Finite Group Databasebeta

← C2×#20261317 #27155071

Subgroups — C2×#20261317 #27155071

Order 12416.

1336 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261317C2116yesmaximal, direct product factor
2C2×#15690805C211yesmaximal, characteristic, complemented
3C2×#15690803C211yesmaximal, characteristic, complemented
4C2×#15690797C211yesmaximal, characteristic, complemented
5C2×#15690796C211yesmaximal, characteristic, complemented
6C2×#15690793C211yesmaximal, characteristic, complemented
7C2×#15690791C211yesmaximal, characteristic, complemented
8C2×#15690788C211yesmaximal, characteristic, complemented
9C2×#15690782C211yesmaximal, characteristic, complemented
10C2×#15690781C211yesmaximal, characteristic, complemented
11C2×#15690778C211yesmaximal, characteristic, complemented
12C2×#15690778C211yesmaximal, characteristic, complemented
13C2×C4×C22⋊D97C211yesmaximal, characteristic, complemented
14C2×#15690770C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×D97C211yesmaximal, characteristic, complemented
16C2×#121×C97C211yesmaximal, characteristic, complemented
17C24×D97C2211yescharacteristic
18C22×Dic194C2211yescharacteristic, complemented
19C22×C22⋊D97C2211yescharacteristic
20C22×C22⋊D97C2211yescharacteristic
21C22×C22⋊D97C2211yescharacteristic
22C22×C22⋊D97C2211yescharacteristic
23C23×Dic97C2211yescharacteristic, complemented
24C22×C4×D97C2211yescharacteristic
25C22×C4×D97C2211yescharacteristic
26#15690805C2218yescomplemented
27#15690803C2218yescomplemented
28#15690797C2218yescomplemented
29#15690796C2218yescomplemented
30#15690793C2218yescomplemented
31#15690791C2218yescomplemented
32#15690788C2218yescomplemented
33#15690782C2218yescomplemented
34#15690781C2218yescomplemented
35#15690778C2218yescomplemented
36#15690778C2218yescomplemented
37C4×C22⋊D97C2218yescomplemented
38#15690770C2218yescomplemented
39C2×C4⋊Dic97C2211yescharacteristic, complemented
40C2×#13095490C2211yescharacteristic, complemented
41C2×#13095490C2211yescharacteristic, complemented
42C2×#13095490C2211yescharacteristic, complemented
43C2×#13095490C2211yescharacteristic, complemented
44C2×C22⋊Dic97C2211yescharacteristic
45C2×C22⋊Dic97C2211yescharacteristic
46C2×C22⋊Dic97C2211yescharacteristic
47C2×C22⋊Dic97C2211yescharacteristic, complemented
48C2×C22⋊Dic97C2211yescharacteristic, complemented
49C2×C22⋊Dic97C2211yescharacteristic, complemented
50C2×#13095488C2211yescharacteristic
51C2×#13095488C2211yescharacteristic
52C2×#13095488C2211yescharacteristic
53C2×#13095488C2211yescharacteristic
54C2×#13095488C2211yescharacteristic
55C2×#13095488C2211yescharacteristic
56C22⋊C4×D97C2218yescomplemented
57C2×C4×Dic97C2211yescharacteristic, complemented
58C2×C4×Dic97C2211yescharacteristic, complemented
59C22×D4×C97C2211yescharacteristic
60C22×D4×C97C2211yescharacteristic
61C22×D4×C97C2211yescharacteristic, complemented
62C23×C4×C97C2211yescharacteristic, complemented
63#121×C97C2218yescomplemented
64C2×C4⋊C4×C97C2211yescharacteristic, complemented
65C2×C22⋊C4×C97C2211yescharacteristic
66C2×C22⋊C4×C97C2211yescharacteristic, complemented
67C23×D97—24no—
68C23×D97—42no—
69C23×D97—42no—
70C23×D97C2311yescharacteristic
71C23×D97C2311yescharacteristic
72C23×D97C2314yes—
73C2×Dic194—42no—
74C2×Dic194C2314yescomplemented
75C2×C22⋊D97—24no—
76C2×C22⋊D97—24no—
77C2×C22⋊D97—42no—
78C2×C22⋊D97—42no—
79C2×C22⋊D97C2314yes—
80C2×C22⋊D97C2314yes—
81C2×C22⋊D97C2314yes—
82C2×C22⋊D97C2314yes—
83C22×Dic97—42no—
84C22×Dic97C2311yescharacteristic
85C22×Dic97C2311yescharacteristic
86C22×Dic97C2311yescharacteristic
87C22×Dic97C2311yescharacteristic
88C22×Dic97C2311yescharacteristic
89C22×Dic97C2311yescharacteristic
90C22×Dic97C2314yescomplemented
91C2×C4×D97—42no—
92C2×C4×D97—42no—
93C2×C4×D97C2314yes—
94C2×C4×D97C2314yes—
95C4⋊Dic97C2314yescomplemented
96#13095490C2314yescomplemented
97#13095490C2314yescomplemented
98#13095490C2314yescomplemented
99#13095490C2314yescomplemented
100C22⋊Dic97C2314yes—
101C22⋊Dic97C2314yes—
102C22⋊Dic97C2314yes—
103C22⋊Dic97C2314yescomplemented
104C22⋊Dic97C2314yescomplemented
105C22⋊Dic97C2314yescomplemented
106#13095488C2314yes—
107#13095488C2314yes—
108#13095488C2314yes—
109#13095488C2314yes—
110#13095488C2314yes—
111#13095488C2314yes—
112C4×Dic97C2314yescomplemented
113C4×Dic97C2314yescomplemented
114C24×C97C2311yescharacteristic
115C24×C97C2311yescharacteristic
116C24×C97C2311yescharacteristic
117C2×D4×C97—42no—
118C2×D4×C97—42no—
119C2×D4×C97—42no—
120C2×D4×C97C2314yes—
121C2×D4×C97C2314yes—
122C2×D4×C97C2314yescomplemented
123C22×C4×C97—42no—
124C22×C4×C97C2311yescharacteristic
125C22×C4×C97C2311yescharacteristic
126C22×C4×C97C2311yescharacteristic
127C22×C4×C97C2311yescharacteristic
128C22×C4×C97C2314yescomplemented
129C4⋊C4×C97C2314yescomplemented
130C22⋊C4×C97C2314yes—
131C22⋊C4×C97C2314yescomplemented
132C23×C97—22no—
133C23×C97—22no—
134C23×C97—22no—
135C23×C97—22no—
136C23×C97—22no—
137C23×C97—22no—
138C23×C97—22no—
139C23×C97—22no—
140C23×C97—22no—
141C23×C97C2412yes—
142C23×C97C2412yes—
143C23×C97C2412yes—
144C23×C97C2411yescharacteristic
145C22×D97—22no—
146C22×D97—22no—
147C22×D97—22no—
148C22×D97—22no—
149C22×D97—22no—
150C22×D97—28no—
151C22×D97—44no—
152C22×D97—44no—
153C22×D97—416no—
154C22×D97#4114yes—
155C22×D97C2412yes—
156C22×D97C2412yes—
157D4×C97—44no—
158D4×C97—44no—
159D4×C97—44no—
160C2×C4×C97—22no—
161C2×C4×C97—22no—
162C2×C4×C97—22no—
163C2×C4×C97—22no—
164C2×C4×C97—44no—
165C2×C4×C97C2412yes—
166C2×C4×C97C2412yes—
167C2×C4×C97C2412yes—
168C2×C4×C97C2412yes—
169C22⋊D97—28no—
170C22⋊D97—28no—
171C22⋊D97—44no—
172C22⋊D97—44no—
173C2×Dic97—22no—
174C2×Dic97—22no—
175C2×Dic97—22no—
176C2×Dic97—22no—
177C2×Dic97—22no—
178C2×Dic97—22no—
179C2×Dic97—44no—
180C2×Dic97C2412yes—
181C2×Dic97C2412yes—
182C2×Dic97C2412yes—
183C2×Dic97C2412yes—
184C2×Dic97C2412yes—
185C2×Dic97C2412yes—
186C4×D97—44no—
187C4×D97—44no—
188Dic194—44no—
189C22×C97—22no—
190C22×C97—22no—
191C22×C97—22no—
192C22×C97—22no—
193C22×C97—22no—
194C22×C97—22no—
195C22×C97—22no—
196C22×C97—22no—
197C22×C97—22no—
198C22×C97—44no—
199C22×C97—44no—
200C22×C97—44no—
201C22×C97C2×#4112yes—
202C22×C97H(5,𝔽2)12yes—
203C22×C97H(5,𝔽2)12yes—
204C22×C97C2511yescharacteristic
205C2×D97—28no—
206C2×D97—22no—
207C2×D97—22no—
208C2×D97—22no—
209C2×D97—22no—
210C2×D97—22no—
211C2×D97—416no—
212C2×D97—44no—
213C2×D97C2×#4114yes—
214C4×C97—22no—
215C4×C97—22no—
216C4×C97—22no—
217C4×C97—22no—
218Dic97—22no—
219Dic97—22no—
220Dic97—22no—
221Dic97—22no—
222Dic97—22no—
223Dic97—22no—
224C2×C97—42no—
225C2×C97—42no—
226C2×C97—42no—
227C2×C97#53814yescomplemented
228C2×C97C22×#4111yescharacteristic
229C2×C97C2×H(5,𝔽2)11yescharacteristic
230C2×C97C2×H(5,𝔽2)11yescharacteristic
231D97—24no—
232D97—42no—
233C2×#538—971nomaximal, Sylow
234C97C2×#53811yescharacteristic, complemented, Sylow
235#538—9716no—
236C2×#126—971no—
237C2×#126—971no—
238C2×#124—971no—
239C2×#124—971no—
240C2×#123—971no—
241C2×#123—971no—
242C2×C22⋊Q8—971no—
243C2×#121—971no—
244C2×#121—971no—
245C2×#121D9711yescharacteristic, complemented
246C2×#120—971no—
247C2×#120—971no—
248C2×C4×D4—971no—
249C2×C4×D4—971no—
250C22×C22⋊C4—971no—
251C25—971no—
252C22×Q8—971no—
253C22×D4—971no—
254C22×D4—971no—
255C22×D4—971no—
256C22×D4—971no—
257C22×D4C2×D9711yescharacteristic
258C22×D4C2×D9711yescharacteristic
259C22×D4C2×D9711yescharacteristic, complemented
260C23×C4—971no—
261C23×C4—971no—
262C23×C4—971no—
263C23×C4C2×D9711yescharacteristic, complemented
264#126—978no—
265#126—978no—
266#124—978no—
267#124—978no—
268#123—978no—
269#123—978no—
270C22⋊Q8—978no—
271#121—978no—
272#121—978no—
273#121C2×D9718yescomplemented
274#120—978no—
275#120—978no—
276C4×D4—978no—
277C4×D4—978no—
278C2×C4⋊C4—971no—
279C2×C4⋊C4—971no—
280C2×C4⋊C4—971no—
281C2×C4⋊C4—971no—
282C2×C4⋊C4—971no—
283C2×C4⋊C4C2×D9711yescharacteristic, complemented
284C2×C22⋊C4—971no—
285C2×C22⋊C4—971no—
286C2×C22⋊C4—971no—
287C2×C22⋊C4—971no—
288C2×C22⋊C4—971no—
289C2×C22⋊C4—971no—
290C2×C22⋊C4—971no—
291C2×C22⋊C4—971no—
292C2×C22⋊C4—971no—
293C2×C22⋊C4—971no—
294C2×C22⋊C4—971no—
295C2×C22⋊C4—971no—
296C2×C22⋊C4—978no—
297C2×C22⋊C4C2×D9711yescharacteristic
298C2×C22⋊C4C2×D9711yescharacteristic, complemented
299C2×C42—971no—
300C2×C42—971no—
301C24—971no—
302C24—974no—
303C24—971no—
304C24—3882no—
305C24—3882no—
306C24—1944no—
307C24C22×D9711yescharacteristic
308C24C22×D9711yescharacteristic
309C24C22×D9711yescharacteristic
310C2×Q8—974no—
311C2×Q8—3882no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—974no—
316C2×D4—974no—
317C2×D4—974no—
318C2×D4—974no—
319C2×D4—3882no—
320C2×D4—3882no—
321C2×D4—1944no—
322C2×D4—1944no—
323C2×D4C22×D9714yes—
324C2×D4C22×D9714yes—
325C2×D4C22×D9714yescomplemented
326C22×C4—42no—
327C22×C4—974no—
328C22×C4—971no—
329C22×C4—971no—
330C22×C4—971no—
331C22×C4—971no—
332C22×C4—971no—
333C22×C4—971no—
334C22×C4—974no—
335C22×C4—974no—
336C22×C4—3882no—
337C22×C4—3882no—
338C22×C4—3882no—
339C22×C4C22×D9711yescharacteristic
340C22×C4C22×D9711yescharacteristic
341C22×C4C22×D9711yescharacteristic
342C22×C4C22×D9711yescharacteristic
343C22×C4C22×D9714yescomplemented
344C4⋊C4—974no—
345C4⋊C4—974no—
346C4⋊C4—974no—
347C4⋊C4—974no—
348C4⋊C4—974no—
349C4⋊C4C22×D9714yescomplemented
350C22⋊C4—974no—
351C22⋊C4—974no—
352C22⋊C4—974no—
353C22⋊C4—974no—
354C22⋊C4—974no—
355C22⋊C4—974no—
356C22⋊C4—974no—
357C22⋊C4—974no—
358C22⋊C4—974no—
359C22⋊C4—974no—
360C22⋊C4—974no—
361C22⋊C4—974no—
362C22⋊C4C22×D9714yes—
363C22⋊C4C22×D9714yescomplemented
364C42—974no—
365C42—974no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—974no—
376C23—972no—
377C23—972no—
378C23—3884no—
379C23—3884no—
380C23—38816no—
381C23—1948no—
382C23—1942no—
383C23—1942no—
384C23—1942no—
385C23—1942no—
386C23—1942no—
387C23C23×D9712yes—
388C23C23×D9712yes—
389C23C23×D9712yes—
390C23C23×D9711yescharacteristic
391Q8—3884no—
392D4—44no—
393D4—44no—
394D4—44no—
395D4—3884no—
396D4—3884no—
397D4—1948no—
398D4—1948no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—44no—
404C2×C4—972no—
405C2×C4—972no—
406C2×C4—972no—
407C2×C4—972no—
408C2×C4—972no—
409C2×C4—972no—
410C2×C4—3884no—
411C2×C4—3884no—
412C2×C4—3884no—
413C2×C4—1942no—
414C2×C4—1942no—
415C2×C4—1942no—
416C2×C4—1942no—
417C2×C4—1942no—
418C2×C4—1942no—
419C2×C4C23×D9712yes—
420C2×C4C23×D9712yes—
421C2×C4C23×D9712yes—
422C2×C4C23×D9712yes—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—44no—
433C22—44no—
434C22—44no—
435C22—974no—
436C22—38816no—
437C22—3884no—
438C22—1942no—
439C22—1942no—
440C22—1942no—
441C22—1942no—
442C22—1942no—
443C22—1948no—
444C22#41×D9712yes—
445C22#1569085512yes—
446C22#1569085512yes—
447C22C24×D9711yescharacteristic
448C4—22no—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—1942no—
453C4—1942no—
454C4—1942no—
455C4—1942no—
456C4—1942no—
457C4—1942no—
458C2—42no—
459C2—42no—
460C2—42no—
461C2—3882no—
462C2—1944no—
463C2#2026131714yesdirect product factor
464C2C2×#41×D9711yescharacteristic
465C2C2×#1569085511yescharacteristic
466C2C2×#1569085511yescharacteristic
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×#121×C97C2
socleC23×C97C24
hypercenterC2×#121D97
hypocenterC97C2×#538
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #20261317C2direct (N × H)
2. C2×#15690805C2semidirect
3. C2×#15690803C2semidirect
4. C2×#15690797C2semidirect
5. C2×#15690796C2semidirect
6. C2×#15690793C2semidirect
7. C2×#15690791C2semidirect
8. C2×#15690788C2semidirect
9. C2×#15690782C2semidirect
10. C2×#15690781C2semidirect
11. C2×#15690778C2semidirect
12. C2×#15690778C2semidirect
13. C2×C4×C22⋊D97C2semidirect
14. C2×#15690770C2semidirect
15. C2×C22⋊C4×D97C2semidirect
16. C2×#121×C97C2semidirect
18. C22×Dic194C22semidirect
23. C23×Dic97C22semidirect
26. #15690805C22semidirect
27. #15690803C22semidirect
28. #15690797C22semidirect
29. #15690796C22semidirect
30. #15690793C22semidirect
31. #15690791C22semidirect
32. #15690788C22semidirect
33. #15690782C22semidirect
34. #15690781C22semidirect
35. #15690778C22semidirect
36. #15690778C22semidirect
37. C4×C22⋊D97C22semidirect
38. #15690770C22semidirect
39. C2×C4⋊Dic97C22semidirect
40. C2×#13095490C22semidirect
41. C2×#13095490C22semidirect
42. C2×#13095490C22semidirect
43. C2×#13095490C22semidirect
47. C2×C22⋊Dic97C22semidirect
48. C2×C22⋊Dic97C22semidirect
49. C2×C22⋊Dic97C22semidirect
56. C22⋊C4×D97C22semidirect
57. C2×C4×Dic97C22semidirect
58. C2×C4×Dic97C22semidirect
61. C22×D4×C97C22semidirect
62. C23×C4×C97C22semidirect
63. #121×C97C22semidirect
64. C2×C4⋊C4×C97C22semidirect
66. C2×C22⋊C4×C97C22semidirect
74. C2×Dic194C23semidirect
90. C22×Dic97C23semidirect
95. C4⋊Dic97C23semidirect
96. #13095490C23semidirect
97. #13095490C23semidirect
98. #13095490C23semidirect
99. #13095490C23semidirect
103. C22⋊Dic97C23semidirect
104. C22⋊Dic97C23semidirect
105. C22⋊Dic97C23semidirect
112. C4×Dic97C23semidirect
113. C4×Dic97C23semidirect
122. C2×D4×C97C23semidirect
128. C22×C4×C97C23semidirect
129. C4⋊C4×C97C23semidirect
131. C22⋊C4×C97C23semidirect
227. C2×C97#538semidirect
234. C97C2×#538semidirect
245. C2×#121D97semidirect
259. C22×D4C2×D97semidirect
263. C23×C4C2×D97semidirect
273. #121C2×D97semidirect
283. C2×C4⋊C4C2×D97semidirect
298. C2×C22⋊C4C2×D97semidirect
325. C2×D4C22×D97semidirect
343. C22×C4C22×D97semidirect
349. C4⋊C4C22×D97semidirect
363. C22⋊C4C22×D97semidirect
463. C2#20261317direct (N × H)

Maximal subgroups

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

subgroupindex
#202613172
C2×#156908052
C2×#156908032
C2×#156907972
C2×#156907962
C2×#156907932
C2×#156907912
C2×#156907882
C2×#156907822
C2×#156907812
C2×#156907782
C2×C4×C22⋊D972
C2×#156907702
C2×C22⋊C4×D972
C2×#121×C972
C2×#53897