Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
3919 conjugacy classes of subgroups.
The subgroup lattice has 127 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #46242 | C2 | 1 | 15 | yes | maximal |
| 2 | #46009 | C2 | 1 | 30 | yes | maximal |
| 3 | #36567 | C2 | 1 | 15 | yes | maximal |
| 4 | C4×#564 | C2 | 1 | 3 | yes | maximal |
| 5 | C2×#564 | C22 | 1 | 1 | yes | characteristic |
| 6 | #2864 | C22 | 1 | 30 | yes | — |
| 7 | #2859 | C22 | 1 | 15 | yes | — |
| 8 | #2845 | C22 | 1 | 5 | yes | — |
| 9 | #2845 | C22 | 1 | 15 | yes | — |
| 10 | #2844 | C22 | 1 | 10 | yes | — |
| 11 | #2802 | C22 | 1 | 60 | yes | — |
| 12 | #2792 | C22 | 1 | 60 | yes | — |
| 13 | #2791 | C22 | 1 | 60 | yes | — |
| 14 | #2590 | C22 | 1 | 60 | yes | — |
| 15 | #2585 | C22 | 1 | 60 | yes | — |
| 16 | #2584 | C22 | 1 | 30 | yes | — |
| 17 | #2507 | C22 | 1 | 30 | yes | — |
| 18 | #2506 | C22 | 1 | 30 | yes | — |
| 19 | #2462 | C22 | 1 | 60 | yes | — |
| 20 | #2382 | C22 | 1 | 30 | yes | — |
| 21 | #2382 | C22 | 1 | 15 | yes | — |
| 22 | #2370 | C22 | 1 | 30 | yes | — |
| 23 | C4×#125 | C22 | 1 | 15 | yes | — |
| 24 | C4×#125 | C22 | 1 | 30 | yes | — |
| 25 | C42×Q8 | C22 | 1 | 5 | yes | — |
| 26 | #564 | Q8 | 1 | 16 | yes | complemented |
| 27 | C2×#125 | C23 | 1 | 30 | yes | — |
| 28 | C2×#125 | C23 | 1 | 15 | yes | — |
| 29 | C2×C4×Q8 | C23 | 1 | 15 | yes | — |
| 30 | #400 | C23 | 1 | 60 | yes | — |
| 31 | #398 | C23 | 1 | 60 | yes | — |
| 32 | #398 | C23 | 1 | 60 | yes | — |
| 33 | #398 | C23 | 1 | 60 | yes | — |
| 34 | #398 | C23 | 1 | 60 | yes | — |
| 35 | #395 | C23 | 1 | 60 | yes | — |
| 36 | #389 | C23 | 1 | 30 | yes | — |
| 37 | #389 | C23 | 1 | 30 | yes | — |
| 38 | #389 | C23 | 1 | 30 | yes | — |
| 39 | #389 | C23 | 1 | 30 | yes | — |
| 40 | #389 | C23 | 1 | 60 | yes | — |
| 41 | #389 | C23 | 1 | 30 | yes | — |
| 42 | #389 | C23 | 1 | 30 | yes | — |
| 43 | #389 | C23 | 1 | 30 | yes | — |
| 44 | #389 | C23 | 1 | 60 | yes | — |
| 45 | #387 | C23 | 1 | 60 | yes | — |
| 46 | #387 | C23 | 1 | 60 | yes | — |
| 47 | #387 | C23 | 1 | 60 | yes | — |
| 48 | #387 | C23 | 1 | 60 | yes | — |
| 49 | #387 | C23 | 1 | 60 | yes | — |
| 50 | #387 | C23 | 1 | 60 | yes | — |
| 51 | #387 | C23 | 1 | 60 | yes | — |
| 52 | #384 | C23 | 1 | 15 | yes | — |
| 53 | #384 | C23 | 1 | 15 | yes | — |
| 54 | #382 | C23 | 1 | 30 | yes | — |
| 55 | #382 | C23 | 1 | 15 | yes | — |
| 56 | C4×C4⋊C4 | C23 | 1 | 30 | yes | — |
| 57 | C4×C4⋊C4 | C23 | 1 | 30 | yes | — |
| 58 | C4×C4⋊C4 | C23 | 1 | 30 | yes | — |
| 59 | C4×C4⋊C4 | C23 | 1 | 30 | yes | — |
| 60 | C4×C4⋊C4 | C23 | 1 | 15 | yes | — |
| 61 | C43 | C23 | 1 | 15 | yes | — |
| 62 | C22×Q8 | C24 | 1 | 5 | yes | — |
| 63 | #125 | — | 4 | 60 | no | — |
| 64 | #125 | C2×Q8 | 1 | 120 | yes | — |
| 65 | C4×Q8 | — | 4 | 30 | no | — |
| 66 | C2×C4⋊C4 | C24 | 1 | 30 | yes | — |
| 67 | C2×C4⋊C4 | C24 | 1 | 15 | yes | — |
| 68 | C2×C4⋊C4 | C24 | 1 | 30 | yes | — |
| 69 | C2×C4⋊C4 | C24 | 1 | 30 | yes | — |
| 70 | C2×C4⋊C4 | C24 | 1 | 60 | yes | — |
| 71 | C2×C4⋊C4 | C24 | 1 | 30 | yes | — |
| 72 | C2×C4⋊C4 | C24 | 1 | 60 | yes | — |
| 73 | C2×C4⋊C4 | C24 | 1 | 30 | yes | — |
| 74 | C2×C42 | C24 | 1 | 15 | yes | — |
| 75 | C2×C42 | C24 | 1 | 30 | yes | — |
| 76 | C2×C42 | C24 | 1 | 30 | yes | — |
| 77 | C2×C42 | C24 | 1 | 15 | yes | — |
| 78 | C2×C42 | C24 | 1 | 5 | yes | — |
| 79 | #95 | C24 | 1 | 1 | yes | characteristic |
| 80 | #95 | C24 | 1 | 15 | yes | — |
| 81 | #95 | C24 | 1 | 10 | yes | — |
| 82 | #95 | C24 | 1 | 60 | yes | — |
| 83 | #95 | C24 | 1 | 60 | yes | — |
| 84 | #95 | C24 | 1 | 60 | yes | — |
| 85 | #95 | C24 | 1 | 60 | yes | — |
| 86 | C2×Q8 | — | 4 | 15 | no | — |
| 87 | C22×C4 | C25 | 1 | 30 | yes | — |
| 88 | C22×C4 | C25 | 1 | 15 | yes | — |
| 89 | C22×C4 | C25 | 1 | 15 | yes | — |
| 90 | C22×C4 | C25 | 1 | 3 | yes | — |
| 91 | C4⋊C4 | — | 2 | 60 | no | — |
| 92 | C4⋊C4 | — | 4 | 30 | no | — |
| 93 | C4⋊C4 | — | 4 | 60 | no | — |
| 94 | C4⋊C4 | — | 4 | 15 | no | — |
| 95 | C4⋊C4 | — | 4 | 30 | no | — |
| 96 | C4⋊C4 | — | 4 | 30 | no | — |
| 97 | C4⋊C4 | — | 4 | 60 | no | — |
| 98 | C4⋊C4 | C22×Q8 | 1 | 120 | yes | — |
| 99 | C42 | — | 2 | 30 | no | — |
| 100 | C42 | — | 4 | 30 | no | — |
| 101 | C42 | — | 4 | 15 | no | — |
| 102 | C42 | — | 4 | 30 | no | — |
| 103 | C42 | C22×Q8 | 1 | 20 | yes | — |
| 104 | C23 | C26 | 1 | 1 | yes | characteristic |
| 105 | Q8 | — | 4 | 20 | no | — |
| 106 | C2×C4 | — | 2 | 15 | no | — |
| 107 | C2×C4 | — | 2 | 6 | no | — |
| 108 | C2×C4 | — | 2 | 15 | no | — |
| 109 | C2×C4 | — | 2 | 15 | no | — |
| 110 | C2×C4 | — | 2 | 30 | no | — |
| 111 | C2×C4 | — | 2 | 30 | no | — |
| 112 | C2×C4 | — | 2 | 30 | no | — |
| 113 | C2×C4 | — | 2 | 30 | no | — |
| 114 | C2×C4 | C23×Q8 | 1 | 30 | yes | — |
| 115 | C2×C4 | #585 | 1 | 6 | yes | — |
| 116 | C22 | C24×Q8 | 1 | 1 | yes | characteristic |
| 117 | C22 | C2×#585 | 1 | 3 | yes | — |
| 118 | C22 | 2−1+6 | 1 | 3 | yes | — |
| 119 | C4 | — | 2 | 30 | no | — |
| 120 | C4 | — | 2 | 6 | no | — |
| 121 | C4 | — | 4 | 30 | no | — |
| 122 | C4 | — | 4 | 15 | no | — |
| 123 | C2 | #62694 | 1 | 1 | yes | characteristic |
| 124 | C2 | #63004 | 1 | 3 | yes | — |
| 125 | C2 | #63068 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 26. #564 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #46242 | 2 |
| #46009 | 2 |
| #36567 | 2 |
| C4×#564 | 2 |