Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
10966 conjugacy classes of subgroups; only the 986 normal subgroups are shown.
This group has 10966 conjugacy classes of subgroups — too many to draw. Only its 986 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 | C22×#392 | C2 | 1 | 10 | yes | maximal, complemented |
| 2 | #17102 | C2 | 1 | 16 | yes | maximal, direct product factor |
| 3 | C2×#1440 | C2 | 1 | 5 | yes | maximal, complemented |
| 4 | C24×D4 | C22 | 1 | 10 | yes | — |
| 5 | C23×C22⋊C4 | C22 | 1 | 10 | yes | — |
| 6 | C23×C22⋊C4 | C22 | 1 | 5 | yes | — |
| 7 | C2×#392 | — | 2 | 80 | no | — |
| 8 | C2×#392 | C22 | 1 | 80 | yes | complemented |
| 9 | C22×#95 | C22 | 1 | 10 | yes | — |
| 10 | #1440 | C22 | 1 | 40 | yes | complemented |
| 11 | C26 | C23 | 1 | 5 | yes | — |
| 12 | C23×D4 | — | 2 | 80 | no | — |
| 13 | C23×D4 | — | 2 | 40 | no | — |
| 14 | C23×D4 | — | 4 | 40 | no | — |
| 15 | C23×D4 | — | 4 | 20 | no | — |
| 16 | C23×D4 | — | 4 | 20 | no | — |
| 17 | C23×D4 | C23 | 1 | 40 | yes | — |
| 18 | C24×C4 | C23 | 1 | 10 | yes | — |
| 19 | C22×C22⋊C4 | — | 2 | 80 | no | — |
| 20 | C22×C22⋊C4 | — | 4 | 20 | no | — |
| 21 | C22×C22⋊C4 | — | 4 | 20 | no | — |
| 22 | C22×C22⋊C4 | D4 | 1 | 40 | yes | complemented |
| 23 | C22×C22⋊C4 | C23 | 1 | 40 | yes | — |
| 24 | C22×C22⋊C4 | C23 | 1 | 20 | yes | — |
| 25 | #392 | — | 2 | 320 | no | — |
| 26 | C2×#95 | — | 2 | 40 | no | — |
| 27 | C2×#95 | C23 | 1 | 40 | yes | — |
| 28 | C25 | — | 2 | 40 | no | — |
| 29 | C25 | — | 2 | 20 | no | — |
| 30 | C25 | — | 2 | 40 | no | — |
| 31 | C25 | — | 2 | 40 | no | — |
| 32 | C25 | C2×D4 | 1 | 20 | yes | — |
| 33 | C25 | C24 | 1 | 10 | yes | — |
| 34 | C25 | C24 | 1 | 1 | yes | characteristic |
| 35 | C22×D4 | — | 2 | 160 | no | — |
| 36 | C22×D4 | — | 2 | 80 | no | — |
| 37 | C22×D4 | — | 4 | 80 | no | — |
| 38 | C22×D4 | — | 4 | 40 | no | — |
| 39 | C22×D4 | — | 4 | 40 | no | — |
| 40 | C22×D4 | — | 4 | 320 | no | — |
| 41 | C22×D4 | — | 4 | 160 | no | — |
| 42 | C22×D4 | — | 4 | 160 | no | — |
| 43 | C22×D4 | — | 4 | 320 | no | — |
| 44 | C22×D4 | — | 4 | 160 | no | — |
| 45 | C23×C4 | — | 2 | 40 | no | — |
| 46 | C23×C4 | — | 2 | 20 | no | — |
| 47 | C23×C4 | — | 2 | 20 | no | — |
| 48 | C23×C4 | — | 2 | 40 | no | — |
| 49 | C23×C4 | C2×D4 | 1 | 40 | yes | — |
| 50 | C23×C4 | C24 | 1 | 20 | yes | — |
| 51 | C2×C22⋊C4 | — | 2 | 160 | no | — |
| 52 | C2×C22⋊C4 | — | 4 | 80 | no | — |
| 53 | C2×C22⋊C4 | — | 4 | 40 | no | — |
| 54 | C2×C22⋊C4 | — | 4 | 40 | no | — |
| 55 | C2×C22⋊C4 | — | 4 | 160 | no | — |
| 56 | C2×C22⋊C4 | C2×D4 | 1 | 80 | yes | complemented |
| 57 | #95 | — | 2 | 80 | no | — |
| 58 | C24 | — | 2 | 40 | no | — |
| 59 | C24 | — | 2 | 80 | no | — |
| 60 | C24 | — | 2 | 160 | no | — |
| 61 | C24 | — | 2 | 40 | no | — |
| 62 | C24 | — | 2 | 20 | no | — |
| 63 | C24 | — | 2 | 40 | no | — |
| 64 | C24 | — | 2 | 40 | no | — |
| 65 | C24 | — | 4 | 80 | no | — |
| 66 | C24 | — | 4 | 80 | no | — |
| 67 | C24 | — | 4 | 80 | no | — |
| 68 | C24 | — | 4 | 40 | no | — |
| 69 | C24 | — | 4 | 80 | no | — |
| 70 | C24 | — | 4 | 80 | no | — |
| 71 | C24 | — | 4 | 40 | no | — |
| 72 | C24 | — | 4 | 80 | no | — |
| 73 | C24 | C22×D4 | 1 | 20 | yes | — |
| 74 | C24 | C22×D4 | 1 | 10 | yes | — |
| 75 | C24 | H(5,𝔽2) | 1 | 20 | yes | — |
| 76 | C24 | C25 | 1 | 1 | yes | characteristic |
| 77 | C2×D4 | — | 4 | 320 | no | — |
| 78 | C2×D4 | — | 4 | 160 | no | — |
| 79 | C2×D4 | — | 4 | 160 | no | — |
| 80 | C2×D4 | — | 4 | 320 | no | — |
| 81 | C2×D4 | — | 4 | 1280 | no | — |
| 82 | C2×D4 | — | 4 | 160 | no | — |
| 83 | C22×C4 | — | 2 | 80 | no | — |
| 84 | C22×C4 | — | 2 | 160 | no | — |
| 85 | C22×C4 | — | 2 | 40 | no | — |
| 86 | C22×C4 | — | 2 | 20 | no | — |
| 87 | C22×C4 | — | 2 | 20 | no | — |
| 88 | C22×C4 | — | 2 | 40 | no | — |
| 89 | C22×C4 | — | 4 | 80 | no | — |
| 90 | C22×C4 | — | 4 | 40 | no | — |
| 91 | C22×C4 | — | 4 | 40 | no | — |
| 92 | C22×C4 | C22×D4 | 1 | 40 | yes | — |
| 93 | C22⋊C4 | — | 4 | 80 | no | — |
| 94 | C22⋊C4 | — | 4 | 160 | no | — |
| 95 | C23 | — | 2 | 20 | no | — |
| 96 | C23 | — | 2 | 40 | no | — |
| 97 | C23 | — | 2 | 80 | no | — |
| 98 | C23 | — | 4 | 320 | no | — |
| 99 | C23 | — | 4 | 160 | no | — |
| 100 | C23 | — | 4 | 40 | no | — |
| 101 | C23 | — | 4 | 40 | no | — |
| 102 | C23 | — | 4 | 40 | no | — |
| 103 | C23 | — | 4 | 20 | no | — |
| 104 | C23 | — | 4 | 40 | no | — |
| 105 | C23 | — | 4 | 40 | no | — |
| 106 | C23 | — | 4 | 40 | no | — |
| 107 | C23 | — | 4 | 20 | no | — |
| 108 | C23 | — | 4 | 80 | no | — |
| 109 | C23 | — | 8 | 160 | no | — |
| 110 | C23 | — | 8 | 160 | no | — |
| 111 | C23 | #534 | 1 | 20 | yes | — |
| 112 | C23 | #535 | 1 | 40 | yes | — |
| 113 | C23 | D42 | 1 | 40 | yes | — |
| 114 | C23 | #560 | 1 | 40 | yes | — |
| 115 | C23 | C23×D4 | 1 | 5 | yes | — |
| 116 | C23 | C2×H(5,𝔽2) | 1 | 10 | yes | — |
| 117 | D4 | — | 4 | 640 | no | — |
| 118 | C2×C4 | — | 2 | 40 | no | — |
| 119 | C2×C4 | — | 2 | 80 | no | — |
| 120 | C2×C4 | — | 4 | 160 | no | — |
| 121 | C2×C4 | — | 4 | 40 | no | — |
| 122 | C2×C4 | — | 4 | 20 | no | — |
| 123 | C2×C4 | — | 4 | 20 | no | — |
| 124 | C22 | — | 4 | 20 | no | — |
| 125 | C22 | — | 4 | 80 | no | — |
| 126 | C22 | — | 4 | 40 | no | — |
| 127 | C22 | — | 8 | 40 | no | — |
| 128 | C22 | — | 8 | 40 | no | — |
| 129 | C22 | — | 8 | 160 | no | — |
| 130 | C22 | #2669 | 1 | 80 | yes | — |
| 131 | C22 | #2815 | 1 | 40 | yes | — |
| 132 | C22 | C2×#534 | 1 | 5 | yes | — |
| 133 | C22 | C2×#535 | 1 | 10 | yes | — |
| 134 | C22 | C2×D42 | 1 | 10 | yes | — |
| 135 | C22 | C2×#560 | 1 | 10 | yes | — |
| 136 | C4 | — | 4 | 40 | no | — |
| 137 | C2 | — | 8 | 20 | no | — |
| 138 | C2 | #17102 | 1 | 16 | yes | direct product factor |
| 139 | C2 | C2×#2815 | 1 | 5 | yes | — |
| 140 | C2 | C2×#2669 | 1 | 10 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#392 | C2 | semidirect |
| 2. #17102 | C2 | direct (N × H) |
| 3. C2×#1440 | C2 | semidirect |
| 8. C2×#392 | C22 | semidirect |
| 10. #1440 | C22 | semidirect |
| 22. C22×C22⋊C4 | D4 | semidirect |
| 56. C2×C22⋊C4 | C2×D4 | semidirect |
| 138. C2 | #17102 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#392 | 2 |
| #17102 | 2 |
| C2×#1440 | 2 |