Finite Group Databasebeta

← #72241908

Subgroups — #72241908

Order 8112.

248 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 53 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#72241904C212yesmaximal, complemented
2#72241901C212yesmaximal, complemented
3C2×#11759818⋊C2C211yesmaximal, characteristic, complemented
4#72241888C211yesmaximal, characteristic, complemented
5C22×#11759816C211yesmaximal, characteristic, complemented
6#72214115C311yesmaximal, characteristic, complemented, Hall
7C22×#92789C2211yescharacteristic, complemented
8C2×#11759818C2212yescomplemented
9C2×#11759816—22no—
10C2×#11759816C2211yescharacteristic
11C2×#11759816C2211yescharacteristic, complemented
12#11759818⋊C2—22no—
13#14523131C2212yescomplemented
14C22×C13⋊D13C2×C311yescharacteristic, complemented
15C2×D132C2×C311yescharacteristic, complemented
16C13×C22⋊D13C2×C312yescomplemented
17#13062579C2×C312yescomplemented
18#13062577C2×C311yescharacteristic, complemented
19C2×#92789—21no—
20C2×#92789C2311yescharacteristic
21#11759818—22no—
22#11759816—21no—
23#11759816D412yescomplemented
24C22×C132C22×C311yescharacteristic, complemented
25C2×C13⋊D13—22no—
26C2×C13⋊D13C22×C311yescharacteristic
27C2×C13⋊D13C22×C311yescharacteristic, complemented
28C2×C13×D13C22×C312yescomplemented
29D132—22no—
30C13×Dic13C22×C312yescomplemented
31D4×D13⋊C3—132nomaximal
32#92789C2×D411yescharacteristic, complemented, Hall
33C2×C132—21no—
34C2×C132C23×C311yescharacteristic
35C13⋊D13—21no—
36C13⋊D13C3×D412yescomplemented
37C13×D13—22no—
38C22×D13⋊C3—132no—
39C22×D13⋊C3—132no—
40D4×C13⋊C3—132no—
41(C22⋊D13)⋊C3—132no—
42(C22⋊D13)⋊C3—262no—
43C4⋊(D13⋊C3)—132no—
44C4×D13⋊C3—132no—
45D4×D13—132no—
46C132C2×C3×D411yescharacteristic, complemented, Sylow
47C22×C13⋊C3—132no—
48C22×C13⋊C3—262no—
49C2×D13⋊C3—132no—
50C2×D13⋊C3—132no—
51C2×D13⋊C3—264no—
52C2×D13⋊C3—264no—
53C2×D13⋊C3—262no—
54C4×C13⋊C3—132no—
55Dic13⋊C3—262no—
56C22×D13—132no—
57C22×D13—132no—
58C22×D13—782no—
59D4×C13—132no—
60C22⋊D13—132no—
61C22⋊D13D13⋊C312yescomplemented
62D52—132no—
63C4×D13—132no—
64C2×C13⋊C3—262no—
65C2×C13⋊C3—262no—
66C2×C13⋊C3—522no—
67D13⋊C3—134no—
68D13⋊C3—262no—
69D13⋊C3—522no—
70C22×C13—62no—
71C22×C13—132no—
72C22×C13C2×D13⋊C312yescomplemented
73C2×D13—132no—
74C2×D13—132no—
75C2×D13—264no—
76C2×D13—264no—
77C2×D13—784no—
78C2×D13—784no—
79C2×D13—782no—
80C2×D13—782no—
81C2×D13C2×D13⋊C312yescomplemented
82C4×C13—132no—
83Dic13C2×D13⋊C312yescomplemented
84C2×C3×D4—1691noHall
85C13⋊C3—262no—
86C2×C13—22no—
87C2×C13—62no—
88C2×C13—62no—
89C2×C13—62no—
90C2×C13—262no—
91C2×C13C22×D13⋊C312yes—
92D13—22no—
93D13—134no—
94D13—262no—
95D13—782no—
96D13—782no—
97D13—784no—
98C23×C3—1691no—
99C23×C3—1691no—
100C3×D4—3382no—
101C3×D4—1692no—
102C2×C3×C4—1691no—
103C2×D4—1691noSylow
104C13—62no—
105C13D4×D13⋊C312yescomplemented
106C22×C3—3382no—
107C22×C3—3382no—
108C22×C3—3382no—
109C22×C3—6761no—
110C22×C3—1691no—
111C22×C3—1691no—
112C3×C4—3382no—
113C23—1691no—
114C23—1691no—
115D4—132no—
116D4—1692no—
117C2×C4—1691no—
118C2×C3—3381no—
119C2×C3—6761no—
120C2×C3—6762no—
121C2×C3—6761no—
122C2×C3—1692no—
123C22—132no—
124C22—3382no—
125C22—3382no—
126C22—1691no—
127C22—1691no—
128C22#11759818⋊C211yescharacteristic, complemented
129C4—132no—
130C3—6761noSylow
131C2—21no—
132C2—262no—
133C2—3381no—
134C2—1692no—
135C2C2×#11759818⋊C211yescharacteristic
subgroupG/H
center, FrattiniC2C2×#11759818⋊C2
commutator, socle, cosocleC2×C132C23×C3
FittingC22×C132C22×C3
hypercenterC22#11759818⋊C2
hypocenterC132C2×C3×D4
LCSG ▷ C2×C132 ▷ C132
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #72241904C2semidirect
2. #72241901C2semidirect
3. C2×#11759818⋊C2C2semidirect
4. #72241888C2semidirect
5. C22×#11759816C2semidirect
6. #72214115C3semidirect
7. C22×#92789C22semidirect
8. C2×#11759818C22semidirect
11. C2×#11759816C22semidirect
13. #14523131C22semidirect
14. C22×C13⋊D13C2×C3semidirect
15. C2×D132C2×C3semidirect
16. C13×C22⋊D13C2×C3semidirect
17. #13062579C2×C3semidirect
18. #13062577C2×C3semidirect
23. #11759816D4semidirect
24. C22×C132C22×C3semidirect
27. C2×C13⋊D13C22×C3semidirect
28. C2×C13×D13C22×C3semidirect
30. C13×Dic13C22×C3semidirect
32. #92789C2×D4semidirect
36. C13⋊D13C3×D4semidirect
46. C132C2×C3×D4semidirect
61. C22⋊D13D13⋊C3semidirect
72. C22×C13C2×D13⋊C3semidirect
81. C2×D13C2×D13⋊C3semidirect
83. Dic13C2×D13⋊C3semidirect
105. C13D4×D13⋊C3semidirect
128. C22#11759818⋊C2semidirect

Maximal subgroups

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

subgroupindex
#722419042
#722419012
C2×#11759818⋊C22
#722418882
C22×#117598162
#722141153
D4×D13⋊C313