Finite Group Databasebeta

← #27157192

Subgroups — #27157192

Order 12416.

270 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C4×#13095521C212yesmaximal, complemented
2#20261591C214yesmaximal, complemented
3#20261286C211yesmaximal, characteristic
4C2×#13095521—22no—
5#15690865C2214yes—
6C4×#11715897—22no—
7C2×#13095512C412yes—
8#15690807C411yescharacteristic
9#15690799C2212yes—
10#15690785—22no—
11#15690772—22no—
12C42×D97C2211yescharacteristic
13C4⋊Q8×C97C411yescharacteristic
14#13095521—28no—
15C2×#11715897—24no—
16#13095512—44no—
17C2×D388—22no—
18C2×D388—22no—
19C2×D388C2×C412yes—
20C2×C4×D97—22no—
21C2×C4×D97C2×C411yescharacteristic
22C2×C4×D97D412yes—
23#13095488—24no—
24C4×Dic97D412yes—
25C4×Dic97C2311yescharacteristic
26C2×Q8×C97C2×C412yes—
27C4⋊C4×C97—22no—
28C42×C97C2×C411yescharacteristic
29C22×D97—22no—
30C22×D97C22⋊C411yescharacteristic
31Q8×C97—42no—
32C2×C4×C97—22no—
33C2×C4×C97C22⋊C412yes—
34C2×C4×C97C22×C411yescharacteristic
35C2×Dic97C22⋊C411yescharacteristic
36C2×Dic97C2×D412yes—
37D388—42no—
38D388—44no—
39D388—44no—
40C4×D97—24no—
41C4×D97—22no—
42C4×D97—44no—
43#11715897—28no—
44C22×C97C2×C22⋊C411yescharacteristic
45C2×D97—21no—
46C2×D97—22no—
47C2×D97—42no—
48C2×D97—44no—
49C4×C97—21no—
50C4×C97—42no—
51C4×C97#10414yes—
52Dic97—22no—
53Dic97#10012yes—
54C2×C97C2×#10011yescharacteristic
55C2×C97C2×#10412yes—
56D97—41no—
57D97—82no—
58#1525—971nomaximal, Sylow
59C97#152511yescharacteristic, complemented, Sylow
60#532—971no—
61C4×M16—972no—
62#329—974no—
63C2×#41—972no—
64C2×M16—1942no—
65C4⋊Q8#8760411yescharacteristic
66#127—971no—
67#124—972no—
68#121—1942no—
69C4×D4—1942no—
70C2×C42—971no—
71#97—974no—
72C4×C8—1942no—
73#41—3884no—
74C2×Q8C2×#8760412yes—
75C2×D4—972no—
76C2×D4—1942no—
77C2×D4—1942no—
78C22×C4—972no—
79C22×C4—971no—
80C22×C4—1942no—
81M16—1948no—
82C2×C8—1944no—
83C4⋊C4—22no—
84C22⋊C4—1944no—
85C42—972no—
86C42—971no—
87C42C2×#8760411yescharacteristic
88C23—971no—
89C23—1942no—
90Q8—42no—
91D4—3882no—
92D4—3884no—
93D4—3884no—
94C2×C4—22no—
95C2×C4—971no—
96C2×C4—972no—
97C2×C4—3884no—
98C2×C4—1942no—
99C2×C4—1944no—
100C2×C4C22⋊#8760412yes—
101C2×C4C22×#8760411yescharacteristic
102C8—1948no—
103C22—3882no—
104C22—3884no—
105C22—1941no—
106C22—1942no—
107C22C2×C22⋊#8760411yescharacteristic
108C4—21no—
109C4—42no—
110C4—972no—
111C4—1942no—
112C4#1569087514yes—
113C2—7762no—
114C2—3881no—
115C2D97⋊#10011yescharacteristic
116C2C2×#1569087512yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261286C2
cosocleC4×Dic97C23
hypercenterC4⋊Q8#87604
hypocenterC97#1525
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C4×#13095521C2semidirect
2. #20261591C2semidirect
59. C97#1525semidirect

Maximal subgroups

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

subgroupindex
C4×#130955212
#202615912
#202612862
#152597