Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1096 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 1096 conjugacy classes of subgroups — too many to draw. Only its 222 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 | C24×C16 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#2216 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #32931 | C2 | 1 | 2 | yes | maximal |
| 4 | #13060 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #12834 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C24×C8 | C22 | 1 | 1 | yes | characteristic |
| 7 | C22×Q32 | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | C23×C16 | — | 2 | 12 | no | — |
| 9 | C23×C16 | C22 | 1 | 3 | yes | — |
| 10 | C23×C16 | C22 | 1 | 3 | yes | — |
| 11 | C2×#467 | C22 | 1 | 6 | yes | — |
| 12 | #2848 | — | 2 | 2 | no | — |
| 13 | #2216 | — | 2 | 12 | no | — |
| 14 | C2×#366 | C22 | 1 | 3 | yes | complemented |
| 15 | C2×#358 | C22 | 1 | 6 | yes | complemented |
| 16 | #1892 | — | 2 | 6 | no | — |
| 17 | #1808 | C22 | 1 | 6 | yes | — |
| 18 | #1380 | C22 | 1 | 6 | yes | complemented |
| 19 | C24×C4 | D4 | 1 | 1 | yes | characteristic |
| 20 | C22×Q16 | C23 | 1 | 2 | yes | — |
| 21 | C23×C8 | — | 2 | 6 | no | — |
| 22 | C23×C8 | — | 2 | 6 | no | — |
| 23 | C23×C8 | D4 | 1 | 3 | yes | — |
| 24 | C23×C8 | C23 | 1 | 3 | yes | — |
| 25 | C2×C22⋊Q8 | — | 2 | 6 | no | — |
| 26 | C2×Q32 | — | 2 | 6 | no | — |
| 27 | C22×C16 | — | 2 | 32 | no | — |
| 28 | C22×C16 | — | 2 | 6 | no | — |
| 29 | C22×C16 | — | 2 | 12 | no | — |
| 30 | C22×C16 | — | 2 | 12 | no | — |
| 31 | C22×C16 | D4 | 1 | 12 | yes | — |
| 32 | C22×C16 | C23 | 1 | 3 | yes | — |
| 33 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 34 | #467 | — | 2 | 12 | no | — |
| 35 | #467 | — | 2 | 12 | no | — |
| 36 | C2×#107 | C23 | 1 | 6 | yes | — |
| 37 | C2×#103 | — | 2 | 6 | no | — |
| 38 | #385 | — | 4 | 6 | no | — |
| 39 | #380 | — | 2 | 6 | no | — |
| 40 | #379 | — | 4 | 2 | no | — |
| 41 | #366 | — | 2 | 6 | no | — |
| 42 | #358 | — | 2 | 12 | no | — |
| 43 | #340 | — | 2 | 6 | no | — |
| 44 | C25 | D8 | 1 | 1 | yes | characteristic |
| 45 | C22×Q8 | — | 2 | 2 | no | — |
| 46 | C23×C4 | — | 2 | 6 | no | — |
| 47 | C23×C4 | — | 2 | 6 | no | — |
| 48 | C23×C4 | D8 | 1 | 3 | yes | — |
| 49 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 50 | C2×Q16 | — | 2 | 6 | no | — |
| 51 | C2×Q16 | — | 2 | 6 | no | — |
| 52 | C22×C8 | — | 2 | 24 | no | — |
| 53 | C22×C8 | — | 2 | 8 | no | — |
| 54 | C22×C8 | — | 2 | 6 | no | — |
| 55 | C22×C8 | — | 2 | 12 | no | — |
| 56 | C22×C8 | — | 2 | 6 | no | — |
| 57 | C22×C8 | — | 2 | 6 | no | — |
| 58 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 59 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 60 | C22×C8 | #41 | 1 | 6 | yes | — |
| 61 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 62 | C22⋊Q8 | — | 4 | 12 | no | — |
| 63 | C22⋊Q8 | — | 4 | 12 | no | — |
| 64 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 65 | C2×C4⋊C4 | — | 4 | 6 | no | — |
| 66 | C2×C22⋊C4 | — | 4 | 6 | no | — |
| 67 | C2×C22⋊C4 | — | 4 | 6 | no | — |
| 68 | Q32 | — | 4 | 4 | no | — |
| 69 | C2×C16 | — | 2 | 24 | no | — |
| 70 | C2×C16 | — | 2 | 6 | no | — |
| 71 | C2×C16 | — | 2 | 24 | no | — |
| 72 | C2×C16 | C2×D4 | 1 | 6 | yes | — |
| 73 | C2×C16 | C2×D4 | 1 | 6 | yes | — |
| 74 | #107 | — | 2 | 6 | no | — |
| 75 | #107 | — | 2 | 6 | no | — |
| 76 | #103 | — | 4 | 12 | no | — |
| 77 | #95 | — | 4 | 6 | no | — |
| 78 | C24 | — | 2 | 6 | no | — |
| 79 | C24 | — | 2 | 6 | no | — |
| 80 | C24 | Q32 | 1 | 4 | yes | complemented |
| 81 | C24 | C2×D8 | 1 | 3 | yes | — |
| 82 | C2×Q8 | — | 4 | 6 | no | — |
| 83 | C2×Q8 | — | 4 | 6 | no | — |
| 84 | C22×C4 | — | 2 | 8 | no | — |
| 85 | C22×C4 | — | 2 | 24 | no | — |
| 86 | C22×C4 | — | 2 | 6 | no | — |
| 87 | C22×C4 | — | 2 | 6 | no | — |
| 88 | C22×C4 | — | 2 | 6 | no | — |
| 89 | C22×C4 | — | 2 | 12 | no | — |
| 90 | C22×C4 | — | 4 | 6 | no | — |
| 91 | C22×C4 | — | 4 | 2 | no | — |
| 92 | C22×C4 | #121 | 1 | 6 | yes | — |
| 93 | C22×C4 | C2×D8 | 1 | 3 | yes | — |
| 94 | C22×C4 | #135 | 1 | 6 | yes | — |
| 95 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 96 | Q16 | — | 4 | 2 | no | — |
| 97 | Q16 | — | 4 | 6 | no | — |
| 98 | C2×C8 | — | 2 | 24 | no | — |
| 99 | C2×C8 | — | 2 | 6 | no | — |
| 100 | C2×C8 | — | 2 | 6 | no | — |
| 101 | C2×C8 | — | 2 | 6 | no | — |
| 102 | C2×C8 | — | 2 | 12 | no | — |
| 103 | C2×C8 | #121 | 1 | 6 | yes | — |
| 104 | C2×C8 | C22×D4 | 1 | 3 | yes | — |
| 105 | C2×C8 | C2×#41 | 1 | 3 | yes | — |
| 106 | C4⋊C4 | — | 4 | 6 | no | — |
| 107 | C4⋊C4 | — | 4 | 6 | no | — |
| 108 | C4⋊C4 | — | 8 | 12 | no | — |
| 109 | C22⋊C4 | — | 8 | 12 | no | — |
| 110 | C22⋊C4 | — | 8 | 12 | no | — |
| 111 | C16 | — | 2 | 6 | no | — |
| 112 | C16 | #120 | 1 | 4 | yes | — |
| 113 | C23 | — | 2 | 8 | no | — |
| 114 | C23 | — | 2 | 24 | no | — |
| 115 | C23 | — | 2 | 6 | no | — |
| 116 | C23 | — | 2 | 24 | no | — |
| 117 | C23 | — | 2 | 6 | no | — |
| 118 | C23 | #466 | 1 | 6 | yes | — |
| 119 | C23 | C2×Q32 | 1 | 6 | yes | — |
| 120 | C23 | #508 | 1 | 6 | yes | — |
| 121 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 122 | Q8 | — | 8 | 2 | no | — |
| 123 | Q8 | — | 8 | 6 | no | — |
| 124 | C2×C4 | — | 2 | 24 | no | — |
| 125 | C2×C4 | — | 2 | 6 | no | — |
| 126 | C2×C4 | — | 2 | 6 | no | — |
| 127 | C2×C4 | — | 2 | 6 | no | — |
| 128 | C2×C4 | — | 2 | 12 | no | — |
| 129 | C2×C4 | — | 8 | 12 | no | — |
| 130 | C2×C4 | — | 8 | 6 | no | — |
| 131 | C2×C4 | — | 8 | 6 | no | — |
| 132 | C2×C4 | #466 | 1 | 6 | yes | — |
| 133 | C2×C4 | C2×#121 | 1 | 3 | yes | — |
| 134 | C2×C4 | C2×#135 | 1 | 3 | yes | — |
| 135 | C8 | — | 2 | 6 | no | — |
| 136 | C8 | C2×#120 | 1 | 1 | yes | characteristic |
| 137 | C8 | #525 | 1 | 3 | yes | — |
| 138 | C22 | — | 2 | 32 | no | — |
| 139 | C22 | — | 2 | 12 | no | — |
| 140 | C22 | — | 2 | 6 | no | — |
| 141 | C22 | — | 2 | 12 | no | — |
| 142 | C22 | — | 2 | 6 | no | — |
| 143 | C22 | #2216 | 1 | 12 | yes | complemented |
| 144 | C22 | C2×#466 | 1 | 3 | yes | — |
| 145 | C22 | C22×Q32 | 1 | 1 | yes | characteristic |
| 146 | C22 | C2×#508 | 1 | 3 | yes | — |
| 147 | C4 | — | 2 | 6 | no | — |
| 148 | C4 | — | 16 | 6 | no | — |
| 149 | C4 | — | 16 | 2 | no | — |
| 150 | C4 | #2847 | 1 | 1 | yes | characteristic |
| 151 | C4 | #3050 | 1 | 3 | yes | — |
| 152 | C2 | — | 2 | 12 | no | — |
| 153 | C2 | #32930 | 1 | 1 | yes | characteristic |
| 154 | C2 | C2×#2216 | 1 | 3 | yes | — |
| 155 | C2 | #33600 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C25 | D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#2216 | C2 | semidirect |
| 4. #13060 | C2 | semidirect |
| 5. #12834 | C2 | semidirect |
| 7. C22×Q32 | C22 | semidirect |
| 14. C2×#366 | C22 | semidirect |
| 15. C2×#358 | C22 | semidirect |
| 18. #1380 | C22 | semidirect |
| 80. C24 | Q32 | semidirect |
| 143. C22 | #2216 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C16 | 2 |
| C2×#2216 | 2 |
| #32931 | 2 |
| #13060 | 2 |
| #12834 | 2 |