Finite Group Databasebeta

← C2×#20261331 #27155086

Subgroups — C2×#20261331 #27155086

Order 12416.

1144 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261331C2116yesmaximal, direct product factor
2C2×#15690805C211yesmaximal, characteristic, complemented
3C2×#15690803C211yesmaximal, characteristic, complemented
4C2×#15690800C211yesmaximal, characteristic, complemented
5C2×#15690800C211yesmaximal, characteristic, complemented
6C2×#15690795C211yesmaximal, characteristic, complemented
7C2×#15690793C211yesmaximal, characteristic, complemented
8C2×#15690792C211yesmaximal, characteristic, complemented
9C2×#15690792C211yesmaximal, characteristic, complemented
10C2×#15690786C211yesmaximal, characteristic, complemented
11C2×#15690786C211yesmaximal, characteristic, complemented
12C2×#15690783C211yesmaximal, characteristic
13C2×#15690772C211yesmaximal, characteristic, complemented
14C2×C4×C22⋊D97C211yesmaximal, characteristic, complemented
15C2×C4⋊C4×D97C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8×C97C211yesmaximal, characteristic, complemented
17C22×D388C2211yescharacteristic
18C22×C22⋊D97C2211yescharacteristic
19C22×C22⋊D97C2211yescharacteristic
20C22×C4×D97C2211yescharacteristic
21C22×C4×D97C2211yescharacteristic
22C22×C4×D97C2211yescharacteristic
23C22×C4×D97C2211yescharacteristic
24#15690805C2218yescomplemented
25#15690803C2218yescomplemented
26#15690800C2218yescomplemented
27#15690800C2218yescomplemented
28#15690795C2218yescomplemented
29#15690793C2218yescomplemented
30#15690792C2218yescomplemented
31#15690792C2218yescomplemented
32#15690786C2218yescomplemented
33#15690786C2218yescomplemented
34#15690783C2218yes—
35#15690772C2218yescomplemented
36C4×C22⋊D97C2218yescomplemented
37C2×C4⋊Dic97C2211yescharacteristic
38C2×C4⋊Dic97C2211yescharacteristic
39C2×C4⋊Dic97C2211yescharacteristic
40C2×#13095490C2211yescharacteristic
41C2×#13095490C2211yescharacteristic
42C2×#13095490C2211yescharacteristic
43C2×#13095490C2211yescharacteristic
44C2×#13095490C2211yescharacteristic
45C2×#13095490C2211yescharacteristic
46C2×#13095490C2211yescharacteristic
47C2×#13095490C2211yescharacteristic
48C4⋊C4×D97C2218yescomplemented
49C2×C22⋊Dic97C2211yescharacteristic
50C2×C22⋊Dic97C2211yescharacteristic
51C2×#13095488C2211yescharacteristic
52C2×#13095488C2211yescharacteristic
53C2×#13095488C2211yescharacteristic
54C2×#13095488C2211yescharacteristic
55C2×#13095488C2211yescharacteristic
56C2×#13095488C2211yescharacteristic
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
68C23×D97C2311yescharacteristic
69C2×D388—24no—
70C2×D388C2314yes—
71C2×C22⋊D97—24no—
72C2×C22⋊D97—42no—
73C2×C22⋊D97C2314yes—
74C2×C22⋊D97C2314yes—
75C22×Dic97C2311yescharacteristic
76C22×Dic97C2311yescharacteristic
77C22×Dic97C2311yescharacteristic
78C22×Dic97C2311yescharacteristic
79C22×Dic97C2311yescharacteristic
80C22×Dic97C2311yescharacteristic
81C2×C4×D97—24no—
82C2×C4×D97—42no—
83C2×C4×D97—42no—
84C2×C4×D97—42no—
85C2×C4×D97C2314yes—
86C2×C4×D97C2314yes—
87C2×C4×D97C2314yes—
88C2×C4×D97C2314yes—
89C4⋊Dic97C2314yes—
90C4⋊Dic97C2314yes—
91C4⋊Dic97C2314yes—
92#13095490C2314yes—
93#13095490C2314yes—
94#13095490C2314yes—
95#13095490C2314yes—
96#13095490C2314yes—
97#13095490C2314yes—
98#13095490C2314yes—
99#13095490C2314yes—
100C22⋊Dic97C2314yes—
101C22⋊Dic97C2314yes—
102#13095488C2314yes—
103#13095488C2314yes—
104#13095488C2314yes—
105#13095488C2314yes—
106#13095488C2314yes—
107#13095488C2314yes—
108C4×Dic97C2314yes—
109C4×Dic97C2314yes—
110C24×C97C2311yescharacteristic
111C2×Q8×C97—42no—
112C2×Q8×C97C2314yes—
113C22×C4×C97—42no—
114C22×C4×C97C2311yescharacteristic
115C22×C4×C97C2311yescharacteristic
116C22×C4×C97C2311yescharacteristic
117C22×C4×C97C2311yescharacteristic
118C22×C4×C97C2311yescharacteristic
119C22×C4×C97C2311yescharacteristic
120C22×C4×C97C2314yes—
121C4⋊C4×C97C2314yes—
122C4⋊C4×C97C2314yes—
123C4⋊C4×C97C2314yes—
124C22⋊C4×C97C2314yes—
125C22⋊C4×C97C2314yes—
126C23×C97—22no—
127C23×C97—22no—
128C23×C97—22no—
129C23×C97C2412yes—
130C23×C97C2411yescharacteristic
131C22×D97—22no—
132C22×D97—22no—
133C22×D97—22no—
134C22×D97—22no—
135C22×D97—22no—
136C22×D97#4114yes—
137C22×D97C2412yes—
138C22×D97C2412yes—
139Q8×C97—44no—
140C2×C4×C97—22no—
141C2×C4×C97—22no—
142C2×C4×C97—22no—
143C2×C4×C97—22no—
144C2×C4×C97—22no—
145C2×C4×C97—22no—
146C2×C4×C97—44no—
147C2×C4×C97C2412yes—
148C2×C4×C97C2412yes—
149C2×C4×C97C2412yes—
150C2×C4×C97C2412yes—
151C2×C4×C97C2412yes—
152C2×C4×C97C2412yes—
153C22⋊D97—28no—
154C22⋊D97—44no—
155C2×Dic97—22no—
156C2×Dic97—22no—
157C2×Dic97—22no—
158C2×Dic97—22no—
159C2×Dic97—22no—
160C2×Dic97—22no—
161C2×Dic97C2412yes—
162C2×Dic97C2412yes—
163C2×Dic97C2412yes—
164C2×Dic97C2412yes—
165C2×Dic97C2412yes—
166C2×Dic97C2412yes—
167D388—28no—
168C4×D97—28no—
169C4×D97—44no—
170C4×D97—44no—
171C4×D97—44no—
172C22×C97—22no—
173C22×C97—22no—
174C22×C97—22no—
175C22×C97—44no—
176C22×C97C2×#4112yes—
177C22×C97H(5,𝔽2)12yes—
178C22×C972−1+412yes—
179C22×C97C2511yescharacteristic
180C2×D97—28no—
181C2×D97—22no—
182C2×D97—22no—
183C2×D97—22no—
184C2×D97—22no—
185C2×D97—22no—
186C2×D97—44no—
187C2×D97C2×#4114yes—
188C4×C97—22no—
189C4×C97—22no—
190C4×C97—22no—
191C4×C97—22no—
192C4×C97—22no—
193C4×C97—22no—
194Dic97—22no—
195Dic97—22no—
196Dic97—22no—
197Dic97—22no—
198Dic97—22no—
199Dic97—22no—
200C2×C97—42no—
201C2×C97#53914yescomplemented
202C2×C97C22×#4111yescharacteristic
203C2×C97C2×H(5,𝔽2)11yescharacteristic
204C2×C97C2×2−1+411yescharacteristic
205D97—24no—
206D97—42no—
207C2×#539—971nomaximal, Sylow
208C97C2×#53911yescharacteristic, complemented, Sylow
209#539—9716no—
210C2×#126—971no—
211C2×#126—971no—
212C2×#125—971no—
213C2×#125—971no—
214C2×#123—971no—
215C2×#123—971no—
216C2×#123—971no—
217C2×#123—971no—
218C2×C22⋊Q8—971no—
219C2×C22⋊Q8—971no—
220C2×C22⋊Q8D9711yescharacteristic, complemented
221C2×#121—971no—
222C2×C4×D4—971no—
223C2×C4×D4—971no—
224C22×C4⋊C4—971no—
225C22×Q8C2×D9711yescharacteristic
226C22×D4—971no—
227C22×D4—971no—
228C22×D4—971no—
229C23×C4—971no—
230C23×C4—971no—
231C23×C4—971no—
232C23×C4—971no—
233C23×C4C2×D9711yescharacteristic
234#126—978no—
235#126—978no—
236#125—978no—
237#125—978no—
238#123—978no—
239#123—978no—
240#123—978no—
241#123—978no—
242C22⋊Q8—978no—
243C22⋊Q8—978no—
244C22⋊Q8C2×D9718yescomplemented
245#121—978no—
246C4×D4—978no—
247C4×D4—978no—
248C2×C4⋊C4—971no—
249C2×C4⋊C4—971no—
250C2×C4⋊C4—971no—
251C2×C4⋊C4—971no—
252C2×C4⋊C4—971no—
253C2×C4⋊C4—971no—
254C2×C4⋊C4—971no—
255C2×C4⋊C4—971no—
256C2×C4⋊C4—971no—
257C2×C4⋊C4—971no—
258C2×C4⋊C4—971no—
259C2×C4⋊C4—978no—
260C2×C4⋊C4C2×D9711yescharacteristic
261C2×C4⋊C4C2×D9711yescharacteristic
262C2×C4⋊C4C2×D9711yescharacteristic
263C2×C22⋊C4—971no—
264C2×C22⋊C4—971no—
265C2×C22⋊C4—971no—
266C2×C22⋊C4—971no—
267C2×C22⋊C4—971no—
268C2×C22⋊C4—971no—
269C2×C22⋊C4—971no—
270C2×C22⋊C4—971no—
271C2×C22⋊C4C2×D9711yescharacteristic
272C2×C22⋊C4C2×D9711yescharacteristic
273C2×C42—971no—
274C2×C42—971no—
275C24—971no—
276C24—971no—
277C24C22×D9711yescharacteristic
278C2×Q8—42no—
279C2×Q8C22×D9714yes—
280C2×D4—974no—
281C2×D4—974no—
282C2×D4—974no—
283C2×D4—3882no—
284C2×D4—1944no—
285C2×D4—1944no—
286C22×C4—42no—
287C22×C4—971no—
288C22×C4—971no—
289C22×C4—971no—
290C22×C4—971no—
291C22×C4—971no—
292C22×C4—971no—
293C22×C4—974no—
294C22×C4—974no—
295C22×C4—974no—
296C22×C4—974no—
297C22×C4—3882no—
298C22×C4—3882no—
299C22×C4—3882no—
300C22×C4—1944no—
301C22×C4C22×D9711yescharacteristic
302C22×C4C22×D9711yescharacteristic
303C22×C4C22×D9711yescharacteristic
304C22×C4C22×D9711yescharacteristic
305C22×C4C22×D9711yescharacteristic
306C22×C4C22×D9711yescharacteristic
307C22×C4C22×D9714yes—
308C4⋊C4—974no—
309C4⋊C4—974no—
310C4⋊C4—974no—
311C4⋊C4—974no—
312C4⋊C4—974no—
313C4⋊C4—974no—
314C4⋊C4—974no—
315C4⋊C4—974no—
316C4⋊C4—974no—
317C4⋊C4—974no—
318C4⋊C4—974no—
319C4⋊C4C22×D9714yes—
320C4⋊C4C22×D9714yes—
321C4⋊C4C22×D9714yes—
322C22⋊C4—974no—
323C22⋊C4—974no—
324C22⋊C4—974no—
325C22⋊C4—974no—
326C22⋊C4—974no—
327C22⋊C4—974no—
328C22⋊C4—974no—
329C22⋊C4—974no—
330C22⋊C4C22×D9714yes—
331C22⋊C4C22×D9714yes—
332C42—974no—
333C42—974no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—974no—
338C23—972no—
339C23—972no—
340C23—1942no—
341C23—1942no—
342C23—1942no—
343C23—1942no—
344C23—1942no—
345C23C23×D9712yes—
346C23C23×D9711yescharacteristic
347Q8—44no—
348D4—3884no—
349D4—1948no—
350D4—1948no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—44no—
358C2×C4—972no—
359C2×C4—972no—
360C2×C4—972no—
361C2×C4—972no—
362C2×C4—972no—
363C2×C4—972no—
364C2×C4—3884no—
365C2×C4—3884no—
366C2×C4—3884no—
367C2×C4—1948no—
368C2×C4—1942no—
369C2×C4—1942no—
370C2×C4—1942no—
371C2×C4—1942no—
372C2×C4—1942no—
373C2×C4—1942no—
374C2×C4C23×D9712yes—
375C2×C4C23×D9712yes—
376C2×C4C23×D9712yes—
377C2×C4C23×D9712yes—
378C2×C4C23×D9712yes—
379C2×C4C23×D9712yes—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—44no—
384C22—974no—
385C22—3884no—
386C22—1942no—
387C22—1942no—
388C22—1942no—
389C22—1942no—
390C22—1942no—
391C22—1948no—
392C22#41×D9712yes—
393C22#1569085512yes—
394C22#1569085812yes—
395C22C24×D9711yescharacteristic
396C4—22no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—1942no—
403C4—1942no—
404C4—1942no—
405C4—1942no—
406C4—1942no—
407C4—1942no—
408C2—42no—
409C2—3882no—
410C2—1944no—
411C2#2026133114yesdirect product factor
412C2C2×#41×D9711yescharacteristic
413C2C2×#1569085511yescharacteristic
414C2C2×#1569085811yescharacteristic
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×C22⋊Q8×C97C2
socleC23×C97C24
prime radicalC2×#15690783C2
hypercenterC2×C22⋊Q8D97
hypocenterC97C2×#539
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20261331C2direct (N × H)
2. C2×#15690805C2semidirect
3. C2×#15690803C2semidirect
4. C2×#15690800C2semidirect
5. C2×#15690800C2semidirect
6. C2×#15690795C2semidirect
7. C2×#15690793C2semidirect
8. C2×#15690792C2semidirect
9. C2×#15690792C2semidirect
10. C2×#15690786C2semidirect
11. C2×#15690786C2semidirect
13. C2×#15690772C2semidirect
14. C2×C4×C22⋊D97C2semidirect
15. C2×C4⋊C4×D97C2semidirect
16. C2×C22⋊Q8×C97C2semidirect
24. #15690805C22semidirect
25. #15690803C22semidirect
26. #15690800C22semidirect
27. #15690800C22semidirect
28. #15690795C22semidirect
29. #15690793C22semidirect
30. #15690792C22semidirect
31. #15690792C22semidirect
32. #15690786C22semidirect
33. #15690786C22semidirect
35. #15690772C22semidirect
36. C4×C22⋊D97C22semidirect
48. C4⋊C4×D97C22semidirect
61. C22⋊Q8×C97C22semidirect
201. C2×C97#539semidirect
208. C97C2×#539semidirect
220. C2×C22⋊Q8D97semidirect
244. C22⋊Q8C2×D97semidirect
411. C2#20261331direct (N × H)

Maximal subgroups

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

subgroupindex
#202613312
C2×#156908052
C2×#156908032
C2×#156908002
C2×#156907952
C2×#156907932
C2×#156907922
C2×#156907862
C2×#156907832
C2×#156907722
C2×C4×C22⋊D972
C2×C4⋊C4×D972
C2×C22⋊Q8×C972
C2×#53997