Finite Group Databasebeta

← C22×#16862291 #30680762

Subgroups — C22×#16862291 #30680762

Order 16768.

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×#16862291C2148yesmaximal, direct product factor
2C22×#14523513C212yesmaximal, complemented
3C22×#14523512C214yesmaximal, complemented
4C23×C22⋊D131C214yesmaximal, complemented
5C22×D4×D131C214yesmaximal, complemented
6C23×D4×C131C211yesmaximal, characteristic, complemented
7C24×D131C2214yescomplemented
8#16862291C221256yesdirect product factor
9C2×#14523513C22148yescomplemented
10C2×#14523512C22196yescomplemented
11C22×Dic262C2212yescomplemented
12C22×D524C2212yescomplemented
13C22×C22⋊D131C2218yescomplemented
14C22×C22⋊D131C2214yescomplemented
15C22×C22⋊D131C22196yescomplemented
16C2×D4×D131C22196yescomplemented
17C23×Dic131C2214yescomplemented
18C22×C4×D131C2214yescomplemented
19C25×C131C2212yescomplemented
20C22×D4×C131C2214yescomplemented
21C22×D4×C131C22124yescomplemented
22C23×C4×C131C2211yescharacteristic, complemented
23C23×D131—416no—
24C23×D131C2314yes—
25C23×D131C23148yescomplemented
26#14523513C231128yescomplemented
27#14523512C231256yescomplemented
28C2×Dic262C23124yescomplemented
29C2×D524C23124yescomplemented
30C2×C22⋊D131C23196yescomplemented
31C2×C22⋊D131C23148yescomplemented
32C2×C22⋊D131C231256yescomplemented
33D4×D131C231256yescomplemented
34C22×Dic131C2314yes—
35C22×Dic131C23148yescomplemented
36C2×C4×D131C23148yescomplemented
37C24×C131—48no—
38C24×C131C2314yes—
39C24×C131C2311yescharacteristic
40C24×C131C23124yescomplemented
41C2×D4×C131C23148yescomplemented
42C2×D4×C131C23164yescomplemented
43C22×C4×C131C2312yes—
44C22×C4×C131C23112yescomplemented
45C22×D131—216no—
46C22×D131—496no—
47C22×D131C24124yes—
48C22×D131C24164yescomplemented
49Dic262C24132yescomplemented
50D524C24132yescomplemented
51C22⋊D131C241128yescomplemented
52C22⋊D131C24164yescomplemented
53C2×Dic131C24124yes—
54C2×Dic131C24164yescomplemented
55C4×D131C24164yescomplemented
56C23×C131—24no—
57C23×C131—216no—
58C23×C131—448no—
59C23×C131C2416yes—
60C23×C131C24124yes—
61C23×C131C2411yescharacteristic
62C23×C131C24132yescomplemented
63D4×C131C24164yescomplemented
64C2×C4×C131C24112yes—
65C2×C4×C131C24116yescomplemented
66C22×C131—212no—
67C22×C131—248no—
68C22×C131—432no—
69C22×C131H(5,𝔽2)14yescomplemented
70C22×C131C2514yes—
71C22×C131C25116yes—
72C22×C131C2513yes—
73C2×D131—248no—
74C2×D131—464no—
75C2×D131C25116yes—
76C4×C131C2518yes—
77Dic131C25116yes—
78C2×C131—216no—
79C2×C131—24no—
80C2×C131C2×H(5,𝔽2)16yescomplemented
81C2×C131C2611yescharacteristic
82D131—216no—
83C131C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
84C22×H(5,𝔽2)—1311nomaximal, Sylow
85C2×H(5,𝔽2)—13148no—
86C22×#41—1314no—
87C22×#41—1312no—
88C23×D4—1314no—
89C23×D4—1314no—
90C23×D4D13111yescharacteristic, complemented
91C25—1314no—
92C25C2×D13112yescomplemented
93H(5,𝔽2)—131256no—
94C2×#41—13196no—
95C2×#41—13148no—
96C22×Q8—1312no—
97C22×D4—1318no—
98C22×D4—1314no—
99C22×D4—13196no—
100C22×D4—13196no—
101C22×D4—1312no—
102C22×D4C2×D13114yescomplemented
103C22×D4C2×D131124yescomplemented
104C23×C4—1314no—
105C23×C4—1314no—
106C23×C4C2×D13111yescharacteristic, complemented
107C24—48no—
108C24—52416no—
109C24—1314no—
110C24—13148no—
111C24C22×D13114yes—
112C24C22×D13111yescharacteristic
113C24C22×D131124yescomplemented
114#41—131256no—
115#41—131128no—
116C2×Q8—13124no—
117C2×D4—13196no—
118C2×D4—13148no—
119C2×D4—131256no—
120C2×D4—131256no—
121C2×D4—13124no—
122C2×D4C22×D131148yescomplemented
123C2×D4C22×D131164yescomplemented
124C22×C4—13148no—
125C22×C4—1314no—
126C22×C4—13148no—
127C22×C4C22×D13112yes—
128C22×C4C22×D131112yescomplemented
129C23—24no—
130C23—216no—
131C23—448no—
132C23—26216no—
133C23—52496no—
134C23—13124no—
135C23—13164no—
136C23C23×D131124yes—
137C23C23×D13116yes—
138C23C23×D13111yescharacteristic
139C23C23×D131132yescomplemented
140Q8—13132no—
141D4—131128no—
142D4—13164no—
143D4—13132no—
144D4C23×D131164yescomplemented
145C2×C4—13124no—
146C2×C4—13164no—
147C2×C4—13164no—
148C2×C4C23×D131112yes—
149C2×C4C23×D131116yescomplemented
150C22—212no—
151C22—248no—
152C22—432no—
153C22—26248no—
154C22—52464no—
155C22—13116no—
156C22#1686229114yesdirect product factor
157C22C24×D131116yes—
158C22C24×D13114yes—
159C22C24×D13113yes—
160C4—13116no—
161C4C24×D13118yes—
162C2—24no—
163C2—216no—
164C2—26216no—
165C2C2×#1686229116yesdirect product factor
166C2C25×D13111yescharacteristic
subgroupG/H
centerC23C23×D131
commutator, cosocleC2×C131C26
FrattiniC2C25×D131
FittingC23×D4×C131C2
socleC23×C131C24
hypercenterC23×D4D131
hypocenterC131C22×H(5,𝔽2)
LCSG ▷ C2×C131 ▷ C131
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16862291C2direct (N × H)
2. C22×#14523513C2semidirect
3. C22×#14523512C2semidirect
4. C23×C22⋊D131C2semidirect
5. C22×D4×D131C2semidirect
6. C23×D4×C131C2semidirect
7. C24×D131C22semidirect
8. #16862291C22direct (N × H)
9. C2×#14523513C22semidirect
10. C2×#14523512C22semidirect
11. C22×Dic262C22semidirect
12. C22×D524C22semidirect
13. C22×C22⋊D131C22semidirect
14. C22×C22⋊D131C22semidirect
15. C22×C22⋊D131C22semidirect
16. C2×D4×D131C22semidirect
17. C23×Dic131C22semidirect
18. C22×C4×D131C22semidirect
19. C25×C131C22semidirect
20. C22×D4×C131C22semidirect
21. C22×D4×C131C22semidirect
22. C23×C4×C131C22semidirect
25. C23×D131C23semidirect
26. #14523513C23semidirect
27. #14523512C23semidirect
28. C2×Dic262C23semidirect
29. C2×D524C23semidirect
30. C2×C22⋊D131C23semidirect
31. C2×C22⋊D131C23semidirect
32. C2×C22⋊D131C23semidirect
33. D4×D131C23semidirect
35. C22×Dic131C23semidirect
36. C2×C4×D131C23semidirect
40. C24×C131C23semidirect
41. C2×D4×C131C23semidirect
42. C2×D4×C131C23semidirect
44. C22×C4×C131C23semidirect
48. C22×D131C24semidirect
49. Dic262C24semidirect
50. D524C24semidirect
51. C22⋊D131C24semidirect
52. C22⋊D131C24semidirect
54. C2×Dic131C24semidirect
55. C4×D131C24semidirect
62. C23×C131C24semidirect
63. D4×C131C24semidirect
65. C2×C4×C131C24semidirect
69. C22×C131H(5,𝔽2)semidirect
80. C2×C131C2×H(5,𝔽2)semidirect
83. C131C22×H(5,𝔽2)semidirect
90. C23×D4D131semidirect
92. C25C2×D131semidirect
102. C22×D4C2×D131semidirect
103. C22×D4C2×D131semidirect
106. C23×C4C2×D131semidirect
113. C24C22×D131semidirect
122. C2×D4C22×D131semidirect
123. C2×D4C22×D131semidirect
128. C22×C4C22×D131semidirect
139. C23C23×D131semidirect
144. D4C23×D131semidirect
149. C2×C4C23×D131semidirect
156. C22#16862291direct (N × H)
165. C2C2×#16862291direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168622912
C22×#145235132
C22×#145235122
C23×C22⋊D1312
C22×D4×D1312
C23×D4×C1312
C22×H(5,𝔽2)131