Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
336 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 59 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 | #26032056 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #26022420 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #26018291 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #26018229 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C89×#1716 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #19152411 | — | 2 | 2 | no | — |
| 7 | #19151870 | — | 2 | 2 | no | — |
| 8 | #19151780 | — | 2 | 1 | no | — |
| 9 | #19151780 | — | 2 | 1 | no | — |
| 10 | #19151716 | C22 | 1 | 2 | yes | — |
| 11 | #19151420 | — | 2 | 2 | no | — |
| 12 | #19151419 | C22 | 1 | 2 | yes | — |
| 13 | #19151407 | — | 2 | 2 | no | — |
| 14 | #498×C89 | — | 2 | 1 | no | — |
| 15 | #498×C89 | C22 | 1 | 1 | yes | characteristic |
| 16 | #494×C89 | C22 | 1 | 1 | yes | characteristic |
| 17 | #487×C89 | — | 2 | 1 | no | — |
| 18 | #322×C89 | C22 | 1 | 1 | yes | characteristic |
| 19 | #15668060 | — | 2 | 1 | no | — |
| 20 | #15668060 | — | 2 | 1 | no | — |
| 21 | #15668044 | — | 4 | 2 | no | — |
| 22 | #15668044 | — | 4 | 2 | no | — |
| 23 | #15668043 | — | 2 | 2 | no | — |
| 24 | #15668043 | — | 2 | 2 | no | — |
| 25 | #15668035 | — | 4 | 2 | no | — |
| 26 | #15668034 | D4 | 1 | 2 | yes | — |
| 27 | Q16×D89 | — | 4 | 2 | no | — |
| 28 | #15668018 | — | 2 | 2 | no | — |
| 29 | #15667952 | — | 2 | 2 | no | — |
| 30 | #15667948 | D4 | 1 | 2 | yes | — |
| 31 | #15667940 | — | 2 | 2 | no | — |
| 32 | #15667931 | — | 2 | 1 | no | — |
| 33 | #15667931 | — | 2 | 1 | no | — |
| 34 | C2×Q16×C89 | — | 2 | 2 | no | — |
| 35 | C4⋊Q8×C89 | — | 2 | 1 | no | — |
| 36 | C4⋊Q8×C89 | D4 | 1 | 1 | yes | characteristic |
| 37 | C4⋊Q8×C89 | D4 | 1 | 1 | yes | characteristic |
| 38 | #106×C89 | — | 2 | 2 | no | — |
| 39 | #106×C89 | — | 4 | 2 | no | — |
| 40 | #103×C89 | — | 4 | 2 | no | — |
| 41 | C4×C8×C89 | — | 2 | 1 | no | — |
| 42 | C4×C8×C89 | C23 | 1 | 1 | yes | characteristic |
| 43 | #13083575 | — | 4 | 2 | no | — |
| 44 | #13083575 | — | 4 | 2 | no | — |
| 45 | #13083574 | — | 2 | 2 | no | — |
| 46 | #13083574 | — | 8 | 1 | no | — |
| 47 | #13083574 | — | 8 | 1 | no | — |
| 48 | Q8×D89 | — | 4 | 2 | no | — |
| 49 | Q8×D89 | — | 4 | 2 | no | — |
| 50 | Dic356 | SD16 | 1 | 2 | yes | — |
| 51 | #13083564 | — | 4 | 2 | no | — |
| 52 | #13083564 | — | 4 | 2 | no | — |
| 53 | #13083563 | — | 2 | 2 | no | — |
| 54 | #13083561 | — | 4 | 2 | no | — |
| 55 | #13083561 | — | 4 | 2 | no | — |
| 56 | D712 | SD16 | 1 | 2 | yes | — |
| 57 | #13083558 | — | 2 | 2 | no | — |
| 58 | #13083557 | — | 4 | 2 | no | — |
| 59 | C8×D89 | — | 4 | 2 | no | — |
| 60 | #13083549 | — | 4 | 2 | no | — |
| 61 | C2×Q8×C89 | — | 2 | 1 | no | — |
| 62 | C2×Q8×C89 | — | 2 | 1 | no | — |
| 63 | C2×Q8×C89 | — | 4 | 1 | no | — |
| 64 | Q16×C89 | — | 4 | 2 | no | — |
| 65 | Q16×C89 | — | 4 | 2 | no | — |
| 66 | C2×C8×C89 | — | 2 | 2 | no | — |
| 67 | C2×C8×C89 | C2×D4 | 1 | 2 | yes | — |
| 68 | C4⋊C4×C89 | — | 4 | 1 | no | — |
| 69 | C4⋊C4×C89 | — | 4 | 1 | no | — |
| 70 | C4⋊C4×C89 | — | 4 | 1 | no | — |
| 71 | C4⋊C4×C89 | — | 4 | 1 | no | — |
| 72 | C42×C89 | C2×D4 | 1 | 1 | yes | characteristic |
| 73 | Q8×C89 | — | 4 | 2 | no | — |
| 74 | Q8×C89 | — | 4 | 2 | no | — |
| 75 | Q8×C89 | — | 8 | 2 | no | — |
| 76 | C2×C4×C89 | — | 2 | 1 | no | — |
| 77 | C2×C4×C89 | — | 4 | 1 | no | — |
| 78 | C2×C4×C89 | — | 4 | 1 | no | — |
| 79 | C2×C4×C89 | — | 8 | 1 | no | — |
| 80 | C2×C4×C89 | #120 | 1 | 1 | yes | characteristic |
| 81 | C22⋊D89 | — | 4 | 2 | no | — |
| 82 | D356 | — | 2 | 2 | no | — |
| 83 | D356 | — | 8 | 1 | no | — |
| 84 | D356 | — | 8 | 1 | no | — |
| 85 | C4×D89 | — | 4 | 2 | no | — |
| 86 | C4×D89 | — | 8 | 2 | no | — |
| 87 | C4×D89 | — | 8 | 2 | no | — |
| 88 | Dic178 | — | 2 | 2 | no | — |
| 89 | Dic178 | — | 8 | 1 | no | — |
| 90 | Dic178 | — | 8 | 1 | no | — |
| 91 | C8×C89 | — | 4 | 2 | no | — |
| 92 | C8×C89 | C2×SD16 | 1 | 2 | yes | — |
| 93 | C8×C89 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 94 | #10623494 | — | 4 | 2 | no | — |
| 95 | C22×C89 | #453 | 1 | 1 | yes | characteristic |
| 96 | C2×D89 | — | 4 | 2 | no | — |
| 97 | C4×C89 | — | 2 | 2 | no | — |
| 98 | C4×C89 | — | 8 | 1 | no | — |
| 99 | C4×C89 | — | 8 | 1 | no | — |
| 100 | C4×C89 | — | 16 | 1 | no | — |
| 101 | C4×C89 | #450 | 1 | 2 | yes | — |
| 102 | Dic89 | — | 4 | 2 | no | — |
| 103 | #12367 | — | 89 | 1 | no | maximal, Sylow |
| 104 | C2×C89 | — | 2 | 1 | no | — |
| 105 | C2×C89 | #1629 | 1 | 1 | yes | characteristic |
| 106 | D89 | — | 8 | 2 | no | — |
| 107 | #3293 | — | 89 | 1 | no | — |
| 108 | #2230 | — | 89 | 2 | no | — |
| 109 | #1716 | D89 | 1 | 1 | yes | characteristic, complemented |
| 110 | #1358 | — | 89 | 1 | no | — |
| 111 | #1336 | — | 89 | 2 | no | — |
| 112 | C89 | #12367 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 113 | #578 | — | 178 | 2 | no | — |
| 114 | #498 | — | 2 | 1 | no | — |
| 115 | #498 | C2×D89 | 1 | 1 | yes | characteristic |
| 116 | #494 | C2×D89 | 1 | 1 | yes | characteristic |
| 117 | #487 | — | 2 | 1 | no | — |
| 118 | #473 | — | 178 | 2 | no | — |
| 119 | #456 | — | 178 | 1 | no | — |
| 120 | #456 | — | 178 | 1 | no | — |
| 121 | #443 | — | 89 | 2 | no | — |
| 122 | #365 | — | 178 | 2 | no | — |
| 123 | #364 | — | 89 | 2 | no | — |
| 124 | #360 | — | 178 | 2 | no | — |
| 125 | #322 | C2×D89 | 1 | 1 | yes | characteristic |
| 126 | 2−1+4 | — | 178 | 1 | no | — |
| 127 | 2−1+4 | — | 178 | 1 | no | — |
| 128 | #137 | — | 356 | 2 | no | — |
| 129 | #137 | — | 356 | 2 | no | — |
| 130 | #137 | — | 178 | 2 | no | — |
| 131 | #137 | — | 178 | 2 | no | — |
| 132 | #135 | — | 89 | 2 | no | — |
| 133 | #135 | — | 356 | 2 | no | — |
| 134 | C2×Q16 | — | 2 | 2 | no | — |
| 135 | C2×Q16 | — | 356 | 2 | no | — |
| 136 | #131 | — | 178 | 2 | no | — |
| 137 | C4⋊Q8 | — | 2 | 1 | no | — |
| 138 | C4⋊Q8 | C22⋊D89 | 1 | 1 | yes | characteristic |
| 139 | C4⋊Q8 | C22⋊D89 | 1 | 1 | yes | characteristic |
| 140 | M32 | — | 178 | 2 | no | — |
| 141 | #108 | — | 89 | 2 | no | — |
| 142 | #106 | — | 2 | 2 | no | — |
| 143 | #106 | — | 4 | 2 | no | — |
| 144 | #104 | — | 178 | 2 | no | — |
| 145 | #103 | — | 4 | 2 | no | — |
| 146 | #101 | — | 178 | 1 | no | — |
| 147 | #101 | — | 178 | 1 | no | — |
| 148 | C4×C8 | — | 2 | 1 | no | — |
| 149 | C4×C8 | C22×D89 | 1 | 1 | yes | characteristic |
| 150 | #41 | — | 356 | 2 | no | — |
| 151 | #41 | — | 356 | 2 | no | — |
| 152 | #41 | — | 178 | 2 | no | — |
| 153 | #41 | — | 712 | 1 | no | — |
| 154 | #41 | — | 712 | 1 | no | — |
| 155 | C2×Q8 | — | 2 | 1 | no | — |
| 156 | C2×Q8 | — | 2 | 1 | no | — |
| 157 | C2×Q8 | — | 4 | 1 | no | — |
| 158 | C2×Q8 | — | 356 | 2 | no | — |
| 159 | C2×Q8 | — | 356 | 2 | no | — |
| 160 | Q16 | — | 4 | 2 | no | — |
| 161 | Q16 | — | 4 | 2 | no | — |
| 162 | Q16 | — | 89 | 2 | no | — |
| 163 | Q16 | — | 356 | 2 | no | — |
| 164 | Q16 | — | 356 | 2 | no | — |
| 165 | SD16 | — | 356 | 2 | no | — |
| 166 | SD16 | — | 356 | 2 | no | — |
| 167 | SD16 | — | 178 | 2 | no | — |
| 168 | D8 | — | 89 | 2 | no | — |
| 169 | M16 | — | 356 | 2 | no | — |
| 170 | M16 | — | 178 | 2 | no | — |
| 171 | C2×C8 | — | 2 | 2 | no | — |
| 172 | C2×C8 | — | 356 | 2 | no | — |
| 173 | C2×C8 | D4×D89 | 1 | 2 | yes | — |
| 174 | C4⋊C4 | — | 4 | 1 | no | — |
| 175 | C4⋊C4 | — | 4 | 1 | no | — |
| 176 | C4⋊C4 | — | 4 | 1 | no | — |
| 177 | C4⋊C4 | — | 4 | 1 | no | — |
| 178 | C42 | C2×C22⋊D89 | 1 | 1 | yes | characteristic |
| 179 | C16 | — | 356 | 2 | no | — |
| 180 | Q8 | — | 4 | 2 | no | — |
| 181 | Q8 | — | 4 | 2 | no | — |
| 182 | Q8 | — | 8 | 2 | no | — |
| 183 | Q8 | — | 178 | 2 | no | — |
| 184 | Q8 | — | 712 | 1 | no | — |
| 185 | Q8 | — | 712 | 1 | no | — |
| 186 | D4 | — | 356 | 2 | no | — |
| 187 | D4 | — | 178 | 2 | no | — |
| 188 | D4 | — | 712 | 1 | no | — |
| 189 | D4 | — | 712 | 1 | no | — |
| 190 | C2×C4 | — | 2 | 1 | no | — |
| 191 | C2×C4 | — | 4 | 1 | no | — |
| 192 | C2×C4 | — | 4 | 1 | no | — |
| 193 | C2×C4 | — | 8 | 1 | no | — |
| 194 | C2×C4 | — | 356 | 2 | no | — |
| 195 | C2×C4 | — | 712 | 2 | no | — |
| 196 | C2×C4 | — | 712 | 2 | no | — |
| 197 | C2×C4 | #15667980 | 1 | 1 | yes | characteristic |
| 198 | C8 | — | 4 | 2 | no | — |
| 199 | C8 | — | 356 | 2 | no | — |
| 200 | C8 | SD16×D89 | 1 | 2 | yes | — |
| 201 | C8 | #15668039 | 1 | 2 | yes | — |
| 202 | C22 | — | 356 | 2 | no | — |
| 203 | C22 | #19151765 | 1 | 1 | yes | characteristic |
| 204 | C4 | — | 2 | 2 | no | — |
| 205 | C4 | — | 8 | 1 | no | — |
| 206 | C4 | — | 8 | 1 | no | — |
| 207 | C4 | — | 16 | 1 | no | — |
| 208 | C4 | — | 356 | 2 | no | — |
| 209 | C4 | #19151747 | 1 | 2 | yes | — |
| 210 | C2 | — | 2 | 1 | no | — |
| 211 | C2 | — | 712 | 2 | no | — |
| 212 | C2 | #26019366 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #26022420 | C2 | semidirect |
| 3. #26018291 | C2 | semidirect |
| 4. #26018229 | C2 | semidirect |
| 5. C89×#1716 | C2 | semidirect |
| 109. #1716 | D89 | semidirect |
| 112. C89 | #12367 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26032056 | 2 |
| #26022420 | 2 |
| #26018291 | 2 |
| #26018229 | 2 |
| C89×#1716 | 2 |
| #12367 | 89 |