Finite Group Databasebeta

← #27144047

Subgroups — #27144047

Order 12416.

112 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261164C211yesmaximal, characteristic, complemented
2D16×D97C211yesmaximal, characteristic, complemented
3#20260574C211yesmaximal, characteristic, complemented
4#20260573C211yesmaximal, characteristic, complemented
5#20260570C211yesmaximal, characteristic, complemented
6#20260568C211yesmaximal, characteristic, complemented
7D32×C97C211yesmaximal, characteristic, complemented
8#15690829—21no—
9D8×D97—21no—
10Dic776C2211yescharacteristic, complemented
11#15690754—21no—
12D1552C2211yescharacteristic
13#15690751—21no—
14C16×D97C2211yescharacteristic
15#15690716C2211yescharacteristic, complemented
16D16×C97C2211yescharacteristic
17D16×C97C2211yescharacteristic, complemented
18C32×C97C2211yescharacteristic, complemented
19#13095510—41no—
20D4×D97—41no—
21Dic388—21no—
22#13095499—41no—
23D776—21no—
24#13095496—41no—
25C8×D97D411yescharacteristic
26#13095486D411yescharacteristic
27D8×C97—21no—
28D8×C97—21no—
29C16×C97C2311yescharacteristic
30C22×D97—81no—
31D4×C97—41no—
32D4×C97—41no—
33C22⋊D97—81no—
34C22⋊D97—81no—
35C2×Dic97—81no—
36D388—41no—
37C4×D97D811yescharacteristic
38Dic194—41no—
39C8×C97C2×D411yescharacteristic
40#11715895D811yescharacteristic
41C22×C97—81no—
42C22×C97—81no—
43C2×D97—81no—
44C2×D97—162no—
45C2×D97D1611yescharacteristic
46C4×C97C2×D811yescharacteristic
47Dic97—81no—
48Dic97D1611yescharacteristic
49C2×C97—161no—
50C2×C97—161no—
51C2×C97C2×D1611yescharacteristic
52D97—21no—
53D97—161no—
54#2263—971nomaximal, Sylow
55C97#226311yescharacteristic, complemented, Sylow
56#508—971no—
57C2×D16—971no—
58SD64—971no—
59SD64—971no—
60D32—971no—
61D32D9711yescharacteristic, complemented
62M64—971no—
63#135—1941no—
64C2×D8—1941no—
65Q32—971no—
66SD32—1941no—
67D16—971no—
68D16—1941no—
69D16C2×D9711yescharacteristic
70D16C2×D9711yescharacteristic, complemented
71C2×C16—971no—
72C32—971no—
73C32C2×D9711yescharacteristic, complemented
74#41—3881no—
75C2×D4—3881no—
76Q16—1941no—
77SD16—3881no—
78D8—21no—
79D8—21no—
80D8—3881no—
81D8—1941no—
82C2×C8—971no—
83C16—971no—
84C16C22×D9711yescharacteristic
85C23—7761no—
86Q8—3881no—
87D4—41no—
88D4—41no—
89D4—7761no—
90D4—7761no—
91D4—3881no—
92C2×C4—7761no—
93C2×C4—971no—
94C8—971no—
95C8D4×D9711yescharacteristic
96C22—81no—
97C22—81no—
98C22—7761no—
99C22—15522no—
100C22—971no—
101C4—7761no—
102C4—971no—
103C4D8×D9711yescharacteristic
104C2—161no—
105C2—161no—
106C2—15521no—
107C2—1941no—
108C2D16×D9711yescharacteristic
subgroupG/H
centerC2D16×D97
commutator, cosocleC16×C97C23
FrattiniC16C22×D97
FittingD32×C97C2
socleC2×C97C2×D16
hypercenterD32D97
hypocenterC97#2263
LCSG ▷ C16×C97 ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ D32

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20261164C2semidirect
2. D16×D97C2semidirect
3. #20260574C2semidirect
4. #20260573C2semidirect
5. #20260570C2semidirect
6. #20260568C2semidirect
7. D32×C97C2semidirect
10. Dic776C22semidirect
15. #15690716C22semidirect
17. D16×C97C22semidirect
18. C32×C97C22semidirect
55. C97#2263semidirect
61. D32D97semidirect
70. D16C2×D97semidirect
73. C32C2×D97semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202611642
D16×D972
#202605742
#202605732
#202605702
#202605682
D32×C972
#226397