Finite Group Databasebeta

← C4×#15690856 #27154943

Subgroups — C4×#15690856 #27154943

Order 12416.

1660 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690856C211yesmaximal, characteristic
2#20261218C211yesmaximal, characteristic, complemented
3#20261217C212yesmaximal, complemented
4#20261215C213yesmaximal, complemented
5#20261209C213yesmaximal, complemented
6#20261207C216yesmaximal, complemented
7C4×#13095512C212yesmaximal, complemented
8C4×#13095511C213yesmaximal, complemented
9C2×C4×D388C213yesmaximal, complemented
10C4×D4×D97C216yesmaximal, complemented
11C4×#41×C97C211yesmaximal, characteristic, complemented
12#15690856C4116yesdirect product factor
13C2×#13095512C2212yes—
14C2×#13095511C2213yes—
15C22×D388C2213yes—
16C2×D4×D97C2216yes—
17C22×C4×D97C2216yescomplemented
18Q8×Dic97C2212yescomplemented
19C4×Dic194C2211yescharacteristic, complemented
20C4×Dic194C2213yescomplemented
21C4×D388C2216yescomplemented
22C4×D388C2213yescomplemented
23C4×D388C2216yescomplemented
24C4×D388C2216yescomplemented
25C4×D388C2213yescomplemented
26D4×Dic97C2216yescomplemented
27#15690772C2216yescomplemented
28#15690772C2216yescomplemented
29C4×C22⋊D97C2216yescomplemented
30C4×C22⋊D97C2216yescomplemented
31#15690770C22112yescomplemented
32#15690769C2213yescomplemented
33#15690768C2216yescomplemented
34#15690767C2216yescomplemented
35C2×C4⋊Dic97C2213yescomplemented
36C4⋊C4×D97C2216yescomplemented
37C2×#13095488C2216yescomplemented
38C22⋊C4×D97C22112yescomplemented
39C42×D97C2216yescomplemented
40C2×#41×C97C2211yescharacteristic
41C4×Q8×C97C2211yescharacteristic, complemented
42C4×Q8×C97C2211yescharacteristic, complemented
43C4×D4×C97C2213yescomplemented
44C4×D4×C97C2213yescomplemented
45#117×C97C2213yescomplemented
46C2×C42×C97C2213yescomplemented
47C23×D97C2316yes—
48#13095512C2×C4116yescomplemented
49#13095511C2×C4124yescomplemented
50C2×Dic194C2311yescharacteristic
51C2×D388C2×C4124yescomplemented
52C2×D388C2316yes—
53C2×D388C2313yes—
54C2×C22⋊D97C2316yes—
55D4×D97C2×C4148yescomplemented
56C22×Dic97C2316yes—
57C2×C4×D97—412no—
58C2×C4×D97C2316yes—
59C2×C4×D97C2316yes—
60C2×C4×D97C23112yes—
61C2×C4×D97C2316yes—
62C4⋊Dic97C2316yes—
63C4⋊Dic97C2313yes—
64C4⋊Dic97C2311yescharacteristic
65#13095490C2316yes—
66#13095490C2312yes—
67C22⋊Dic97C2316yes—
68#13095488C2316yes—
69#13095488C23112yes—
70#13095488C2316yes—
71C4×Dic97C2312yes—
72C4×Dic97C2316yes—
73C4×Dic97C2316yes—
74#41×C97C2×C418yescomplemented
75C2×Q8×C97C2311yescharacteristic
76C2×D4×C97C2313yes—
77C22×C4×C97C2313yes—
78C22×C4×C97C2313yes—
79C22×C4×C97C2313yes—
80C4⋊C4×C97C2313yes—
81C4⋊C4×C97C2313yes—
82C22⋊C4×C97C2316yes—
83C42×C97C2313yes—
84C42×C97C2313yes—
85C42×C97C2311yescharacteristic
86C42×C97C2313yes—
87C23×C97C2413yes—
88C22×D97—46no—
89C22×D97—46no—
90C22×D97C22×C4124yescomplemented
91C22×D97C2416yes—
92Q8×C97C22×C414yescomplemented
93D4×C97C22×C4112yescomplemented
94C2×C4×C97—26no—
95C2×C4×C97—43no—
96C2×C4×C97C22×C4112yescomplemented
97C2×C4×C97C2411yescharacteristic
98C2×C4×C97C2413yes—
99C2×C4×C97C2413yes—
100C2×C4×C97C2411yescharacteristic
101C2×C4×C97C2413yes—
102C2×C4×C97C2411yescharacteristic
103C22⋊D97C22×C4124yescomplemented
104C2×Dic97—46no—
105C2×Dic97C2412yes—
106C2×Dic97C2416yes—
107C2×Dic97C2412yes—
108D388C22×C4124yescomplemented
109D388C22×C4112yescomplemented
110C4×D97—212no—
111C4×D97—412no—
112C4×D97—46no—
113C4×D97C22×C4124yescomplemented
114Dic194C22×C414yescomplemented
115C22×C97—23no—
116C22×C97—23no—
117C22×C97C23×C416yes—
118C22×C97C2511yescharacteristic
119C2×D97—26no—
120C2×D97—26no—
121C2×D97—424no—
122C2×D97C23×C4112yes—
123C4×C97—23no—
124C4×C97—23no—
125C4×C97—21no—
126C4×C97C23×C412yes—
127C4×C97C23×C416yes—
128C4×C97H(5,𝔽2)12yescomplemented
129Dic97—26no—
130Dic97—22no—
131Dic97C23×C414yes—
132C2×C97—26no—
133C2×C97C24×C411yescharacteristic
134C2×C97C2×H(5,𝔽2)11yescharacteristic
135C2×C97#58511yescharacteristic
136D97—212no—
137C4×H(5,𝔽2)—971nomaximal, Sylow
138C97C4×H(5,𝔽2)11yescharacteristic, complemented, Sylow
139C2×H(5,𝔽2)—971no—
140#520—971no—
141#520—972no—
142#520—973no—
143#518—976no—
144#518—973no—
145C4×#41—972no—
146C4×#41—973no—
147C4×#41D9711yescharacteristic, complemented
148C2×C4×D4—976no—
149C2×C4×D4—973no—
150H(5,𝔽2)—9716no—
151C2×#41—972no—
152C2×#41—973no—
153C2×#41C2×D9711yescharacteristic
154C22×D4—976no—
155C22×D4—973no—
156C23×C4—976no—
157C4×Q8—972no—
158C4×Q8—971no—
159C4×Q8—973no—
160C4×Q8C2×D9711yescharacteristic, complemented
161C4×Q8C2×D9711yescharacteristic, complemented
162C4×D4—976no—
163C4×D4—976no—
164C4×D4—976no—
165C4×D4—976no—
166C4×D4—976no—
167C4×D4—9712no—
168C4×D4—976no—
169C4×D4—973no—
170C4×D4—976no—
171C4×D4—976no—
172C4×D4—973no—
173C4×D4C2×D9713yescomplemented
174C4×D4C2×D9713yescomplemented
175#117—973no—
176#117—976no—
177#117—976no—
178#117C2×D9713yescomplemented
179C2×C4⋊C4—973no—
180C2×C4⋊C4—976no—
181C2×C22⋊C4—976no—
182C2×C22⋊C4—9712no—
183C2×C42—976no—
184C2×C42C2×D9713yescomplemented
185C24—976no—
186#41—9716no—
187#41—9724no—
188#41C4×D9718yescomplemented
189C2×Q8—971no—
190C2×Q8C22×D9711yescharacteristic
191C2×D4—976no—
192C2×D4—9748no—
193C2×D4—9724no—
194C2×D4—976no—
195C2×D4—973no—
196C2×D4C22×D9713yes—
197C22×C4—976no—
198C22×C4—976no—
199C22×C4—976no—
200C22×C4—9712no—
201C22×C4—976no—
202C22×C4—38812no—
203C22×C4C22×D9713yes—
204C22×C4C22×D9713yes—
205C22×C4C22×D9713yes—
206C4⋊C4—976no—
207C4⋊C4—972no—
208C4⋊C4—976no—
209C4⋊C4—973no—
210C4⋊C4—971no—
211C4⋊C4C22×D9713yes—
212C4⋊C4C22×D9713yes—
213C22⋊C4—976no—
214C22⋊C4—976no—
215C22⋊C4—9712no—
216C22⋊C4—976no—
217C22⋊C4C22×D9716yes—
218C42—976no—
219C42—976no—
220C42—972no—
221C42C22×D9713yes—
222C42C22×D9713yes—
223C42C22×D9713yes—
224C42C22×D9711yescharacteristic
225C23—9724no—
226C23—976no—
227C23—3886no—
228C23—3886no—
229C23C23×D9713yes—
230Q8—974no—
231Q8C2×C4×D9714yescomplemented
232D4—9724no—
233D4—9724no—
234D4—9712no—
235D4C2×C4×D97112yescomplemented
236C2×C4—26no—
237C2×C4—43no—
238C2×C4—972no—
239C2×C4—9724no—
240C2×C4—972no—
241C2×C4—976no—
242C2×C4—38812no—
243C2×C4—3886no—
244C2×C4—3886no—
245C2×C4—19412no—
246C2×C4C2×C4×D97112yescomplemented
247C2×C4C23×D9713yes—
248C2×C4C23×D9711yescharacteristic
249C2×C4C23×D9713yes—
250C2×C4C23×D9713yes—
251C2×C4C23×D9711yescharacteristic
252C2×C4C23×D9711yescharacteristic
253C22—23no—
254C22—23no—
255C22—9712no—
256C22—38824no—
257C22—1946no—
258C22—1946no—
259C22C22×C4×D9716yes—
260C22C24×D9711yescharacteristic
261C4—23no—
262C4—21no—
263C4—23no—
264C4—974no—
265C4—1946no—
266C4—1942no—
267C4C22×C4×D9716yes—
268C4C22×C4×D9712yes—
269C4#1569085612yesdirect product factor
270C2—26no—
271C2—19412no—
272C2C23×C4×D9711yescharacteristic
273C2C2×#1569085611yescharacteristic
274C2#2026157111yescharacteristic
subgroupG/H
centerC2×C4C23×D97
commutatorC2×C97C24×C4
FrattiniC22C24×D97
FittingC4×#41×C97C2
socle, cosocleC22×C97C25
prime radicalC2×#15690856C2
hypercenterC4×#41D97
hypocenterC97C4×H(5,𝔽2)
LCSG ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
2. #20261218C2semidirect
3. #20261217C2semidirect
4. #20261215C2semidirect
5. #20261209C2semidirect
6. #20261207C2semidirect
7. C4×#13095512C2semidirect
8. C4×#13095511C2semidirect
9. C2×C4×D388C2semidirect
10. C4×D4×D97C2semidirect
11. C4×#41×C97C2semidirect
12. #15690856C4direct (N × H)
17. C22×C4×D97C22semidirect
18. Q8×Dic97C22semidirect
19. C4×Dic194C22semidirect
20. C4×Dic194C22semidirect
21. C4×D388C22semidirect
22. C4×D388C22semidirect
23. C4×D388C22semidirect
24. C4×D388C22semidirect
25. C4×D388C22semidirect
26. D4×Dic97C22semidirect
27. #15690772C22semidirect
28. #15690772C22semidirect
29. C4×C22⋊D97C22semidirect
30. C4×C22⋊D97C22semidirect
31. #15690770C22semidirect
32. #15690769C22semidirect
33. #15690768C22semidirect
34. #15690767C22semidirect
35. C2×C4⋊Dic97C22semidirect
36. C4⋊C4×D97C22semidirect
37. C2×#13095488C22semidirect
38. C22⋊C4×D97C22semidirect
39. C42×D97C22semidirect
41. C4×Q8×C97C22semidirect
42. C4×Q8×C97C22semidirect
43. C4×D4×C97C22semidirect
44. C4×D4×C97C22semidirect
45. #117×C97C22semidirect
46. C2×C42×C97C22semidirect
48. #13095512C2×C4semidirect
49. #13095511C2×C4semidirect
51. C2×D388C2×C4semidirect
55. D4×D97C2×C4semidirect
74. #41×C97C2×C4semidirect
90. C22×D97C22×C4semidirect
92. Q8×C97C22×C4semidirect
93. D4×C97C22×C4semidirect
96. C2×C4×C97C22×C4semidirect
103. C22⋊D97C22×C4semidirect
108. D388C22×C4semidirect
109. D388C22×C4semidirect
113. C4×D97C22×C4semidirect
114. Dic194C22×C4semidirect
128. C4×C97H(5,𝔽2)semidirect
138. C97C4×H(5,𝔽2)semidirect
147. C4×#41D97semidirect
160. C4×Q8C2×D97semidirect
161. C4×Q8C2×D97semidirect
173. C4×D4C2×D97semidirect
174. C4×D4C2×D97semidirect
178. #117C2×D97semidirect
184. C2×C42C2×D97semidirect
188. #41C4×D97semidirect
231. Q8C2×C4×D97semidirect
235. D4C2×C4×D97semidirect
246. C2×C4C2×C4×D97semidirect
269. C4#15690856direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908562
#202612182
#202612172
#202612152
#202612092
#202612072
C4×#130955122
C4×#130955112
C2×C4×D3882
C4×D4×D972
C4×#41×C972
C4×H(5,𝔽2)97