Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
10820 conjugacy classes of subgroups; only the 650 normal subgroups are shown.
This group has 10820 conjugacy classes of subgroups — too many to draw. Only its 650 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 | #7555 | C2 | 1 | 5 | yes | maximal, complemented |
| 3 | C24×D4 | C22 | 1 | 10 | yes | — |
| 4 | C23×C22⋊C4 | C22 | 1 | 15 | yes | — |
| 5 | C2×#392 | — | 2 | 120 | no | — |
| 6 | C22×#95 | C22 | 1 | 10 | yes | — |
| 7 | C26 | C23 | 1 | 5 | yes | — |
| 8 | C23×D4 | — | 2 | 120 | no | — |
| 9 | C23×D4 | — | 2 | 60 | no | — |
| 10 | C23×D4 | — | 4 | 60 | no | — |
| 11 | C24×C4 | C23 | 1 | 10 | yes | — |
| 12 | C22×C22⋊C4 | — | 2 | 120 | no | — |
| 13 | C22×C22⋊C4 | — | 4 | 30 | no | — |
| 14 | C22×C22⋊C4 | D4 | 1 | 60 | yes | complemented |
| 15 | #392 | — | 2 | 320 | no | — |
| 16 | C2×#95 | — | 2 | 60 | no | — |
| 17 | C25 | — | 2 | 60 | no | — |
| 18 | C25 | — | 2 | 15 | no | — |
| 19 | C25 | — | 2 | 60 | no | — |
| 20 | C25 | — | 2 | 5 | no | — |
| 21 | C25 | C2×D4 | 1 | 30 | yes | — |
| 22 | C25 | C24 | 1 | 1 | yes | characteristic |
| 23 | C22×D4 | — | 2 | 240 | no | — |
| 24 | C22×D4 | — | 4 | 480 | no | — |
| 25 | C22×D4 | — | 4 | 120 | no | — |
| 26 | C22×D4 | — | 4 | 240 | no | — |
| 27 | C22×D4 | — | 4 | 240 | no | — |
| 28 | C22×D4 | — | 4 | 80 | no | — |
| 29 | C22×D4 | — | 4 | 80 | no | — |
| 30 | C22×D4 | — | 4 | 40 | no | — |
| 31 | C23×C4 | — | 2 | 60 | no | — |
| 32 | C23×C4 | — | 2 | 60 | no | — |
| 33 | C23×C4 | C2×D4 | 1 | 60 | yes | — |
| 34 | C2×C22⋊C4 | — | 2 | 240 | no | — |
| 35 | C2×C22⋊C4 | — | 4 | 60 | no | — |
| 36 | C2×C22⋊C4 | — | 4 | 240 | no | — |
| 37 | #95 | — | 2 | 80 | no | — |
| 38 | C24 | — | 2 | 60 | no | — |
| 39 | C24 | — | 2 | 30 | no | — |
| 40 | C24 | — | 2 | 60 | no | — |
| 41 | C24 | — | 2 | 30 | no | — |
| 42 | C24 | — | 2 | 240 | no | — |
| 43 | C24 | — | 4 | 20 | no | — |
| 44 | C24 | — | 4 | 20 | no | — |
| 45 | C24 | — | 4 | 60 | no | — |
| 46 | C24 | — | 4 | 60 | no | — |
| 47 | C24 | — | 4 | 60 | no | — |
| 48 | C24 | — | 4 | 60 | no | — |
| 49 | C24 | — | 4 | 60 | no | — |
| 50 | C24 | — | 4 | 120 | no | — |
| 51 | C24 | — | 4 | 40 | no | — |
| 52 | C24 | — | 4 | 60 | no | — |
| 53 | C24 | #127 | 1 | 20 | yes | — |
| 54 | C24 | C22×D4 | 1 | 15 | yes | — |
| 55 | C24 | H(5,𝔽2) | 1 | 15 | yes | — |
| 56 | C24 | 2−1+4 | 1 | 1 | yes | characteristic |
| 57 | C2×D4 | — | 4 | 480 | no | — |
| 58 | C2×D4 | — | 4 | 960 | no | — |
| 59 | C2×D4 | — | 4 | 960 | no | — |
| 60 | C22×C4 | — | 2 | 60 | no | — |
| 61 | C22×C4 | — | 2 | 60 | no | — |
| 62 | C22×C4 | — | 2 | 240 | no | — |
| 63 | C22×C4 | — | 4 | 20 | no | — |
| 64 | C22×C4 | — | 4 | 60 | no | — |
| 65 | C22×C4 | — | 4 | 60 | no | — |
| 66 | C22×C4 | — | 4 | 20 | no | — |
| 67 | C22×C4 | #120 | 1 | 40 | yes | — |
| 68 | C22⋊C4 | — | 4 | 240 | no | — |
| 69 | C23 | — | 2 | 20 | no | — |
| 70 | C23 | — | 2 | 120 | no | — |
| 71 | C23 | — | 4 | 60 | no | — |
| 72 | C23 | — | 4 | 30 | no | — |
| 73 | C23 | — | 4 | 120 | no | — |
| 74 | C23 | — | 4 | 240 | no | — |
| 75 | C23 | — | 4 | 120 | no | — |
| 76 | C23 | — | 4 | 120 | no | — |
| 77 | C23 | — | 4 | 120 | no | — |
| 78 | C23 | — | 4 | 30 | no | — |
| 79 | C23 | — | 8 | 20 | no | — |
| 80 | C23 | — | 8 | 20 | no | — |
| 81 | C23 | — | 8 | 120 | no | — |
| 82 | C23 | — | 8 | 40 | no | — |
| 83 | C23 | — | 8 | 60 | no | — |
| 84 | C23 | — | 8 | 60 | no | — |
| 85 | C23 | C2×#120 | 1 | 10 | yes | — |
| 86 | C23 | C2×#127 | 1 | 5 | yes | — |
| 87 | C23 | #534 | 1 | 15 | yes | — |
| 88 | C23 | #535 | 1 | 30 | yes | — |
| 89 | C23 | #537 | 1 | 15 | yes | — |
| 90 | C23 | D42 | 1 | 60 | yes | — |
| 91 | C23 | #560 | 1 | 20 | yes | — |
| 92 | D4 | — | 4 | 640 | no | — |
| 93 | C2×C4 | — | 2 | 120 | no | — |
| 94 | C2×C4 | — | 4 | 120 | no | — |
| 95 | C2×C4 | — | 4 | 120 | no | — |
| 96 | C22 | — | 4 | 60 | no | — |
| 97 | C22 | — | 4 | 80 | no | — |
| 98 | C22 | — | 8 | 120 | no | — |
| 99 | C22 | — | 8 | 60 | no | — |
| 100 | C22 | — | 8 | 60 | no | — |
| 101 | C22 | #2403 | 1 | 15 | yes | — |
| 102 | C22 | #2433 | 1 | 30 | yes | — |
| 103 | C22 | #2615 | 1 | 10 | yes | — |
| 104 | C22 | #2656 | 1 | 5 | yes | — |
| 105 | C22 | #2669 | 1 | 60 | yes | — |
| 106 | C22 | #2675 | 1 | 20 | yes | — |
| 107 | C22 | #2815 | 1 | 15 | yes | — |
| 108 | C4 | — | 4 | 40 | no | — |
| 109 | C2 | — | 8 | 20 | no | — |
| 110 | C2 | #15258 | 1 | 10 | yes | — |
| 111 | C2 | #16252 | 1 | 15 | yes | — |
| 112 | C2 | #17102 | 1 | 6 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#392 | C2 | semidirect |
| 2. #7555 | C2 | semidirect |
| 14. C22×C22⋊C4 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#392 | 2 |
| #7555 | 2 |