Finite Group Databasebeta

← C2×#20261344 #27155101

Subgroups — C2×#20261344 #27155101

Order 12416.

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#20261344C2116yesmaximal, direct product factor
2C2×#15690809C211yesmaximal, characteristic, complemented
3C2×#15690805C211yesmaximal, characteristic
4C2×#15690805C211yesmaximal, characteristic
5C2×#15690805C211yesmaximal, characteristic
6C2×#15690804C211yesmaximal, characteristic
7C2×#15690801C211yesmaximal, characteristic, complemented
8C2×#15690800C211yesmaximal, characteristic, complemented
9C2×#15690800C211yesmaximal, characteristic, complemented
10C2×#15690800C211yesmaximal, characteristic, complemented
11C2×#15690788C211yesmaximal, characteristic
12C2×#15690788C211yesmaximal, characteristic
13C2×C4×Dic194C211yesmaximal, characteristic, complemented
14C2×#15690775C211yesmaximal, characteristic, complemented
15C2×#15690768C211yesmaximal, characteristic
16C2×#125×C97C211yesmaximal, characteristic, complemented
17C22×Dic194C2211yescharacteristic
18C22×Dic194C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20#15690809C2218yescomplemented
21#15690805C2218yes—
22#15690805C2218yes—
23#15690805C2218yes—
24#15690804C2218yes—
25#15690801C2218yescomplemented
26#15690800C2218yescomplemented
27#15690800C2218yescomplemented
28#15690800C2218yescomplemented
29#15690788C2218yes—
30#15690788C2218yes—
31C4×Dic194C2218yescomplemented
32#15690775C2218yescomplemented
33#15690768C2218yes—
34C2×C4⋊Dic97C2211yescharacteristic
35C2×C4⋊Dic97C2211yescharacteristic
36C2×C4⋊Dic97C2211yescharacteristic
37C2×C4⋊Dic97C2211yescharacteristic
38C2×#13095490C2211yescharacteristic
39C2×#13095490C2211yescharacteristic
40C2×#13095490C2211yescharacteristic
41C2×#13095490C2211yescharacteristic
42C2×#13095490C2211yescharacteristic
43C2×#13095490C2211yescharacteristic
44C2×#13095490C2211yescharacteristic
45C2×#13095490C2211yescharacteristic
46C2×#13095490C2211yescharacteristic
47C2×#13095490C2211yescharacteristic
48C2×#13095488C2211yescharacteristic
49C2×#13095488C2211yescharacteristic
50C2×#13095488C2211yescharacteristic
51C2×#13095488C2211yescharacteristic
52C2×#13095488C2211yescharacteristic
53C2×#13095488C2211yescharacteristic
54C2×C4×Dic97C2211yescharacteristic
55C2×C4×Dic97C2211yescharacteristic
56C2×C4×Dic97C2211yescharacteristic
57C2×C4×Dic97C2211yescharacteristic
58C2×C4×Dic97C2211yescharacteristic
59#125×C97C2218yescomplemented
60C2×C4⋊C4×C97C2211yescharacteristic
61C2×C4⋊C4×C97C2211yescharacteristic
62C2×C4⋊C4×C97C2211yescharacteristic
63C2×C4⋊C4×C97C2211yescharacteristic
64C2×C4⋊C4×C97C2211yescharacteristic
65C2×C4⋊C4×C97C2211yescharacteristic
66C2×C42×C97C2211yescharacteristic
67C23×D97C2311yescharacteristic
68C2×Dic194—24no—
69C2×Dic194—24no—
70C2×Dic194C2314yes—
71C2×Dic194C2314yes—
72C22×Dic97C2311yescharacteristic
73C22×Dic97C2311yescharacteristic
74C22×Dic97C2311yescharacteristic
75C22×Dic97C2311yescharacteristic
76C22×Dic97C2311yescharacteristic
77C22×Dic97C2311yescharacteristic
78C22×Dic97C2311yescharacteristic
79C2×C4×D97—24no—
80C2×C4×D97C2314yes—
81C4⋊Dic97C2314yes—
82C4⋊Dic97C2314yes—
83C4⋊Dic97C2314yes—
84C4⋊Dic97C2314yes—
85#13095490C2314yes—
86#13095490C2314yes—
87#13095490C2314yes—
88#13095490C2314yes—
89#13095490C2314yes—
90#13095490C2314yes—
91#13095490C2314yes—
92#13095490C2314yes—
93#13095490C2314yes—
94#13095490C2314yes—
95#13095488C2314yes—
96#13095488C2314yes—
97#13095488C2314yes—
98#13095488C2314yes—
99#13095488C2314yes—
100#13095488C2314yes—
101C4×Dic97C2314yes—
102C4×Dic97C2314yes—
103C4×Dic97C2314yes—
104C4×Dic97C2314yes—
105C4×Dic97C2314yes—
106C22×C4×C97C2311yescharacteristic
107C22×C4×C97C2311yescharacteristic
108C22×C4×C97C2311yescharacteristic
109C22×C4×C97C2311yescharacteristic
110C22×C4×C97C2311yescharacteristic
111C22×C4×C97C2311yescharacteristic
112C22×C4×C97C2311yescharacteristic
113C4⋊C4×C97C2314yes—
114C4⋊C4×C97C2314yes—
115C4⋊C4×C97C2314yes—
116C4⋊C4×C97C2314yes—
117C4⋊C4×C97C2314yes—
118C4⋊C4×C97C2314yes—
119C42×C97C2314yes—
120C23×C97C2411yescharacteristic
121C22×D97—22no—
122C22×D97—22no—
123C22×D97—22no—
124C22×D97C2412yes—
125C2×C4×C97—22no—
126C2×C4×C97—22no—
127C2×C4×C97—22no—
128C2×C4×C97—22no—
129C2×C4×C97—22no—
130C2×C4×C97—22no—
131C2×C4×C97—22no—
132C2×C4×C97C2412yes—
133C2×C4×C97C2412yes—
134C2×C4×C97C2412yes—
135C2×C4×C97C2412yes—
136C2×C4×C97C2412yes—
137C2×C4×C97C2412yes—
138C2×C4×C97C2412yes—
139C2×Dic97—22no—
140C2×Dic97—22no—
141C2×Dic97—22no—
142C2×Dic97—22no—
143C2×Dic97—22no—
144C2×Dic97—22no—
145C2×Dic97#4114yes—
146C2×Dic97C2412yes—
147C2×Dic97C2412yes—
148C2×Dic97C2412yes—
149C2×Dic97C2412yes—
150C2×Dic97C2412yes—
151C2×Dic97C2412yes—
152C2×Dic97C2412yes—
153C4×D97—28no—
154Dic194—28no—
155Dic194—28no—
156C22×C97C2×#4112yes—
157C22×C972−1+412yes—
158C22×C972−1+412yes—
159C22×C97C2511yescharacteristic
160C2×D97—22no—
161C2×D97—22no—
162C2×D97—22no—
163C2×D97—44no—
164C4×C97—22no—
165C4×C97—22no—
166C4×C97—22no—
167C4×C97—22no—
168C4×C97—22no—
169C4×C97—22no—
170C4×C97—22no—
171Dic97—22no—
172Dic97—22no—
173Dic97—22no—
174Dic97—22no—
175Dic97—22no—
176Dic97—22no—
177Dic97C2×#4114yes—
178C2×C97#54114yescomplemented
179C2×C97C22×#4111yescharacteristic
180C2×C97C2×2−1+411yescharacteristic
181C2×C97C2×2−1+411yescharacteristic
182D97—42no—
183C2×#541—971nomaximal, Sylow
184C97C2×#54111yescharacteristic, complemented, Sylow
185#541—9716no—
186C2×C4⋊Q8—971no—
187C2×#126—971no—
188C2×#126—971no—
189C2×#126—971no—
190C2×#126—971no—
191C2×#125—971no—
192C2×#125—971no—
193C2×#125—971no—
194C2×#125—971no—
195C2×#125D9711yescharacteristic, complemented
196C2×C22⋊Q8—971no—
197C2×C22⋊Q8—971no—
198C2×C4×Q8—971no—
199C2×C4×Q8—971no—
200C2×#117—971no—
201C22×Q8—971no—
202C22×Q8—971no—
203C23×C4—971no—
204C4⋊Q8—978no—
205#126—978no—
206#126—978no—
207#126—978no—
208#126—978no—
209#125—978no—
210#125—978no—
211#125—978no—
212#125—978no—
213#125C2×D9718yescomplemented
214C22⋊Q8—978no—
215C22⋊Q8—978no—
216C4×Q8—978no—
217C4×Q8—978no—
218#117—978no—
219C2×C4⋊C4—971no—
220C2×C4⋊C4—971no—
221C2×C4⋊C4—971no—
222C2×C4⋊C4—971no—
223C2×C4⋊C4—971no—
224C2×C4⋊C4—971no—
225C2×C4⋊C4—971no—
226C2×C4⋊C4—971no—
227C2×C4⋊C4—971no—
228C2×C4⋊C4—971no—
229C2×C4⋊C4—971no—
230C2×C4⋊C4—971no—
231C2×C4⋊C4—971no—
232C2×C4⋊C4—971no—
233C2×C4⋊C4C2×D9711yescharacteristic
234C2×C4⋊C4C2×D9711yescharacteristic
235C2×C4⋊C4C2×D9711yescharacteristic
236C2×C4⋊C4C2×D9711yescharacteristic
237C2×C4⋊C4C2×D9711yescharacteristic
238C2×C4⋊C4C2×D9711yescharacteristic
239C2×C22⋊C4—971no—
240C2×C22⋊C4—971no—
241C2×C22⋊C4—971no—
242C2×C22⋊C4—971no—
243C2×C22⋊C4—971no—
244C2×C22⋊C4—971no—
245C2×C42—971no—
246C2×C42—971no—
247C2×C42—971no—
248C2×C42—971no—
249C2×C42—971no—
250C2×C42C2×D9711yescharacteristic
251C24—971no—
252C2×Q8—974no—
253C2×Q8—974no—
254C2×Q8—1944no—
255C2×Q8—1944no—
256C22×C4—971no—
257C22×C4—971no—
258C22×C4—971no—
259C22×C4—971no—
260C22×C4—971no—
261C22×C4—971no—
262C22×C4—971no—
263C22×C4—974no—
264C22×C4—1944no—
265C22×C4C22×D9711yescharacteristic
266C22×C4C22×D9711yescharacteristic
267C22×C4C22×D9711yescharacteristic
268C22×C4C22×D9711yescharacteristic
269C22×C4C22×D9711yescharacteristic
270C22×C4C22×D9711yescharacteristic
271C22×C4C22×D9711yescharacteristic
272C4⋊C4—974no—
273C4⋊C4—974no—
274C4⋊C4—974no—
275C4⋊C4—974no—
276C4⋊C4—974no—
277C4⋊C4—974no—
278C4⋊C4—974no—
279C4⋊C4—974no—
280C4⋊C4—974no—
281C4⋊C4—974no—
282C4⋊C4—974no—
283C4⋊C4—974no—
284C4⋊C4—974no—
285C4⋊C4—974no—
286C4⋊C4C22×D9714yes—
287C4⋊C4C22×D9714yes—
288C4⋊C4C22×D9714yes—
289C4⋊C4C22×D9714yes—
290C4⋊C4C22×D9714yes—
291C4⋊C4C22×D9714yes—
292C22⋊C4—974no—
293C22⋊C4—974no—
294C22⋊C4—974no—
295C22⋊C4—974no—
296C22⋊C4—974no—
297C22⋊C4—974no—
298C42—974no—
299C42—974no—
300C42—974no—
301C42—974no—
302C42—974no—
303C42C22×D9714yes—
304C23—972no—
305C23—1942no—
306C23—1942no—
307C23—1942no—
308C23C23×D9711yescharacteristic
309Q8—1948no—
310Q8—1948no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—974no—
319C2×C4—972no—
320C2×C4—972no—
321C2×C4—972no—
322C2×C4—972no—
323C2×C4—972no—
324C2×C4—972no—
325C2×C4—972no—
326C2×C4—1948no—
327C2×C4—1942no—
328C2×C4—1942no—
329C2×C4—1942no—
330C2×C4—1942no—
331C2×C4—1942no—
332C2×C4—1942no—
333C2×C4C23×D9712yes—
334C2×C4C23×D9712yes—
335C2×C4C23×D9712yes—
336C2×C4C23×D9712yes—
337C2×C4C23×D9712yes—
338C2×C4C23×D9712yes—
339C2×C4C23×D9712yes—
340C22—3884no—
341C22—1942no—
342C22—1942no—
343C22—1942no—
344C22#41×D9712yes—
345C22#1569085712yes—
346C22#1569085812yes—
347C22C24×D9711yescharacteristic
348C4—22no—
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—974no—
356C4—1942no—
357C4—1942no—
358C4—1942no—
359C4—1942no—
360C4—1942no—
361C4—1942no—
362C2—3882no—
363C2#2026134414yesdirect product factor
364C2C2×#41×D9711yescharacteristic
365C2C2×#1569085711yescharacteristic
366C2C2×#1569085811yescharacteristic
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×#125×C97C2
socleC23×C97C24
prime radicalC23×D97C23
hypercenterC2×#125D97
hypocenterC97C2×#541
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. #20261344C2direct (N × H)
2. C2×#15690809C2semidirect
7. C2×#15690801C2semidirect
8. C2×#15690800C2semidirect
9. C2×#15690800C2semidirect
10. C2×#15690800C2semidirect
13. C2×C4×Dic194C2semidirect
14. C2×#15690775C2semidirect
16. C2×#125×C97C2semidirect
20. #15690809C22semidirect
25. #15690801C22semidirect
26. #15690800C22semidirect
27. #15690800C22semidirect
28. #15690800C22semidirect
31. C4×Dic194C22semidirect
32. #15690775C22semidirect
59. #125×C97C22semidirect
178. C2×C97#541semidirect
184. C97C2×#541semidirect
195. C2×#125D97semidirect
213. #125C2×D97semidirect
363. C2#20261344direct (N × H)

Maximal subgroups

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

subgroupindex
#202613442
C2×#156908092
C2×#156908052
C2×#156908042
C2×#156908012
C2×#156908002
C2×#156907882
C2×C4×Dic1942
C2×#156907752
C2×#156907682
C2×#125×C972
C2×#54197