Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
5110 conjugacy classes of subgroups.
The subgroup lattice has 176 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #59875 | C2 | 1 | 32 | yes | maximal, direct product factor |
| 2 | C2×#2810 | C2 | 1 | 4 | yes | maximal |
| 3 | C2×#2568 | C2 | 1 | 6 | yes | maximal |
| 4 | C2×#2514 | C2 | 1 | 12 | yes | maximal |
| 5 | C2×#2355 | C2 | 1 | 6 | yes | maximal |
| 6 | C2×C4×C4⋊Q8 | C2 | 1 | 3 | yes | maximal |
| 7 | C22×C4⋊Q8 | C22 | 1 | 3 | yes | — |
| 8 | C22×C4×Q8 | C22 | 1 | 6 | yes | — |
| 9 | #2810 | C22 | 1 | 64 | yes | — |
| 10 | #2568 | C22 | 1 | 96 | yes | — |
| 11 | #2514 | C22 | 1 | 192 | yes | — |
| 12 | C2×#401 | C22 | 1 | 8 | yes | — |
| 13 | C2×#400 | C22 | 1 | 24 | yes | — |
| 14 | #2355 | C22 | 1 | 96 | yes | — |
| 15 | C4×C4⋊Q8 | C22 | 1 | 48 | yes | — |
| 16 | C2×#391 | C22 | 1 | 12 | yes | — |
| 17 | C2×#391 | C22 | 1 | 12 | yes | — |
| 18 | C2×#391 | C22 | 1 | 6 | yes | — |
| 19 | C2×#389 | C22 | 1 | 6 | yes | — |
| 20 | C2×#389 | C22 | 1 | 6 | yes | — |
| 21 | C2×#387 | C22 | 1 | 24 | yes | — |
| 22 | C2×#384 | C22 | 1 | 3 | yes | — |
| 23 | C2×#383 | C22 | 1 | 8 | yes | — |
| 24 | C2×#382 | C22 | 1 | 12 | yes | — |
| 25 | C2×C4×C4⋊C4 | C22 | 1 | 12 | yes | — |
| 26 | C2×C4×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 27 | C2×C4×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 28 | C2×C43 | C22 | 1 | 1 | yes | characteristic |
| 29 | C23×Q8 | C23 | 1 | 6 | yes | — |
| 30 | C2×C4⋊Q8 | — | 2 | 24 | no | — |
| 31 | C2×C4⋊Q8 | C23 | 1 | 24 | yes | — |
| 32 | C2×C4×Q8 | — | 2 | 48 | no | — |
| 33 | C2×C4×Q8 | C23 | 1 | 48 | yes | — |
| 34 | C22×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 35 | C22×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 36 | C22×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 37 | C22×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 38 | C22×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 39 | C22×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 40 | C22×C42 | C23 | 1 | 12 | yes | — |
| 41 | C22×C42 | C23 | 1 | 3 | yes | — |
| 42 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 43 | C22×C42 | C23 | 1 | 8 | yes | — |
| 44 | C22×C42 | C23 | 1 | 3 | yes | — |
| 45 | #401 | C23 | 1 | 64 | yes | — |
| 46 | #400 | C23 | 1 | 192 | yes | — |
| 47 | #391 | C23 | 1 | 48 | yes | — |
| 48 | #391 | C23 | 1 | 96 | yes | — |
| 49 | #391 | C23 | 1 | 96 | yes | — |
| 50 | #389 | C23 | 1 | 48 | yes | — |
| 51 | #389 | C23 | 1 | 48 | yes | — |
| 52 | #387 | C23 | 1 | 192 | yes | — |
| 53 | #384 | C23 | 1 | 24 | yes | — |
| 54 | #383 | C23 | 1 | 64 | yes | — |
| 55 | #382 | C23 | 1 | 96 | yes | — |
| 56 | C4×C4⋊C4 | C23 | 1 | 48 | yes | — |
| 57 | C4×C4⋊C4 | C23 | 1 | 48 | yes | — |
| 58 | C4×C4⋊C4 | C23 | 1 | 96 | yes | — |
| 59 | C2×#95 | C23 | 1 | 8 | yes | — |
| 60 | C2×#95 | C23 | 1 | 12 | yes | — |
| 61 | C2×#95 | C23 | 1 | 24 | yes | — |
| 62 | C2×#95 | C23 | 1 | 24 | yes | — |
| 63 | C2×#95 | C23 | 1 | 12 | yes | — |
| 64 | C43 | C23 | 1 | 8 | yes | — |
| 65 | C22×Q8 | — | 2 | 24 | no | — |
| 66 | C22×Q8 | — | 2 | 24 | no | — |
| 67 | C22×Q8 | — | 2 | 24 | no | — |
| 68 | C22×Q8 | C24 | 1 | 24 | yes | — |
| 69 | C23×C4 | C24 | 1 | 12 | yes | — |
| 70 | C23×C4 | C24 | 1 | 12 | yes | — |
| 71 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 72 | C23×C4 | C24 | 1 | 3 | yes | — |
| 73 | C23×C4 | C24 | 1 | 3 | yes | — |
| 74 | C4⋊Q8 | — | 2 | 96 | no | — |
| 75 | C4×Q8 | — | 2 | 192 | no | — |
| 76 | C2×C4⋊C4 | — | 2 | 24 | no | — |
| 77 | C2×C4⋊C4 | — | 2 | 48 | no | — |
| 78 | C2×C4⋊C4 | — | 2 | 24 | no | — |
| 79 | C2×C4⋊C4 | — | 2 | 24 | no | — |
| 80 | C2×C4⋊C4 | — | 2 | 24 | no | — |
| 81 | C2×C4⋊C4 | — | 4 | 12 | no | — |
| 82 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 83 | C2×C4⋊C4 | C24 | 1 | 48 | yes | — |
| 84 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 85 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 86 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 87 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 88 | C2×C42 | — | 2 | 48 | no | — |
| 89 | C2×C42 | — | 2 | 12 | no | — |
| 90 | C2×C42 | — | 4 | 2 | no | — |
| 91 | C2×C42 | — | 4 | 16 | no | — |
| 92 | C2×C42 | #41 | 1 | 24 | yes | — |
| 93 | C2×C42 | C24 | 1 | 48 | yes | — |
| 94 | C2×C42 | C24 | 1 | 12 | yes | — |
| 95 | C2×C42 | C24 | 1 | 4 | yes | — |
| 96 | C2×C42 | C24 | 1 | 32 | yes | — |
| 97 | C2×C42 | C24 | 1 | 12 | yes | — |
| 98 | #95 | C24 | 1 | 96 | yes | — |
| 99 | #95 | C24 | 1 | 48 | yes | — |
| 100 | #95 | C24 | 1 | 48 | yes | — |
| 101 | #95 | C24 | 1 | 96 | yes | — |
| 102 | #95 | C24 | 1 | 32 | yes | — |
| 103 | C24 | C25 | 1 | 1 | yes | characteristic |
| 104 | C2×Q8 | — | 2 | 48 | no | — |
| 105 | C2×Q8 | — | 2 | 48 | no | — |
| 106 | C2×Q8 | — | 2 | 48 | no | — |
| 107 | C2×Q8 | — | 4 | 96 | no | — |
| 108 | C22×C4 | — | 2 | 6 | no | — |
| 109 | C22×C4 | — | 2 | 6 | no | — |
| 110 | C22×C4 | — | 2 | 24 | no | — |
| 111 | C22×C4 | — | 2 | 24 | no | — |
| 112 | C22×C4 | — | 2 | 24 | no | — |
| 113 | C22×C4 | — | 2 | 6 | no | — |
| 114 | C22×C4 | — | 2 | 24 | no | — |
| 115 | C22×C4 | — | 2 | 24 | no | — |
| 116 | C22×C4 | — | 2 | 24 | no | — |
| 117 | C22×C4 | C2×#41 | 1 | 12 | yes | — |
| 118 | C22×C4 | C2×#41 | 1 | 12 | yes | — |
| 119 | C22×C4 | C2×#41 | 1 | 12 | yes | — |
| 120 | C22×C4 | 2−1+4 | 1 | 12 | yes | — |
| 121 | C22×C4 | C25 | 1 | 2 | yes | — |
| 122 | C22×C4 | C25 | 1 | 24 | yes | — |
| 123 | C22×C4 | C25 | 1 | 24 | yes | — |
| 124 | C22×C4 | C25 | 1 | 6 | yes | — |
| 125 | C22×C4 | C25 | 1 | 6 | yes | — |
| 126 | C4⋊C4 | — | 2 | 48 | no | — |
| 127 | C4⋊C4 | — | 2 | 96 | no | — |
| 128 | C4⋊C4 | — | 2 | 48 | no | — |
| 129 | C4⋊C4 | — | 2 | 48 | no | — |
| 130 | C4⋊C4 | — | 2 | 48 | no | — |
| 131 | C4⋊C4 | — | 4 | 24 | no | — |
| 132 | C42 | — | 2 | 96 | no | — |
| 133 | C42 | — | 2 | 24 | no | — |
| 134 | C42 | — | 4 | 4 | no | — |
| 135 | C42 | — | 4 | 32 | no | — |
| 136 | C42 | C2×#41 | 1 | 48 | yes | — |
| 137 | C23 | C22×#41 | 1 | 6 | yes | — |
| 138 | C23 | C2×2−1+4 | 1 | 6 | yes | — |
| 139 | C23 | #585 | 1 | 2 | yes | — |
| 140 | C23 | C26 | 1 | 1 | yes | characteristic |
| 141 | Q8 | — | 4 | 96 | no | — |
| 142 | C2×C4 | — | 2 | 6 | no | — |
| 143 | C2×C4 | — | 2 | 6 | no | — |
| 144 | C2×C4 | — | 2 | 24 | no | — |
| 145 | C2×C4 | — | 2 | 24 | no | — |
| 146 | C2×C4 | — | 2 | 24 | no | — |
| 147 | C2×C4 | — | 2 | 24 | no | — |
| 148 | C2×C4 | — | 2 | 6 | no | — |
| 149 | C2×C4 | — | 2 | 24 | no | — |
| 150 | C2×C4 | — | 2 | 24 | no | — |
| 151 | C2×C4 | — | 2 | 24 | no | — |
| 152 | C2×C4 | — | 4 | 48 | no | — |
| 153 | C2×C4 | — | 4 | 48 | no | — |
| 154 | C2×C4 | — | 4 | 4 | no | — |
| 155 | C2×C4 | #556 | 1 | 48 | yes | — |
| 156 | C2×C4 | C22×#41 | 1 | 12 | yes | — |
| 157 | C2×C4 | C22×#41 | 1 | 12 | yes | — |
| 158 | C2×C4 | C22×#41 | 1 | 12 | yes | — |
| 159 | C2×C4 | C2×2−1+4 | 1 | 12 | yes | — |
| 160 | C22 | C2×#556 | 1 | 12 | yes | — |
| 161 | C22 | #3507 | 1 | 12 | yes | — |
| 162 | C22 | #3556 | 1 | 4 | yes | — |
| 163 | C22 | C23×#41 | 1 | 3 | yes | — |
| 164 | C22 | C22×2−1+4 | 1 | 3 | yes | — |
| 165 | C22 | C2×#585 | 1 | 1 | yes | characteristic |
| 166 | C4 | — | 2 | 12 | no | — |
| 167 | C4 | — | 4 | 2 | no | — |
| 168 | C4 | — | 4 | 24 | no | — |
| 169 | C4 | — | 4 | 24 | no | — |
| 170 | C4 | C2×#556 | 1 | 24 | yes | — |
| 171 | C2 | #59875 | 1 | 8 | yes | direct product factor |
| 172 | C2 | C22×#556 | 1 | 3 | yes | — |
| 173 | C2 | C2×#3556 | 1 | 1 | yes | characteristic |
| 174 | C2 | C2×#3507 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #59875 | C2 | direct (N × H) |
| 171. C2 | #59875 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #59875 | 2 |
| C2×#2810 | 2 |
| C2×#2568 | 2 |
| C2×#2514 | 2 |
| C2×#2355 | 2 |
| C2×C4×C4⋊Q8 | 2 |