Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
950 conjugacy classes of subgroups; only the 449 normal subgroups are shown.
This group has 950 conjugacy classes of subgroups — too many to draw. Only its 449 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 | C7×#564 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #90391 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #90389 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #90386 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #90384 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | #90383 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #90382 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C7×#125 | C22 | 1 | 3 | yes | — |
| 9 | C7×#125 | C22 | 1 | 6 | yes | — |
| 10 | C7×#125 | C22 | 1 | 6 | yes | — |
| 11 | #6461 | C22 | 1 | 6 | yes | — |
| 12 | #6461 | C22 | 1 | 6 | yes | — |
| 13 | #6460 | C22 | 1 | 3 | yes | — |
| 14 | #6459 | C22 | 1 | 6 | yes | — |
| 15 | #6459 | C22 | 1 | 6 | yes | — |
| 16 | #6458 | C22 | 1 | 3 | yes | — |
| 17 | #6458 | C22 | 1 | 6 | yes | — |
| 18 | #6458 | C22 | 1 | 6 | yes | — |
| 19 | #6458 | C22 | 1 | 6 | yes | — |
| 20 | #6458 | C22 | 1 | 6 | yes | — |
| 21 | #6457 | C22 | 1 | 6 | yes | — |
| 22 | #6457 | C22 | 1 | 6 | yes | — |
| 23 | #6457 | C22 | 1 | 6 | yes | — |
| 24 | #6457 | C22 | 1 | 6 | yes | — |
| 25 | #6457 | C22 | 1 | 6 | yes | — |
| 26 | #6456 | C22 | 1 | 6 | yes | — |
| 27 | #6456 | C22 | 1 | 3 | yes | — |
| 28 | #6456 | C22 | 1 | 6 | yes | — |
| 29 | #6455 | C22 | 1 | 6 | yes | — |
| 30 | D7×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 31 | #6452 | C22 | 1 | 6 | yes | — |
| 32 | #6450 | C22 | 1 | 3 | yes | — |
| 33 | #6439 | C22 | 1 | 2 | yes | — |
| 34 | #6438 | C22 | 1 | 3 | yes | — |
| 35 | #6437 | C22 | 1 | 2 | yes | — |
| 36 | C4×D28 | C22 | 1 | 6 | yes | — |
| 37 | #6435 | C22 | 1 | 6 | yes | — |
| 38 | C42×D7 | C22 | 1 | 1 | yes | characteristic |
| 39 | #3567 | — | 7 | 1 | no | maximal, Sylow |
| 40 | C2×D28 | C23 | 1 | 6 | yes | — |
| 41 | C2×D28 | C23 | 1 | 6 | yes | — |
| 42 | C2×D28 | C23 | 1 | 6 | yes | — |
| 43 | C2×D28 | C23 | 1 | 3 | yes | — |
| 44 | C2×C4×D7 | C23 | 1 | 3 | yes | — |
| 45 | C2×C4×D7 | C23 | 1 | 6 | yes | — |
| 46 | C2×C4×D7 | C23 | 1 | 6 | yes | — |
| 47 | C2×C4×D7 | C23 | 1 | 6 | yes | — |
| 48 | C2×Dic14 | C23 | 1 | 3 | yes | — |
| 49 | C7×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 50 | C7×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 51 | C7×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 52 | C7×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 53 | C7×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 54 | C7×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 55 | C42×C7 | C23 | 1 | 1 | yes | characteristic |
| 56 | C42×C7 | C23 | 1 | 2 | yes | — |
| 57 | C42×C7 | C23 | 1 | 2 | yes | — |
| 58 | (C2×D7)⋊C4 | C23 | 1 | 6 | yes | — |
| 59 | (C2×D7)⋊C4 | C23 | 1 | 6 | yes | — |
| 60 | (C2×D7)⋊C4 | C23 | 1 | 6 | yes | — |
| 61 | (C2×D7)⋊C4 | C23 | 1 | 6 | yes | — |
| 62 | (C2×D7)⋊C4 | C23 | 1 | 6 | yes | — |
| 63 | (C2×D7)⋊C4 | C23 | 1 | 6 | yes | — |
| 64 | (C2×D7)⋊C4 | C23 | 1 | 6 | yes | — |
| 65 | C4⋊Dic7 | C23 | 1 | 6 | yes | — |
| 66 | Dic7⋊C4 | C23 | 1 | 6 | yes | — |
| 67 | Dic7⋊C4 | C23 | 1 | 6 | yes | — |
| 68 | Dic7⋊C4 | C23 | 1 | 6 | yes | — |
| 69 | C4×Dic7 | C23 | 1 | 3 | yes | — |
| 70 | C4×Dic7 | C23 | 1 | 6 | yes | — |
| 71 | #564 | D7 | 1 | 1 | yes | characteristic, complemented |
| 72 | #562 | — | 7 | 3 | no | — |
| 73 | #553 | — | 7 | 6 | no | — |
| 74 | #552 | — | 7 | 6 | no | — |
| 75 | #540 | — | 7 | 6 | no | — |
| 76 | #539 | — | 7 | 6 | no | — |
| 77 | #529 | — | 7 | 3 | no | — |
| 78 | C22×D7 | C24 | 1 | 6 | yes | — |
| 79 | C22×D7 | C24 | 1 | 1 | yes | characteristic |
| 80 | C2×C4×C7 | C24 | 1 | 6 | yes | — |
| 81 | C2×C4×C7 | C24 | 1 | 3 | yes | — |
| 82 | C2×C4×C7 | C24 | 1 | 6 | yes | — |
| 83 | C2×Dic7 | C24 | 1 | 3 | yes | — |
| 84 | C2×Dic7 | C24 | 1 | 6 | yes | — |
| 85 | D28 | — | 4 | 3 | no | — |
| 86 | D28 | — | 4 | 6 | no | — |
| 87 | D28 | — | 4 | 6 | no | — |
| 88 | D28 | — | 4 | 6 | no | — |
| 89 | C4×D7 | — | 4 | 6 | no | — |
| 90 | C4×D7 | — | 4 | 6 | no | — |
| 91 | C4×D7 | — | 4 | 3 | no | — |
| 92 | C4×D7 | — | 4 | 6 | no | — |
| 93 | Dic14 | — | 4 | 3 | no | — |
| 94 | #127 | — | 7 | 2 | no | — |
| 95 | #126 | — | 7 | 6 | no | — |
| 96 | #126 | — | 7 | 6 | no | — |
| 97 | #126 | — | 7 | 2 | no | — |
| 98 | #125 | — | 7 | 6 | no | — |
| 99 | #125 | C2×D7 | 1 | 3 | yes | — |
| 100 | #125 | C2×D7 | 1 | 6 | yes | — |
| 101 | #125 | C2×D7 | 1 | 6 | yes | — |
| 102 | #124 | — | 7 | 3 | no | — |
| 103 | #123 | — | 7 | 6 | no | — |
| 104 | #123 | — | 7 | 6 | no | — |
| 105 | #123 | — | 7 | 6 | no | — |
| 106 | #123 | — | 7 | 6 | no | — |
| 107 | #123 | — | 7 | 6 | no | — |
| 108 | C22⋊Q8 | — | 7 | 3 | no | — |
| 109 | C22⋊Q8 | — | 7 | 6 | no | — |
| 110 | C22⋊Q8 | — | 7 | 6 | no | — |
| 111 | #121 | — | 7 | 3 | no | — |
| 112 | #121 | — | 7 | 6 | no | — |
| 113 | #121 | — | 7 | 6 | no | — |
| 114 | #121 | — | 7 | 6 | no | — |
| 115 | #121 | — | 7 | 6 | no | — |
| 116 | C4×Q8 | — | 7 | 3 | no | — |
| 117 | C4×D4 | — | 7 | 6 | no | — |
| 118 | C4×D4 | — | 7 | 3 | no | — |
| 119 | C4×D4 | — | 7 | 6 | no | — |
| 120 | C4×D4 | — | 7 | 6 | no | — |
| 121 | #117 | — | 7 | 6 | no | — |
| 122 | #117 | — | 7 | 6 | no | — |
| 123 | C2×C4⋊C4 | — | 7 | 6 | no | — |
| 124 | C2×C42 | — | 7 | 1 | no | — |
| 125 | C22×C7 | C25 | 1 | 1 | yes | characteristic |
| 126 | C2×D7 | — | 2 | 6 | no | — |
| 127 | C2×D7 | — | 2 | 6 | no | — |
| 128 | C2×D7 | — | 2 | 3 | no | — |
| 129 | C2×D7 | — | 2 | 6 | no | — |
| 130 | C4×C7 | — | 2 | 3 | no | — |
| 131 | C4×C7 | — | 2 | 6 | no | — |
| 132 | C4×C7 | — | 2 | 6 | no | — |
| 133 | Dic7 | — | 2 | 6 | no | — |
| 134 | Dic7 | — | 2 | 3 | no | — |
| 135 | C2×Q8 | — | 7 | 3 | no | — |
| 136 | C2×D4 | — | 7 | 6 | no | — |
| 137 | C2×D4 | — | 7 | 3 | no | — |
| 138 | C2×D4 | — | 7 | 6 | no | — |
| 139 | C2×D4 | — | 7 | 6 | no | — |
| 140 | C22×C4 | — | 7 | 3 | no | — |
| 141 | C22×C4 | — | 7 | 6 | no | — |
| 142 | C22×C4 | — | 7 | 6 | no | — |
| 143 | C22×C4 | — | 7 | 6 | no | — |
| 144 | C4⋊C4 | — | 7 | 6 | no | — |
| 145 | C4⋊C4 | — | 7 | 6 | no | — |
| 146 | C4⋊C4 | — | 7 | 6 | no | — |
| 147 | C4⋊C4 | — | 7 | 6 | no | — |
| 148 | C4⋊C4 | C22×D7 | 1 | 6 | yes | — |
| 149 | C4⋊C4 | C22×D7 | 1 | 6 | yes | — |
| 150 | C4⋊C4 | C22×D7 | 1 | 6 | yes | — |
| 151 | C4⋊C4 | C22×D7 | 1 | 6 | yes | — |
| 152 | C4⋊C4 | C22×D7 | 1 | 3 | yes | — |
| 153 | C4⋊C4 | C22×D7 | 1 | 3 | yes | — |
| 154 | C22⋊C4 | — | 7 | 6 | no | — |
| 155 | C22⋊C4 | — | 7 | 6 | no | — |
| 156 | C22⋊C4 | — | 7 | 6 | no | — |
| 157 | C22⋊C4 | — | 7 | 6 | no | — |
| 158 | C22⋊C4 | — | 7 | 6 | no | — |
| 159 | C22⋊C4 | — | 7 | 6 | no | — |
| 160 | C22⋊C4 | — | 7 | 6 | no | — |
| 161 | C42 | — | 7 | 3 | no | — |
| 162 | C42 | — | 7 | 6 | no | — |
| 163 | C42 | C22×D7 | 1 | 1 | yes | characteristic |
| 164 | C42 | C22×D7 | 1 | 2 | yes | — |
| 165 | C42 | C22×D7 | 1 | 2 | yes | — |
| 166 | C2×C7 | #585 | 1 | 3 | yes | — |
| 167 | D7 | — | 4 | 6 | no | — |
| 168 | D7 | — | 4 | 1 | no | — |
| 169 | C23 | — | 7 | 6 | no | — |
| 170 | C23 | — | 7 | 1 | no | — |
| 171 | Q8 | — | 28 | 3 | no | — |
| 172 | D4 | — | 28 | 6 | no | — |
| 173 | D4 | — | 28 | 6 | no | — |
| 174 | D4 | — | 28 | 6 | no | — |
| 175 | D4 | — | 28 | 3 | no | — |
| 176 | C2×C4 | — | 7 | 3 | no | — |
| 177 | C2×C4 | — | 7 | 6 | no | — |
| 178 | C2×C4 | — | 28 | 6 | no | — |
| 179 | C2×C4 | — | 28 | 6 | no | — |
| 180 | C2×C4 | — | 28 | 6 | no | — |
| 181 | C2×C4 | — | 28 | 3 | no | — |
| 182 | C2×C4 | C23×D7 | 1 | 3 | yes | — |
| 183 | C2×C4 | C23×D7 | 1 | 6 | yes | — |
| 184 | C2×C4 | C23×D7 | 1 | 6 | yes | — |
| 185 | C7 | #3567 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 186 | C22 | — | 14 | 6 | no | — |
| 187 | C22 | — | 14 | 6 | no | — |
| 188 | C22 | — | 14 | 3 | no | — |
| 189 | C22 | — | 14 | 6 | no | — |
| 190 | C22 | C24×D7 | 1 | 1 | yes | characteristic |
| 191 | C4 | — | 2 | 6 | no | — |
| 192 | C4 | — | 2 | 6 | no | — |
| 193 | C4 | — | 2 | 3 | no | — |
| 194 | C4 | — | 14 | 3 | no | — |
| 195 | C4 | — | 14 | 6 | no | — |
| 196 | C2 | — | 28 | 6 | no | — |
| 197 | C2 | — | 28 | 1 | no | — |
| 198 | C2 | #90620 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D7 |
| commutator, socle, cosocle | C22×C7 | C25 |
| Fitting | C7×#564 | C2 |
| hypercenter | #564 | D7 |
| hypocenter | C7 | #3567 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #564 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#564 | C2 | semidirect |
| 2. #90391 | C2 | semidirect |
| 3. #90389 | C2 | semidirect |
| 4. #90386 | C2 | semidirect |
| 5. #90384 | C2 | semidirect |
| 6. #90383 | C2 | semidirect |
| 7. #90382 | C2 | semidirect |
| 71. #564 | D7 | semidirect |
| 185. C7 | #3567 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#564 | 2 |
| #90391 | 2 |
| #90389 | 2 |
| #90386 | 2 |
| #90384 | 2 |
| #90383 | 2 |
| #90382 | 2 |
| #3567 | 7 |