Finite Group Databasebeta

← C2×#14521209 #16854918

Subgroups — C2×#14521209 #16854918

Order 3968.

760 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 165 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×#92495C211yesmaximal, characteristic
2C22×#92482C211yesmaximal, characteristic, complemented
3C22×#92481C211yesmaximal, characteristic, complemented
4#14521209C218yesmaximal, direct product factor
5C2×#11758398C211yesmaximal, characteristic, complemented
6C2×#11758397C211yesmaximal, characteristic, complemented
7C2×C22⋊(C31⋊C8)C211yesmaximal, characteristic, complemented
8C2×C31×C22⋊Q8C211yesmaximal, characteristic, complemented
9C22×Q8×C31C2211yescharacteristic
10C23×C4×C31C2211yescharacteristic
11C22×C22⋊D31—21no—
12C2×#92495—22no—
13C2×#92495—24no—
14C2×#92495—24no—
15C2×#92495C2214yes—
16C22×D124C2211yescharacteristic
17C22×C4×D31—21no—
18C22×Dic62C2211yescharacteristic
19C31×C22⋊Q8—22no—
20C31×C22⋊Q8C2214yescomplemented
21C2×C4⋊C4×C31—21no—
22C2×C4⋊C4×C31C2211yescharacteristic
23C2×C22⋊C4×C31—21no—
24C2×#92482—24no—
25C2×#92482C2214yescomplemented
26C2×#92481—24no—
27C2×#92481C2214yescomplemented
28C22×C31⋊C8C2211yescharacteristic
29C22×C31⋊C8C2211yescharacteristic
30C22⋊(C31⋊C8)C2214yescomplemented
31#11758398C2214yescomplemented
32#11758397C2214yescomplemented
33C24×C31D411yescharacteristic
34C23×D31—21no—
35C2×Q8×C31—21no—
36C2×Q8×C31—22no—
37C2×Q8×C31—22no—
38C2×Q8×C31C2312yes—
39C22×C4×C31—22no—
40C22×C4×C31—22no—
41C22×C4×C31—21no—
42C22×C4×C31—21no—
43C22×C4×C31D411yescharacteristic
44C22×C4×C31D412yes—
45C22×C4×C31C2312yes—
46C22×C4×C31C2311yescharacteristic
47C2×C22⋊D31—28no—
48C2×C22⋊D31—24no—
49C22×Dic31—21no—
50#92495—216no—
51#92495—28no—
52#92495—28no—
53C2×D124—21no—
54C2×D124—22no—
55C2×D124D414yes—
56C2×D124C2312yes—
57C2×C4×D31—28no—
58C2×C4×D31—24no—
59C2×Dic62—21no—
60C2×Dic62—22no—
61C2×Dic62D414yes—
62C2×Dic62C2312yes—
63C4⋊C4×C31—24no—
64C4⋊C4×C31—21no—
65C4⋊C4×C31C2312yes—
66C22⋊C4×C31—24no—
67#92482—28no—
68#92481—28no—
69C2×C31⋊C8—22no—
70C2×C31⋊C8—22no—
71C2×C31⋊C8C2312yes—
72C2×C31⋊C8C2312yes—
73C23×C31—22no—
74C23×C31—22no—
75C23×C31—22no—
76C23×C31C2×D412yes—
77C23×C31C2×D411yescharacteristic
78C22×D31—24no—
79C22×D31—22no—
80C22×D31—42no—
81C22×D31—42no—
82Q8×C31—22no—
83Q8×C31—22no—
84Q8×C31—42no—
85C2×C4×C31—22no—
86C2×C4×C31—24no—
87C2×C4×C31—22no—
88C2×C4×C31—22no—
89C2×C4×C31—22no—
90C2×C4×C31—22no—
91C2×C4×C31—42no—
92C2×C4×C31—42no—
93C2×C4×C31C2×D412yes—
94C2×C4×C31C2×D412yes—
95C2×C4×C31C2×D412yes—
96C2×C4×C31C2×D411yescharacteristic
97C2×C4×C31C2411yescharacteristic
98C22⋊D31—216no—
99C2×Dic31—24no—
100C2×Dic31—22no—
101D124—24no—
102D124—22no—
103D124C2×D414yes—
104C4×D31—216no—
105Dic62—24no—
106Dic62—22no—
107Dic62C2×D414yes—
108C31⋊C8—22no—
109C31⋊C8—22no—
110C2×#452—311nomaximal, Sylow
111C22×C31—22no—
112C22×C31—22no—
113C22×C31—22no—
114C22×C31—44no—
115C22×C31#12012yes—
116C22×C31#13512yes—
117C22×C31#13712yes—
118C22×C31C22×D411yescharacteristic
119C2×D31—24no—
120C2×D31—42no—
121C2×D31—42no—
122C2×D31—48no—
123C4×C31—22no—
124C4×C31—42no—
125C4×C31—42no—
126C4×C31#12012yes—
127C4×C31C22×D411yescharacteristic
128C4×C31C22×D411yescharacteristic
129Dic31—24no—
130C22×#41—311no—
131C22×Q16—311no—
132C22×SD16—311no—
133C2×C22⋊Q8D3111yescharacteristic, complemented
134#452—318no—
135C2×#103—311no—
136C2×#102—311no—
137C2×C22⋊C8—311no—
138C2×C31—42no—
139C2×C31#45214yescomplemented
140C2×C31C2×#12011yescharacteristic
141C2×C31C2×#13511yescharacteristic
142C2×C31C2×#13711yescharacteristic
143D31—44no—
144C2×#41—314no—
145C2×#41—624no—
146C2×#41—624no—
147C2×#41—622no—
148C22×Q8—311no—
149C22×Q8C2×D3111yescharacteristic
150C22×D4—311no—
151C22×D4—621no—
152C23×C4—621no—
153C23×C4C2×D3111yescharacteristic
154C2×Q16—314no—
155C2×Q16—624no—
156C2×SD16—314no—
157C2×SD16—624no—
158C22×C8—311no—
159C22×C8—311no—
160C22⋊Q8—22no—
161C22⋊Q8C2×D3114yescomplemented
162C2×C4⋊C4—21no—
163C2×C4⋊C4C2×D3111yescharacteristic
164C2×C22⋊C4—21no—
165#103—314no—
166#102—314no—
167C22⋊C8—314no—
168C31C2×#45211yescharacteristic, complemented, Sylow
169C24—621no—
170C24C22⋊D3111yescharacteristic
171#41—628no—
172#41—628no—
173#41—6216no—
174C2×Q8—21no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—314no—
178C2×Q8—312no—
179C2×Q8—622no—
180C2×Q8—621no—
181C2×Q8C22×D3112yes—
182C2×D4—314no—
183C2×D4—312no—
184C2×D4—628no—
185C2×D4—624no—
186C2×D4—622no—
187C2×D4—621no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—628no—
193C22×C4—621no—
194C22×C4—624no—
195C22×C4C22⋊D3112yes—
196C22×C4C22⋊D3111yescharacteristic
197C22×C4C22×D3112yes—
198C22×C4C22×D3111yescharacteristic
199Q16—628no—
200SD16—628no—
201C2×C8—312no—
202C2×C8—312no—
203C2×C8—622no—
204C2×C8—622no—
205C4⋊C4—24no—
206C4⋊C4—21no—
207C4⋊C4C22×D3112yes—
208C22⋊C4—24no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—624no—
213C23—622no—
214C23—1242no—
215C23—1242no—
216C23C2×C22⋊D3112yes—
217C23C2×C22⋊D3111yescharacteristic
218Q8—22no—
219Q8—22no—
220Q8—42no—
221Q8—314no—
222Q8—624no—
223Q8—622no—
224D4—314no—
225D4—6216no—
226D4—624no—
227D4—622no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—6216no—
237C2×C4—624no—
238C2×C4—622no—
239C2×C4D4×D3112yes—
240C2×C4D4×D3112yes—
241C2×C4C2×C22⋊D3112yes—
242C2×C4C2×C22⋊D3111yescharacteristic
243C2×C4C23×D3111yescharacteristic
244C8—622no—
245C8—622no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—44no—
250C22—624no—
251C22—1248no—
252C22—1242no—
253C22—1242no—
254C22#1175851412yes—
255C22#1175851912yes—
256C22#1175852712yes—
257C22C22×C22⋊D3111yescharacteristic
258C4—22no—
259C4—42no—
260C4—42no—
261C4—624no—
262C4#1175851412yes—
263C4C2×D4×D3111yescharacteristic
264C4C2×D4×D3111yescharacteristic
265C2—42no—
266C2—1244no—
267C2#1452120914yesdirect product factor
268C2C2×#1175851411yescharacteristic
269C2C2×#1175852711yescharacteristic
270C2C2×#1175851911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×C22⋊Q8C2
socleC23×C31C2×D4
prime radicalC22×#92495C2
hypercenterC2×C22⋊Q8D31
hypocenterC31C2×#452
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C22×#92482C2semidirect
3. C22×#92481C2semidirect
4. #14521209C2direct (N × H)
5. C2×#11758398C2semidirect
6. C2×#11758397C2semidirect
7. C2×C22⋊(C31⋊C8)C2semidirect
8. C2×C31×C22⋊Q8C2semidirect
20. C31×C22⋊Q8C22semidirect
25. C2×#92482C22semidirect
27. C2×#92481C22semidirect
30. C22⋊(C31⋊C8)C22semidirect
31. #11758398C22semidirect
32. #11758397C22semidirect
133. C2×C22⋊Q8D31semidirect
139. C2×C31#452semidirect
161. C22⋊Q8C2×D31semidirect
168. C31C2×#452semidirect
267. C2#14521209direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#924952
C22×#924822
C22×#924812
#145212092
C2×#117583982
C2×#117583972
C2×C22⋊(C31⋊C8)2
C2×C31×C22⋊Q82
C2×#45231