Finite Group Databasebeta

← C2×#14521217 #16855035

Subgroups — C2×#14521217 #16855035

Order 3968.

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

normal subgroups only

This group has 844 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
1C2×#11758526C212yesmaximal, complemented
2C2×#11758517C211yesmaximal, characteristic, complemented
3#14521217C218yesmaximal, direct product factor
4C2×#11758425C212yesmaximal, complemented
5C2×#11758421C211yesmaximal, characteristic, complemented
6C2×C31×H(5,𝔽2)C211yesmaximal, characteristic, complemented
7C31×H(5,𝔽2)—24no—
8C31×H(5,𝔽2)—22no—
9C31×H(5,𝔽2)C2214yescomplemented
10C2×#41×C31—22no—
11C2×#41×C31C2212yes—
12C22×D4×C31—22no—
13C22×D4×C31—22no—
14C22×D4×C31C2211yescharacteristic
15C22×C22⋊D31—22no—
16C22×D124C2211yescharacteristic, complemented
17#11758526—24no—
18#11758526C2218yescomplemented
19C2×#92481—22no—
20#11758517—22no—
21#11758517C2214yescomplemented
22C2×#92479—22no—
23C2×C4×Dic31C2211yescharacteristic, complemented
24C2×#92474C2212yescomplemented
25#11758425C2218yescomplemented
26#11758421C2214yescomplemented
27C24×C31—22no—
28C24×C31D412yes—
29C23×D31—21no—
30#41×C31—28no—
31#41×C31—28no—
32#41×C31—22no—
33#41×C31—22no—
34#41×C31D414yescomplemented
35#41×C31C2314yes—
36C2×Q8×C31D412yescomplemented
37C2×D4×C31—24no—
38C2×D4×C31—28no—
39C2×D4×C31—28no—
40C2×D4×C31—22no—
41C2×D4×C31—28no—
42C2×D4×C31—28no—
43C2×D4×C31—22no—
44C2×D4×C31—22no—
45C2×D4×C31D412yes—
46C2×D4×C31D412yescomplemented
47C2×D4×C31C2312yes—
48C22×C4×C31—22no—
49C22×C4×C31—22no—
50C22×C4×C31C2311yescharacteristic
51C2×C22⋊D31—28no—
52C2×C22⋊D31—48no—
53C22×Dic31—21no—
54C2×D124—22no—
55C2×D124—22no—
56C2×D124—21no—
57C2×D124C2312yescomplemented
58#92481—28no—
59#92479—28no—
60C4×Dic31—21no—
61C4×Dic31C2312yescomplemented
62#92474—22no—
63#92474C2314yescomplemented
64C2×C31⋊C8—22no—
65C23×C31—28no—
66C23×C31—22no—
67C23×C31—22no—
68C23×C31—22no—
69C23×C31—44no—
70C23×C31—84no—
71C23×C31C2×D411yescharacteristic
72C23×C31C2×D414yes—
73C22×D31—22no—
74C22×D31—44no—
75C22×D31—41no—
76C22×D31—41no—
77Q8×C31—22no—
78Q8×C31C2×D414yescomplemented
79D4×C31—216no—
80D4×C31—24no—
81D4×C31—24no—
82D4×C31—24no—
83D4×C31—22no—
84D4×C31—24no—
85D4×C31C2×D414yescomplemented
86C2×C4×C31—28no—
87C2×C4×C31—22no—
88C2×C4×C31—24no—
89C2×C4×C31—24no—
90C2×C4×C31C2×D411yescharacteristic
91C2×C4×C31C2×D412yes—
92C2×C4×C31C2×D412yes—
93C2×C4×C31C2411yescharacteristic
94C22⋊D31—416no—
95C2×Dic31—24no—
96C2×Dic31—22no—
97D124—24no—
98D124—42no—
99C31⋊C8—24no—
100C2×#453—311nomaximal, Sylow
101C22×C31—22no—
102C22×C31—24no—
103C22×C31—24no—
104C22×C31—44no—
105C22×C31—44no—
106C22×C31—44no—
107C22×C31—82no—
108C22×C31—88no—
109C22×C31#12011yescharacteristic
110C22×C31#12011yescharacteristic
111C22×C31C22×D411yescharacteristic
112C2×D31—44no—
113C2×D31—42no—
114C2×D31—82no—
115C2×D31—82no—
116C4×C31—24no—
117C4×C31#12012yes—
118C4×C31C22×D412yes—
119Dic31—24no—
120C2×H(5,𝔽2)D3111yescharacteristic, complemented
121C2×AGL(1,ℤ/8)—312no—
122C2×#127—311no—
123#453—318no—
124C2×#104—312no—
125C2×#100—311no—
126C2×C31—21no—
127C2×C31—21no—
128C2×C31—44no—
129C2×C31—44no—
130C2×C31#45312yescomplemented
131C2×C31C2×#12011yescharacteristic
132D31—82no—
133H(5,𝔽2)—24no—
134H(5,𝔽2)—22no—
135H(5,𝔽2)C2×D3114yescomplemented
136C2×#41—22no—
137C2×#41C2×D3112yes—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—311no—
141C22×D4—622no—
142C22×D4C2×D3111yescharacteristic
143AGL(1,ℤ/8)—318no—
144AGL(1,ℤ/8)—624no—
145C2×SD16—622no—
146C2×D8—622no—
147C2×M16—312no—
148#127—314no—
149#127—622no—
150C2×C42—311no—
151#104—318no—
152#100—314no—
153C31C2×#45311yescharacteristic, complemented, Sylow
154C24—22no—
155C24—621no—
156C24C22⋊D3112yes—
157#41—28no—
158#41—28no—
159#41—22no—
160#41—22no—
161#41C22⋊D3114yescomplemented
162#41C22×D3114yes—
163C2×Q8C22⋊D3112yescomplemented
164C2×D4—24no—
165C2×D4—28no—
166C2×D4—28no—
167C2×D4—22no—
168C2×D4—28no—
169C2×D4—28no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—312no—
173C2×D4—628no—
174C2×D4—622no—
175C2×D4—622no—
176C2×D4—621no—
177C2×D4—1248no—
178C2×D4C22⋊D3112yes—
179C2×D4C22⋊D3112yescomplemented
180C2×D4C22×D3112yes—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—621no—
184C22×C4C22×D3111yescharacteristic
185SD16—628no—
186D8—628no—
187M16—314no—
188M16—622no—
189C2×C8—622no—
190C42—312no—
191C42—621no—
192C23—28no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—44no—
197C23—84no—
198C23—622no—
199C23—1244no—
200C23—1241no—
201C23—1241no—
202C23C2×C22⋊D3111yescharacteristic
203C23C2×C22⋊D3114yes—
204Q8—22no—
205Q8C2×C22⋊D3114yescomplemented
206D4—216no—
207D4—24no—
208D4—24no—
209D4—24no—
210D4—22no—
211D4—24no—
212D4—624no—
213D4—12416no—
214D4—1242no—
215D4C2×C22⋊D3114yescomplemented
216C2×C4—28no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—624no—
221C2×C4—622no—
222C2×C4C2×C22⋊D3111yescharacteristic
223C2×C4C2×C22⋊D3112yes—
224C2×C4C2×C22⋊D3112yes—
225C2×C4C23×D3111yescharacteristic
226C8—624no—
227C22—22no—
228C22—24no—
229C22—24no—
230C22—44no—
231C22—44no—
232C22—44no—
233C22—82no—
234C22—88no—
235C22—1244no—
236C22—1242no—
237C22—2482no—
238C22—2482no—
239C22#1175853011yescharacteristic
240C22#1175853011yescharacteristic
241C22C22×C22⋊D3111yescharacteristic
242C4—24no—
243C4—624no—
244C4#1175853012yes—
245C4C22×C22⋊D3112yes—
246C2—21no—
247C2—21no—
248C2—44no—
249C2—44no—
250C2—2482no—
251C2#1452121712yesdirect product factor
252C2C2×#1175853011yescharacteristic
subgroupG/H
centerC22#11758530
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×H(5,𝔽2)C2
socleC22×C31#120
hypercenterC2×H(5,𝔽2)D31
hypocenterC31C2×#453
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#11758526C2semidirect
2. C2×#11758517C2semidirect
3. #14521217C2direct (N × H)
4. C2×#11758425C2semidirect
5. C2×#11758421C2semidirect
6. C2×C31×H(5,𝔽2)C2semidirect
9. C31×H(5,𝔽2)C22semidirect
16. C22×D124C22semidirect
18. #11758526C22semidirect
21. #11758517C22semidirect
23. C2×C4×Dic31C22semidirect
24. C2×#92474C22semidirect
25. #11758425C22semidirect
26. #11758421C22semidirect
34. #41×C31D4semidirect
36. C2×Q8×C31D4semidirect
46. C2×D4×C31D4semidirect
57. C2×D124C23semidirect
61. C4×Dic31C23semidirect
63. #92474C23semidirect
78. Q8×C31C2×D4semidirect
85. D4×C31C2×D4semidirect
120. C2×H(5,𝔽2)D31semidirect
130. C2×C31#453semidirect
135. H(5,𝔽2)C2×D31semidirect
153. C31C2×#453semidirect
161. #41C22⋊D31semidirect
163. C2×Q8C22⋊D31semidirect
179. C2×D4C22⋊D31semidirect
205. Q8C2×C22⋊D31semidirect
215. D4C2×C22⋊D31semidirect
251. C2#14521217direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117585262
C2×#117585172
#145212172
C2×#117584252
C2×#117584212
C2×C31×H(5,𝔽2)2
C2×#45331