Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
994 conjugacy classes of subgroups; only the 262 normal subgroups are shown.
This group has 994 conjugacy classes of subgroups — too many to draw. Only its 262 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 | #60191 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #23566 | C2 | 1 | 2 | yes | maximal |
| 3 | #21155 | C2 | 1 | 3 | yes | maximal |
| 4 | #11225 | C2 | 1 | 6 | yes | maximal |
| 5 | #10866 | C2 | 1 | 3 | yes | maximal |
| 6 | #3458 | C22 | 1 | 1 | yes | characteristic |
| 7 | #2961 | — | 2 | 6 | no | — |
| 8 | #2960 | — | 2 | 6 | no | — |
| 9 | C2×#427 | C22 | 1 | 3 | yes | — |
| 10 | C22×#106 | C22 | 1 | 1 | yes | characteristic |
| 11 | #2905 | C22 | 1 | 3 | yes | — |
| 12 | C2×#405 | C22 | 1 | 3 | yes | — |
| 13 | #2317 | C22 | 1 | 6 | yes | — |
| 14 | #2062 | — | 2 | 8 | no | — |
| 15 | #1943 | — | 2 | 6 | no | — |
| 16 | #1941 | — | 2 | 6 | no | — |
| 17 | #1740 | C22 | 1 | 6 | yes | — |
| 18 | #1736 | C22 | 1 | 6 | yes | — |
| 19 | C2×H(3,ℤ/4) | C22 | 1 | 6 | yes | — |
| 20 | C22×M16 | C23 | 1 | 3 | yes | — |
| 21 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 22 | #544 | — | 2 | 6 | no | — |
| 23 | #543 | D4 | 1 | 4 | yes | — |
| 24 | #533 | — | 2 | 6 | no | — |
| 25 | C2×#117 | C23 | 1 | 6 | yes | — |
| 26 | C2×#117 | C23 | 1 | 3 | yes | — |
| 27 | C22×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 28 | #446 | — | 2 | 24 | no | — |
| 29 | C8×Q8 | — | 2 | 8 | no | — |
| 30 | #427 | — | 2 | 6 | no | — |
| 31 | #427 | Q8 | 1 | 12 | yes | complemented |
| 32 | C2×#107 | — | 2 | 3 | no | — |
| 33 | C2×#106 | — | 2 | 3 | no | — |
| 34 | C2×#106 | — | 2 | 3 | no | — |
| 35 | C2×#106 | — | 2 | 3 | no | — |
| 36 | #424 | — | 2 | 12 | no | — |
| 37 | #423 | — | 2 | 6 | no | — |
| 38 | #405 | — | 2 | 6 | no | — |
| 39 | #405 | — | 2 | 6 | no | — |
| 40 | C2×#97 | D4 | 1 | 6 | yes | — |
| 41 | C2×C4×C8 | D4 | 1 | 6 | yes | — |
| 42 | #389 | — | 2 | 12 | no | — |
| 43 | #385 | — | 2 | 12 | no | — |
| 44 | #384 | — | 2 | 6 | no | — |
| 45 | #382 | — | 2 | 6 | no | — |
| 46 | #342 | — | 2 | 12 | no | — |
| 47 | #340 | — | 2 | 12 | no | — |
| 48 | H(3,ℤ/4) | — | 4 | 6 | no | — |
| 49 | C23×C4 | — | 2 | 2 | no | — |
| 50 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 51 | C2×M16 | — | 2 | 6 | no | — |
| 52 | C2×M16 | — | 2 | 3 | no | — |
| 53 | C2×M16 | — | 2 | 6 | no | — |
| 54 | C2×M16 | — | 2 | 3 | no | — |
| 55 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 56 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 57 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 58 | C22×C8 | C2×Q8 | 1 | 6 | yes | — |
| 59 | C22×C8 | #41 | 1 | 2 | yes | — |
| 60 | C4⋊Q8 | — | 2 | 12 | no | — |
| 61 | #125 | — | 2 | 12 | no | — |
| 62 | C22⋊Q8 | — | 2 | 24 | no | — |
| 63 | C4×Q8 | — | 2 | 8 | no | — |
| 64 | #117 | — | 2 | 6 | no | — |
| 65 | #117 | — | 2 | 6 | no | — |
| 66 | #117 | — | 2 | 6 | no | — |
| 67 | #117 | — | 2 | 6 | no | — |
| 68 | #117 | — | 2 | 6 | no | — |
| 69 | #117 | C2×Q8 | 1 | 12 | yes | — |
| 70 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 71 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 72 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 73 | C2×C4⋊C4 | — | 4 | 12 | no | — |
| 74 | C2×C4⋊C4 | #41 | 1 | 6 | yes | — |
| 75 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 76 | C2×C22⋊C4 | C2×D4 | 1 | 6 | yes | — |
| 77 | C2×C42 | — | 2 | 6 | no | — |
| 78 | C2×C42 | C2×D4 | 1 | 6 | yes | — |
| 79 | #107 | — | 2 | 12 | no | — |
| 80 | #106 | — | 2 | 12 | no | — |
| 81 | #106 | — | 4 | 1 | no | — |
| 82 | #106 | — | 4 | 3 | no | — |
| 83 | C4⋊C8 | — | 2 | 24 | no | — |
| 84 | #97 | — | 2 | 12 | no | — |
| 85 | C4×C8 | — | 2 | 6 | no | — |
| 86 | C4×C8 | — | 2 | 6 | no | — |
| 87 | #95 | — | 4 | 12 | no | — |
| 88 | C24 | C22×D4 | 1 | 1 | yes | characteristic |
| 89 | C2×Q8 | — | 2 | 8 | no | — |
| 90 | C22×C4 | — | 4 | 6 | no | — |
| 91 | C22×C4 | — | 4 | 6 | no | — |
| 92 | C22×C4 | — | 16 | 2 | no | — |
| 93 | C22×C4 | #121 | 1 | 12 | yes | — |
| 94 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 95 | C22×C4 | C22×Q8 | 1 | 3 | yes | — |
| 96 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 97 | C22×C4 | C2×#41 | 1 | 3 | yes | — |
| 98 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 99 | C22×C4 | 2−1+4 | 1 | 3 | yes | — |
| 100 | M16 | — | 4 | 6 | no | — |
| 101 | M16 | — | 4 | 6 | no | — |
| 102 | C2×C8 | — | 2 | 6 | no | — |
| 103 | C2×C8 | — | 2 | 6 | no | — |
| 104 | C2×C8 | — | 2 | 6 | no | — |
| 105 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 106 | C2×C8 | #120 | 1 | 3 | yes | — |
| 107 | C2×C8 | C22⋊Q8 | 1 | 6 | yes | — |
| 108 | C2×C8 | C22⋊Q8 | 1 | 6 | yes | — |
| 109 | C2×C8 | C4⋊Q8 | 1 | 6 | yes | — |
| 110 | C2×C8 | C4⋊Q8 | 1 | 6 | yes | — |
| 111 | C4⋊C4 | — | 2 | 24 | no | — |
| 112 | C4⋊C4 | — | 2 | 12 | no | — |
| 113 | C4⋊C4 | — | 2 | 12 | no | — |
| 114 | C4⋊C4 | — | 4 | 6 | no | — |
| 115 | C4⋊C4 | — | 4 | 2 | no | — |
| 116 | C4⋊C4 | — | 4 | 6 | no | — |
| 117 | C4⋊C4 | — | 16 | 12 | no | — |
| 118 | C22⋊C4 | — | 2 | 12 | no | — |
| 119 | C22⋊C4 | — | 2 | 12 | no | — |
| 120 | C22⋊C4 | — | 4 | 6 | no | — |
| 121 | C42 | — | 2 | 6 | no | — |
| 122 | C42 | — | 2 | 6 | no | — |
| 123 | C42 | — | 2 | 12 | no | — |
| 124 | C42 | — | 4 | 6 | no | — |
| 125 | C23 | — | 4 | 2 | no | — |
| 126 | C23 | C2×#121 | 1 | 3 | yes | — |
| 127 | C23 | D4×Q8 | 1 | 3 | yes | — |
| 128 | C23 | #550 | 1 | 1 | yes | characteristic |
| 129 | Q8 | — | 8 | 8 | no | — |
| 130 | C2×C4 | — | 2 | 12 | no | — |
| 131 | C2×C4 | — | 4 | 12 | no | — |
| 132 | C2×C4 | — | 4 | 12 | no | — |
| 133 | C2×C4 | — | 8 | 6 | no | — |
| 134 | C2×C4 | — | 8 | 6 | no | — |
| 135 | C2×C4 | — | 8 | 12 | no | — |
| 136 | C2×C4 | — | 8 | 2 | no | — |
| 137 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 138 | C2×C4 | C2×C22⋊Q8 | 1 | 3 | yes | — |
| 139 | C2×C4 | #525 | 1 | 3 | yes | — |
| 140 | C2×C4 | C2×C4⋊Q8 | 1 | 3 | yes | — |
| 141 | C2×C4 | #536 | 1 | 3 | yes | — |
| 142 | C2×C4 | #537 | 1 | 3 | yes | — |
| 143 | C2×C4 | #554 | 1 | 3 | yes | — |
| 144 | C2×C4 | #556 | 1 | 3 | yes | — |
| 145 | C2×C4 | #557 | 1 | 6 | yes | — |
| 146 | C8 | — | 4 | 6 | no | — |
| 147 | C8 | #395 | 1 | 2 | yes | — |
| 148 | C8 | #395 | 1 | 6 | yes | — |
| 149 | C22 | — | 2 | 6 | no | — |
| 150 | C22 | — | 4 | 3 | no | — |
| 151 | C22 | — | 4 | 1 | no | — |
| 152 | C22 | #2538 | 1 | 3 | yes | — |
| 153 | C22 | #2546 | 1 | 3 | yes | — |
| 154 | C22 | #2847 | 1 | 1 | yes | characteristic |
| 155 | C4 | — | 4 | 12 | no | — |
| 156 | C4 | — | 8 | 12 | no | — |
| 157 | C4 | — | 16 | 2 | no | — |
| 158 | C4 | C2×#395 | 1 | 1 | yes | characteristic |
| 159 | C4 | #2581 | 1 | 3 | yes | — |
| 160 | C4 | #2584 | 1 | 3 | yes | — |
| 161 | C4 | #2616 | 1 | 1 | yes | characteristic |
| 162 | C2 | — | 2 | 3 | no | — |
| 163 | C2 | — | 2 | 3 | no | — |
| 164 | C2 | #15158 | 1 | 1 | yes | characteristic |
| 165 | C2 | #32947 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 31. #427 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60191 | 2 |
| #23566 | 2 |
| #21155 | 2 |
| #11225 | 2 |
| #10866 | 2 |