Finite Group Databasebeta

← C2×#14521226 #16855040

Subgroups — C2×#14521226 #16855040

Order 3968.

760 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 149 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×#11758561C211yesmaximal, characteristic
2C2×#11758519C212yesmaximal, complemented
3#14521226C218yesmaximal, direct product factor
4C2×#11758393C212yesmaximal, complemented
5C2×#11758421C211yesmaximal, characteristic, complemented
6C2×C31×#124C211yesmaximal, characteristic, complemented
7C22×Q8×C31C2211yescharacteristic, complemented
8C22×D4×C31C2211yescharacteristic
9C22×C22⋊D31—22no—
10#11758561—22no—
11#11758561—24no—
12#11758561C2214yes—
13C2×#92497—22no—
14C2×D4×D31—22no—
15C2×#92495C2212yes—
16C31×#124—22no—
17C31×#124C2214yescomplemented
18C2×C22⋊C4×C31—22no—
19C2×C42×C31C2211yescharacteristic, complemented
20C2×#92482—22no—
21#11758519—24no—
22#11758519C2218yescomplemented
23C2×#92481—22no—
24C2×#92474C2212yescomplemented
25#11758421C2214yescomplemented
26#11758393C2218yescomplemented
27C24×C31D412yes—
28C23×D31—22no—
29C2×Q8×C31—21no—
30C2×Q8×C31—22no—
31C2×Q8×C31—22no—
32C2×Q8×C31C2312yescomplemented
33C2×D4×C31—22no—
34C2×D4×C31—22no—
35C2×D4×C31D412yes—
36C2×D4×C31C2312yes—
37C22×C4×C31—21no—
38C22×C4×C31—21no—
39C22×C4×C31C2311yescharacteristic
40C2×C22⋊D31—24no—
41C2×C22⋊D31—28no—
42C2×C22⋊D31—22no—
43C2×C22⋊D31—28no—
44C22×Dic31—22no—
45#92497—28no—
46#92497—28no—
47D4×D31—28no—
48D4×D31—28no—
49#92495—22no—
50#92495—22no—
51#92495D414yes—
52#92495C2314yes—
53C2×D124D412yes—
54C2×C4×D31—22no—
55C2×Dic62D412yes—
56C22⋊C4×C31—28no—
57C42×C31—21no—
58C42×C31C2312yescomplemented
59#92482—28no—
60#92481—28no—
61#92474—22no—
62#92474C2314yescomplemented
63C2×C31⋊C8—22no—
64C23×C31—22no—
65C23×C31—22no—
66C23×C31—44no—
67C23×C31C2×D411yescharacteristic
68C23×C31C2×D414yes—
69C22×D31—22no—
70C22×D31—28no—
71C22×D31—84no—
72Q8×C31—24no—
73Q8×C31—42no—
74D4×C31—24no—
75D4×C31—24no—
76C2×C4×C31—22no—
77C2×C4×C31—22no—
78C2×C4×C31—42no—
79C2×C4×C31—41no—
80C2×C4×C31—41no—
81C2×C4×C31C2×D412yes—
82C2×C4×C31C2×D412yes—
83C2×C4×C31C2×D411yescharacteristic
84C2×C4×C31C2411yescharacteristic
85C22⋊D31—216no—
86C22⋊D31—24no—
87C22⋊D31—24no—
88C2×Dic31—28no—
89C2×Dic31—22no—
90D124—22no—
91D124C2×D414yes—
92C4×D31—24no—
93C4×D31—24no—
94Dic62—22no—
95Dic62C2×D414yes—
96C31⋊C8—24no—
97C2×#455—311nomaximal, Sylow
98C22×C31—22no—
99C22×C31—24no—
100C22×C31—44no—
101C22×C31—44no—
102C22×C31—82no—
103C22×C31#12011yescharacteristic
104C22×C31#12011yescharacteristic
105C22×C31C22×D411yescharacteristic
106C2×D31—24no—
107C2×D31—44no—
108C2×D31—88no—
109C4×C31—42no—
110C4×C31—42no—
111C4×C31#12012yes—
112C4×C31C22×D412yes—
113Dic31—24no—
114C2×H(5,𝔽2)—311no—
115C2×#137—312no—
116C2×#124D3111yescharacteristic, complemented
117#455—318no—
118C2×#104—312no—
119C2×#100—311no—
120C2×C31—21no—
121C2×C31—21no—
122C2×C31—44no—
123C2×C31#45512yescomplemented
124C2×C31C2×#12011yescharacteristic
125D31—44no—
126H(5,𝔽2)—314no—
127H(5,𝔽2)—624no—
128H(5,𝔽2)—622no—
129C2×#41—312no—
130C2×#41—622no—
131C22×Q8C2×D3111yescharacteristic, complemented
132C22×D4—622no—
133C22×D4—622no—
134C22×D4C2×D3111yescharacteristic
135#137—318no—
136#137—624no—
137C2×Q16—622no—
138C2×SD16—622no—
139C2×M16—312no—
140#124—22no—
141#124C2×D3114yescomplemented
142C2×C22⋊C4—22no—
143C2×C42C2×D3111yescharacteristic, complemented
144#104—318no—
145#100—314no—
146C31C2×#45511yescharacteristic, complemented, Sylow
147C24—622no—
148C24C22⋊D3112yes—
149#41—314no—
150#41—314no—
151#41—628no—
152#41—628no—
153#41—622no—
154#41—622no—
155C2×Q8—21no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—312no—
159C2×Q8C22×D3112yescomplemented
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—312no—
163C2×D4—628no—
164C2×D4—628no—
165C2×D4—624no—
166C2×D4—628no—
167C2×D4—622no—
168C2×D4—628no—
169C2×D4C22⋊D3112yes—
170C2×D4C22×D3112yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—622no—
174C22×C4—622no—
175C22×C4C22×D3111yescharacteristic
176Q16—628no—
177SD16—628no—
178M16—314no—
179M16—622no—
180C2×C8—622no—
181C22⋊C4—28no—
182C42—21no—
183C42C22×D3112yescomplemented
184C23—22no—
185C23—22no—
186C23—44no—
187C23—628no—
188C23—622no—
189C23—2484no—
190C23C2×C22⋊D3111yescharacteristic
191C23C2×C22⋊D3114yes—
192Q8—24no—
193Q8—42no—
194Q8—314no—
195Q8—622no—
196D4—24no—
197D4—24no—
198D4—314no—
199D4—6216no—
200D4—624no—
201D4—624no—
202D4—622no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—628no—
209C2×C4—622no—
210C2×C4—624no—
211C2×C4—624no—
212C2×C4D4×D3112yes—
213C2×C4D4×D3112yes—
214C2×C4C2×C22⋊D3111yescharacteristic
215C2×C4C23×D3111yescharacteristic
216C8—624no—
217C22—22no—
218C22—24no—
219C22—44no—
220C22—44no—
221C22—82no—
222C22—624no—
223C22—1244no—
224C22—2488no—
225C22#1175851411yescharacteristic
226C22#1175851411yescharacteristic
227C22C22×C22⋊D3111yescharacteristic
228C4—42no—
229C4—42no—
230C4—624no—
231C4#1175851412yes—
232C4C2×D4×D3112yes—
233C2—21no—
234C2—21no—
235C2—44no—
236C2—1244no—
237C2#1452122612yesdirect product factor
238C2C2×#1175851411yescharacteristic
subgroupG/H
centerC22#11758514
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×#124C2
socleC22×C31#120
prime radicalC2×#11758561C2
hypercenterC2×#124D31
hypocenterC31C2×#455
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#11758519C2semidirect
3. #14521226C2direct (N × H)
4. C2×#11758393C2semidirect
5. C2×#11758421C2semidirect
6. C2×C31×#124C2semidirect
7. C22×Q8×C31C22semidirect
17. C31×#124C22semidirect
19. C2×C42×C31C22semidirect
22. #11758519C22semidirect
24. C2×#92474C22semidirect
25. #11758421C22semidirect
26. #11758393C22semidirect
32. C2×Q8×C31C23semidirect
58. C42×C31C23semidirect
62. #92474C23semidirect
116. C2×#124D31semidirect
123. C2×C31#455semidirect
131. C22×Q8C2×D31semidirect
141. #124C2×D31semidirect
143. C2×C42C2×D31semidirect
146. C31C2×#455semidirect
159. C2×Q8C22×D31semidirect
183. C42C22×D31semidirect
237. C2#14521226direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117585612
C2×#117585192
#145212262
C2×#117583932
C2×#117584212
C2×C31×#1242
C2×#45531