no | — |
| 82 | C23×C7 | — | 3 | 90 | no | — |
| 83 | C23×C7 | — | 3 | 90 | no | — |
| 84 | C23×C7 | — | 3 | 120 | no | — |
| 85 | C23×C7 | — | 3 | 90 | no | — |
| 86 | C23×C7 | — | 3 | 30 | no | — |
| 87 | C23×C7 | — | 3 | 15 | no | — |
| 88 | C23×C7 | — | 3 | 5 | no | — |
| 89 | C23×C7 | — | 3 | 5 | no | — |
| 90 | C23×C7 | C22×A4 | 1 | 15 | yes | complemented |
| 91 | C22×D7 | — | 3 | 180 | no | — |
| 92 | C22×D7 | — | 3 | 120 | no | — |
| 93 | C22×D7 | — | 3 | 720 | no | — |
| 94 | C22×D7 | — | 3 | 720 | no | — |
| 95 | C22×D7 | — | 3 | 360 | no | — |
| 96 | C22×D7 | — | 3 | 120 | no | — |
| 97 | C22×D7 | — | 3 | 180 | no | — |
| 98 | C22×D7 | — | 3 | 30 | no | — |
| 99 | C22×D7 | — | 3 | 180 | no | — |
| 100 | C22×D7 | — | 3 | 240 | no | — |
| 101 | C22×D7 | — | 3 | 40 | no | — |
| 102 | C22×D7 | — | 3 | 360 | no | — |
| 103 | C22×D7 | — | 3 | 40 | no | — |
| 104 | C22×D7 | — | 3 | 20 | no | — |
| 105 | C22×D7 | — | 3 | 5 | no | — |
| 106 | C22×D7 | — | 3 | 30 | no | — |
| 107 | C22×D7 | — | 3 | 30 | no | — |
| 108 | C22×D7 | — | 3 | 30 | no | — |
| 109 | C22×D7 | — | 3 | 60 | no | — |
| 110 | C22×D7 | C22×A4 | 1 | 20 | yes | complemented |
| 111 | C22×D7 | C22⋊A4 | 1 | 1 | yes | characteristic, complemented |
| 112 | C22⋊A4 | — | 7 | 1 | no | — |
| 113 | C22×A4 | — | 28 | 30 | no | — |
| 114 | C22×A4 | — | 28 | 5 | no | — |
| 115 | C2×ΓL(1,𝔽8) | — | 16 | 3 | no | — |
| 116 | AGL(1,𝔽7) | — | 16 | 4 | no | — |
| 117 | C25 | — | 3 | 15 | no | — |
| 118 | C25 | — | 3 | 5 | no | — |
| 119 | C25 | — | 7 | 5 | no | — |
| 120 | C25 | — | 7 | 4 | no | — |
| 121 | C25 | — | 21 | 30 | no | — |
| 122 | C25 | — | 21 | 30 | no | — |
| 123 | C25 | — | 21 | 60 | no | — |
| 124 | C25 | — | 21 | 30 | no | — |
| 125 | C25 | — | 21 | 10 | no | — |
| 126 | C25 | — | 21 | 40 | no | — |
| 127 | C25 | — | 21 | 180 | no | — |
| 128 | C25 | — | 21 | 30 | no | — |
| 129 | C25 | — | 21 | 180 | no | — |
| 130 | C25 | — | 21 | 240 | no | — |
| 131 | C25 | — | 21 | 30 | no | — |
| 132 | C25 | — | 21 | 5 | no | — |
| 133 | C25 | C2×AGL(1,𝔽7) | 1 | 3 | yes | complemented |
| 134 | C22×C7 | — | 3 | 15 | no | — |
| 135 | C22×C7 | — | 3 | 10 | no | — |
| 136 | C22×C7 | — | 3 | 60 | no | — |
| 137 | C22×C7 | — | 3 | 90 | no | — |
| 138 | C22×C7 | — | 3 | 10 | no | — |
| 139 | C22×C7 | — | 3 | 15 | no | — |
| 140 | C22×C7 | — | 3 | 15 | no | — |
| 141 | C22×C7 | C23×A4 | 1 | 5 | yes | complemented |
| 142 | C22×C7 | C2×C22⋊A4 | 1 | 1 | yes | characteristic, complemented |
| 143 | C2×D7 | — | 3 | 360 | no | — |
| 144 | C2×D7 | — | 3 | 60 | no | — |
| 145 | C2×D7 | — | 3 | 120 | no | — |
| 146 | C2×D7 | — | 3 | 30 | no | — |
| 147 | C2×D7 | — | 3 | 60 | no | — |
| 148 | C2×D7 | — | 3 | 20 | no | — |
| 149 | C2×D7 | — | 3 | 20 | no | — |
| 150 | C2×D7 | C2×C22⋊A4 | 1 | 6 | yes | complemented |
| 151 | C23×C3 | — | 112 | 1 | no | — |
| 152 | C2×A4 | — | 28 | 20 | no | — |
| 153 | C2×A4 | — | 28 | 15 | no | — |
| 154 | ΓL(1,𝔽8) | — | 16 | 1 | no | Hall |
| 155 | C24 | — | 3 | 15 | no | — |
| 156 | C24 | — | 3 | 90 | no | — |
| 157 | C24 | — | 3 | 60 | no | — |
| 158 | C24 | — | 3 | 10 | no | — |
| 159 | C24 | — | 3 | 10 | no | — |
| 160 | C24 | — | 3 | 15 | no | — |
| 161 | C24 | — | 3 | 15 | no | — |
| 162 | C24 | — | 7 | 30 | no | — |
| 163 | C24 | — | 21 | 20 | no | — |
| 164 | C24 | — | 21 | 5 | no | — |
| 165 | C24 | — | 21 | 60 | no | — |
| 166 | C24 | — | 21 | 30 | no | — |
| 167 | C24 | — | 21 | 20 | no | — |
| 168 | C24 | — | 21 | 5 | no | — |
| 169 | C24 | — | 21 | 90 | no | — |
| 170 | C24 | — | 21 | 60 | no | — |
| 171 | C24 | — | 21 | 180 | no | — |
| 172 | C24 | — | 21 | 360 | no | — |
| 173 | C24 | — | 21 | 180 | no | — |
| 174 | C24 | — | 21 | 360 | no | — |
| 175 | C24 | — | 21 | 180 | no | — |
| 176 | C24 | — | 21 | 360 | no | — |
| 177 | C24 | — | 21 | 360 | no | — |
| 178 | C24 | — | 21 | 480 | no | — |
| 179 | C24 | — | 21 | 480 | no | — |
| 180 | C24 | — | 21 | 360 | no | — |
| 181 | C24 | — | 21 | 60 | no | — |
| 182 | C24 | — | 21 | 30 | no | — |
| 183 | C24 | — | 21 | 30 | no | — |
| 184 | C24 | C22×AGL(1,𝔽7) | 1 | 1 | yes | characteristic, complemented |
| 185 | C24 | D7⋊A4 | 1 | 5 | yes | complemented |
| 186 | C2×C7 | — | 3 | 15 | no | — |
| 187 | C2×C7 | — | 3 | 5 | no | — |
| 188 | C2×C7 | C22×C22⋊A4 | 1 | 3 | yes | complemented |
| 189 | D7 | — | 3 | 20 | no | — |
| 190 | D7 | C22×C22⋊A4 | 1 | 4 | yes | complemented |
| 191 | C22×C3 | — | 112 | 6 | no | — |
| 192 | C22×C3 | — | 112 | 1 | no | — |
| 193 | A4 | — | 28 | 5 | no | — |
| 194 | C23 | — | 3 | 15 | no | — |
| 195 | C23 | — | 3 | 90 | no | — |
| 196 | C23 | — | 3 | 90 | no | — |
| 197 | C23 | — | 3 | 120 | no | — |
| 198 | C23 | — | 3 | 30 | no | — |
| 199 | C23 | — | 3 | 90 | no | — |
| 200 | C23 | — | 3 | 15 | no | — |
| 201 | C23 | — | 3 | 5 | no | — |
| 202 | C23 | — | 3 | 5 | no | — |
| 203 | C23 | — | 7 | 1 | no | — |
| 204 | C23 | — | 7 | 20 | no | — |
| 205 | C23 | — | 21 | 30 | no | — |
| 206 | C23 | — | 21 | 30 | no | — |
| 207 | C23 | — | 21 | 60 | no | — |
| 208 | C23 | — | 21 | 30 | no | — |
| 209 | C23 | — | 21 | 180 | no | — |
| 210 | C23 | — | 21 | 40 | no | — |
| 211 | C23 | — | 21 | 360 | no | — |
| 212 | C23 | — | 21 | 40 | no | — |
| 213 | C23 | — | 21 | 240 | no | — |
| 214 | C23 | — | 21 | 120 | no | — |
| 215 | C23 | — | 21 | 360 | no | — |
| 216 | C23 | — | 21 | 720 | no | — |
| 217 | C23 | — | 21 | 120 | no | — |
| 218 | C23 | — | 21 | 720 | no | — |
| 219 | C23 | — | 21 | 180 | no | — |
| 220 | C23 | — | 21 | 30 | no | — |
| 221 | C23 | — | 21 | 180 | no | — |
| 222 | C23 | — | 21 | 20 | no | — |
| 223 | C23 | — | 21 | 5 | no | — |
| 224 | C23 | C2×D7⋊A4 | 1 | 15 | yes | complemented |
| 225 | C7 | C23×C22⋊A4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 226 | C2×C3 | — | 112 | 4 | no | — |
| 227 | C2×C3 | — | 112 | 3 | no | — |
| 228 | C22 | — | 3 | 15 | no | — |
| 229 | C22 | — | 3 | 10 | no | — |
| 230 | C22 | — | 3 | 90 | no | — |
| 231 | C22 | — | 3 | 10 | no | — |
| 232 | C22 | — | 3 | 60 | no | — |
| 233 | C22 | — | 3 | 15 | no | — |
| 234 | C22 | — | 3 | 15 | no | — |
| 235 | C22 | — | 7 | 6 | no | — |
| 236 | C22 | — | 21 | 20 | no | — |
| 237 | C22 | — | 21 | 60 | no | — |
| 238 | C22 | — | 21 | 20 | no | — |
| 239 | C22 | — | 21 | 120 | no | — |
| 240 | C22 | — | 21 | 60 | no | — |
| 241 | C22 | — | 21 | 360 | no | — |
| 242 | C22 | — | 21 | 30 | no | — |
| 243 | C22 | C22×D7⋊A4 | 1 | 5 | yes | complemented |
| 244 | C22 | D7⋊(C22⋊A4) | 1 | 1 | yes | characteristic, direct product factor |
| 245 | C3 | — | 112 | 1 | no | Sylow |
| 246 | C2 | — | 3 | 5 | no | — |
| 247 | C2 | — | 3 | 15 | no | — |
| 248 | C2 | — | 7 | 4 | no | — |
| 249 | C2 | — | 21 | 20 | no | — |
| 250 | C2 | C2×D7⋊(C22⋊A4) | 1 | 3 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D7⋊(C22⋊A4) |
| commutator, cosocle, hypocenter | C24×C7 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C26×C7 | C2×C3 |
| LCS | G ▷ C24×C7 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C22⋊(C22⋊ΓL(1,𝔽8)) | C2 | semidirect |
| 2. C2×D7⋊(C22⋊A4) | C2 | direct (N × H) |
| 3. C26×D7 | C3 | semidirect |
| 4. C2×C22⋊(C22⋊ΓL(1,𝔽8)) | C22 | semidirect |
| 5. D7⋊(C22⋊A4) | C22 | direct (N × H) |
| 7. C26×C7 | C2×C3 | semidirect |
| 11. C25×D7 | C2×C3 | semidirect |
| 13. C22⋊(C22⋊ΓL(1,𝔽8)) | C23 | semidirect |
| 18. C25×C7 | C22×C3 | semidirect |
| 31. C24×D7 | A4 | semidirect |
| 32. C24×D7 | C22×C3 | semidirect |
| 46. C24×C7 | C2×A4 | semidirect |
| 47. C24×C7 | C23×C3 | semidirect |
| 69. C23×D7 | C2×A4 | semidirect |
| 80. C26 | AGL(1,𝔽7) | semidirect |
| 90. C23×C7 | C22×A4 | semidirect |
| 110. C22×D7 | C22×A4 | semidirect |
| 111. C22×D7 | C22⋊A4 | semidirect |
| 133. C25 | C2×AGL(1,𝔽7) | semidirect |
| 141. C22×C7 | C23×A4 | semidirect |
| 142. C22×C7 | C2×C22⋊A4 | semidirect |
| 150. C2×D7 | C2×C22⋊A4 | semidirect |
| 184. C24 | C22×AGL(1,𝔽7) | semidirect |
| 185. C24 | D7⋊A4 | semidirect |
| 188. C2×C7 | C22×C22⋊A4 | semidirect |
| 190. D7 | C22×C22⋊A4 | semidirect |
| 224. C23 | C2×D7⋊A4 | semidirect |
| 225. C7 | C23×C22⋊A4 | semidirect |
| 243. C22 | C22×D7⋊A4 | semidirect |
| 244. C22 | D7⋊(C22⋊A4) | direct (N × H) |
| 250. C2 | C2×D7⋊(C22⋊A4) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C22⋊(C22⋊ΓL(1,𝔽8)) | 2 |
| C2×D7⋊(C22⋊A4) | 2 |
| C26×D7 | 3 |
| C22×D7⋊A4 | 4 |
| C23×C22⋊A4 | 7 |