Finite Group Databasebeta

← C4×#15690855 #27154941

Subgroups — C4×#15690855 #27154941

Order 12416.

1660 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690855C211yesmaximal, characteristic
2#20261216C212yesmaximal, complemented
3#20261214C214yesmaximal, complemented
4#20261210C211yesmaximal, characteristic, complemented
5#20261208C214yesmaximal, complemented
6#20261207C214yesmaximal, complemented
7C4×#13095511C212yesmaximal, complemented
8C4×#13095510C214yesmaximal, complemented
9C2×C4×C22⋊D97C214yesmaximal, complemented
10C4×D4×D97C214yesmaximal, complemented
11C2×C4×D4×C97C211yesmaximal, characteristic, complemented
12#15690855C4116yesdirect product factor
13C2×#13095511C2212yes—
14C2×#13095510C2214yes—
15C22×C22⋊D97C2214yes—
16C2×D4×D97C2214yes—
17C22×C4×D97C2214yescomplemented
18C4×Dic194C2212yescomplemented
19C4×Dic194C2212yescomplemented
20#15690775C2214yescomplemented
21C4×D388C2212yescomplemented
22C4×D388C2212yescomplemented
23D4×Dic97C2214yescomplemented
24D4×Dic97C2214yescomplemented
25#15690772C2214yescomplemented
26C4×C22⋊D97C2218yescomplemented
27C4×C22⋊D97C2214yescomplemented
28C4×C22⋊D97C2218yescomplemented
29C4×C22⋊D97C2218yescomplemented
30C4×C22⋊D97C2214yescomplemented
31#15690770C2218yescomplemented
32#15690770C2218yescomplemented
33#15690769C2212yescomplemented
34#15690768C2214yescomplemented
35#15690767C2214yescomplemented
36#15690766C2218yescomplemented
37C2×#13095490C2214yescomplemented
38C4⋊C4×D97C2214yescomplemented
39C2×C22⋊Dic97C2214yescomplemented
40C2×#13095488C2214yescomplemented
41C22⋊C4×D97C2218yescomplemented
42C2×C4×Dic97C2214yescomplemented
43C42×D97C2214yescomplemented
44C22×D4×C97C2211yescharacteristic
45C23×C4×C97C2212yescomplemented
46C4×D4×C97C2214yescomplemented
47C4×D4×C97C2214yescomplemented
48C2×C4⋊C4×C97C2211yescharacteristic, complemented
49C2×C22⋊C4×C97C2212yescomplemented
50C2×C42×C97C2211yescharacteristic, complemented
51C23×D97C2314yes—
52#13095511C2×C4116yescomplemented
53#13095510C2×C4132yescomplemented
54C2×Dic194C2312yes—
55C2×D388C2312yes—
56C2×C22⋊D97C2×C4132yescomplemented
57C2×C22⋊D97C2318yes—
58C2×C22⋊D97C2314yes—
59D4×D97C2×C4132yescomplemented
60C22×Dic97C2314yes—
61C22×Dic97C2314yes—
62C22×Dic97C2314yes—
63C2×C4×D97—48no—
64C2×C4×D97C2314yes—
65C2×C4×D97C2314yes—
66C2×C4×D97C2318yes—
67C2×C4×D97C2314yes—
68C4⋊Dic97C2312yes—
69C4⋊Dic97C2312yes—
70#13095490C2318yes—
71#13095490C2314yes—
72#13095490C2314yes—
73C22⋊Dic97C2318yes—
74C22⋊Dic97C2314yes—
75#13095488C2314yes—
76#13095488C2318yes—
77#13095488C2314yes—
78C4×Dic97C2314yes—
79C4×Dic97C2314yes—
80C4×Dic97C2314yes—
81C4×Dic97C2318yes—
82C24×C97C2312yes—
83C2×D4×C97C2×C418yescomplemented
84C2×D4×C97C2314yes—
85C22×C4×C97—44no—
86C22×C4×C97C2314yes—
87C22×C4×C97C2311yescharacteristic
88C22×C4×C97C2314yes—
89C22×C4×C97C2311yescharacteristic
90C22×C4×C97C2312yes—
91C22×C4×C97C2311yescharacteristic
92C4⋊C4×C97C2314yes—
93C22⋊C4×C97C2318yes—
94C42×C97C2312yes—
95C42×C97C2312yes—
96C23×C97—42no—
97C23×C97—42no—
98C23×C97C22×C418yescomplemented
99C23×C97C2414yes—
100C23×C97C2411yescharacteristic
101C22×D97—44no—
102C22×D97—44no—
103C22×D97C22×C4116yescomplemented
104C22×D97C2414yes—
105D4×C97C22×C4116yescomplemented
106C2×C4×C97—22no—
107C2×C4×C97—28no—
108C2×C4×C97—42no—
109C2×C4×C97—44no—
110C2×C4×C97—41no—
111C2×C4×C97C22×C414yescomplemented
112C2×C4×C97C2412yes—
113C2×C4×C97C2411yescharacteristic
114C2×C4×C97C2414yes—
115C2×C4×C97C2412yes—
116C2×C4×C97C2411yescharacteristic
117C22⋊D97C22×C4132yescomplemented
118C22⋊D97C22×C4116yescomplemented
119C2×Dic97—44no—
120C2×Dic97—44no—
121C2×Dic97C22×C4116yescomplemented
122C2×Dic97C2414yes—
123C2×Dic97C2414yes—
124C2×Dic97C2414yes—
125D388C22×C418yescomplemented
126C4×D97—28no—
127C4×D97—48no—
128C4×D97—44no—
129C4×D97C22×C4116yescomplemented
130Dic194C22×C418yescomplemented
131C22×C97—24no—
132C22×C97—21no—
133C22×C97—24no—
134C22×C97—21no—
135C22×C97—48no—
136C22×C97C23×C418yes—
137C22×C97C23×C412yes—
138C22×C97C2511yescharacteristic
139C2×D97—24no—
140C2×D97—24no—
141C2×D97—416no—
142C2×D97C23×C418yes—
143C4×C97—24no—
144C4×C97—21no—
145C4×C97—22no—
146C4×C97C23×C414yes—
147C4×C97H(5,𝔽2)12yescomplemented
148Dic97—24no—
149Dic97—24no—
150Dic97C23×C418yes—
151C2×C97—28no—
152C2×C97—22no—
153C2×C97C24×C411yescharacteristic
154C2×C97C2×H(5,𝔽2)11yescharacteristic
155C2×C97#58511yescharacteristic
156D97—28no—
157C4×H(5,𝔽2)—971nomaximal, Sylow
158C97C4×H(5,𝔽2)11yescharacteristic, complemented, Sylow
159C2×H(5,𝔽2)—971no—
160#520—974no—
161#520—972no—
162#518—971no—
163#518—974no—
164#518—974no—
165C4×#41—974no—
166C4×#41—972no—
167C2×C4×D4—974no—
168C2×C4×D4—974no—
169C2×C4×D4D9711yescharacteristic, complemented
170H(5,𝔽2)—9716no—
171C2×#41—974no—
172C2×#41—972no—
173C22×D4—974no—
174C22×D4—974no—
175C22×D4C2×D9711yescharacteristic
176C23×C4—974no—
177C23×C4C2×D9712yescomplemented
178C4×Q8—974no—
179C4×Q8—972no—
180C4×Q8—972no—
181C4×D4—974no—
182C4×D4—974no—
183C4×D4—978no—
184C4×D4—974no—
185C4×D4—978no—
186C4×D4—978no—
187C4×D4—974no—
188C4×D4—974no—
189C4×D4—978no—
190C4×D4—978no—
191C4×D4—972no—
192C4×D4—972no—
193C4×D4C2×D9714yescomplemented
194C4×D4C2×D9714yescomplemented
195#117—972no—
196#117—974no—
197#117—978no—
198#117—974no—
199C2×C4⋊C4—974no—
200C2×C4⋊C4—974no—
201C2×C4⋊C4C2×D9711yescharacteristic, complemented
202C2×C22⋊C4—974no—
203C2×C22⋊C4—974no—
204C2×C22⋊C4—978no—
205C2×C22⋊C4C2×D9712yescomplemented
206C2×C42—974no—
207C2×C42—974no—
208C2×C42C2×D9711yescharacteristic, complemented
209C24—974no—
210C24C22×D9712yes—
211#41—9732no—
212#41—9716no—
213C2×Q8—972no—
214C2×D4—9732no—
215C2×D4—974no—
216C2×D4—978no—
217C2×D4—9732no—
218C2×D4—972no—
219C2×D4C4×D9718yescomplemented
220C2×D4C22×D9714yes—
221C22×C4—44no—
222C22×C4—974no—
223C22×C4—974no—
224C22×C4—974no—
225C22×C4—974no—
226C22×C4—974no—
227C22×C4—978no—
228C22×C4—974no—
229C22×C4—3888no—
230C22×C4C22×D9714yes—
231C22×C4C22×D9711yescharacteristic
232C22×C4C22×D9711yescharacteristic
233C22×C4C22×D9714yes—
234C22×C4C22×D9712yes—
235C22×C4C22×D9711yescharacteristic
236C4⋊C4—978no—
237C4⋊C4—974no—
238C4⋊C4—974no—
239C4⋊C4—972no—
240C4⋊C4—972no—
241C4⋊C4C22×D9714yes—
242C22⋊C4—974no—
243C22⋊C4—978no—
244C22⋊C4—978no—
245C22⋊C4—974no—
246C22⋊C4—974no—
247C22⋊C4C22×D9718yes—
248C42—978no—
249C42—974no—
250C42—974no—
251C42—974no—
252C42C22×D9712yes—
253C42C22×D9712yes—
254C23—42no—
255C23—42no—
256C23—9716no—
257C23—974no—
258C23—3884no—
259C23—3884no—
260C23C2×C4×D9718yescomplemented
261C23C23×D9711yescharacteristic
262C23C23×D9714yes—
263Q8—978no—
264D4—9732no—
265D4—9716no—
266D4—978no—
267D4C2×C4×D97116yescomplemented
268C2×C4—22no—
269C2×C4—28no—
270C2×C4—44no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—9716no—
274C2×C4—974no—
275C2×C4—9716no—
276C2×C4—974no—
277C2×C4—974no—
278C2×C4—3888no—
279C2×C4—3884no—
280C2×C4—3884no—
281C2×C4—3884no—
282C2×C4—1948no—
283C2×C4C2×C4×D9714yescomplemented
284C2×C4C23×D9712yes—
285C2×C4C23×D9714yes—
286C2×C4C23×D9711yescharacteristic
287C2×C4C23×D9712yes—
288C2×C4C23×D9711yescharacteristic
289C22—21no—
290C22—24no—
291C22—21no—
292C22—24no—
293C22—48no—
294C22—978no—
295C22—38816no—
296C22—1944no—
297C22—1944no—
298C22C22×C4×D9718yes—
299C22C22×C4×D9712yes—
300C22C24×D9711yescharacteristic
301C4—21no—
302C4—24no—
303C4—22no—
304C4—978no—
305C4—1944no—
306C4—1944no—
307C4C22×C4×D9714yes—
308C4#1569085512yesdirect product factor
309C2—22no—
310C2—28no—
311C2—1948no—
312C2C23×C4×D9711yescharacteristic
313C2C2×#1569085511yescharacteristic
314C2#2026157111yescharacteristic
subgroupG/H
centerC2×C4C23×D97
commutatorC2×C97C24×C4
FrattiniC22C24×D97
FittingC2×C4×D4×C97C2
socle, cosocleC22×C97C25
prime radicalC2×#15690855C2
hypercenterC2×C4×D4D97
hypocenterC97C4×H(5,𝔽2)
LCSG ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. #20261216C2semidirect
3. #20261214C2semidirect
4. #20261210C2semidirect
5. #20261208C2semidirect
6. #20261207C2semidirect
7. C4×#13095511C2semidirect
8. C4×#13095510C2semidirect
9. C2×C4×C22⋊D97C2semidirect
10. C4×D4×D97C2semidirect
11. C2×C4×D4×C97C2semidirect
12. #15690855C4direct (N × H)
17. C22×C4×D97C22semidirect
18. C4×Dic194C22semidirect
19. C4×Dic194C22semidirect
20. #15690775C22semidirect
21. C4×D388C22semidirect
22. C4×D388C22semidirect
23. D4×Dic97C22semidirect
24. D4×Dic97C22semidirect
25. #15690772C22semidirect
26. C4×C22⋊D97C22semidirect
27. C4×C22⋊D97C22semidirect
28. C4×C22⋊D97C22semidirect
29. C4×C22⋊D97C22semidirect
30. C4×C22⋊D97C22semidirect
31. #15690770C22semidirect
32. #15690770C22semidirect
33. #15690769C22semidirect
34. #15690768C22semidirect
35. #15690767C22semidirect
36. #15690766C22semidirect
37. C2×#13095490C22semidirect
38. C4⋊C4×D97C22semidirect
39. C2×C22⋊Dic97C22semidirect
40. C2×#13095488C22semidirect
41. C22⋊C4×D97C22semidirect
42. C2×C4×Dic97C22semidirect
43. C42×D97C22semidirect
45. C23×C4×C97C22semidirect
46. C4×D4×C97C22semidirect
47. C4×D4×C97C22semidirect
48. C2×C4⋊C4×C97C22semidirect
49. C2×C22⋊C4×C97C22semidirect
50. C2×C42×C97C22semidirect
52. #13095511C2×C4semidirect
53. #13095510C2×C4semidirect
56. C2×C22⋊D97C2×C4semidirect
59. D4×D97C2×C4semidirect
83. C2×D4×C97C2×C4semidirect
98. C23×C97C22×C4semidirect
103. C22×D97C22×C4semidirect
105. D4×C97C22×C4semidirect
111. C2×C4×C97C22×C4semidirect
117. C22⋊D97C22×C4semidirect
118. C22⋊D97C22×C4semidirect
121. C2×Dic97C22×C4semidirect
125. D388C22×C4semidirect
129. C4×D97C22×C4semidirect
130. Dic194C22×C4semidirect
147. C4×C97H(5,𝔽2)semidirect
158. C97C4×H(5,𝔽2)semidirect
169. C2×C4×D4D97semidirect
177. C23×C4C2×D97semidirect
193. C4×D4C2×D97semidirect
194. C4×D4C2×D97semidirect
201. C2×C4⋊C4C2×D97semidirect
205. C2×C22⋊C4C2×D97semidirect
208. C2×C42C2×D97semidirect
219. C2×D4C4×D97semidirect
260. C23C2×C4×D97semidirect
267. D4C2×C4×D97semidirect
283. C2×C4C2×C4×D97semidirect
308. C4#15690855direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908552
#202612162
#202612142
#202612102
#202612082
#202612072
C4×#130955112
C4×#130955102
C2×C4×C22⋊D972
C4×D4×D972
C2×C4×D4×C972
C4×H(5,𝔽2)97