Finite Group Databasebeta

← C2×#27161686 #38756075

Subgroups — C2×#27161686 #38756075

Order 25216.

1004 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161686C2116yesmaximal, direct product factor
2C2×#20263325C211yesmaximal, characteristic, complemented
3C2×#20263319C211yesmaximal, characteristic
4C2×#20263319C211yesmaximal, characteristic
5C2×#20263319C211yesmaximal, characteristic
6C2×#20263318C211yesmaximal, characteristic, complemented
7C2×#20263317C211yesmaximal, characteristic, complemented
8C2×#20263316C211yesmaximal, characteristic, complemented
9C2×#20263316C211yesmaximal, characteristic, complemented
10C2×#20263316C211yesmaximal, characteristic, complemented
11C2×#20263303C211yesmaximal, characteristic
12C2×#20263303C211yesmaximal, characteristic
13C2×C4×Dic394C211yesmaximal, characteristic, complemented
14C2×#20263291C211yesmaximal, characteristic, complemented
15C2×#20263282C211yesmaximal, characteristic
16C2×#126×C197C211yesmaximal, characteristic
17C22×Dic394C2211yescharacteristic
18C22×Dic394C2211yescharacteristic
19C23×Dic197C2211yescharacteristic
20#20263325C2218yescomplemented
21#20263319C2218yes—
22#20263319C2218yes—
23#20263319C2218yes—
24#20263318C2218yescomplemented
25#20263317C2218yescomplemented
26#20263316C2218yescomplemented
27#20263316C2218yescomplemented
28#20263316C2218yescomplemented
29#20263303C2218yes—
30#20263303C2218yes—
31C4×Dic394C2218yescomplemented
32#20263291C2218yescomplemented
33#20263282C2218yes—
34C2×C4⋊Dic197C2211yescharacteristic
35C2×C4⋊Dic197C2211yescharacteristic
36C2×C4⋊Dic197C2211yescharacteristic
37C2×C4⋊Dic197C2211yescharacteristic
38C2×C4⋊Dic197C2211yescharacteristic
39C2×#15691337C2211yescharacteristic
40C2×#15691337C2211yescharacteristic
41C2×#15691337C2211yescharacteristic
42C2×#15691337C2211yescharacteristic
43C2×#15691337C2211yescharacteristic
44C2×#15691337C2211yescharacteristic
45C2×#15691337C2211yescharacteristic
46C2×#15691337C2211yescharacteristic
47C2×#15691337C2211yescharacteristic
48C2×#15691337C2211yescharacteristic
49C2×#15691337C2211yescharacteristic
50C2×#15691337C2211yescharacteristic
51C2×C22⋊Dic197C2211yescharacteristic
52C2×C22⋊Dic197C2211yescharacteristic
53C2×C22⋊Dic197C2211yescharacteristic
54C2×C4×Dic197C2211yescharacteristic
55C2×C4×Dic197C2211yescharacteristic
56C2×C4×Dic197C2211yescharacteristic
57C2×C4×Dic197C2211yescharacteristic
58C2×C4×Dic197C2211yescharacteristic
59#126×C197C2218yes—
60C2×C4⋊C4×C197C2211yescharacteristic
61C2×C4⋊C4×C197C2211yescharacteristic
62C2×C4⋊C4×C197C2211yescharacteristic
63C2×C22⋊C4×C197C2211yescharacteristic
64C2×C22⋊C4×C197C2211yescharacteristic
65C2×C22⋊C4×C197C2211yescharacteristic
66C2×C42×C197C2211yescharacteristic
67C2×Dic394—24no—
68C2×Dic394—24no—
69C2×Dic394C2314yes—
70C2×Dic394C2314yes—
71C22×Dic197—24no—
72C22×Dic197C2311yescharacteristic
73C22×Dic197C2311yescharacteristic
74C22×Dic197C2311yescharacteristic
75C22×Dic197C2311yescharacteristic
76C22×Dic197C2311yescharacteristic
77C22×Dic197C2311yescharacteristic
78C22×Dic197C2311yescharacteristic
79C22×Dic197C2314yes—
80C22×Dic197C2311yescharacteristic
81C4⋊Dic197C2314yes—
82C4⋊Dic197C2314yes—
83C4⋊Dic197C2314yes—
84C4⋊Dic197C2314yes—
85C4⋊Dic197C2314yes—
86#15691337C2314yes—
87#15691337C2314yes—
88#15691337C2314yes—
89#15691337C2314yes—
90#15691337C2314yes—
91#15691337C2314yes—
92#15691337C2314yes—
93#15691337C2314yes—
94#15691337C2314yes—
95#15691337C2314yes—
96#15691337C2314yes—
97#15691337C2314yes—
98C22⋊Dic197C2314yes—
99C22⋊Dic197C2314yes—
100C22⋊Dic197C2314yes—
101C4×Dic197C2314yes—
102C4×Dic197C2314yes—
103C4×Dic197C2314yes—
104C4×Dic197C2314yes—
105C4×Dic197C2314yes—
106C24×C197C2311yescharacteristic
107C22×C4×C197C2311yescharacteristic
108C22×C4×C197C2311yescharacteristic
109C22×C4×C197C2311yescharacteristic
110C22×C4×C197C2311yescharacteristic
111C22×C4×C197C2311yescharacteristic
112C22×C4×C197C2311yescharacteristic
113C4⋊C4×C197C2314yes—
114C4⋊C4×C197C2314yes—
115C4⋊C4×C197C2314yes—
116C22⋊C4×C197C2314yes—
117C22⋊C4×C197C2314yes—
118C22⋊C4×C197C2314yes—
119C42×C197C2314yes—
120Dic394—28no—
121Dic394—28no—
122C2×Dic197—22no—
123C2×Dic197—22no—
124C2×Dic197—22no—
125C2×Dic197—22no—
126C2×Dic197—22no—
127C2×Dic197—22no—
128C2×Dic197—22no—
129C2×Dic197—28no—
130C2×Dic197#4114yes—
131C2×Dic197C2412yes—
132C2×Dic197C2412yes—
133C2×Dic197C2412yes—
134C2×Dic197C2412yes—
135C2×Dic197C2412yes—
136C2×Dic197C2412yes—
137C2×Dic197C2412yes—
138C2×Dic197C2412yes—
139C23×C197—22no—
140C23×C197—22no—
141C23×C197—22no—
142C23×C197C2412yes—
143C23×C197C2411yescharacteristic
144C2×C4×C197—22no—
145C2×C4×C197—22no—
146C2×C4×C197—22no—
147C2×C4×C197—22no—
148C2×C4×C197—22no—
149C2×C4×C197—22no—
150C2×C4×C197C2412yes—
151C2×C4×C197C2412yes—
152C2×C4×C197C2412yes—
153C2×C4×C197C2412yes—
154C2×C4×C197C2412yes—
155C2×C4×C197C2412yes—
156C22×C197—22no—
157C22×C197—22no—
158C22×C197—22no—
159C22×C197—44no—
160C22×C197C2×#4112yes—
161C22×C1972−1+412yes—
162C22×C1972−1+412yes—
163C22×C197C2511yescharacteristic
164C4×C197—22no—
165C4×C197—22no—
166C4×C197—22no—
167C4×C197—22no—
168C4×C197—22no—
169C4×C197—22no—
170Dic197—22no—
171Dic197—22no—
172Dic197—22no—
173Dic197—22no—
174Dic197—22no—
175Dic197—22no—
176Dic197—22no—
177Dic197C2×#4114yes—
178C2×C197—42no—
179C2×C197#54114yescomplemented
180C2×C197C22×#4111yescharacteristic
181C2×C197C2×2−1+411yescharacteristic
182C2×C197C2×2−1+411yescharacteristic
183C197C2×#54111yescharacteristic, complemented, Sylow
184C2×#541—1971nomaximal, Sylow
185#541—19716no—
186C2×C4⋊Q8—1971no—
187C2×#126—1971no—
188C2×#126—1971no—
189C2×#126—1971no—
190C2×#126D19711yescharacteristic
191C2×#125—1971no—
192C2×#125—1971no—
193C2×#125—1971no—
194C2×#125—1971no—
195C2×#125—1971no—
196C2×C22⋊Q8—1971no—
197C2×C22⋊Q8—1971no—
198C2×C4×Q8—1971no—
199C2×C4×Q8—1971no—
200C2×#117—1971no—
201C22×Q8—1971no—
202C22×Q8—1971no—
203C23×C4—1971no—
204C4⋊Q8—1978no—
205#126—1978no—
206#126—1978no—
207#126—1978no—
208#126C2×D19718yes—
209#125—1978no—
210#125—1978no—
211#125—1978no—
212#125—1978no—
213#125—1978no—
214C22⋊Q8—1978no—
215C22⋊Q8—1978no—
216C4×Q8—1978no—
217C4×Q8—1978no—
218#117—1978no—
219C2×C4⋊C4—1971no—
220C2×C4⋊C4—1971no—
221C2×C4⋊C4—1971no—
222C2×C4⋊C4—1971no—
223C2×C4⋊C4—1971no—
224C2×C4⋊C4—1971no—
225C2×C4⋊C4—1971no—
226C2×C4⋊C4—1971no—
227C2×C4⋊C4—1971no—
228C2×C4⋊C4—1971no—
229C2×C4⋊C4—1971no—
230C2×C4⋊C4—1971no—
231C2×C4⋊C4—1971no—
232C2×C4⋊C4—1971no—
233C2×C4⋊C4—1971no—
234C2×C4⋊C4—1971no—
235C2×C4⋊C4—1971no—
236C2×C4⋊C4C2×D19711yescharacteristic
237C2×C4⋊C4C2×D19711yescharacteristic
238C2×C4⋊C4C2×D19711yescharacteristic
239C2×C22⋊C4—1971no—
240C2×C22⋊C4—1971no—
241C2×C22⋊C4—1971no—
242C2×C22⋊C4C2×D19711yescharacteristic
243C2×C22⋊C4C2×D19711yescharacteristic
244C2×C22⋊C4C2×D19711yescharacteristic
245C2×C42—1971no—
246C2×C42—1971no—
247C2×C42—1971no—
248C2×C42—1971no—
249C2×C42—1971no—
250C2×C42C2×D19711yescharacteristic
251C24C22×D19711yescharacteristic
252C2×Q8—3944no—
253C2×Q8—3944no—
254C2×Q8—1974no—
255C2×Q8—1974no—
256C22×C4—3944no—
257C22×C4—1974no—
258C22×C4—1971no—
259C22×C4—1971no—
260C22×C4—1971no—
261C22×C4—1971no—
262C22×C4—1971no—
263C22×C4—1971no—
264C22×C4—1971no—
265C22×C4—1971no—
266C22×C4C22×D19711yescharacteristic
267C22×C4C22×D19711yescharacteristic
268C22×C4C22×D19711yescharacteristic
269C22×C4C22×D19711yescharacteristic
270C22×C4C22×D19711yescharacteristic
271C22×C4C22×D19711yescharacteristic
272C4⋊C4—1974no—
273C4⋊C4—1974no—
274C4⋊C4—1974no—
275C4⋊C4—1974no—
276C4⋊C4—1974no—
277C4⋊C4—1974no—
278C4⋊C4—1974no—
279C4⋊C4—1974no—
280C4⋊C4—1974no—
281C4⋊C4—1974no—
282C4⋊C4—1974no—
283C4⋊C4—1974no—
284C4⋊C4—1974no—
285C4⋊C4—1974no—
286C4⋊C4—1974no—
287C4⋊C4—1974no—
288C4⋊C4—1974no—
289C4⋊C4C22×D19714yes—
290C4⋊C4C22×D19714yes—
291C4⋊C4C22×D19714yes—
292C22⋊C4—1974no—
293C22⋊C4—1974no—
294C22⋊C4—1974no—
295C22⋊C4C22×D19714yes—
296C22⋊C4C22×D19714yes—
297C22⋊C4C22×D19714yes—
298C42—1974no—
299C42—1974no—
300C42—1974no—
301C42—1974no—
302C42—1974no—
303C42C22×D19714yes—
304C23—22no—
305C23—22no—
306C23—22no—
307C23C23×D19712yes—
308C23C23×D19711yescharacteristic
309Q8—3948no—
310Q8—3948no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—3942no—
318C2×C4—3942no—
319C2×C4—3942no—
320C2×C4—3942no—
321C2×C4—3942no—
322C2×C4—3942no—
323C2×C4—3942no—
324C2×C4—3948no—
325C2×C4—1972no—
326C2×C4—1972no—
327C2×C4—1972no—
328C2×C4—1972no—
329C2×C4—1972no—
330C2×C4—1972no—
331C2×C4—1972no—
332C2×C4—1974no—
333C2×C4—1972no—
334C2×C4C23×D19712yes—
335C2×C4C23×D19712yes—
336C2×C4C23×D19712yes—
337C2×C4C23×D19712yes—
338C2×C4C23×D19712yes—
339C2×C4C23×D19712yes—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—44no—
344C22#41×D19712yes—
345C22#2026337312yes—
346C22#2026337312yes—
347C22C24×D19711yescharacteristic
348C4—22no—
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—3942no—
355C4—3942no—
356C4—3942no—
357C4—3942no—
358C4—3942no—
359C4—3942no—
360C4—3942no—
361C4—1974no—
362C2—42no—
363C2#2716168614yesdirect product factor
364C2C2×#41×D19711yescharacteristic
365C2C2×#2026337311yescharacteristic
366C2C2×#2026337311yescharacteristic
subgroupG/H
centerC23C23×D197
commutator, cosocleC22×C197C25
FrattiniC22C24×D197
FittingC2×#126×C197C2
socleC23×C197C24
prime radicalC24×C197C23
hypercenterC2×#126D197
hypocenterC197C2×#541
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #27161686C2direct (N × H)
2. C2×#20263325C2semidirect
6. C2×#20263318C2semidirect
7. C2×#20263317C2semidirect
8. C2×#20263316C2semidirect
9. C2×#20263316C2semidirect
10. C2×#20263316C2semidirect
13. C2×C4×Dic394C2semidirect
14. C2×#20263291C2semidirect
20. #20263325C22semidirect
24. #20263318C22semidirect
25. #20263317C22semidirect
26. #20263316C22semidirect
27. #20263316C22semidirect
28. #20263316C22semidirect
31. C4×Dic394C22semidirect
32. #20263291C22semidirect
179. C2×C197#541semidirect
183. C197C2×#541semidirect
363. C2#27161686direct (N × H)

Maximal subgroups

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

subgroupindex
#271616862
C2×#202633252
C2×#202633192
C2×#202633182
C2×#202633172
C2×#202633162
C2×#202633032
C2×C4×Dic3942
C2×#202632912
C2×#202632822
C2×#126×C1972
C2×#541197