Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
800 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 234 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 | C2×#1998 | C2 | 1 | 3 | yes | maximal |
| 2 | #21177 | C2 | 1 | 3 | yes | maximal |
| 3 | #11312 | C2 | 1 | 2 | yes | maximal |
| 4 | #11252 | C2 | 1 | 6 | yes | maximal |
| 5 | C4×#338 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C22×#125 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#430 | C22 | 1 | 6 | yes | — |
| 8 | C2×#423 | C22 | 1 | 3 | yes | — |
| 9 | C2×C4×M16 | C22 | 1 | 3 | yes | — |
| 10 | #1998 | — | 2 | 12 | no | — |
| 11 | #1950 | — | 2 | 12 | no | — |
| 12 | C2×#343 | C22 | 1 | 6 | yes | — |
| 13 | C2×#343 | C22 | 1 | 6 | yes | — |
| 14 | #1738 | C22 | 1 | 6 | yes | — |
| 15 | C2×#338 | C22 | 1 | 3 | yes | — |
| 16 | C2×#338 | C22 | 1 | 1 | yes | characteristic |
| 17 | C22×M16 | C23 | 1 | 6 | yes | — |
| 18 | C22×M16 | C23 | 1 | 3 | yes | — |
| 19 | C22×M16 | C23 | 1 | 3 | yes | — |
| 20 | C2×#125 | — | 2 | 6 | no | — |
| 21 | C2×#125 | — | 2 | 3 | no | — |
| 22 | C2×#125 | — | 2 | 3 | no | — |
| 23 | C22×C4⋊C4 | Q8 | 1 | 4 | yes | — |
| 24 | C22×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 25 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 26 | #430 | — | 2 | 12 | no | — |
| 27 | #430 | — | 4 | 6 | no | — |
| 28 | C2×#108 | — | 2 | 12 | no | — |
| 29 | #423 | — | 2 | 3 | no | — |
| 30 | #423 | — | 2 | 3 | no | — |
| 31 | #423 | D4 | 1 | 12 | yes | — |
| 32 | C2×C4⋊C8 | — | 2 | 6 | no | — |
| 33 | C4×M16 | — | 2 | 6 | no | — |
| 34 | C4×M16 | — | 2 | 6 | no | — |
| 35 | C2×#97 | — | 2 | 3 | no | — |
| 36 | C2×C4×C8 | — | 2 | 3 | no | — |
| 37 | #343 | — | 2 | 12 | no | — |
| 38 | #343 | — | 2 | 12 | no | — |
| 39 | #341 | — | 2 | 12 | no | — |
| 40 | #338 | — | 2 | 6 | no | — |
| 41 | #338 | Q8 | 1 | 4 | yes | — |
| 42 | C23×C4 | C2×Q8 | 1 | 4 | yes | — |
| 43 | C23×C4 | C2×Q8 | 1 | 2 | yes | — |
| 44 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 45 | C2×M16 | — | 2 | 3 | no | — |
| 46 | C2×M16 | — | 2 | 3 | no | — |
| 47 | C2×M16 | — | 2 | 6 | no | — |
| 48 | C2×M16 | — | 2 | 3 | no | — |
| 49 | C2×M16 | — | 2 | 3 | no | — |
| 50 | C2×M16 | — | 2 | 3 | no | — |
| 51 | C2×M16 | — | 2 | 3 | no | — |
| 52 | C2×M16 | — | 2 | 6 | no | — |
| 53 | C2×M16 | — | 2 | 6 | no | — |
| 54 | C2×M16 | — | 2 | 3 | no | — |
| 55 | C2×M16 | — | 2 | 6 | no | — |
| 56 | C2×M16 | — | 2 | 3 | no | — |
| 57 | C2×M16 | — | 2 | 3 | no | — |
| 58 | C2×M16 | — | 2 | 3 | no | — |
| 59 | C2×M16 | — | 2 | 6 | no | — |
| 60 | C2×M16 | C2×D4 | 1 | 12 | yes | — |
| 61 | C2×M16 | C2×Q8 | 1 | 6 | yes | — |
| 62 | C2×M16 | #41 | 1 | 6 | yes | — |
| 63 | C22×C8 | — | 2 | 6 | no | — |
| 64 | C22×C8 | — | 2 | 3 | no | — |
| 65 | C22×C8 | — | 2 | 3 | no | — |
| 66 | #125 | — | 4 | 6 | no | — |
| 67 | #125 | — | 4 | 6 | no | — |
| 68 | #125 | — | 4 | 1 | no | — |
| 69 | #125 | — | 4 | 3 | no | — |
| 70 | C2×C4⋊C4 | — | 2 | 12 | no | — |
| 71 | C2×C4⋊C4 | — | 2 | 12 | no | — |
| 72 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 73 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 74 | C2×C42 | C2×D4 | 1 | 6 | yes | — |
| 75 | C2×C42 | #41 | 1 | 3 | yes | — |
| 76 | C2×C42 | #41 | 1 | 3 | yes | — |
| 77 | #108 | — | 4 | 24 | no | — |
| 78 | C4⋊C8 | — | 2 | 12 | no | — |
| 79 | C4⋊C8 | — | 2 | 12 | no | — |
| 80 | #97 | — | 4 | 6 | no | — |
| 81 | C4×C8 | — | 4 | 6 | no | — |
| 82 | C24 | C22×Q8 | 1 | 1 | yes | characteristic |
| 83 | C22×C4 | — | 2 | 12 | no | — |
| 84 | C22×C4 | — | 8 | 2 | no | — |
| 85 | C22×C4 | — | 8 | 4 | no | — |
| 86 | C22×C4 | C22⋊Q8 | 1 | 6 | yes | — |
| 87 | C22×C4 | C22⋊Q8 | 1 | 6 | yes | — |
| 88 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 89 | C22×C4 | C22×Q8 | 1 | 1 | yes | characteristic |
| 90 | C22×C4 | C2×#41 | 1 | 3 | yes | — |
| 91 | C22×C4 | C2×#41 | 1 | 3 | yes | — |
| 92 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 93 | C22×C4 | 2−1+4 | 1 | 3 | yes | — |
| 94 | M16 | — | 2 | 6 | no | — |
| 95 | M16 | — | 2 | 12 | no | — |
| 96 | M16 | — | 2 | 12 | no | — |
| 97 | M16 | — | 2 | 6 | no | — |
| 98 | M16 | — | 2 | 6 | no | — |
| 99 | M16 | — | 2 | 6 | no | — |
| 100 | M16 | — | 4 | 6 | no | — |
| 101 | M16 | — | 4 | 6 | no | — |
| 102 | M16 | — | 4 | 3 | no | — |
| 103 | M16 | — | 4 | 3 | no | — |
| 104 | M16 | — | 4 | 3 | no | — |
| 105 | M16 | — | 4 | 3 | no | — |
| 106 | C2×C8 | — | 2 | 12 | no | — |
| 107 | C2×C8 | — | 2 | 6 | no | — |
| 108 | C2×C8 | — | 2 | 6 | no | — |
| 109 | C2×C8 | — | 4 | 3 | no | — |
| 110 | C2×C8 | — | 4 | 3 | no | — |
| 111 | C2×C8 | — | 4 | 3 | no | — |
| 112 | C2×C8 | — | 4 | 3 | no | — |
| 113 | C2×C8 | — | 4 | 6 | no | — |
| 114 | C2×C8 | — | 4 | 6 | no | — |
| 115 | C4⋊C4 | — | 4 | 6 | no | — |
| 116 | C4⋊C4 | — | 4 | 2 | no | — |
| 117 | C4⋊C4 | — | 4 | 4 | no | — |
| 118 | C4⋊C4 | — | 4 | 12 | no | — |
| 119 | C42 | #121 | 1 | 6 | yes | — |
| 120 | C42 | #121 | 1 | 6 | yes | — |
| 121 | C42 | #126 | 1 | 1 | yes | characteristic |
| 122 | C42 | #126 | 1 | 3 | yes | — |
| 123 | C23 | — | 4 | 2 | no | — |
| 124 | C23 | C2×C22⋊Q8 | 1 | 3 | yes | — |
| 125 | C23 | #557 | 1 | 3 | yes | — |
| 126 | C23 | Q82 | 1 | 1 | yes | characteristic |
| 127 | C2×C4 | — | 4 | 6 | no | — |
| 128 | C2×C4 | — | 4 | 6 | no | — |
| 129 | C2×C4 | — | 4 | 4 | no | — |
| 130 | C2×C4 | — | 8 | 12 | no | — |
| 131 | C2×C4 | #396 | 1 | 2 | yes | — |
| 132 | C2×C4 | #396 | 1 | 6 | yes | — |
| 133 | C2×C4 | C2×#121 | 1 | 3 | yes | — |
| 134 | C2×C4 | C2×#126 | 1 | 1 | yes | characteristic |
| 135 | C2×C4 | #529 | 1 | 3 | yes | — |
| 136 | C2×C4 | #533 | 1 | 3 | yes | — |
| 137 | C2×C4 | #541 | 1 | 3 | yes | — |
| 138 | C2×C4 | #542 | 1 | 3 | yes | — |
| 139 | C2×C4 | #546 | 1 | 3 | yes | — |
| 140 | C2×C4 | #547 | 1 | 3 | yes | — |
| 141 | C2×C4 | #548 | 1 | 3 | yes | — |
| 142 | C2×C4 | D4×Q8 | 1 | 3 | yes | — |
| 143 | C8 | — | 4 | 12 | no | — |
| 144 | C8 | — | 4 | 6 | no | — |
| 145 | C8 | — | 4 | 6 | no | — |
| 146 | C22 | — | 2 | 6 | no | — |
| 147 | C22 | — | 4 | 3 | no | — |
| 148 | C22 | — | 4 | 1 | no | — |
| 149 | C22 | C2×#396 | 1 | 1 | yes | characteristic |
| 150 | C22 | #2451 | 1 | 3 | yes | — |
| 151 | C22 | #2453 | 1 | 3 | yes | — |
| 152 | C4 | — | 2 | 2 | no | — |
| 153 | C4 | — | 2 | 6 | no | — |
| 154 | C4 | — | 8 | 4 | no | — |
| 155 | C4 | #2427 | 1 | 3 | yes | — |
| 156 | C4 | #2429 | 1 | 3 | yes | — |
| 157 | C4 | #2645 | 1 | 1 | yes | characteristic |
| 158 | C4 | #2863 | 1 | 1 | yes | characteristic |
| 159 | C2 | — | 2 | 3 | no | — |
| 160 | C2 | — | 2 | 3 | no | — |
| 161 | C2 | #15189 | 1 | 1 | yes | characteristic |
| 162 | C2 | #23622 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#396 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1998 | 2 |
| #21177 | 2 |
| #11312 | 2 |
| #11252 | 2 |
| C4×#338 | 2 |