Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
772 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 772 conjugacy classes of subgroups — too many to draw. Only its 145 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 | #90616 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #90521 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #90216 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C7×#410 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #89778 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #89513 | C2 | 1 | 8 | yes | maximal, complemented |
| 7 | C2×C7×#41 | C22 | 1 | 1 | yes | characteristic |
| 8 | #6554 | — | 2 | 2 | no | — |
| 9 | #6553 | — | 2 | 2 | no | — |
| 10 | D7×#41 | — | 2 | 4 | no | — |
| 11 | #6551 | — | 2 | 1 | no | — |
| 12 | #6548 | — | 2 | 1 | no | — |
| 13 | #6548 | C22 | 1 | 4 | yes | — |
| 14 | C2×#1157 | — | 2 | 2 | no | — |
| 15 | C2×#1157 | C22 | 1 | 2 | yes | — |
| 16 | C22×D28 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | C7×#117 | C22 | 1 | 2 | yes | complemented |
| 18 | #6510 | — | 2 | 1 | no | — |
| 19 | #6503 | — | 2 | 1 | no | — |
| 20 | #6500 | C22 | 1 | 4 | yes | complemented |
| 21 | #6493 | — | 2 | 2 | no | — |
| 22 | #6489 | C22 | 1 | 1 | yes | characteristic, complemented |
| 23 | #6458 | — | 2 | 4 | no | — |
| 24 | #6445 | C22 | 1 | 8 | yes | complemented |
| 25 | #6438 | — | 2 | 2 | no | — |
| 26 | #6437 | — | 2 | 2 | no | — |
| 27 | C7×#99 | C22 | 1 | 4 | yes | complemented |
| 28 | #6380 | C22 | 1 | 8 | yes | complemented |
| 29 | #3027 | — | 7 | 1 | no | maximal, Sylow |
| 30 | C23×D7 | C23 | 1 | 2 | yes | — |
| 31 | C7×#41 | — | 2 | 2 | no | — |
| 32 | C7×#41 | — | 2 | 2 | no | — |
| 33 | C2×C7×Q8 | D4 | 1 | 1 | yes | characteristic, complemented |
| 34 | C2×C7×D4 | D4 | 1 | 1 | yes | characteristic, complemented |
| 35 | C2×C7×D4 | C23 | 1 | 2 | yes | — |
| 36 | C22×C4×C7 | D4 | 1 | 2 | yes | complemented |
| 37 | C22×C4×C7 | C23 | 1 | 1 | yes | characteristic |
| 38 | C2×C22⋊D7 | — | 2 | 4 | no | — |
| 39 | C2×C22⋊D7 | — | 2 | 4 | no | — |
| 40 | C2×C22⋊D7 | C23 | 1 | 4 | yes | — |
| 41 | #1161 | — | 2 | 4 | no | — |
| 42 | Q8×D7 | — | 2 | 4 | no | — |
| 43 | #1159 | — | 2 | 4 | no | — |
| 44 | #1159 | — | 2 | 8 | no | — |
| 45 | D4×D7 | — | 2 | 4 | no | — |
| 46 | D4×D7 | — | 2 | 8 | no | — |
| 47 | #1157 | — | 2 | 4 | no | — |
| 48 | #1157 | — | 2 | 4 | no | — |
| 49 | #1157 | — | 2 | 2 | no | — |
| 50 | #1157 | — | 2 | 4 | no | — |
| 51 | #1157 | — | 2 | 2 | no | — |
| 52 | #1157 | — | 2 | 2 | no | — |
| 53 | #1157 | — | 2 | 2 | no | — |
| 54 | C2×D28 | — | 2 | 2 | no | — |
| 55 | C2×D28 | — | 2 | 2 | no | — |
| 56 | C2×D28 | — | 2 | 2 | no | — |
| 57 | C2×D28 | — | 2 | 1 | no | — |
| 58 | C2×D28 | — | 2 | 1 | no | — |
| 59 | C2×D28 | — | 2 | 8 | no | — |
| 60 | C2×D28 | D4 | 1 | 2 | yes | complemented |
| 61 | C2×C4×D7 | — | 2 | 4 | no | — |
| 62 | C2×C4×D7 | D4 | 1 | 4 | yes | complemented |
| 63 | C2×Dic14 | — | 2 | 2 | no | — |
| 64 | C2×Dic14 | D4 | 1 | 2 | yes | complemented |
| 65 | C7×C4⋊C4 | — | 2 | 2 | no | — |
| 66 | C7×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 67 | C42×C7 | — | 2 | 2 | no | — |
| 68 | C22⋊Dic7 | C23 | 1 | 2 | yes | — |
| 69 | (C2×D7)⋊C4 | — | 2 | 4 | no | — |
| 70 | (C2×D7)⋊C4 | — | 2 | 8 | no | — |
| 71 | C4⋊Dic7 | — | 2 | 1 | no | — |
| 72 | C4×Dic7 | — | 2 | 1 | no | — |
| 73 | #585 | — | 7 | 1 | no | — |
| 74 | #535 | — | 7 | 1 | no | — |
| 75 | #535 | — | 7 | 2 | no | — |
| 76 | B(4,𝔽2) | — | 7 | 8 | no | — |
| 77 | #410 | — | 7 | 2 | no | — |
| 78 | #410 | D7 | 1 | 1 | yes | characteristic, complemented |
| 79 | C23×C7 | C2×D4 | 1 | 2 | yes | — |
| 80 | C23×C7 | C24 | 1 | 1 | yes | characteristic |
| 81 | C22×D7 | — | 2 | 4 | no | — |
| 82 | C22×D7 | — | 2 | 4 | no | — |
| 83 | C22×D7 | — | 2 | 2 | no | — |
| 84 | C22×D7 | — | 8 | 2 | no | — |
| 85 | C22×D7 | C2×D4 | 1 | 4 | yes | — |
| 86 | C7×Q8 | — | 2 | 2 | no | — |
| 87 | C7×D4 | — | 2 | 4 | no | — |
| 88 | C7×D4 | — | 2 | 2 | no | — |
| 89 | C2×C4×C7 | — | 2 | 2 | no | — |
| 90 | C2×C4×C7 | — | 2 | 2 | no | — |
| 91 | C2×C4×C7 | — | 2 | 4 | no | — |
| 92 | C2×C4×C7 | C2×D4 | 1 | 2 | yes | — |
| 93 | C2×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | C2×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | C2×C4×C7 | C2×D4 | 1 | 2 | yes | — |
| 96 | C2×C4×C7 | C2×D4 | 1 | 2 | yes | — |
| 97 | C22⋊D7 | — | 2 | 8 | no | — |
| 98 | C22⋊D7 | — | 2 | 4 | no | — |
| 99 | C22⋊D7 | — | 2 | 4 | no | — |
| 100 | C22⋊D7 | — | 8 | 4 | no | — |
| 101 | C2×Dic7 | — | 2 | 4 | no | — |
| 102 | C2×Dic7 | — | 2 | 2 | no | — |
| 103 | C2×Dic7 | C2×D4 | 1 | 4 | yes | — |
| 104 | D28 | — | 2 | 4 | no | — |
| 105 | D28 | — | 2 | 2 | no | — |
| 106 | D28 | — | 2 | 2 | no | — |
| 107 | D28 | — | 4 | 1 | no | — |
| 108 | D28 | — | 4 | 2 | no | — |
| 109 | D28 | — | 4 | 1 | no | — |
| 110 | D28 | — | 8 | 8 | no | — |
| 111 | C4×D7 | — | 2 | 8 | no | — |
| 112 | C4×D7 | — | 2 | 4 | no | — |
| 113 | C4×D7 | — | 2 | 4 | no | — |
| 114 | Dic14 | — | 2 | 4 | no | — |
| 115 | Dic14 | — | 2 | 2 | no | — |
| 116 | Dic14 | — | 2 | 2 | no | — |
| 117 | 2−1+4 | — | 14 | 2 | no | — |
| 118 | 2−1+4 | — | 14 | 1 | no | — |
| 119 | H(5,𝔽2) | — | 7 | 4 | no | — |
| 120 | H(5,𝔽2) | — | 14 | 2 | no | — |
| 121 | H(5,𝔽2) | — | 14 | 1 | no | — |
| 122 | C2×#41 | — | 7 | 2 | no | — |
| 123 | C2×#41 | — | 14 | 4 | no | — |
| 124 | C2×#41 | — | 14 | 2 | no | — |
| 125 | C2×#41 | C2×D7 | 1 | 1 | yes | characteristic |
| 126 | C22×D4 | — | 7 | 1 | no | — |
| 127 | #127 | — | 14 | 2 | no | — |
| 128 | #127 | — | 14 | 1 | no | — |
| 129 | #124 | — | 14 | 2 | no | — |
| 130 | #124 | — | 14 | 1 | no | — |
| 131 | #121 | — | 14 | 4 | no | — |
| 132 | #121 | — | 14 | 2 | no | — |
| 133 | #120 | — | 7 | 4 | no | — |
| 134 | #120 | — | 7 | 8 | no | — |
| 135 | #117 | — | 7 | 1 | no | — |
| 136 | #117 | C2×D7 | 1 | 2 | yes | complemented |
| 137 | #99 | — | 7 | 8 | no | — |
| 138 | #99 | C2×D7 | 1 | 4 | yes | complemented |
| 139 | C22×C7 | — | 2 | 2 | no | — |
| 140 | C22×C7 | — | 4 | 1 | no | — |
| 141 | C22×C7 | — | 4 | 2 | no | — |
| 142 | C22×C7 | C22×D4 | 1 | 1 | yes | characteristic |
| 143 | C22×C7 | C22×D4 | 1 | 2 | yes | — |
| 144 | C2×D7 | — | 2 | 4 | no | — |
| 145 | C2×D7 | — | 4 | 4 | no | — |
| 146 | C2×D7 | — | 4 | 8 | no | — |
| 147 | C2×D7 | — | 4 | 4 | no | — |
| 148 | C2×D7 | — | 4 | 2 | no | — |
| 149 | C2×D7 | — | 8 | 4 | no | — |
| 150 | C4×C7 | — | 2 | 2 | no | — |
| 151 | C4×C7 | — | 4 | 4 | no | — |
| 152 | C4×C7 | #120 | 1 | 1 | yes | characteristic |
| 153 | C4×C7 | #120 | 1 | 2 | yes | — |
| 154 | C4×C7 | #120 | 1 | 1 | yes | characteristic |
| 155 | Dic7 | — | 2 | 4 | no | — |
| 156 | Dic7 | — | 4 | 2 | no | — |
| 157 | C24 | — | 7 | 2 | no | — |
| 158 | #41 | — | 2 | 2 | no | — |
| 159 | #41 | — | 2 | 2 | no | — |
| 160 | #41 | — | 14 | 4 | no | — |
| 161 | #41 | — | 14 | 4 | no | — |
| 162 | #41 | — | 14 | 8 | no | — |
| 163 | #41 | — | 14 | 4 | no | — |
| 164 | #41 | — | 14 | 4 | no | — |
| 165 | #41 | — | 14 | 2 | no | — |
| 166 | #41 | — | 14 | 2 | no | — |
| 167 | #41 | — | 14 | 4 | no | — |
| 168 | #41 | — | 14 | 2 | no | — |
| 169 | #41 | — | 14 | 2 | no | — |
| 170 | C2×Q8 | — | 7 | 2 | no | — |
| 171 | C2×Q8 | — | 14 | 2 | no | — |
| 172 | C2×Q8 | — | 14 | 4 | no | — |
| 173 | C2×Q8 | D28 | 1 | 1 | yes | characteristic, complemented |
| 174 | C2×D4 | — | 7 | 4 | no | — |
| 175 | C2×D4 | — | 7 | 2 | no | — |
| 176 | C2×D4 | — | 14 | 2 | no | — |
| 177 | C2×D4 | — | 14 | 8 | no | — |
| 178 | C2×D4 | — | 14 | 8 | no | — |
| 179 | C2×D4 | — | 14 | 4 | no | — |
| 180 | C2×D4 | — | 14 | 4 | no | — |
| 181 | C2×D4 | — | 14 | 4 | no | — |
| 182 | C2×D4 | — | 14 | 1 | no | — |
| 183 | C2×D4 | — | 14 | 1 | no | — |
| 184 | C2×D4 | — | 14 | 2 | no | — |
| 185 | C2×D4 | — | 14 | 2 | no | — |
| 186 | C2×D4 | D28 | 1 | 1 | yes | characteristic, complemented |
| 187 | C2×D4 | C22×D7 | 1 | 2 | yes | — |
| 188 | C22×C4 | — | 7 | 4 | no | — |
| 189 | C22×C4 | — | 14 | 4 | no | — |
| 190 | C22×C4 | D28 | 1 | 2 | yes | complemented |
| 191 | C22×C4 | C22×D7 | 1 | 1 | yes | characteristic |
| 192 | C4⋊C4 | — | 2 | 2 | no | — |
| 193 | C4⋊C4 | — | 14 | 1 | no | — |
| 194 | C22⋊C4 | — | 7 | 2 | no | — |
| 195 | C22⋊C4 | — | 14 | 8 | no | — |
| 196 | C22⋊C4 | — | 14 | 4 | no | — |
| 197 | C22⋊C4 | C22×D7 | 1 | 4 | yes | — |
| 198 | C42 | — | 2 | 2 | no | — |
| 199 | C42 | — | 14 | 1 | no | — |
| 200 | C2×C7 | — | 2 | 1 | no | — |
| 201 | C2×C7 | — | 2 | 2 | no | — |
| 202 | C2×C7 | — | 4 | 2 | no | — |
| 203 | C2×C7 | C2×#120 | 1 | 1 | yes | characteristic |
| 204 | D7 | — | 4 | 4 | no | — |
| 205 | D7 | — | 8 | 2 | no | — |
| 206 | C23 | — | 7 | 4 | no | — |
| 207 | C23 | — | 14 | 4 | no | — |
| 208 | C23 | — | 14 | 2 | no | — |
| 209 | C23 | — | 14 | 4 | no | — |
| 210 | C23 | — | 56 | 2 | no | — |
| 211 | C23 | C2×D28 | 1 | 2 | yes | — |
| 212 | C23 | C23×D7 | 1 | 1 | yes | characteristic |
| 213 | Q8 | — | 2 | 2 | no | — |
| 214 | Q8 | — | 14 | 4 | no | — |
| 215 | Q8 | — | 14 | 2 | no | — |
| 216 | Q8 | — | 14 | 2 | no | — |
| 217 | D4 | — | 2 | 4 | no | — |
| 218 | D4 | — | 2 | 2 | no | — |
| 219 | D4 | — | 14 | 4 | no | — |
| 220 | D4 | — | 14 | 8 | no | — |
| 221 | D4 | — | 14 | 4 | no | — |
| 222 | D4 | — | 14 | 4 | no | — |
| 223 | D4 | — | 14 | 2 | no | — |
| 224 | D4 | — | 14 | 2 | no | — |
| 225 | D4 | — | 28 | 1 | no | — |
| 226 | D4 | — | 28 | 1 | no | — |
| 227 | D4 | — | 28 | 2 | no | — |
| 228 | D4 | — | 56 | 8 | no | — |
| 229 | D4 | — | 56 | 4 | no | — |
| 230 | C2×C4 | — | 2 | 2 | no | — |
| 231 | C2×C4 | — | 2 | 2 | no | — |
| 232 | C2×C4 | — | 2 | 4 | no | — |
| 233 | C2×C4 | — | 7 | 4 | no | — |
| 234 | C2×C4 | — | 14 | 8 | no | — |
| 235 | C2×C4 | — | 14 | 4 | no | — |
| 236 | C2×C4 | — | 14 | 4 | no | — |
| 237 | C2×C4 | — | 14 | 4 | no | — |
| 238 | C2×C4 | — | 14 | 2 | no | — |
| 239 | C2×C4 | C2×D28 | 1 | 2 | yes | — |
| 240 | C2×C4 | C2×D28 | 1 | 1 | yes | characteristic |
| 241 | C2×C4 | C2×D28 | 1 | 1 | yes | characteristic |
| 242 | C2×C4 | D4×D7 | 1 | 2 | yes | — |
| 243 | C2×C4 | D4×D7 | 1 | 2 | yes | — |
| 244 | C7 | #3027 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 245 | C22 | — | 2 | 2 | no | — |
| 246 | C22 | — | 4 | 1 | no | — |
| 247 | C22 | — | 4 | 2 | no | — |
| 248 | C22 | — | 14 | 4 | no | — |
| 249 | C22 | — | 28 | 4 | no | — |
| 250 | C22 | — | 28 | 8 | no | — |
| 251 | C22 | — | 28 | 4 | no | — |
| 252 | C22 | — | 28 | 2 | no | — |
| 253 | C22 | — | 56 | 4 | no | — |
| 254 | C22 | C22×D28 | 1 | 1 | yes | characteristic |
| 255 | C22 | C2×D4×D7 | 1 | 2 | yes | — |
| 256 | C4 | — | 2 | 2 | no | — |
| 257 | C4 | — | 4 | 4 | no | — |
| 258 | C4 | — | 14 | 4 | no | — |
| 259 | C4 | — | 28 | 2 | no | — |
| 260 | C4 | #6445 | 1 | 1 | yes | characteristic |
| 261 | C4 | #6445 | 1 | 1 | yes | characteristic |
| 262 | C4 | #6445 | 1 | 2 | yes | — |
| 263 | C2 | — | 2 | 2 | no | — |
| 264 | C2 | — | 2 | 1 | no | — |
| 265 | C2 | — | 4 | 2 | no | — |
| 266 | C2 | — | 28 | 4 | no | — |
| 267 | C2 | — | 56 | 2 | no | — |
| 268 | C2 | C2×#6445 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#6445 |
| commutator, cosocle | C23×C7 | C24 |
| Frattini | C23 | C23×D7 |
| Fitting | C7×#410 | C2 |
| socle | C2×C7 | C2×#120 |
| hypercenter | #410 | D7 |
| hypocenter | C7 | #3027 |
| LCS | G ▷ C23×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #410 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #90616 | C2 | semidirect |
| 2. #90521 | C2 | semidirect |
| 3. #90216 | C2 | semidirect |
| 4. C7×#410 | C2 | semidirect |
| 5. #89778 | C2 | semidirect |
| 6. #89513 | C2 | semidirect |
| 16. C22×D28 | C22 | semidirect |
| 17. C7×#117 | C22 | semidirect |
| 20. #6500 | C22 | semidirect |
| 22. #6489 | C22 | semidirect |
| 24. #6445 | C22 | semidirect |
| 27. C7×#99 | C22 | semidirect |
| 28. #6380 | C22 | semidirect |
| 33. C2×C7×Q8 | D4 | semidirect |
| 34. C2×C7×D4 | D4 | semidirect |
| 36. C22×C4×C7 | D4 | semidirect |
| 60. C2×D28 | D4 | semidirect |
| 62. C2×C4×D7 | D4 | semidirect |
| 64. C2×Dic14 | D4 | semidirect |
| 78. #410 | D7 | semidirect |
| 136. #117 | C2×D7 | semidirect |
| 138. #99 | C2×D7 | semidirect |
| 173. C2×Q8 | D28 | semidirect |
| 186. C2×D4 | D28 | semidirect |
| 190. C22×C4 | D28 | semidirect |
| 244. C7 | #3027 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #90616 | 2 |
| #90521 | 2 |
| #90216 | 2 |
| C7×#410 | 2 |
| #89778 | 2 |
| #89513 | 2 |
| #3027 | 7 |