Finite Group Databasebeta

← #16855108

Subgroups — #16855108

Order 3968.

800 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 161 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#14521608C213yesmaximal, complemented
2#14521239C214yesmaximal, complemented
3#14521236C214yesmaximal, complemented
4C2×#11758422C213yesmaximal, complemented
5C2×C31×H(5,𝔽2)C211yesmaximal, characteristic, complemented
6C31×H(5,𝔽2)—26no—
7C31×H(5,𝔽2)C2214yes—
8C2×#41×C31—23no—
9C22×D4×C31—23no—
10C22×D4×C31C2213yes—
11C22×C22⋊D31C2211yescharacteristic, complemented
12#11758530—23no—
13C2×C22⋊Dic31C2213yescomplemented
14#11758516—26no—
15#11758514C2216yescomplemented
16#11758512C2216yescomplemented
17#11758506—23no—
18#11758422C22112yescomplemented
19C24×C31—21no—
20C24×C31D413yescomplemented
21C24×C31C2311yescharacteristic
22C23×D31C2311yescharacteristic, complemented
23#41×C31—212no—
24#41×C31—212no—
25C2×Q8×C31—21no—
26C2×D4×C31—23no—
27C2×D4×C31—212no—
28C2×D4×C31—26no—
29C2×D4×C31—212no—
30C2×D4×C31—26no—
31C2×D4×C31D416yescomplemented
32C2×D4×C31C2316yes—
33C22×C4×C31—23no—
34C22×C4×C31D413yescomplemented
35C2×C22⋊D31—23no—
36C2×C22⋊D31—23no—
37C2×C22⋊D31—26no—
38C22×Dic31—23no—
39C22×Dic31C2311yescharacteristic, complemented
40C22⋊Dic31—23no—
41C22⋊Dic31—26no—
42C22⋊Dic31C2316yescomplemented
43(C2×D31)⋊C4—26no—
44Dic31⋊C4—26no—
45C23×C31—24no—
46C23×C31—23no—
47C23×C31—23no—
48C23×C31—81no—
49C23×C31—83no—
50C23×C31—82no—
51C23×C31C2×D416yes—
52C23×C31C2×D413yes—
53C23×C31C2×D413yes—
54C23×C31C2411yescharacteristic
55C22×D31—23no—
56C22×D31—81no—
57Q8×C31—24no—
58D4×C31—212no—
59D4×C31—212no—
60D4×C31—212no—
61C2×C4×C31—212no—
62C2×C4×C31—23no—
63C2×C4×C31—23no—
64C2×C4×C31C2×D416yes—
65C22⋊D31—41no—
66C22⋊D31—43no—
67C22⋊D31—86no—
68C2×Dic31—23no—
69C2×Dic31—26no—
70C2×Dic31—83no—
71#3026—311nomaximal, Sylow
72C22×C31—26no—
73C22×C31—43no—
74C22×C31—41no—
75C22×C31—43no—
76C22×C31—46no—
77C22×C31—46no—
78C22×C31—83no—
79C22×C31—84no—
80C22×C31#12011yescharacteristic
81C22×C31#12013yes—
82C22×C31C22×D413yes—
83C2×D31—46no—
84C2×D31—41no—
85C4×C31—26no—
86Dic31—46no—
87Dic31—41no—
88C2×H(5,𝔽2)D3111yescharacteristic, complemented
89#534—313no—
90#458—314no—
91B(4,𝔽2)—314no—
92C2×#99—313no—
93C2×C31—23no—
94C2×C31—21no—
95C2×C31—23no—
96C2×C31—46no—
97C2×C31C2×#12011yescharacteristic
98D31—81no—
99H(5,𝔽2)—26no—
100H(5,𝔽2)C2×D3114yes—
101C2×#41—23no—
102C22×D4—23no—
103C22×D4—311no—
104C22×D4C2×D3113yes—
105#123—316no—
106#123—623no—
107#121—626no—
108#120—316no—
109#120—623no—
110C2×C22⋊C4—313no—
111#99—3112no—
112C31#302611yescharacteristic, complemented, Sylow
113C24—21no—
114C24—311no—
115C24C22⋊D3113yescomplemented
116C24C22×D3111yescharacteristic
117#41—212no—
118#41—212no—
119C2×Q8—21no—
120C2×D4—23no—
121C2×D4—212no—
122C2×D4—26no—
123C2×D4—212no—
124C2×D4—26no—
125C2×D4—623no—
126C2×D4—623no—
127C2×D4—626no—
128C2×D4C22⋊D3116yescomplemented
129C2×D4C22×D3116yes—
130C22×C4—23no—
131C22×C4—311no—
132C22×C4—623no—
133C22×C4C22⋊D3113yescomplemented
134C4⋊C4—626no—
135C22⋊C4—316no—
136C22⋊C4—623no—
137C22⋊C4—626no—
138C22⋊C4—626no—
139C23—24no—
140C23—23no—
141C23—23no—
142C23—81no—
143C23—83no—
144C23—82no—
145C23—623no—
146C23—2481no—
147C23C2×C22⋊D3116yes—
148C23C2×C22⋊D3113yes—
149C23C2×C22⋊D3113yes—
150C23C23×D3111yescharacteristic
151Q8—24no—
152D4—212no—
153D4—212no—
154D4—212no—
155D4—1241no—
156D4—1243no—
157D4—2486no—
158C2×C4—212no—
159C2×C4—23no—
160C2×C4—23no—
161C2×C4—626no—
162C2×C4—623no—
163C2×C4—2483no—
164C2×C4C2×C22⋊D3116yes—
165C22—26no—
166C22—43no—
167C22—43no—
168C22—41no—
169C22—46no—
170C22—46no—
171C22—83no—
172C22—84no—
173C22—1246no—
174C22—1241no—
175C22#1175853011yescharacteristic
176C22#1175853013yes—
177C22C22×C22⋊D3113yes—
178C4—26no—
179C4—1246no—
180C4—1241no—
181C2—23no—
182C2—21no—
183C2—23no—
184C2—46no—
185C2—2481no—
186C2C2×#1175853011yescharacteristic
subgroupG/H
centerC2C2×#11758530
commutator, cosocleC23×C31C24
FrattiniC23C23×D31
FittingC2×C31×H(5,𝔽2)C2
socleC2×C31C2×#120
hypercenterC2×H(5,𝔽2)D31
hypocenterC31#3026
LCSG ▷ C23×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C24 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #14521608C2semidirect
2. #14521239C2semidirect
3. #14521236C2semidirect
4. C2×#11758422C2semidirect
5. C2×C31×H(5,𝔽2)C2semidirect
11. C22×C22⋊D31C22semidirect
13. C2×C22⋊Dic31C22semidirect
15. #11758514C22semidirect
16. #11758512C22semidirect
18. #11758422C22semidirect
20. C24×C31D4semidirect
22. C23×D31C23semidirect
31. C2×D4×C31D4semidirect
34. C22×C4×C31D4semidirect
39. C22×Dic31C23semidirect
42. C22⋊Dic31C23semidirect
88. C2×H(5,𝔽2)D31semidirect
112. C31#3026semidirect
115. C24C22⋊D31semidirect
128. C2×D4C22⋊D31semidirect
133. C22×C4C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
#145216082
#145212392
#145212362
C2×#117584222
C2×C31×H(5,𝔽2)2
#302631