← #34193
Subgroups — #34193
Order 256.
1207 conjugacy classes of subgroups.
The subgroup lattice has 165 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C4×#41 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #2354 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | Q8×C4⋊C4 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #2347 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #2341 | C2 | 1 | 2 | yes | maximal |
| 6 | #2319 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #2285 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #2283 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | #2276 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | C22×#41 | C22 | 1 | 1 | yes | characteristic |
| 11 | C4×#41 | C4 | 1 | 16 | yes | complemented |
| 12 | C2×C4×Q8 | C22 | 1 | 4 | yes | — |
| 13 | C2×C4×Q8 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×C4×Q8 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×C4×D4 | C22 | 1 | 3 | yes | — |
| 16 | C2×C4×D4 | C22 | 1 | 3 | yes | — |
| 17 | C2×#117 | C22 | 1 | 6 | yes | — |
| 18 | C2×#117 | C22 | 1 | 3 | yes | — |
| 19 | C22×C42 | C22 | 1 | 3 | yes | — |
| 20 | #390 | C22 | 1 | 12 | yes | — |
| 21 | #389 | C22 | 1 | 12 | yes | — |
| 22 | #388 | C22 | 1 | 24 | yes | — |
| 23 | #384 | C22 | 1 | 6 | yes | — |
| 24 | #383 | C22 | 1 | 6 | yes | — |
| 25 | #382 | C22 | 1 | 1 | yes | characteristic |
| 26 | #382 | C22 | 1 | 3 | yes | — |
| 27 | #381 | C22 | 1 | 6 | yes | — |
| 28 | #380 | C22 | 1 | 6 | yes | — |
| 29 | #380 | C22 | 1 | 6 | yes | — |
| 30 | #380 | C22 | 1 | 6 | yes | — |
| 31 | C4×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 32 | C4×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 33 | C4×C4⋊C4 | C22 | 1 | 12 | yes | — |
| 34 | C4×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 35 | C4×C22⋊C4 | C22 | 1 | 6 | yes | — |
| 36 | #376 | C22 | 1 | 6 | yes | — |
| 37 | C2×#41 | — | 2 | 8 | no | — |
| 38 | C2×#41 | C2×C4 | 1 | 8 | yes | — |
| 39 | C22×Q8 | C23 | 1 | 1 | yes | characteristic |
| 40 | C22×D4 | C23 | 1 | 3 | yes | — |
| 41 | C23×C4 | C23 | 1 | 3 | yes | — |
| 42 | C23×C4 | C23 | 1 | 3 | yes | — |
| 43 | C23×C4 | C23 | 1 | 3 | yes | — |
| 44 | C4×Q8 | — | 2 | 16 | no | — |
| 45 | C4×Q8 | C2×C4 | 1 | 8 | yes | — |
| 46 | C4×Q8 | C2×C4 | 1 | 8 | yes | — |
| 47 | C4×D4 | C2×C4 | 1 | 24 | yes | — |
| 48 | C4×D4 | C2×C4 | 1 | 24 | yes | — |
| 49 | #117 | — | 4 | 12 | no | — |
| 50 | #117 | C2×C4 | 1 | 24 | yes | — |
| 51 | C2×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 52 | C2×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 53 | C2×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 54 | C2×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 55 | C2×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 56 | C2×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 57 | C2×C22⋊C4 | C23 | 1 | 12 | yes | — |
| 58 | C2×C22⋊C4 | C23 | 1 | 12 | yes | — |
| 59 | C2×C22⋊C4 | C23 | 1 | 6 | yes | — |
| 60 | C2×C42 | C2×C4 | 1 | 24 | yes | — |
| 61 | C2×C42 | C23 | 1 | 12 | yes | — |
| 62 | C2×C42 | C23 | 1 | 12 | yes | — |
| 63 | C2×C42 | C23 | 1 | 3 | yes | — |
| 64 | C2×C42 | C23 | 1 | 3 | yes | — |
| 65 | C2×C42 | C23 | 1 | 3 | yes | — |
| 66 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 67 | #95 | C23 | 1 | 2 | yes | — |
| 68 | #95 | C23 | 1 | 6 | yes | — |
| 69 | #95 | C23 | 1 | 12 | yes | — |
| 70 | #95 | C23 | 1 | 2 | yes | — |
| 71 | #95 | C23 | 1 | 6 | yes | — |
| 72 | #95 | C23 | 1 | 12 | yes | — |
| 73 | C24 | C24 | 1 | 3 | yes | — |
| 74 | #41 | — | 2 | 32 | no | — |
| 75 | C2×Q8 | C22×C4 | 1 | 4 | yes | — |
| 76 | C2×Q8 | #41 | 1 | 8 | yes | — |
| 77 | C2×D4 | — | 2 | 12 | no | — |
| 78 | C2×D4 | C22×C4 | 1 | 12 | yes | — |
| 79 | C22×C4 | — | 2 | 12 | no | — |
| 80 | C22×C4 | — | 2 | 6 | no | — |
| 81 | C22×C4 | — | 2 | 6 | no | — |
| 82 | C22×C4 | — | 4 | 6 | no | — |
| 83 | C22×C4 | C22×C4 | 1 | 12 | yes | — |
| 84 | C22×C4 | C22×C4 | 1 | 12 | yes | — |
| 85 | C22×C4 | C22×C4 | 1 | 12 | yes | — |
| 86 | C22×C4 | C24 | 1 | 4 | yes | — |
| 87 | C22×C4 | C24 | 1 | 12 | yes | — |
| 88 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 89 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 90 | C22×C4 | C24 | 1 | 3 | yes | — |
| 91 | C22×C4 | C24 | 1 | 3 | yes | — |
| 92 | C22×C4 | C24 | 1 | 3 | yes | — |
| 93 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 94 | C4⋊C4 | — | 2 | 24 | no | — |
| 95 | C4⋊C4 | — | 2 | 12 | no | — |
| 96 | C4⋊C4 | — | 2 | 4 | no | — |
| 97 | C4⋊C4 | — | 4 | 12 | no | — |
| 98 | C4⋊C4 | C22×C4 | 1 | 12 | yes | — |
| 99 | C4⋊C4 | C22×C4 | 1 | 12 | yes | — |
| 100 | C22⋊C4 | — | 4 | 12 | no | — |
| 101 | C22⋊C4 | — | 4 | 12 | no | — |
| 102 | C22⋊C4 | C22×C4 | 1 | 24 | yes | — |
| 103 | C42 | — | 2 | 24 | no | — |
| 104 | C42 | — | 4 | 12 | no | — |
| 105 | C42 | C22×C4 | 1 | 4 | yes | — |
| 106 | C42 | C22×C4 | 1 | 12 | yes | — |
| 107 | C42 | C22×C4 | 1 | 12 | yes | — |
| 108 | C42 | C22×C4 | 1 | 12 | yes | — |
| 109 | C23 | — | 2 | 6 | no | — |
| 110 | C23 | — | 2 | 3 | no | — |
| 111 | C23 | — | 2 | 3 | no | — |
| 112 | C23 | — | 2 | 6 | no | — |
| 113 | C23 | C23×C4 | 1 | 6 | yes | — |
| 114 | C23 | C25 | 1 | 1 | yes | characteristic |
| 115 | Q8 | #117 | 1 | 16 | yes | complemented |
| 116 | D4 | — | 2 | 24 | no | — |
| 117 | C2×C4 | — | 2 | 2 | no | — |
| 118 | C2×C4 | — | 2 | 3 | no | — |
| 119 | C2×C4 | — | 2 | 3 | no | — |
| 120 | C2×C4 | — | 2 | 6 | no | — |
| 121 | C2×C4 | — | 2 | 2 | no | — |
| 122 | C2×C4 | — | 2 | 24 | no | — |
| 123 | C2×C4 | — | 2 | 24 | no | — |
| 124 | C2×C4 | — | 2 | 4 | no | — |
| 125 | C2×C4 | — | 2 | 12 | no | — |
| 126 | C2×C4 | — | 2 | 12 | no | — |
| 127 | C2×C4 | — | 2 | 4 | no | — |
| 128 | C2×C4 | — | 2 | 4 | no | — |
| 129 | C2×C4 | — | 2 | 12 | no | — |
| 130 | C2×C4 | — | 4 | 12 | no | — |
| 131 | C2×C4 | C23×C4 | 1 | 2 | yes | — |
| 132 | C2×C4 | C23×C4 | 1 | 2 | yes | — |
| 133 | C2×C4 | C23×C4 | 1 | 6 | yes | — |
| 134 | C2×C4 | C23×C4 | 1 | 6 | yes | — |
| 135 | C2×C4 | C23×C4 | 1 | 6 | yes | — |
| 136 | C2×C4 | C23×C4 | 1 | 2 | yes | — |
| 137 | C2×C4 | C2×#41 | 1 | 12 | yes | — |
| 138 | C2×C4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 139 | C2×C4 | 2−1+4 | 1 | 2 | yes | — |
| 140 | C22 | — | 2 | 6 | no | — |
| 141 | C22 | — | 2 | 6 | no | — |
| 142 | C22 | — | 2 | 6 | no | — |
| 143 | C22 | — | 4 | 6 | no | — |
| 144 | C22 | — | 4 | 6 | no | — |
| 145 | C22 | C24×C4 | 1 | 1 | yes | characteristic |
| 146 | C22 | C22×#41 | 1 | 2 | yes | — |
| 147 | C22 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 148 | C22 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 149 | C22 | #585 | 1 | 2 | yes | — |
| 150 | C4 | — | 2 | 2 | no | — |
| 151 | C4 | — | 2 | 6 | no | — |
| 152 | C4 | — | 2 | 2 | no | — |
| 153 | C4 | — | 2 | 6 | no | — |
| 154 | C4 | — | 4 | 12 | no | — |
| 155 | C4 | — | 4 | 4 | no | — |
| 156 | C4 | C2×#117 | 1 | 12 | yes | — |
| 157 | C4 | #520 | 1 | 4 | yes | — |
| 158 | C2 | — | 4 | 6 | no | — |
| 159 | C2 | C22×#117 | 1 | 1 | yes | characteristic |
| 160 | C2 | C2×#520 | 1 | 1 | yes | characteristic |
| 161 | C2 | #3428 | 1 | 1 | yes | characteristic |
| 162 | C2 | #3517 | 1 | 2 | yes | — |
| 163 | C2 | #3518 | 1 | 2 | yes | — |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta