Finite Group Databasebeta

← #120×Dic31 #16848214

Subgroups — #120×Dic31 #16848214

Order 3968.

1224 conjugacy classes of subgroups; only the 289 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 289 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
1C24×Dic31C211yesmaximal, characteristic, complemented
2C2×D4×Dic31C213yesmaximal, complemented
3#14520929C213yesmaximal, complemented
4#14520924C213yesmaximal, complemented
5#14520905C211yesmaximal, characteristic, complemented
6C22⋊C4×Dic31C213yesmaximal, complemented
7C2×C31×#120C211yesmaximal, characteristic
8C25×C31C2211yescharacteristic
9C22×D4×C31C2213yes—
10C23×Dic31—212no—
11C23×Dic31C2213yes—
12C23×Dic31C2213yes—
13C23×Dic31C2211yescharacteristic, complemented
14C31×#120C418yescomplemented
15C2×C22⋊C4×C31C2213yes—
16C2×C22⋊Dic31C2213yes—
17C2×C22⋊Dic31C2213yes—
18C2×C22⋊Dic31C2213yescomplemented
19D4×Dic31—212no—
20C2×C4⋊Dic31C2213yescomplemented
21C2×C4×Dic31C2213yescomplemented
22C31⋊#95C2216yescomplemented
23C24×C31—26no—
24C24×C31—26no—
25C24×C31C2×C414yescomplemented
26C24×C31C2311yescharacteristic
27C24×C31C2313yes—
28C2×D4×C31—26no—
29C2×D4×C31—26no—
30C2×D4×C31C2×C4112yescomplemented
31C22×C4×C31C2313yes—
32C22×Dic31—232no—
33C22×Dic31—26no—
34C22×Dic31—212no—
35C22×Dic31—212no—
36C22×Dic31—26no—
37C22×Dic31D4112yescomplemented
38C22×Dic31C2313yes—
39C22×Dic31C2311yescharacteristic
40C22×Dic31C2313yes—
41C22×Dic31C2311yescharacteristic
42C22⋊C4×C31C2×C4112yescomplemented
43C22⋊Dic31—26no—
44C22⋊Dic31—26no—
45C22⋊Dic31—26no—
46C4⋊Dic31—26no—
47C4×Dic31—26no—
48C23×C31—28no—
49C23×C31—224no—
50C23×C31—224no—
51C23×C31—26no—
52C23×C31—26no—
53C23×C31—23no—
54C23×C31—23no—
55C23×C31C22×C416yes—
56C23×C31C22×C412yescomplemented
57C23×C31C2×D416yes—
58C23×C31#4116yes—
59C23×C31C2411yescharacteristic
60D4×C31—224no—
61C2×C4×C31—23no—
62C2×C4×C31—23no—
63C2×C4×C31C22×C416yescomplemented
64C2×Dic31—224no—
65C2×Dic31—26no—
66C2×Dic31—224no—
67C2×Dic31—23no—
68C2×Dic31—23no—
69C2×Dic31—26no—
70C2×Dic31—44no—
71C2×Dic31C2×D416yes—
72C2×Dic31C2×D416yes—
73C4×#120—311nomaximal, Sylow
74C22×C31—232no—
75C22×C31—212no—
76C22×C31—26no—
77C22×C31—26no—
78C22×C31—212no—
79C22×C31—26no—
80C22×C31—41no—
81C22×C31—43no—
82C22×C31C4×D4112yescomplemented
83C22×C31C23×C411yescharacteristic
84C22×C31C22×D413yes—
85C22×C31C2×#4113yes—
86C4×C31—26no—
87Dic31—26no—
88Dic31—41no—
89Dic31—43no—
90Dic31#12014yesdirect product factor
91C24×C4—311no—
92C2×#120D3111yescharacteristic
93C2×C4×D4—313no—
94#386—313no—
95#385—313no—
96#379—311no—
97C4×C22⋊C4—313no—
98C2×C31—212no—
99C2×C31—42no—
100C2×C31C2×C4×D413yes—
101C2×C31C2×#12011yescharacteristic
102C2×C31#52513yes—
103C25C2×D3111yescharacteristic
104C22×D4C2×D3113yes—
105C23×C4—313no—
106C23×C4—311no—
107C23×C4—313no—
108C23×C4—6212no—
109#120Dic3118yesdirect product factor
110C4×D4—6212no—
111C2×C4⋊C4—313no—
112C2×C22⋊C4—313no—
113C2×C22⋊C4—313no—
114C2×C22⋊C4—313no—
115C2×C22⋊C4C2×D3113yes—
116C2×C42—313no—
117#95—316no—
118C31C4×#12011yescharacteristic, complemented, Sylow
119C24—26no—
120C24—26no—
121C24C2×Dic3114yescomplemented
122C24C22×D3111yescharacteristic
123C24C22×D3113yes—
124C2×D4—26no—
125C2×D4—26no—
126C2×D4C2×Dic31112yescomplemented
127C22×C4—311no—
128C22×C4—3112no—
129C22×C4—313no—
130C22×C4—313no—
131C22×C4—311no—
132C22×C4—6212no—
133C22×C4—6212no—
134C22×C4—626no—
135C22×C4—6232no—
136C22×C4—626no—
137C22×C4C22×D3113yes—
138C4⋊C4—626no—
139C22⋊C4—626no—
140C22⋊C4—626no—
141C22⋊C4—626no—
142C22⋊C4C2×Dic31112yescomplemented
143C42—626no—
144C23—28no—
145C23—224no—
146C23—224no—
147C23—26no—
148C23—26no—
149C23—23no—
150C23—23no—
151C23D4×D3116yes—
152C23#9249716yes—
153C23C22×Dic3116yes—
154C23C22×Dic3112yescomplemented
155C23C23×D3111yescharacteristic
156D4—224no—
157C2×C4—23no—
158C2×C4—23no—
159C2×C4—316no—
160C2×C4—316no—
161C2×C4—6224no—
162C2×C4—6224no—
163C2×C4—626no—
164C2×C4—623no—
165C2×C4—623no—
166C2×C4—626no—
167C2×C4—1244no—
168C2×C4C22×Dic3116yescomplemented
169C22—232no—
170C22—212no—
171C22—26no—
172C22—26no—
173C22—212no—
174C22—26no—
175C22—41no—
176C22—43no—
177C22D4×Dic31112yescomplemented
178C22C2×D4×D3113yes—
179C22C2×#9249713yes—
180C22C23×Dic3111yescharacteristic
181C4—26no—
182C4—314no—
183C4—626no—
184C4—1241no—
185C4—1243no—
186C2—212no—
187C2—42no—
188C2C2×D4×Dic3113yes—
189C2#120×D3111yescharacteristic
190C2#1452155513yes—
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
Fitting, prime radicalC2×C31×#120C2
socle, cosocleC23×C31C24
hypercenterC2×#120D31
hypocenterC31C4×#120
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C24×Dic31C2semidirect
2. C2×D4×Dic31C2semidirect
3. #14520929C2semidirect
4. #14520924C2semidirect
5. #14520905C2semidirect
6. C22⋊C4×Dic31C2semidirect
13. C23×Dic31C22semidirect
14. C31×#120C4semidirect
18. C2×C22⋊Dic31C22semidirect
20. C2×C4⋊Dic31C22semidirect
21. C2×C4×Dic31C22semidirect
22. C31⋊#95C22semidirect
25. C24×C31C2×C4semidirect
30. C2×D4×C31C2×C4semidirect
37. C22×Dic31D4semidirect
42. C22⋊C4×C31C2×C4semidirect
56. C23×C31C22×C4semidirect
63. C2×C4×C31C22×C4semidirect
82. C22×C31C4×D4semidirect
90. Dic31#120direct (N × H)
109. #120Dic31direct (N × H)
118. C31C4×#120semidirect
121. C24C2×Dic31semidirect
126. C2×D4C2×Dic31semidirect
142. C22⋊C4C2×Dic31semidirect
154. C23C22×Dic31semidirect
168. C2×C4C22×Dic31semidirect
177. C22D4×Dic31semidirect

Maximal subgroups

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

subgroupindex
C24×Dic312
C2×D4×Dic312
#145209292
#145209242
#145209052
C22⋊C4×Dic312
C2×C31×#1202
C4×#12031