Finite Group Databasebeta

← C2×#20261486 #27155867

Subgroups — C2×#20261486 #27155867

Order 12416.

1024 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261486C2116yesmaximal, direct product factor
2C2×#15690809C211yesmaximal, characteristic, complemented
3C2×#15690808C211yesmaximal, characteristic, complemented
4C2×#15690805C211yesmaximal, characteristic, complemented
5C2×#15690805C211yesmaximal, characteristic, complemented
6C2×#15690803C211yesmaximal, characteristic, complemented
7C2×#15690803C211yesmaximal, characteristic, complemented
8C2×#15690801C211yesmaximal, characteristic, complemented
9C2×#15690800C211yesmaximal, characteristic, complemented
10C2×#15690796C211yesmaximal, characteristic
11C2×#15690793C211yesmaximal, characteristic
12C2×#15690792C211yesmaximal, characteristic
13C2×#15690789C211yesmaximal, characteristic, complemented
14C2×#15690788C211yesmaximal, characteristic, complemented
15C2×#15690786C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8×C97C211yesmaximal, characteristic, complemented
17C22×Dic194C2211yescharacteristic
18C22×Dic194C2211yescharacteristic
19C22×C22⋊D97C2211yescharacteristic
20C22×C4×D97C2211yescharacteristic
21#15690809C2218yescomplemented
22#15690808C2218yescomplemented
23#15690805C2218yescomplemented
24#15690805C2218yescomplemented
25#15690803C2218yescomplemented
26#15690803C2218yescomplemented
27#15690801C2218yescomplemented
28#15690800C2218yescomplemented
29#15690796C2218yes—
30#15690793C2218yes—
31#15690792C2218yes—
32#15690789C2218yescomplemented
33#15690788C2218yescomplemented
34#15690786C2218yescomplemented
35C2×C4⋊Dic97C2211yescharacteristic
36C2×C4⋊Dic97C2211yescharacteristic
37C2×C4⋊Dic97C2211yescharacteristic
38C2×C4⋊Dic97C2211yescharacteristic
39C2×#13095490C2211yescharacteristic
40C2×#13095490C2211yescharacteristic
41C2×#13095490C2211yescharacteristic
42C2×#13095490C2211yescharacteristic
43C2×#13095490C2211yescharacteristic
44C2×#13095490C2211yescharacteristic
45C2×#13095490C2211yescharacteristic
46C2×#13095490C2211yescharacteristic
47C2×#13095490C2211yescharacteristic
48C2×C22⋊Dic97C2211yescharacteristic
49C2×C22⋊Dic97C2211yescharacteristic
50C2×C22⋊Dic97C2211yescharacteristic
51C2×#13095488C2211yescharacteristic
52C2×#13095488C2211yescharacteristic
53C2×#13095488C2211yescharacteristic
54C2×#13095488C2211yescharacteristic
55C2×#13095488C2211yescharacteristic
56C2×C4×Dic97C2211yescharacteristic
57C2×C4×Dic97C2211yescharacteristic
58C2×C4×Dic97C2211yescharacteristic
59C22×Q8×C97C2211yescharacteristic
60C23×C4×C97C2211yescharacteristic
61C22⋊Q8×C97C2218yescomplemented
62C2×C4⋊C4×C97C2211yescharacteristic
63C2×C4⋊C4×C97C2211yescharacteristic
64C2×C4⋊C4×C97C2211yescharacteristic
65C2×C22⋊C4×C97C2211yescharacteristic
66C2×C22⋊C4×C97C2211yescharacteristic
67C23×D97C2311yescharacteristic
68C2×Dic194—42no—
69C2×Dic194—42no—
70C2×Dic194C2314yes—
71C2×Dic194C2314yes—
72C2×C22⋊D97—42no—
73C2×C22⋊D97C2314yes—
74C22×Dic97C2311yescharacteristic
75C22×Dic97C2311yescharacteristic
76C22×Dic97C2311yescharacteristic
77C22×Dic97C2311yescharacteristic
78C22×Dic97C2311yescharacteristic
79C22×Dic97C2311yescharacteristic
80C22×Dic97C2311yescharacteristic
81C2×C4×D97—42no—
82C2×C4×D97C2314yes—
83C4⋊Dic97C2314yes—
84C4⋊Dic97C2314yes—
85C4⋊Dic97C2314yes—
86C4⋊Dic97C2314yes—
87#13095490C2314yes—
88#13095490C2314yes—
89#13095490C2314yes—
90#13095490C2314yes—
91#13095490C2314yes—
92#13095490C2314yes—
93#13095490C2314yes—
94#13095490C2314yes—
95#13095490C2314yes—
96C22⋊Dic97C2314yes—
97C22⋊Dic97C2314yes—
98C22⋊Dic97C2314yes—
99#13095488C2314yes—
100#13095488C2314yes—
101#13095488C2314yes—
102#13095488C2314yes—
103#13095488C2314yes—
104C4×Dic97C2314yes—
105C4×Dic97C2314yes—
106C4×Dic97C2314yes—
107C24×C97C2311yescharacteristic
108C2×Q8×C97—42no—
109C2×Q8×C97C2314yes—
110C22×C4×C97—42no—
111C22×C4×C97C2311yescharacteristic
112C22×C4×C97C2311yescharacteristic
113C22×C4×C97C2311yescharacteristic
114C22×C4×C97C2311yescharacteristic
115C22×C4×C97C2311yescharacteristic
116C22×C4×C97C2311yescharacteristic
117C22×C4×C97C2314yes—
118C4⋊C4×C97C2314yes—
119C4⋊C4×C97C2314yes—
120C4⋊C4×C97C2314yes—
121C22⋊C4×C97C2314yes—
122C22⋊C4×C97C2314yes—
123C23×C97—22no—
124C23×C97—22no—
125C23×C97—22no—
126C23×C97C2412yes—
127C23×C97C2411yescharacteristic
128C22×D97—22no—
129C22×D97—22no—
130C22×D97—22no—
131C22×D97C2412yes—
132Q8×C97—44no—
133C2×C4×C97—22no—
134C2×C4×C97—22no—
135C2×C4×C97—22no—
136C2×C4×C97—22no—
137C2×C4×C97—22no—
138C2×C4×C97—22no—
139C2×C4×C97—44no—
140C2×C4×C97C2412yes—
141C2×C4×C97C2412yes—
142C2×C4×C97C2412yes—
143C2×C4×C97C2412yes—
144C2×C4×C97C2412yes—
145C2×C4×C97C2412yes—
146C22⋊D97—44no—
147C2×Dic97—22no—
148C2×Dic97—22no—
149C2×Dic97—22no—
150C2×Dic97—22no—
151C2×Dic97—22no—
152C2×Dic97—22no—
153C2×Dic97—22no—
154C2×Dic97C2412yes—
155C2×Dic97C2412yes—
156C2×Dic97C2412yes—
157C2×Dic97C2412yes—
158C2×Dic97C2412yes—
159C2×Dic97C2412yes—
160C2×Dic97C2412yes—
161C4×D97—44no—
162Dic194—44no—
163Dic194—44no—
164C22×C97—22no—
165C22×C97—22no—
166C22×C97—22no—
167C22×C97—44no—
168C22×C97H(5,𝔽2)12yes—
169C22×C972−1+412yes—
170C22×C972−1+412yes—
171C22×C97C2511yescharacteristic
172C2×D97—22no—
173C2×D97—22no—
174C2×D97—22no—
175C2×D97—44no—
176C4×C97—22no—
177C4×C97—22no—
178C4×C97—22no—
179C4×C97—22no—
180C4×C97—22no—
181C4×C97—22no—
182Dic97—22no—
183Dic97—22no—
184Dic97—22no—
185Dic97—22no—
186Dic97—22no—
187Dic97—22no—
188Dic97—22no—
189C2×C97—42no—
190C2×C97#56314yescomplemented
191C2×C97C2×H(5,𝔽2)11yescharacteristic
192C2×C97C2×2−1+411yescharacteristic
193C2×C97C2×2−1+411yescharacteristic
194D97—42no—
195C2×#563—971nomaximal, Sylow
196C97C2×#56311yescharacteristic, complemented, Sylow
197#563—9716no—
198C2×C4⋊Q8—971no—
199C2×C4⋊Q8—971no—
200C2×#126—971no—
201C2×#126—971no—
202C2×#126—971no—
203C2×#126—971no—
204C2×#125—971no—
205C2×#125—971no—
206C2×#124—971no—
207C2×#123—971no—
208C2×#123—971no—
209C2×C22⋊Q8—971no—
210C2×C22⋊Q8—971no—
211C2×C22⋊Q8—971no—
212C2×C22⋊Q8D9711yescharacteristic, complemented
213C22×Q8—971no—
214C22×Q8—971no—
215C22×Q8C2×D9711yescharacteristic
216C22×D4—971no—
217C23×C4—971no—
218C23×C4C2×D9711yescharacteristic
219C4⋊Q8—978no—
220C4⋊Q8—978no—
221#126—978no—
222#126—978no—
223#126—978no—
224#126—978no—
225#125—978no—
226#125—978no—
227#124—978no—
228#123—978no—
229#123—978no—
230C22⋊Q8—978no—
231C22⋊Q8—978no—
232C22⋊Q8—978no—
233C22⋊Q8C2×D9718yescomplemented
234C2×C4⋊C4—971no—
235C2×C4⋊C4—971no—
236C2×C4⋊C4—971no—
237C2×C4⋊C4—971no—
238C2×C4⋊C4—971no—
239C2×C4⋊C4—971no—
240C2×C4⋊C4—971no—
241C2×C4⋊C4—971no—
242C2×C4⋊C4—971no—
243C2×C4⋊C4—971no—
244C2×C4⋊C4—971no—
245C2×C4⋊C4—971no—
246C2×C4⋊C4—971no—
247C2×C4⋊C4C2×D9711yescharacteristic
248C2×C4⋊C4C2×D9711yescharacteristic
249C2×C4⋊C4C2×D9711yescharacteristic
250C2×C22⋊C4—971no—
251C2×C22⋊C4—971no—
252C2×C22⋊C4—971no—
253C2×C22⋊C4—971no—
254C2×C22⋊C4—971no—
255C2×C22⋊C4—971no—
256C2×C22⋊C4—971no—
257C2×C22⋊C4—971no—
258C2×C22⋊C4C2×D9711yescharacteristic
259C2×C22⋊C4C2×D9711yescharacteristic
260C2×C42—971no—
261C2×C42—971no—
262C2×C42—971no—
263C24—971no—
264C24C22×D9711yescharacteristic
265C2×Q8—42no—
266C2×Q8—974no—
267C2×Q8—974no—
268C2×Q8—3882no—
269C2×Q8—3882no—
270C2×Q8C22×D9714yes—
271C2×D4—974no—
272C2×D4—3882no—
273C22×C4—42no—
274C22×C4—971no—
275C22×C4—971no—
276C22×C4—971no—
277C22×C4—971no—
278C22×C4—971no—
279C22×C4—971no—
280C22×C4—971no—
281C22×C4—974no—
282C22×C4—3882no—
283C22×C4C22×D9711yescharacteristic
284C22×C4C22×D9711yescharacteristic
285C22×C4C22×D9711yescharacteristic
286C22×C4C22×D9711yescharacteristic
287C22×C4C22×D9711yescharacteristic
288C22×C4C22×D9711yescharacteristic
289C22×C4C22×D9714yes—
290C4⋊C4—974no—
291C4⋊C4—974no—
292C4⋊C4—974no—
293C4⋊C4—974no—
294C4⋊C4—974no—
295C4⋊C4—974no—
296C4⋊C4—974no—
297C4⋊C4—974no—
298C4⋊C4—974no—
299C4⋊C4—974no—
300C4⋊C4—974no—
301C4⋊C4—974no—
302C4⋊C4—974no—
303C4⋊C4C22×D9714yes—
304C4⋊C4C22×D9714yes—
305C4⋊C4C22×D9714yes—
306C22⋊C4—974no—
307C22⋊C4—974no—
308C22⋊C4—974no—
309C22⋊C4—974no—
310C22⋊C4—974no—
311C22⋊C4—974no—
312C22⋊C4—974no—
313C22⋊C4—974no—
314C22⋊C4C22×D9714yes—
315C22⋊C4C22×D9714yes—
316C42—974no—
317C42—974no—
318C42—974no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—972no—
323C23—1942no—
324C23—1942no—
325C23—1942no—
326C23C23×D9712yes—
327C23C23×D9711yescharacteristic
328Q8—44no—
329Q8—3884no—
330Q8—3884no—
331D4—3884no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—44no—
339C2×C4—972no—
340C2×C4—972no—
341C2×C4—972no—
342C2×C4—972no—
343C2×C4—972no—
344C2×C4—972no—
345C2×C4—972no—
346C2×C4—3884no—
347C2×C4—1942no—
348C2×C4—1942no—
349C2×C4—1942no—
350C2×C4—1942no—
351C2×C4—1942no—
352C2×C4—1942no—
353C2×C4—1942no—
354C2×C4C23×D9712yes—
355C2×C4C23×D9712yes—
356C2×C4C23×D9712yes—
357C2×C4C23×D9712yes—
358C2×C4C23×D9712yes—
359C2×C4C23×D9712yes—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—44no—
364C22—3884no—
365C22—1942no—
366C22—1942no—
367C22—1942no—
368C22#1569085512yes—
369C22#1569085712yes—
370C22#1569085812yes—
371C22C24×D9711yescharacteristic
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—1942no—
379C4—1942no—
380C4—1942no—
381C4—1942no—
382C4—1942no—
383C4—1942no—
384C4—1942no—
385C2—42no—
386C2—3882no—
387C2#2026148614yesdirect product factor
388C2C2×#1569085511yescharacteristic
389C2C2×#1569085711yescharacteristic
390C2C2×#1569085811yescharacteristic
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×C22⋊Q8×C97C2
socleC23×C97C24
prime radicalC22×C22⋊D97C22
hypercenterC2×C22⋊Q8D97
hypocenterC97C2×#563
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20261486C2direct (N × H)
2. C2×#15690809C2semidirect
3. C2×#15690808C2semidirect
4. C2×#15690805C2semidirect
5. C2×#15690805C2semidirect
6. C2×#15690803C2semidirect
7. C2×#15690803C2semidirect
8. C2×#15690801C2semidirect
9. C2×#15690800C2semidirect
13. C2×#15690789C2semidirect
14. C2×#15690788C2semidirect
15. C2×#15690786C2semidirect
16. C2×C22⋊Q8×C97C2semidirect
21. #15690809C22semidirect
22. #15690808C22semidirect
23. #15690805C22semidirect
24. #15690805C22semidirect
25. #15690803C22semidirect
26. #15690803C22semidirect
27. #15690801C22semidirect
28. #15690800C22semidirect
32. #15690789C22semidirect
33. #15690788C22semidirect
34. #15690786C22semidirect
61. C22⋊Q8×C97C22semidirect
190. C2×C97#563semidirect
196. C97C2×#563semidirect
212. C2×C22⋊Q8D97semidirect
233. C22⋊Q8C2×D97semidirect
387. C2#20261486direct (N × H)

Maximal subgroups

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

subgroupindex
#202614862
C2×#156908092
C2×#156908082
C2×#156908052
C2×#156908032
C2×#156908012
C2×#156908002
C2×#156907962
C2×#156907932
C2×#156907922
C2×#156907892
C2×#156907882
C2×#156907862
C2×C22⋊Q8×C972
C2×#56397