Finite Group Databasebeta

← C2×#14521706 #16859175

Subgroups — C2×#14521706 #16859175

Order 3968.

1620 conjugacy classes of subgroups; only the 573 normal subgroups are shown.

normal subgroups only

This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 573 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
1C22×Q8×D31C211yesmaximal, characteristic, complemented
2#14521706C2116yesmaximal, direct product factor
3C2×#11758513C213yesmaximal, complemented
4C2×#11758501C213yesmaximal, complemented
5C2×#11758515C213yesmaximal, complemented
6C2×Q8×Dic31C211yesmaximal, characteristic, complemented
7C2×C4×C22⋊D31C213yesmaximal, complemented
8C22×#41×C31C211yesmaximal, characteristic, complemented
9C2×#41×C31—28no—
10C2×#41×C31C2218yescomplemented
11C22×Q8×C31C2211yescharacteristic
12C22×D4×C31C2213yes—
13C23×C4×C31C2213yes—
14C22×C22⋊D31C2213yes—
15C2×Q8×D31—28no—
16C2×Q8×D31C2218yescomplemented
17C22×C4×D31C2213yes—
18C22×Dic62C2213yes—
19C2×C22⋊Dic31C2213yes—
20C2×C22⋊Dic31C2216yes—
21Q8×Dic31C2218yescomplemented
22#11758515C22124yescomplemented
23#11758513C22124yescomplemented
24C4×C22⋊D31C22124yescomplemented
25C2×(C2×D31)⋊C4C2211yescharacteristic
26C2×C4⋊Dic31C2213yes—
27#11758501C22124yescomplemented
28C2×Dic31⋊C4C2213yes—
29C2×C4×Dic31C2213yes—
30C24×C31C2313yes—
31C23×D31C2311yescharacteristic
32#41×C31—232no—
33C2×Q8×C31D418yescomplemented
34C2×Q8×C31C2314yes—
35C2×D4×C31—212no—
36C2×D4×C31C23112yes—
37C22×C4×C31—212no—
38C22×C4×C31C2311yescharacteristic
39C22×C4×C31C23112yes—
40C22×C4×C31C2313yes—
41C2×C22⋊D31—212no—
42C2×C22⋊D31C23112yes—
43C22×Dic31C2311yescharacteristic
44C22×Dic31C2313yes—
45C22×Dic31C2313yes—
46Q8×D31—232no—
47C2×C4×D31—212no—
48C2×C4×D31C23112yes—
49C2×Dic62—212no—
50C2×Dic62C23112yes—
51C22⋊Dic31C23124yes—
52C22⋊Dic31C23112yes—
53(C2×D31)⋊C4C2314yes—
54C4⋊Dic31C23112yes—
55Dic31⋊C4C23112yes—
56C4×Dic31C23112yes—
57C23×C31—26no—
58C23×C31—26no—
59C23×C31—26no—
60C23×C31C2416yes—
61C23×C31C2411yescharacteristic
62C22×D31—22no—
63C22×D31—22no—
64C22×D31#4114yescomplemented
65C22×D31C2412yes—
66Q8×C31C2×D4116yescomplemented
67D4×C31—224no—
68C2×C4×C31—22no—
69C2×C4×C31—224no—
70C2×C4×C31C2×D4112yes—
71C2×C4×C31C2412yes—
72C2×C4×C31C2416yes—
73C22⋊D31—224no—
74C2×Dic31—26no—
75C2×Dic31—26no—
76C2×Dic31—22no—
77C2×Dic31C2412yes—
78C2×Dic31C2416yes—
79C2×Dic31C2416yes—
80C4×D31—224no—
81Dic62—224no—
82C2×#548—311nomaximal, Sylow
83C22×C31—26no—
84C22×C31—26no—
85C22×C31—26no—
86C22×C31—412no—
87C22×C31C22×D412yes—
88C22×C31C2×#4112yes—
89C22×C312−1+412yes—
90C22×C31C2511yescharacteristic
91C2×D31—28no—
92C2×D31—22no—
93C2×D31—22no—
94C2×D31C2×#4114yescomplemented
95C4×C31—22no—
96C4×C31C22×D4112yes—
97Dic31—26no—
98Dic31—26no—
99Dic31—22no—
100C22×#41D3111yescharacteristic, complemented
101C23×Q8—311no—
102#548—3116no—
103C2×#124—313no—
104C2×C22⋊Q8—313no—
105C2×#121—313no—
106C2×C4×Q8—311no—
107C2×C4×D4—313no—
108C2×C31—46no—
109C2×C31#54814yescomplemented
110C2×C31C23×D411yescharacteristic
111C2×C31C22×#4111yescharacteristic
112C2×C31C2×2−1+411yescharacteristic
113D31—24no—
114C2×#41—28no—
115C2×#41C2×D3118yescomplemented
116C22×Q8—318no—
117C22×Q8—313no—
118C22×Q8—628no—
119C22×Q8C2×D3111yescharacteristic
120C22×D4—313no—
121C22×D4C2×D3113yes—
122C23×C4—313no—
123C23×C4C2×D3113yes—
124#124—3124no—
125C22⋊Q8—3124no—
126#121—3124no—
127C4×Q8—318no—
128C4×D4—3124no—
129C2×C4⋊C4—313no—
130C2×C4⋊C4—313no—
131C2×C22⋊C4—313no—
132C2×C22⋊C4—316no—
133C2×C22⋊C4—311no—
134C2×C42—313no—
135C31C2×#54811yescharacteristic, complemented, Sylow
136C24—311no—
137C24C22×D3113yes—
138#41—232no—
139C2×Q8—3112no—
140C2×Q8—6232no—
141C2×Q8—6212no—
142C2×Q8C22⋊D3118yescomplemented
143C2×Q8C22×D3114yes—
144C2×D4—212no—
145C2×D4—3112no—
146C2×D4—6212no—
147C2×D4C22×D31112yes—
148C22×C4—212no—
149C22×C4—311no—
150C22×C4—313no—
151C22×C4—313no—
152C22×C4—3112no—
153C22×C4—6212no—
154C22×C4C22×D31112yes—
155C22×C4C22×D3111yescharacteristic
156C22×C4C22×D3113yes—
157C4⋊C4—3112no—
158C4⋊C4—3112no—
159C22⋊C4—3112no—
160C22⋊C4—3124no—
161C22⋊C4—314no—
162C42—3112no—
163C23—26no—
164C23—26no—
165C23—26no—
166C23—314no—
167C23—312no—
168C23—622no—
169C23—622no—
170C23C23×D3116yes—
171C23C23×D3111yescharacteristic
172Q8—6224no—
173Q8C2×C22⋊D31116yescomplemented
174D4—224no—
175D4—6224no—
176C2×C4—22no—
177C2×C4—224no—
178C2×C4—312no—
179C2×C4—316no—
180C2×C4—316no—
181C2×C4—626no—
182C2×C4—622no—
183C2×C4—626no—
184C2×C4—6224no—
185C2×C4C2×C22⋊D31112yes—
186C2×C4C23×D3112yes—
187C2×C4C23×D3116yes—
188C22—26no—
189C22—26no—
190C22—26no—
191C22—412no—
192C22—314no—
193C22—622no—
194C22—622no—
195C22—628no—
196C22#41×D3112yes—
197C22#1175856712yes—
198C22C22×C22⋊D3112yes—
199C22C24×D3111yescharacteristic
200C4—22no—
201C4—626no—
202C4—626no—
203C4—622no—
204C4C22×C22⋊D31112yes—
205C2—46no—
206C2—624no—
207C2#1452170614yesdirect product factor
208C2C23×C22⋊D3111yescharacteristic
209C2C2×#41×D3111yescharacteristic
210C2C2×#1175856711yescharacteristic
subgroupG/H
centerC23C23×D31
commutator, cosocleC22×C31C25
FrattiniC22C24×D31
FittingC22×#41×C31C2
socleC23×C31C24
hypercenterC22×#41D31
hypocenterC31C2×#548
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D31C2semidirect
2. #14521706C2direct (N × H)
3. C2×#11758513C2semidirect
4. C2×#11758501C2semidirect
5. C2×#11758515C2semidirect
6. C2×Q8×Dic31C2semidirect
7. C2×C4×C22⋊D31C2semidirect
8. C22×#41×C31C2semidirect
10. C2×#41×C31C22semidirect
16. C2×Q8×D31C22semidirect
21. Q8×Dic31C22semidirect
22. #11758515C22semidirect
23. #11758513C22semidirect
24. C4×C22⋊D31C22semidirect
27. #11758501C22semidirect
33. C2×Q8×C31D4semidirect
64. C22×D31#41semidirect
66. Q8×C31C2×D4semidirect
94. C2×D31C2×#41semidirect
100. C22×#41D31semidirect
109. C2×C31#548semidirect
115. C2×#41C2×D31semidirect
135. C31C2×#548semidirect
142. C2×Q8C22⋊D31semidirect
173. Q8C2×C22⋊D31semidirect
207. C2#14521706direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×D312
#145217062
C2×#117585132
C2×#117585012
C2×#117585152
C2×Q8×Dic312
C2×C4×C22⋊D312
C22×#41×C312
C2×#54831