← C22×#22548721 #44565069
Subgroups — C22×#22548721 #44565069
Order 32128.
1504 conjugacy classes of subgroups.
The subgroup lattice has 154 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C22×#16860810 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#22548721 | C2 | 1 | 24 | yes | maximal, direct product factor |
| 3 | C22×#16860794 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C22×#16860793 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C22×C8×D251 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C23×C8×C251 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C2×#16860810 | C4 | 1 | 12 | yes | — |
| 8 | C2×#16860810 | C22 | 1 | 12 | yes | — |
| 9 | C22×Dic502 | C4 | 1 | 1 | yes | characteristic |
| 10 | C22×D1004 | C4 | 1 | 1 | yes | characteristic |
| 11 | C22×C22⋊D251 | C4 | 1 | 2 | yes | — |
| 12 | C22×C4×D251 | C22 | 1 | 2 | yes | — |
| 13 | #22548721 | C22 | 1 | 64 | yes | direct product factor |
| 14 | C2×#16860794 | C22 | 1 | 12 | yes | complemented |
| 15 | C2×#16860793 | C22 | 1 | 24 | yes | complemented |
| 16 | C22×C251⋊C8 | C22 | 1 | 2 | yes | — |
| 17 | C2×C8×D251 | C22 | 1 | 24 | yes | complemented |
| 18 | C23×C4×C251 | C22 | 1 | 1 | yes | characteristic |
| 19 | C22×C8×C251 | C22 | 1 | 1 | yes | characteristic |
| 20 | C22×C8×C251 | C22 | 1 | 1 | yes | characteristic |
| 21 | C22×C8×C251 | C22 | 1 | 12 | yes | complemented |
| 22 | C23×D251 | C2×C4 | 1 | 2 | yes | — |
| 23 | #16860810 | C2×C4 | 1 | 48 | yes | — |
| 24 | #16860810 | C23 | 1 | 16 | yes | — |
| 25 | C2×Dic502 | C2×C4 | 1 | 12 | yes | — |
| 26 | C2×D1004 | C2×C4 | 1 | 12 | yes | — |
| 27 | C2×C22⋊D251 | C2×C4 | 1 | 24 | yes | — |
| 28 | C22×Dic251 | C2×C4 | 1 | 2 | yes | — |
| 29 | C2×C4×D251 | C2×C4 | 1 | 12 | yes | — |
| 30 | C2×C4×D251 | C23 | 1 | 12 | yes | — |
| 31 | #16860794 | C23 | 1 | 16 | yes | complemented |
| 32 | #16860793 | C23 | 1 | 32 | yes | complemented |
| 33 | C2×C251⋊C8 | C23 | 1 | 12 | yes | — |
| 34 | C8×D251 | C23 | 1 | 32 | yes | complemented |
| 35 | C24×C251 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | C22×C4×C251 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C22×C4×C251 | C2×C4 | 1 | 6 | yes | — |
| 38 | C22×C4×C251 | C23 | 1 | 1 | yes | characteristic |
| 39 | C22×C4×C251 | C23 | 1 | 6 | yes | — |
| 40 | C2×C8×C251 | C23 | 1 | 6 | yes | — |
| 41 | C2×C8×C251 | C23 | 1 | 6 | yes | — |
| 42 | C2×C8×C251 | C23 | 1 | 16 | yes | complemented |
| 43 | C22×D251 | — | 2 | 8 | no | — |
| 44 | C22×D251 | C22×C4 | 1 | 12 | yes | — |
| 45 | Dic502 | C22×C4 | 1 | 16 | yes | — |
| 46 | D1004 | C22×C4 | 1 | 16 | yes | — |
| 47 | C22⋊D251 | C22×C4 | 1 | 32 | yes | — |
| 48 | C2×Dic251 | C22×C4 | 1 | 12 | yes | — |
| 49 | C4×D251 | C22×C4 | 1 | 24 | yes | — |
| 50 | C4×D251 | C24 | 1 | 8 | yes | — |
| 51 | C251⋊C8 | C24 | 1 | 8 | yes | — |
| 52 | C23×C251 | — | 2 | 4 | no | — |
| 53 | C23×C251 | C22×C4 | 1 | 6 | yes | — |
| 54 | C23×C251 | C22×C4 | 1 | 1 | yes | characteristic |
| 55 | C2×C4×C251 | C22×C4 | 1 | 3 | yes | — |
| 56 | C2×C4×C251 | C22×C4 | 1 | 6 | yes | — |
| 57 | C2×C4×C251 | C22×C4 | 1 | 12 | yes | — |
| 58 | C2×C4×C251 | C24 | 1 | 3 | yes | — |
| 59 | C2×C4×C251 | C24 | 1 | 4 | yes | — |
| 60 | C8×C251 | C24 | 1 | 4 | yes | — |
| 61 | C8×C251 | C24 | 1 | 4 | yes | — |
| 62 | C22×C251 | — | 2 | 12 | no | — |
| 63 | C22×C251 | #131 | 1 | 4 | yes | complemented |
| 64 | C22×C251 | C23×C4 | 1 | 4 | yes | — |
| 65 | C22×C251 | C23×C4 | 1 | 3 | yes | — |
| 66 | C2×D251 | — | 2 | 24 | no | — |
| 67 | C2×D251 | C23×C4 | 1 | 8 | yes | — |
| 68 | C4×C251 | C23×C4 | 1 | 3 | yes | — |
| 69 | C4×C251 | C23×C4 | 1 | 4 | yes | — |
| 70 | C4×C251 | C25 | 1 | 1 | yes | characteristic |
| 71 | Dic251 | C23×C4 | 1 | 8 | yes | — |
| 72 | C2×C251 | — | 2 | 4 | no | — |
| 73 | C2×C251 | C2×#131 | 1 | 6 | yes | complemented |
| 74 | C2×C251 | C24×C4 | 1 | 1 | yes | characteristic |
| 75 | D251 | — | 2 | 8 | no | — |
| 76 | C251 | C22×#131 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 77 | C22×#131 | — | 251 | 1 | no | maximal, Sylow |
| 78 | C22×#41 | — | 251 | 1 | no | — |
| 79 | C2×#131 | — | 251 | 24 | no | — |
| 80 | C22×M16 | — | 251 | 1 | no | — |
| 81 | C22×M16 | — | 251 | 2 | no | — |
| 82 | C23×C8 | — | 251 | 2 | no | — |
| 83 | C23×C8 | D251 | 1 | 1 | yes | characteristic, complemented |
| 84 | C2×#41 | — | 251 | 12 | no | — |
| 85 | C2×#41 | — | 251 | 12 | no | — |
| 86 | C22×Q8 | — | 251 | 1 | no | — |
| 87 | C22×D4 | — | 251 | 2 | no | — |
| 88 | C22×D4 | — | 251 | 1 | no | — |
| 89 | C23×C4 | — | 251 | 2 | no | — |
| 90 | C23×C4 | C2×D251 | 1 | 1 | yes | characteristic |
| 91 | #131 | — | 251 | 64 | no | — |
| 92 | C2×M16 | — | 251 | 12 | no | — |
| 93 | C2×M16 | — | 251 | 24 | no | — |
| 94 | C22×C8 | — | 251 | 2 | no | — |
| 95 | C22×C8 | — | 251 | 24 | no | — |
| 96 | C22×C8 | C2×D251 | 1 | 1 | yes | characteristic |
| 97 | C22×C8 | C2×D251 | 1 | 1 | yes | characteristic |
| 98 | C22×C8 | C2×D251 | 1 | 12 | yes | complemented |
| 99 | C24 | — | 251 | 2 | no | — |
| 100 | C24 | C4×D251 | 1 | 1 | yes | characteristic |
| 101 | #41 | — | 251 | 48 | no | — |
| 102 | #41 | — | 251 | 16 | no | — |
| 103 | C2×Q8 | — | 251 | 12 | no | — |
| 104 | C2×D4 | — | 251 | 24 | no | — |
| 105 | C2×D4 | — | 251 | 12 | no | — |
| 106 | C22×C4 | — | 251 | 2 | no | — |
| 107 | C22×C4 | — | 251 | 12 | no | — |
| 108 | C22×C4 | — | 251 | 12 | no | — |
| 109 | C22×C4 | C4×D251 | 1 | 1 | yes | characteristic |
| 110 | C22×C4 | C4×D251 | 1 | 6 | yes | — |
| 111 | C22×C4 | C22×D251 | 1 | 1 | yes | characteristic |
| 112 | C22×C4 | C22×D251 | 1 | 6 | yes | — |
| 113 | M16 | — | 251 | 16 | no | — |
| 114 | M16 | — | 251 | 32 | no | — |
| 115 | C2×C8 | — | 251 | 12 | no | — |
| 116 | C2×C8 | — | 251 | 32 | no | — |
| 117 | C2×C8 | C22×D251 | 1 | 6 | yes | — |
| 118 | C2×C8 | C22×D251 | 1 | 6 | yes | — |
| 119 | C2×C8 | C22×D251 | 1 | 16 | yes | complemented |
| 120 | C23 | — | 2 | 4 | no | — |
| 121 | C23 | — | 502 | 8 | no | — |
| 122 | C23 | — | 251 | 12 | no | — |
| 123 | C23 | C2×C4×D251 | 1 | 6 | yes | — |
| 124 | C23 | C2×C4×D251 | 1 | 1 | yes | characteristic |
| 125 | Q8 | — | 251 | 16 | no | — |
| 126 | D4 | — | 251 | 32 | no | — |
| 127 | D4 | — | 251 | 16 | no | — |
| 128 | C2×C4 | — | 251 | 12 | no | — |
| 129 | C2×C4 | — | 251 | 24 | no | — |
| 130 | C2×C4 | — | 251 | 8 | no | — |
| 131 | C2×C4 | C2×C4×D251 | 1 | 3 | yes | — |
| 132 | C2×C4 | C2×C4×D251 | 1 | 6 | yes | — |
| 133 | C2×C4 | C2×C4×D251 | 1 | 12 | yes | — |
| 134 | C2×C4 | C23×D251 | 1 | 3 | yes | — |
| 135 | C2×C4 | C23×D251 | 1 | 4 | yes | — |
| 136 | C8 | — | 251 | 8 | no | — |
| 137 | C8 | C23×D251 | 1 | 4 | yes | — |
| 138 | C8 | C23×D251 | 1 | 4 | yes | — |
| 139 | C22 | — | 2 | 12 | no | — |
| 140 | C22 | — | 502 | 24 | no | — |
| 141 | C22 | — | 251 | 8 | no | — |
| 142 | C22 | #22548721 | 1 | 4 | yes | direct product factor |
| 143 | C22 | C22×C4×D251 | 1 | 4 | yes | — |
| 144 | C22 | C22×C4×D251 | 1 | 3 | yes | — |
| 145 | C4 | — | 251 | 8 | no | — |
| 146 | C4 | C22×C4×D251 | 1 | 3 | yes | — |
| 147 | C4 | C22×C4×D251 | 1 | 4 | yes | — |
| 148 | C4 | C24×D251 | 1 | 1 | yes | characteristic |
| 149 | C2 | — | 2 | 4 | no | — |
| 150 | C2 | — | 502 | 8 | no | — |
| 151 | C2 | C2×#22548721 | 1 | 6 | yes | direct product factor |
| 152 | C2 | C23×C4×D251 | 1 | 1 | yes | characteristic |
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