Finite Group Databasebeta

← C22×#11758561 #16860512

Subgroups — C22×#11758561 #16860512

Order 3968.

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×#11758561C2148yesmaximal, direct product factor
2C22×#92495C212yesmaximal, complemented
3C22×#92497C214yesmaximal, complemented
4C23×C22⋊D31C214yesmaximal, complemented
5C22×D4×D31C214yesmaximal, complemented
6C23×D4×C31C211yesmaximal, characteristic, complemented
7C25×C31C2212yescomplemented
8C24×D31C2214yescomplemented
9C22×D4×C31C2214yescomplemented
10C22×D4×C31C22124yescomplemented
11C23×C4×C31C2211yescharacteristic, complemented
12C22×C22⋊D31C2218yescomplemented
13C22×C22⋊D31C2214yescomplemented
14C22×C22⋊D31C22196yescomplemented
15C23×Dic31C2214yescomplemented
16#11758561C221256yesdirect product factor
17C2×#92497C22196yescomplemented
18C2×D4×D31C22196yescomplemented
19C2×#92495C22148yescomplemented
20C22×D124C2212yescomplemented
21C22×C4×D31C2214yescomplemented
22C22×Dic62C2212yescomplemented
23C24×C31—48no—
24C24×C31C2314yes—
25C24×C31C2311yescharacteristic
26C24×C31C23124yescomplemented
27C23×D31—416no—
28C23×D31C2314yes—
29C23×D31C23148yescomplemented
30C2×D4×C31C23148yescomplemented
31C2×D4×C31C23164yescomplemented
32C22×C4×C31C2312yes—
33C22×C4×C31C23112yescomplemented
34C2×C22⋊D31C23148yescomplemented
35C2×C22⋊D31C23196yescomplemented
36C2×C22⋊D31C231256yescomplemented
37C22×Dic31C2314yes—
38C22×Dic31C23148yescomplemented
39#92497C231256yescomplemented
40D4×D31C231256yescomplemented
41#92495C231128yescomplemented
42C2×D124C23124yescomplemented
43C2×C4×D31C23148yescomplemented
44C2×Dic62C23124yescomplemented
45C23×C31—216no—
46C23×C31—24no—
47C23×C31—448no—
48C23×C31C24124yes—
49C23×C31C2416yes—
50C23×C31C2411yescharacteristic
51C23×C31C24132yescomplemented
52C22×D31—216no—
53C22×D31—496no—
54C22×D31C24124yes—
55C22×D31C24164yescomplemented
56D4×C31C24164yescomplemented
57C2×C4×C31C24112yes—
58C2×C4×C31C24116yescomplemented
59C22⋊D31C24164yescomplemented
60C22⋊D31C241128yescomplemented
61C2×Dic31C24124yes—
62C2×Dic31C24164yescomplemented
63D124C24132yescomplemented
64C4×D31C24164yescomplemented
65Dic62C24132yescomplemented
66C22×H(5,𝔽2)—311nomaximal, Sylow
67C22×C31—248no—
68C22×C31—212no—
69C22×C31—432no—
70C22×C31H(5,𝔽2)14yescomplemented
71C22×C31C25116yes—
72C22×C31C2514yes—
73C22×C31C2513yes—
74C2×D31—248no—
75C2×D31—464no—
76C2×D31C25116yes—
77C4×C31C2518yes—
78Dic31C25116yes—
79C2×H(5,𝔽2)—3148no—
80C22×#41—314no—
81C22×#41—312no—
82C23×D4—314no—
83C23×D4—314no—
84C23×D4D3111yescharacteristic, complemented
85C2×C31—216no—
86C2×C31—24no—
87C2×C31C2×H(5,𝔽2)16yescomplemented
88C2×C31C2611yescharacteristic
89D31—216no—
90C25—314no—
91C25C2×D3112yescomplemented
92H(5,𝔽2)—31256no—
93C2×#41—3196no—
94C2×#41—3148no—
95C22×Q8—312no—
96C22×D4—318no—
97C22×D4—314no—
98C22×D4—3196no—
99C22×D4—3196no—
100C22×D4—312no—
101C22×D4C2×D3114yescomplemented
102C22×D4C2×D31124yescomplemented
103C23×C4—314no—
104C23×C4—314no—
105C23×C4C2×D3111yescharacteristic, complemented
106C31C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
107C24—48no—
108C24—314no—
109C24—3148no—
110C24—12416no—
111C24C22×D3114yes—
112C24C22×D3111yescharacteristic
113C24C22×D31124yescomplemented
114#41—31256no—
115#41—31128no—
116C2×Q8—3124no—
117C2×D4—3196no—
118C2×D4—3148no—
119C2×D4—31256no—
120C2×D4—31256no—
121C2×D4—3124no—
122C2×D4C22×D31148yescomplemented
123C2×D4C22×D31164yescomplemented
124C22×C4—3148no—
125C22×C4—314no—
126C22×C4—3148no—
127C22×C4C22×D3112yes—
128C22×C4C22×D31112yescomplemented
129C23—24no—
130C23—216no—
131C23—448no—
132C23—3124no—
133C23—3164no—
134C23—6216no—
135C23—12496no—
136C23C23×D31124yes—
137C23C23×D3116yes—
138C23C23×D3111yescharacteristic
139C23C23×D31132yescomplemented
140Q8—3132no—
141D4—31128no—
142D4—3164no—
143D4—3132no—
144D4C23×D31164yescomplemented
145C2×C4—3124no—
146C2×C4—3164no—
147C2×C4—3164no—
148C2×C4C23×D31112yes—
149C2×C4C23×D31116yescomplemented
150C22—212no—
151C22—248no—
152C22—432no—
153C22—3116no—
154C22—6248no—
155C22—12464no—
156C22#1175856114yesdirect product factor
157C22C24×D31116yes—
158C22C24×D3114yes—
159C22C24×D3113yes—
160C4—3116no—
161C4C24×D3118yes—
162C2—24no—
163C2—216no—
164C2—6216no—
165C2C2×#1175856116yesdirect product factor
166C2C25×D3111yescharacteristic
subgroupG/H
centerC23C23×D31
commutator, cosocleC2×C31C26
FrattiniC2C25×D31
FittingC23×D4×C31C2
socleC23×C31C24
hypercenterC23×D4D31
hypocenterC31C22×H(5,𝔽2)
LCSG ▷ C2×C31 ▷ C31
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#11758561C2direct (N × H)
2. C22×#92495C2semidirect
3. C22×#92497C2semidirect
4. C23×C22⋊D31C2semidirect
5. C22×D4×D31C2semidirect
6. C23×D4×C31C2semidirect
7. C25×C31C22semidirect
8. C24×D31C22semidirect
9. C22×D4×C31C22semidirect
10. C22×D4×C31C22semidirect
11. C23×C4×C31C22semidirect
12. C22×C22⋊D31C22semidirect
13. C22×C22⋊D31C22semidirect
14. C22×C22⋊D31C22semidirect
15. C23×Dic31C22semidirect
16. #11758561C22direct (N × H)
17. C2×#92497C22semidirect
18. C2×D4×D31C22semidirect
19. C2×#92495C22semidirect
20. C22×D124C22semidirect
21. C22×C4×D31C22semidirect
22. C22×Dic62C22semidirect
26. C24×C31C23semidirect
29. C23×D31C23semidirect
30. C2×D4×C31C23semidirect
31. C2×D4×C31C23semidirect
33. C22×C4×C31C23semidirect
34. C2×C22⋊D31C23semidirect
35. C2×C22⋊D31C23semidirect
36. C2×C22⋊D31C23semidirect
38. C22×Dic31C23semidirect
39. #92497C23semidirect
40. D4×D31C23semidirect
41. #92495C23semidirect
42. C2×D124C23semidirect
43. C2×C4×D31C23semidirect
44. C2×Dic62C23semidirect
51. C23×C31C24semidirect
55. C22×D31C24semidirect
56. D4×C31C24semidirect
58. C2×C4×C31C24semidirect
59. C22⋊D31C24semidirect
60. C22⋊D31C24semidirect
62. C2×Dic31C24semidirect
63. D124C24semidirect
64. C4×D31C24semidirect
65. Dic62C24semidirect
70. C22×C31H(5,𝔽2)semidirect
84. C23×D4D31semidirect
87. C2×C31C2×H(5,𝔽2)semidirect
91. C25C2×D31semidirect
101. C22×D4C2×D31semidirect
102. C22×D4C2×D31semidirect
105. C23×C4C2×D31semidirect
106. C31C22×H(5,𝔽2)semidirect
113. C24C22×D31semidirect
122. C2×D4C22×D31semidirect
123. C2×D4C22×D31semidirect
128. C22×C4C22×D31semidirect
139. C23C23×D31semidirect
144. D4C23×D31semidirect
149. C2×C4C23×D31semidirect
156. C22#11758561direct (N × H)
165. C2C2×#11758561direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117585612
C22×#924952
C22×#924972
C23×C22⋊D312
C22×D4×D312
C23×D4×C312
C22×H(5,𝔽2)31