Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
815 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 815 conjugacy classes of subgroups — too many to draw. Only its 136 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 | #37642 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #27917 | C2 | 1 | 3 | yes | maximal |
| 3 | #10583 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #10358 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #8721 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C2×#482 | — | 2 | 6 | no | — |
| 7 | #2856 | C22 | 1 | 1 | yes | characteristic |
| 8 | #2844 | C22 | 1 | 4 | yes | — |
| 9 | #2628 | — | 2 | 6 | no | — |
| 10 | #2571 | — | 2 | 3 | no | — |
| 11 | #2551 | — | 2 | 3 | no | — |
| 12 | #2285 | C22 | 1 | 3 | yes | — |
| 13 | #2094 | — | 2 | 3 | no | — |
| 14 | #2093 | — | 2 | 3 | no | — |
| 15 | #1961 | C22 | 1 | 6 | yes | — |
| 16 | #1847 | C22 | 1 | 6 | yes | — |
| 17 | #1749 | C22 | 1 | 3 | yes | — |
| 18 | #1295 | C22 | 1 | 12 | yes | — |
| 19 | C2×C4⋊Q8 | — | 2 | 6 | no | — |
| 20 | C2×#127 | C23 | 1 | 1 | yes | characteristic |
| 21 | C2×#121 | — | 2 | 3 | no | — |
| 22 | C2×#117 | D4 | 1 | 6 | yes | — |
| 23 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 24 | #482 | — | 4 | 24 | no | — |
| 25 | C2×#107 | — | 2 | 6 | no | — |
| 26 | C2×#106 | — | 2 | 6 | no | — |
| 27 | C2×#102 | — | 2 | 6 | no | — |
| 28 | C2×#102 | — | 2 | 6 | no | — |
| 29 | C2×C22⋊C8 | — | 2 | 3 | no | — |
| 30 | C2×#97 | C23 | 1 | 6 | yes | — |
| 31 | #400 | — | 2 | 12 | no | — |
| 32 | #399 | — | 2 | 4 | no | — |
| 33 | #395 | — | 2 | 12 | no | — |
| 34 | #393 | — | 2 | 4 | no | — |
| 35 | #390 | — | 2 | 3 | no | — |
| 36 | #389 | — | 2 | 12 | no | — |
| 37 | #384 | D4 | 1 | 3 | yes | — |
| 38 | #384 | D4 | 1 | 3 | yes | — |
| 39 | #384 | C23 | 1 | 1 | yes | characteristic |
| 40 | #382 | — | 2 | 6 | no | — |
| 41 | #382 | C23 | 1 | 6 | yes | — |
| 42 | #380 | — | 2 | 6 | no | — |
| 43 | C4×C4⋊C4 | — | 2 | 6 | no | — |
| 44 | #336 | — | 2 | 3 | no | — |
| 45 | C22×Q8 | — | 2 | 4 | no | — |
| 46 | C22×D4 | — | 2 | 3 | no | — |
| 47 | C22×D4 | — | 4 | 1 | no | — |
| 48 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 49 | C22×C8 | — | 2 | 6 | no | — |
| 50 | C4⋊Q8 | — | 8 | 12 | no | — |
| 51 | #127 | — | 2 | 4 | no | — |
| 52 | #121 | — | 4 | 12 | no | — |
| 53 | #117 | — | 4 | 12 | no | — |
| 54 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 55 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 56 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 57 | C2×C4⋊C4 | — | 2 | 12 | no | — |
| 58 | C2×C4⋊C4 | C2×D4 | 1 | 6 | yes | — |
| 59 | C2×C4⋊C4 | C2×D4 | 1 | 6 | yes | — |
| 60 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 61 | C2×C22⋊C4 | — | 4 | 3 | no | — |
| 62 | C2×C42 | — | 2 | 4 | no | — |
| 63 | C2×C42 | — | 2 | 6 | no | — |
| 64 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 65 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 66 | #107 | — | 4 | 12 | no | — |
| 67 | #106 | — | 4 | 12 | no | — |
| 68 | #102 | — | 4 | 12 | no | — |
| 69 | #102 | — | 4 | 12 | no | — |
| 70 | C22⋊C8 | — | 2 | 12 | no | — |
| 71 | #97 | — | 2 | 12 | no | — |
| 72 | #95 | — | 2 | 4 | no | — |
| 73 | #95 | — | 2 | 12 | no | — |
| 74 | #95 | — | 2 | 12 | no | — |
| 75 | C24 | — | 4 | 1 | no | — |
| 76 | C24 | #120 | 1 | 1 | yes | characteristic |
| 77 | C2×Q8 | — | 8 | 12 | no | — |
| 78 | C2×D4 | — | 4 | 6 | no | — |
| 79 | C2×D4 | — | 4 | 6 | no | — |
| 80 | C2×D4 | — | 4 | 6 | no | — |
| 81 | C2×D4 | — | 8 | 6 | no | — |
| 82 | C22×C4 | — | 2 | 6 | no | — |
| 83 | C22×C4 | — | 2 | 6 | no | — |
| 84 | C22×C4 | — | 2 | 6 | no | — |
| 85 | C22×C4 | — | 2 | 6 | no | — |
| 86 | C22×C4 | — | 4 | 3 | no | — |
| 87 | C22×C4 | — | 4 | 1 | no | — |
| 88 | C22×C4 | #120 | 1 | 3 | yes | — |
| 89 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 90 | C2×C8 | — | 2 | 12 | no | — |
| 91 | C2×C8 | — | 4 | 6 | no | — |
| 92 | C2×C8 | — | 4 | 6 | no | — |
| 93 | C4⋊C4 | — | 4 | 6 | no | — |
| 94 | C4⋊C4 | — | 4 | 6 | no | — |
| 95 | C4⋊C4 | — | 4 | 12 | no | — |
| 96 | C4⋊C4 | — | 4 | 6 | no | — |
| 97 | C4⋊C4 | — | 4 | 12 | no | — |
| 98 | C4⋊C4 | — | 8 | 12 | no | — |
| 99 | C22⋊C4 | — | 8 | 6 | no | — |
| 100 | C22⋊C4 | — | 8 | 6 | no | — |
| 101 | C42 | — | 2 | 6 | no | — |
| 102 | C42 | — | 4 | 12 | no | — |
| 103 | C42 | 2−1+4 | 1 | 4 | yes | — |
| 104 | C23 | — | 2 | 1 | no | — |
| 105 | C23 | — | 2 | 3 | no | — |
| 106 | C23 | — | 2 | 3 | no | — |
| 107 | C23 | — | 8 | 1 | no | — |
| 108 | C23 | — | 8 | 3 | no | — |
| 109 | C23 | — | 8 | 3 | no | — |
| 110 | C23 | C2×#120 | 1 | 1 | yes | characteristic |
| 111 | Q8 | — | 16 | 8 | no | — |
| 112 | D4 | — | 8 | 12 | no | — |
| 113 | D4 | — | 16 | 4 | no | — |
| 114 | C2×C4 | — | 2 | 3 | no | — |
| 115 | C2×C4 | — | 2 | 3 | no | — |
| 116 | C2×C4 | — | 2 | 3 | no | — |
| 117 | C2×C4 | — | 2 | 3 | no | — |
| 118 | C2×C4 | — | 4 | 12 | no | — |
| 119 | C2×C4 | — | 4 | 6 | no | — |
| 120 | C2×C4 | — | 4 | 6 | no | — |
| 121 | C2×C4 | — | 4 | 12 | no | — |
| 122 | C2×C4 | — | 4 | 12 | no | — |
| 123 | C2×C4 | — | 8 | 3 | no | — |
| 124 | C2×C4 | #453 | 1 | 6 | yes | — |
| 125 | C2×C4 | #536 | 1 | 6 | yes | — |
| 126 | C8 | — | 4 | 12 | no | — |
| 127 | C22 | — | 4 | 3 | no | — |
| 128 | C22 | — | 4 | 1 | no | — |
| 129 | C22 | — | 4 | 3 | no | — |
| 130 | C22 | — | 16 | 3 | no | — |
| 131 | C22 | — | 16 | 1 | no | — |
| 132 | C22 | — | 16 | 3 | no | — |
| 133 | C22 | #2614 | 1 | 1 | yes | characteristic |
| 134 | C22 | C2×#453 | 1 | 3 | yes | — |
| 135 | C22 | #3020 | 1 | 3 | yes | — |
| 136 | C4 | — | 2 | 6 | no | — |
| 137 | C4 | — | 2 | 6 | no | — |
| 138 | C4 | — | 8 | 6 | no | — |
| 139 | C4 | — | 8 | 6 | no | — |
| 140 | C4 | — | 16 | 1 | no | — |
| 141 | C2 | — | 8 | 1 | no | — |
| 142 | C2 | — | 32 | 1 | no | — |
| 143 | C2 | #19054 | 1 | 3 | yes | — |
| 144 | C2 | #19073 | 1 | 1 | yes | characteristic |
| 145 | C2 | #29650 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #2856 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #37642 | C2 | semidirect |
| 3. #10583 | C2 | semidirect |
| 4. #10358 | C2 | semidirect |
| 5. #8721 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #37642 | 2 |
| #27917 | 2 |
| #10583 | 2 |
| #10358 | 2 |
| #8721 | 2 |