Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
441 conjugacy classes of subgroups; only the 6 normal subgroups are shown.
This group has 441 conjugacy classes of subgroups — too many to draw. Only its 6 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 | H(7,𝔽2) | C7 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 2 | C2×AGL(1,𝔽8) | — | 8 | 2 | no | maximal |
| 3 | #585 | — | 7 | 1 | no | — |
| 4 | #585 | — | 7 | 3 | no | — |
| 5 | C2×H(5,𝔽2) | — | 7 | 3 | no | — |
| 6 | C2×H(5,𝔽2) | — | 7 | 2 | no | — |
| 7 | AGL(1,𝔽8) | — | 8 | 2 | no | — |
| 8 | 2−1+4 | — | 7 | 2 | no | — |
| 9 | 2−1+4 | — | 7 | 3 | no | — |
| 10 | 2−1+4 | — | 7 | 3 | no | — |
| 11 | H(5,𝔽2) | — | 7 | 3 | no | — |
| 12 | H(5,𝔽2) | — | 7 | 6 | no | — |
| 13 | H(5,𝔽2) | — | 7 | 6 | no | — |
| 14 | H(5,𝔽2) | — | 7 | 6 | no | — |
| 15 | H(5,𝔽2) | — | 7 | 6 | no | — |
| 16 | H(5,𝔽2) | — | 7 | 6 | no | — |
| 17 | H(5,𝔽2) | — | 7 | 1 | no | — |
| 18 | H(5,𝔽2) | — | 7 | 3 | no | — |
| 19 | H(5,𝔽2) | — | 7 | 3 | no | — |
| 20 | C2×#41 | — | 7 | 3 | no | — |
| 21 | C2×#41 | — | 7 | 3 | no | — |
| 22 | C2×#41 | — | 7 | 6 | no | — |
| 23 | C2×#41 | — | 7 | 6 | no | — |
| 24 | C2×#41 | — | 7 | 6 | no | — |
| 25 | C2×#41 | — | 7 | 3 | no | — |
| 26 | C2×#41 | — | 7 | 3 | no | — |
| 27 | C22×D4 | — | 7 | 1 | no | — |
| 28 | C22×D4 | — | 7 | 6 | no | — |
| 29 | C22×D4 | — | 7 | 3 | no | — |
| 30 | C22×D4 | — | 7 | 2 | no | — |
| 31 | C22×D4 | — | 7 | 3 | no | — |
| 32 | C24 | — | 7 | 2 | no | — |
| 33 | C24 | — | 7 | 2 | no | — |
| 34 | C24 | AGL(1,𝔽8) | 1 | 2 | yes | complemented |
| 35 | #41 | — | 7 | 6 | no | — |
| 36 | #41 | — | 7 | 6 | no | — |
| 37 | #41 | — | 7 | 6 | no | — |
| 38 | #41 | — | 7 | 3 | no | — |
| 39 | #41 | — | 7 | 3 | no | — |
| 40 | #41 | — | 7 | 6 | no | — |
| 41 | #41 | — | 7 | 6 | no | — |
| 42 | #41 | — | 7 | 2 | no | — |
| 43 | #41 | — | 7 | 6 | no | — |
| 44 | #41 | — | 7 | 6 | no | — |
| 45 | #41 | — | 7 | 6 | no | — |
| 46 | #41 | — | 7 | 6 | no | — |
| 47 | #41 | — | 7 | 6 | no | — |
| 48 | #41 | — | 7 | 6 | no | — |
| 49 | #41 | — | 7 | 3 | no | — |
| 50 | #41 | — | 7 | 3 | no | — |
| 51 | C2×Q8 | — | 7 | 3 | no | — |
| 52 | C2×Q8 | — | 7 | 3 | no | — |
| 53 | C2×Q8 | — | 7 | 2 | no | — |
| 54 | C2×Q8 | — | 7 | 2 | no | — |
| 55 | C2×D4 | — | 7 | 3 | no | — |
| 56 | C2×D4 | — | 7 | 6 | no | — |
| 57 | C2×D4 | — | 7 | 3 | no | — |
| 58 | C2×D4 | — | 7 | 6 | no | — |
| 59 | C2×D4 | — | 7 | 6 | no | — |
| 60 | C2×D4 | — | 7 | 6 | no | — |
| 61 | C2×D4 | — | 7 | 3 | no | — |
| 62 | C2×D4 | — | 7 | 6 | no | — |
| 63 | C2×D4 | — | 7 | 3 | no | — |
| 64 | C2×D4 | — | 7 | 6 | no | — |
| 65 | C2×D4 | — | 7 | 6 | no | — |
| 66 | C2×D4 | — | 7 | 6 | no | — |
| 67 | C2×D4 | — | 7 | 6 | no | — |
| 68 | C2×D4 | — | 7 | 6 | no | — |
| 69 | C2×D4 | — | 7 | 6 | no | — |
| 70 | C2×D4 | — | 7 | 3 | no | — |
| 71 | C2×D4 | — | 7 | 6 | no | — |
| 72 | C2×D4 | — | 7 | 3 | no | — |
| 73 | C22×C4 | — | 7 | 1 | no | — |
| 74 | C22×C4 | — | 7 | 2 | no | — |
| 75 | C22×C4 | — | 7 | 6 | no | — |
| 76 | C22×C4 | — | 7 | 3 | no | — |
| 77 | C22×C4 | — | 7 | 3 | no | — |
| 78 | C2×C7 | — | 64 | 1 | no | — |
| 79 | C23 | — | 7 | 3 | no | — |
| 80 | C23 | — | 7 | 6 | no | — |
| 81 | C23 | — | 7 | 1 | no | — |
| 82 | C23 | — | 7 | 3 | no | — |
| 83 | C23 | — | 7 | 2 | no | — |
| 84 | C23 | — | 8 | 2 | no | — |
| 85 | C23 | — | 56 | 2 | no | — |
| 86 | C23 | — | 56 | 2 | no | — |
| 87 | Q8 | — | 7 | 3 | no | — |
| 88 | Q8 | — | 7 | 3 | no | — |
| 89 | Q8 | — | 7 | 2 | no | — |
| 90 | D4 | — | 7 | 3 | no | — |
| 91 | D4 | — | 7 | 3 | no | — |
| 92 | D4 | — | 7 | 1 | no | — |
| 93 | D4 | — | 7 | 6 | no | — |
| 94 | D4 | — | 7 | 3 | no | — |
| 95 | D4 | — | 7 | 6 | no | — |
| 96 | D4 | — | 7 | 6 | no | — |
| 97 | D4 | — | 7 | 6 | no | — |
| 98 | D4 | — | 7 | 6 | no | — |
| 99 | C2×C4 | — | 7 | 6 | no | — |
| 100 | C2×C4 | — | 7 | 3 | no | — |
| 101 | C2×C4 | — | 7 | 6 | no | — |
| 102 | C2×C4 | — | 7 | 6 | no | — |
| 103 | C2×C4 | — | 7 | 3 | no | — |
| 104 | C2×C4 | — | 7 | 3 | no | — |
| 105 | C2×C4 | — | 7 | 3 | no | — |
| 106 | C7 | — | 64 | 1 | no | Sylow |
| 107 | C22 | — | 7 | 2 | no | — |
| 108 | C22 | — | 7 | 3 | no | — |
| 109 | C22 | — | 28 | 2 | no | — |
| 110 | C22 | — | 28 | 3 | no | — |
| 111 | C22 | — | 28 | 3 | no | — |
| 112 | C22 | — | 28 | 6 | no | — |
| 113 | C22 | — | 28 | 1 | no | — |
| 114 | C4 | — | 7 | 1 | no | — |
| 115 | C4 | — | 7 | 3 | no | — |
| 116 | C2 | — | 14 | 2 | no | — |
| 117 | C2 | — | 14 | 3 | no | — |
| 118 | C2 | #90632 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2 | #90632 |
| commutator, Fitting, cosocle, hypocenter | H(7,𝔽2) | C7 |
| LCS | G ▷ H(7,𝔽2) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. H(7,𝔽2) | C7 | semidirect |
| 34. C24 | AGL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(7,𝔽2) | 7 |
| C2×AGL(1,𝔽8) | 8 |