Finite Group Databasebeta

← C22×#22575498 #46825686

Subgroups — C22×#22575498 #46825686

Order 34432.

6560 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.

normal subgroups only

This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3221 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
1C2×#22575498C2148yesmaximal, direct product factor
2C22×#16864402C212yesmaximal, complemented
3C22×#16864401C214yesmaximal, complemented
4C23×C22⋊D269C214yesmaximal, complemented
5C22×D4×D269C214yesmaximal, complemented
6C23×D4×C269C211yesmaximal, characteristic, complemented
7C24×D269C2214yescomplemented
8#22575498C221256yesdirect product factor
9C2×#16864402C22148yescomplemented
10C2×#16864401C22196yescomplemented
11C22×Dic538C2212yescomplemented
12C22×D1076C2212yescomplemented
13C22×C22⋊D269C2218yescomplemented
14C22×C22⋊D269C2214yescomplemented
15C22×C22⋊D269C22196yescomplemented
16C2×D4×D269C22196yescomplemented
17C23×Dic269C2214yescomplemented
18C22×C4×D269C2214yescomplemented
19C25×C269C2212yescomplemented
20C22×D4×C269C2214yescomplemented
21C22×D4×C269C22124yescomplemented
22C23×C4×C269C2211yescharacteristic, complemented
23C23×D269—416no—
24C23×D269C2314yes—
25C23×D269C23148yescomplemented
26#16864402C231128yescomplemented
27#16864401C231256yescomplemented
28C2×Dic538C23124yescomplemented
29C2×D1076C23124yescomplemented
30C2×C22⋊D269C23196yescomplemented
31C2×C22⋊D269C23148yescomplemented
32C2×C22⋊D269C231256yescomplemented
33D4×D269C231256yescomplemented
34C22×Dic269C2314yes—
35C22×Dic269C23148yescomplemented
36C2×C4×D269C23148yescomplemented
37C24×C269—48no—
38C24×C269C2314yes—
39C24×C269C2311yescharacteristic
40C24×C269C23124yescomplemented
41C2×D4×C269C23148yescomplemented
42C2×D4×C269C23164yescomplemented
43C22×C4×C269C2312yes—
44C22×C4×C269C23112yescomplemented
45C22×D269—216no—
46C22×D269—496no—
47C22×D269C24124yes—
48C22×D269C24164yescomplemented
49Dic538C24132yescomplemented
50D1076C24132yescomplemented
51C22⋊D269C241128yescomplemented
52C22⋊D269C24164yescomplemented
53C2×Dic269C24124yes—
54C2×Dic269C24164yescomplemented
55C4×D269C24164yescomplemented
56C23×C269—24no—
57C23×C269—216no—
58C23×C269—448no—
59C23×C269C2416yes—
60C23×C269C24124yes—
61C23×C269C2411yescharacteristic
62C23×C269C24132yescomplemented
63D4×C269C24164yescomplemented
64C2×C4×C269C24112yes—
65C2×C4×C269C24116yescomplemented
66C22×C269—212no—
67C22×C269—248no—
68C22×C269—432no—
69C22×C269H(5,𝔽2)14yescomplemented
70C22×C269C2514yes—
71C22×C269C25116yes—
72C22×C269C2513yes—
73C2×D269—248no—
74C2×D269—464no—
75C2×D269C25116yes—
76C4×C269C2518yes—
77Dic269C25116yes—
78C2×C269—216no—
79C2×C269—24no—
80C2×C269C2×H(5,𝔽2)16yescomplemented
81C2×C269C2611yescharacteristic
82D269—216no—
83C269C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
84C22×H(5,𝔽2)—2691nomaximal, Sylow
85C2×H(5,𝔽2)—26948no—
86C22×#41—2694no—
87C22×#41—2692no—
88C23×D4—2694no—
89C23×D4—2694no—
90C23×D4D26911yescharacteristic, complemented
91C25—2694no—
92C25C2×D26912yescomplemented
93H(5,𝔽2)—269256no—
94C2×#41—26996no—
95C2×#41—26948no—
96C22×Q8—2692no—
97C22×D4—2698no—
98C22×D4—2694no—
99C22×D4—26996no—
100C22×D4—26996no—
101C22×D4—2692no—
102C22×D4C2×D26914yescomplemented
103C22×D4C2×D269124yescomplemented
104C23×C4—2694no—
105C23×C4—2694no—
106C23×C4C2×D26911yescharacteristic, complemented
107C24—48no—
108C24—2694no—
109C24—26948no—
110C24—107616no—
111C24C22×D26914yes—
112C24C22×D26911yescharacteristic
113C24C22×D269124yescomplemented
114#41—269256no—
115#41—269128no—
116C2×Q8—26924no—
117C2×D4—26996no—
118C2×D4—26948no—
119C2×D4—269256no—
120C2×D4—269256no—
121C2×D4—26924no—
122C2×D4C22×D269148yescomplemented
123C2×D4C22×D269164yescomplemented
124C22×C4—26948no—
125C22×C4—2694no—
126C22×C4—26948no—
127C22×C4C22×D26912yes—
128C22×C4C22×D269112yescomplemented
129C23—24no—
130C23—216no—
131C23—448no—
132C23—26924no—
133C23—26964no—
134C23—53816no—
135C23—107696no—
136C23C23×D269124yes—
137C23C23×D26916yes—
138C23C23×D26911yescharacteristic
139C23C23×D269132yescomplemented
140Q8—26932no—
141D4—269128no—
142D4—26964no—
143D4—26932no—
144D4C23×D269164yescomplemented
145C2×C4—26924no—
146C2×C4—26964no—
147C2×C4—26964no—
148C2×C4C23×D269112yes—
149C2×C4C23×D269116yescomplemented
150C22—212no—
151C22—248no—
152C22—432no—
153C22—26916no—
154C22—53848no—
155C22—107664no—
156C22#2257549814yesdirect product factor
157C22C24×D269116yes—
158C22C24×D26914yes—
159C22C24×D26913yes—
160C4—26916no—
161C4C24×D26918yes—
162C2—24no—
163C2—216no—
164C2—53816no—
165C2C2×#2257549816yesdirect product factor
166C2C25×D26911yescharacteristic
subgroupG/H
centerC23C23×D269
commutator, cosocleC2×C269C26
FrattiniC2C25×D269
FittingC23×D4×C269C2
socleC23×C269C24
hypercenterC23×D4D269
hypocenterC269C22×H(5,𝔽2)
LCSG ▷ C2×C269 ▷ C269
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#22575498C2direct (N × H)
2. C22×#16864402C2semidirect
3. C22×#16864401C2semidirect
4. C23×C22⋊D269C2semidirect
5. C22×D4×D269C2semidirect
6. C23×D4×C269C2semidirect
7. C24×D269C22semidirect
8. #22575498C22direct (N × H)
9. C2×#16864402C22semidirect
10. C2×#16864401C22semidirect
11. C22×Dic538C22semidirect
12. C22×D1076C22semidirect
13. C22×C22⋊D269C22semidirect
14. C22×C22⋊D269C22semidirect
15. C22×C22⋊D269C22semidirect
16. C2×D4×D269C22semidirect
17. C23×Dic269C22semidirect
18. C22×C4×D269C22semidirect
19. C25×C269C22semidirect
20. C22×D4×C269C22semidirect
21. C22×D4×C269C22semidirect
22. C23×C4×C269C22semidirect
25. C23×D269C23semidirect
26. #16864402C23semidirect
27. #16864401C23semidirect
28. C2×Dic538C23semidirect
29. C2×D1076C23semidirect
30. C2×C22⋊D269C23semidirect
31. C2×C22⋊D269C23semidirect
32. C2×C22⋊D269C23semidirect
33. D4×D269C23semidirect
35. C22×Dic269C23semidirect
36. C2×C4×D269C23semidirect
40. C24×C269C23semidirect
41. C2×D4×C269C23semidirect
42. C2×D4×C269C23semidirect
44. C22×C4×C269C23semidirect
48. C22×D269C24semidirect
49. Dic538C24semidirect
50. D1076C24semidirect
51. C22⋊D269C24semidirect
52. C22⋊D269C24semidirect
54. C2×Dic269C24semidirect
55. C4×D269C24semidirect
62. C23×C269C24semidirect
63. D4×C269C24semidirect
65. C2×C4×C269C24semidirect
69. C22×C269H(5,𝔽2)semidirect
80. C2×C269C2×H(5,𝔽2)semidirect
83. C269C22×H(5,𝔽2)semidirect
90. C23×D4D269semidirect
92. C25C2×D269semidirect
102. C22×D4C2×D269semidirect
103. C22×D4C2×D269semidirect
106. C23×C4C2×D269semidirect
113. C24C22×D269semidirect
122. C2×D4C22×D269semidirect
123. C2×D4C22×D269semidirect
128. C22×C4C22×D269semidirect
139. C23C23×D269semidirect
144. D4C23×D269semidirect
149. C2×C4C23×D269semidirect
156. C22#22575498direct (N × H)
165. C2C2×#22575498direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225754982
C22×#168644022
C22×#168644012
C23×C22⋊D2692
C22×D4×D2692
C23×D4×C2692
C22×H(5,𝔽2)269