Finite Group Databasebeta

← C2×#14521216 #16855034

Subgroups — C2×#14521216 #16855034

Order 3968.

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

normal subgroups only

This group has 844 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, complemented
2C2×#11758509C212yesmaximal, complemented
3#14521216C218yesmaximal, direct product factor
4C2×#11758393C212yesmaximal, complemented
5C2×#11758421C211yesmaximal, characteristic, complemented
6C2×C31×#127C211yesmaximal, characteristic, complemented
7C22×D4×C31—22no—
8C22×D4×C31C2211yescharacteristic
9C22×D4×C31C2211yescharacteristic, complemented
10C22×C22⋊D31—22no—
11#11758561—22no—
12#11758561—24no—
13#11758561C2214yescomplemented
14C2×#92497—22no—
15C2×D4×D31—22no—
16C2×#92495C2212yes—
17C31×#127—22no—
18C31×#127C2214yescomplemented
19C2×C42×C31C2211yescharacteristic, complemented
20C2×#92480—22no—
21#11758509—24no—
22#11758509C2218yescomplemented
23C2×#92479—22no—
24C2×#92474C2212yescomplemented
25#11758421C2214yescomplemented
26#11758393C2218yescomplemented
27C24×C31—21no—
28C24×C31D412yes—
29C23×D31—22no—
30C2×D4×C31—28no—
31C2×D4×C31—22no—
32C2×D4×C31—22no—
33C2×D4×C31—22no—
34C2×D4×C31—22no—
35C2×D4×C31—21no—
36C2×D4×C31—48no—
37C2×D4×C31D412yes—
38C2×D4×C31C2312yes—
39C2×D4×C31C2312yescomplemented
40C22×C4×C31—21no—
41C22×C4×C31C2311yescharacteristic
42C2×C22⋊D31—24no—
43C2×C22⋊D31—28no—
44C2×C22⋊D31—22no—
45C2×C22⋊D31—28no—
46C22×Dic31—22no—
47#92497—28no—
48#92497—28no—
49D4×D31—28no—
50D4×D31—28no—
51#92495—22no—
52#92495—22no—
53#92495D414yescomplemented
54#92495C2314yes—
55C2×D124D412yescomplemented
56C2×C4×D31—22no—
57C2×Dic62D412yescomplemented
58C42×C31—21no—
59C42×C31C2312yescomplemented
60#92480—28no—
61#92479—28no—
62#92474—22no—
63#92474C2314yescomplemented
64C2×C31⋊C8—22no—
65C23×C31—22no—
66C23×C31—22no—
67C23×C31—22no—
68C23×C31—44no—
69C23×C31—44no—
70C23×C31—41no—
71C23×C31—41no—
72C23×C31C2×D411yescharacteristic
73C23×C31C2×D414yes—
74C22×D31—22no—
75C22×D31—28no—
76C22×D31—84no—
77D4×C31—24no—
78D4×C31—24no—
79D4×C31—24no—
80D4×C31—416no—
81D4×C31—42no—
82C2×C4×C31—24no—
83C2×C4×C31—22no—
84C2×C4×C31C2×D412yes—
85C2×C4×C31C2×D412yes—
86C2×C4×C31C2×D411yescharacteristic
87C2×C4×C31C2411yescharacteristic
88C22⋊D31—216no—
89C22⋊D31—24no—
90C22⋊D31—24no—
91C2×Dic31—28no—
92C2×Dic31—22no—
93D124—22no—
94D124C2×D414yescomplemented
95C4×D31—24no—
96C4×D31—24no—
97Dic62—22no—
98Dic62C2×D414yescomplemented
99C31⋊C8—24no—
100C2×#453—311nomaximal, Sylow
101C22×C31—22no—
102C22×C31—24no—
103C22×C31—44no—
104C22×C31—44no—
105C22×C31—44no—
106C22×C31—42no—
107C22×C31—82no—
108C22×C31—82no—
109C22×C31—82no—
110C22×C31#12011yescharacteristic
111C22×C31#12011yescharacteristic
112C22×C31C22×D411yescharacteristic
113C2×D31—24no—
114C2×D31—44no—
115C2×D31—88no—
116C4×C31—24no—
117C4×C31#12012yes—
118C4×C31C22×D412yes—
119Dic31—24no—
120C2×H(5,𝔽2)—311no—
121C2×AGL(1,ℤ/8)—312no—
122C2×#127D3111yescharacteristic, complemented
123#453—318no—
124C2×#104—312no—
125C2×#100—311no—
126C2×C31—21no—
127C2×C31—21no—
128C2×C31—44no—
129C2×C31—82no—
130C2×C31#45312yescomplemented
131C2×C31C2×#12011yescharacteristic
132D31—44no—
133H(5,𝔽2)—314no—
134H(5,𝔽2)—624no—
135H(5,𝔽2)—622no—
136C2×#41—312no—
137C2×#41—622no—
138C22×D4—22no—
139C22×D4—622no—
140C22×D4—622no—
141C22×D4C2×D3111yescharacteristic
142C22×D4C2×D3111yescharacteristic, complemented
143AGL(1,ℤ/8)—318no—
144AGL(1,ℤ/8)—624no—
145C2×SD16—622no—
146C2×D8—622no—
147C2×M16—312no—
148#127—22no—
149#127C2×D3114yescomplemented
150C2×C42C2×D3111yescharacteristic, complemented
151#104—318no—
152#100—314no—
153C31C2×#45311yescharacteristic, complemented, Sylow
154C24—21no—
155C24—622no—
156C24C22⋊D3112yes—
157#41—314no—
158#41—314no—
159#41—628no—
160#41—628no—
161#41—622no—
162#41—622no—
163C2×Q8—312no—
164C2×D4—28no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—21no—
170C2×D4—48no—
171C2×D4—312no—
172C2×D4—628no—
173C2×D4—628no—
174C2×D4—624no—
175C2×D4—628no—
176C2×D4—622no—
177C2×D4—628no—
178C2×D4C22⋊D3112yes—
179C2×D4C22×D3112yes—
180C2×D4C22×D3112yescomplemented
181C22×C4—21no—
182C22×C4—622no—
183C22×C4—622no—
184C22×C4C22×D3111yescharacteristic
185SD16—628no—
186D8—628no—
187M16—314no—
188M16—622no—
189C2×C8—622no—
190C42—21no—
191C42C22×D3112yescomplemented
192C23—22no—
193C23—22no—
194C23—22no—
195C23—44no—
196C23—44no—
197C23—41no—
198C23—41no—
199C23—628no—
200C23—622no—
201C23—2484no—
202C23C2×C22⋊D3111yescharacteristic
203C23C2×C22⋊D3114yes—
204Q8—314no—
205Q8—622no—
206D4—24no—
207D4—24no—
208D4—24no—
209D4—416no—
210D4—42no—
211D4—314no—
212D4—6216no—
213D4—624no—
214D4—624no—
215D4—622no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—628no—
219C2×C4—622no—
220C2×C4—624no—
221C2×C4—624no—
222C2×C4D4×D3112yes—
223C2×C4D4×D3112yes—
224C2×C4C2×C22⋊D3111yescharacteristic
225C2×C4C23×D3111yescharacteristic
226C8—624no—
227C22—22no—
228C22—24no—
229C22—44no—
230C22—44no—
231C22—44no—
232C22—42no—
233C22—82no—
234C22—82no—
235C22—82no—
236C22—624no—
237C22—1244no—
238C22—2488no—
239C22#1175851411yescharacteristic
240C22#1175851411yescharacteristic
241C22C22×C22⋊D3111yescharacteristic
242C4—24no—
243C4—624no—
244C4#1175851412yes—
245C4C2×D4×D3112yes—
246C2—21no—
247C2—21no—
248C2—44no—
249C2—82no—
250C2—1244no—
251C2#1452121612yesdirect product factor
252C2C2×#1175851411yescharacteristic
subgroupG/H
centerC22#11758514
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×#127C2
socleC22×C31#120
hypercenterC2×#127D31
hypocenterC31C2×#453
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#11758561C2semidirect
2. C2×#11758509C2semidirect
3. #14521216C2direct (N × H)
4. C2×#11758393C2semidirect
5. C2×#11758421C2semidirect
6. C2×C31×#127C2semidirect
9. C22×D4×C31C22semidirect
13. #11758561C22semidirect
18. C31×#127C22semidirect
19. C2×C42×C31C22semidirect
22. #11758509C22semidirect
24. C2×#92474C22semidirect
25. #11758421C22semidirect
26. #11758393C22semidirect
39. C2×D4×C31C23semidirect
53. #92495D4semidirect
55. C2×D124D4semidirect
57. C2×Dic62D4semidirect
59. C42×C31C23semidirect
63. #92474C23semidirect
94. D124C2×D4semidirect
98. Dic62C2×D4semidirect
122. C2×#127D31semidirect
130. C2×C31#453semidirect
142. C22×D4C2×D31semidirect
149. #127C2×D31semidirect
150. C2×C42C2×D31semidirect
153. C31C2×#453semidirect
180. C2×D4C22×D31semidirect
191. C42C22×D31semidirect
251. C2#14521216direct (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×#117585092
#145212162
C2×#117583932
C2×#117584212
C2×C31×#1272
C2×#45331