Finite Group Databasebeta

← C2×#20257733 #27123963

Subgroups — C2×#20257733 #27123963

Order 12032.

7086 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257733C2132yesmaximal, direct product factor
2C2×#15689872C214yesmaximal, complemented
3C2×#15689839C211yesmaximal, characteristic, complemented
4C2×#15689833C212yesmaximal, complemented
5C2×#15689814C214yesmaximal, complemented
6C2×#15689776C214yesmaximal, complemented
7C2×#15689750C214yesmaximal, complemented
8C2×#15689745C214yesmaximal, complemented
9C2×#15689698C214yesmaximal, complemented
10C2×C4⋊Q8×D47C211yesmaximal, characteristic, complemented
11C2×#15689660C212yesmaximal, complemented
12C2×Q82×C47C211yesmaximal, characteristic, complemented
13C22×#10625354C2214yes—
14C22×Q8×D47C2212yes—
15#15689872C22164yescomplemented
16#15689839C22116yescomplemented
17#15689833C22132yescomplemented
18#15689814C22164yescomplemented
19#15689776C22164yescomplemented
20#15689750C22164yescomplemented
21#15689745C22164yescomplemented
22#15689698C22164yescomplemented
23C2×#13094024C2211yescharacteristic
24C2×#13094023C2214yes—
25C2×#13094022C2214yes—
26C2×#13094022C2212yes—
27C4⋊Q8×D47C22116yescomplemented
28#15689660C22132yescomplemented
29C2×#13094019C2214yes—
30C2×#13094018C2218yes—
31C2×#13094016C2214yes—
32C2×#13094015C2212yes—
33C2×#13094014C2218yes—
34C2×#13094013C2212yes—
35C2×#13094013C2214yes—
36C2×#13094012C2214yes—
37C2×#13094009C2218yes—
38C2×#13094009C2218yes—
39C2×#13094009C2214yes—
40C2×#13094002C2218yes—
41C2×#13094002C2218yes—
42C2×#13094000C2218yes—
43C2×#13094000C2212yes—
44C2×#13094000C2214yes—
45C2×#13093999C2212yes—
46C2×#13093999C2214yes—
47C2×C4×D188C2214yes—
48C2×C4×D188C2214yes—
49C2×C4×D188C2212yes—
50C2×#13093982C2214yes—
51C2×#13093981C2214yes—
52C2×#13093981C2214yes—
53C2×C4⋊C4×D47C2214yes—
54C2×C4⋊C4×D47C2214yes—
55C2×C42×D47C2211yescharacteristic
56Q82×C47C22116yescomplemented
57C2×C4⋊Q8×C47C2214yes—
58C2×C4⋊Q8×C47C2211yescharacteristic
59C2×C4⋊Q8×C47C2214yes—
60C2×C4×Q8×C47C2214yes—
61C2×C4×Q8×C47C2212yes—
62C2×#10625354—416no—
63C2×#10625354C23132yes—
64C22×Dic94C2312yes—
65C22×Dic94C2314yes—
66C2×Q8×D47—48no—
67C2×Q8×D47C23116yes—
68C22×D188C2312yes—
69C22×D188C2314yes—
70C22×D188C2314yes—
71C22×C4×D47C2312yes—
72C22×C4×D47C2311yescharacteristic
73C22×C4×D47C2314yes—
74C22×C4×D47C2314yes—
75C22×C4×D47C2318yes—
76#13094024C2318yes—
77#13094023C23132yes—
78#13094022C23132yes—
79#13094022C23116yes—
80#13094019C23132yes—
81#13094018C23164yes—
82#13094016C23132yes—
83#13094015C23116yes—
84#13094014C23164yes—
85#13094013C23116yes—
86#13094013C23132yes—
87#13094012C23132yes—
88#13094009C23164yes—
89#13094009C23164yes—
90#13094009C23132yes—
91#13094002C23164yes—
92#13094002C23164yes—
93#13094000C23132yes—
94#13094000C23164yes—
95#13094000C23116yes—
96#13093999C23132yes—
97#13093999C23116yes—
98C4×D188C23132yes—
99C4×D188C23132yes—
100C4×D188C23116yes—
101#13093982C23132yes—
102#13093981C23132yes—
103#13093981C23132yes—
104C2×C4⋊Dic47C2314yes—
105C2×C4⋊Dic47C2312yes—
106C2×C4⋊Dic47C2314yes—
107C2×Dic47⋊C4C2318yes—
108C2×Dic47⋊C4C2318yes—
109C2×Dic47⋊C4C2318yes—
110C2×Dic47⋊C4C2314yes—
111C2×Dic47⋊C4C2314yes—
112C4⋊C4×D47C23132yes—
113C4⋊C4×D47C23132yes—
114C2×(C2×D47)⋊C4C2314yes—
115C2×(C2×D47)⋊C4C2314yes—
116C2×(C2×D47)⋊C4C2314yes—
117C2×(C2×D47)⋊C4C2318yes—
118C2×(C2×D47)⋊C4C2318yes—
119C2×(C2×D47)⋊C4C2314yes—
120C2×(C2×D47)⋊C4C2314yes—
121C2×C4×Dic47C2312yes—
122C2×C4×Dic47C2311yescharacteristic
123C2×C4×Dic47C2314yes—
124C42×D47C2318yes—
125C22×Q8×C47C2312yes—
126C22×Q8×C47C2314yes—
127C22×Q8×C47C2312yes—
128C4⋊Q8×C47C23132yes—
129C4⋊Q8×C47C2318yes—
130C4⋊Q8×C47C23132yes—
131C4×Q8×C47C23132yes—
132C4×Q8×C47C23116yes—
133C2×C4⋊C4×C47C2312yes—
134C2×C4⋊C4×C47C2314yes—
135C2×C4⋊C4×C47C2314yes—
136C2×C4⋊C4×C47C2318yes—
137C2×C42×C47C2311yescharacteristic
138C2×C42×C47C2314yes—
139C2×C42×C47C2314yes—
140C23×D47C2414yes—
141C23×D47C2411yescharacteristic
142C2×Q8×C47—28no—
143C2×Q8×C47—44no—
144C2×Q8×C47—48no—
145C2×Q8×C47C2418yes—
146C2×Q8×C47C24116yes—
147C2×Q8×C47C2418yes—
148C22×C4×C47C2414yes—
149C22×C4×C47C2411yescharacteristic
150C22×C4×C47C2414yes—
151C22×C4×C47C2414yes—
152C22×C4×C47C2412yes—
153C22×Dic47C2412yes—
154C22×Dic47C2414yes—
155C22×Dic47C2411yescharacteristic
156C22×Dic47C2414yes—
157#10625354—464no—
158Q8×D47—432no—
159C2×D188—216no—
160C2×D188—28no—
161C2×D188—48no—
162C2×D188C2418yes—
163C2×D188C24116yes—
164C2×D188C24116yes—
165C2×C4×D47—24no—
166C2×C4×D47—216no—
167C2×C4×D47—28no—
168C2×C4×D47—216no—
169C2×C4×D47—416no—
170C2×C4×D47C2418yes—
171C2×C4×D47C2414yes—
172C2×C4×D47C24116yes—
173C2×C4×D47C24116yes—
174C2×C4×D47C24132yes—
175C2×Dic94—48no—
176C2×Dic94—44no—
177C2×Dic94C2418yes—
178C2×Dic94C24116yes—
179C4⋊C4×C47C2418yes—
180C4⋊C4×C47C24116yes—
181C4⋊C4×C47C24116yes—
182C4⋊C4×C47C24132yes—
183C42×C47C2414yes—
184C42×C47C24116yes—
185C42×C47C24116yes—
186(C2×D47)⋊C4C24116yes—
187(C2×D47)⋊C4C24116yes—
188(C2×D47)⋊C4C24116yes—
189(C2×D47)⋊C4C24132yes—
190(C2×D47)⋊C4C24132yes—
191(C2×D47)⋊C4C24116yes—
192(C2×D47)⋊C4C24116yes—
193C4⋊Dic47C24116yes—
194C4⋊Dic47C2418yes—
195C4⋊Dic47C24116yes—
196Dic47⋊C4C24132yes—
197Dic47⋊C4C24132yes—
198Dic47⋊C4C24132yes—
199Dic47⋊C4C24116yes—
200Dic47⋊C4C24116yes—
201C4×Dic47C2418yes—
202C4×Dic47C2414yes—
203C4×Dic47C24116yes—
204C23×C47C2511yescharacteristic
205C22×D47—24no—
206C22×D47—28no—
207C22×D47—28no—
208C22×D47—28no—
209C22×D47H(5,𝔽2)14yes—
210C22×D47C2518yes—
211C22×D47C2512yes—
212Q8×C47—216no—
213Q8×C47—48no—
214Q8×C47—416no—
215C2×C4×C47—22no—
216C2×C4×C47—28no—
217C2×C4×C47—28no—
218C2×C4×C47—28no—
219C2×C4×C472−1+418yes—
220C2×C4×C47C2518yes—
221C2×C4×C47C2512yes—
222C2×C4×C47C2518yes—
223C2×C4×C47C2518yes—
224C2×C4×C47C2514yes—
225C2×Dic47—22no—
226C2×Dic47—28no—
227C2×Dic47—24no—
228C2×Dic47—28no—
229C2×Dic47C2514yes—
230C2×Dic47C2518yes—
231C2×Dic47C2512yes—
232C2×Dic47C2518yes—
233D188—232no—
234D188—216no—
235D188—416no—
236C4×D47—28no—
237C4×D47—232no—
238C4×D47—216no—
239C4×D47—232no—
240C4×D47—432no—
241Dic94—416no—
242Dic94—48no—
243C2×#3544—471nomaximal, Sylow
244C22×C47C2×H(5,𝔽2)12yes—
245C22×C47C2×2−1+414yes—
246C22×C47C2611yescharacteristic
247C2×D47—24no—
248C2×D47—28no—
249C2×D47—28no—
250C2×D47—28no—
251C2×D47—28no—
252C2×D47—416no—
253C2×D47C2×H(5,𝔽2)14yes—
254C4×C47—22no—
255C4×C47—28no—
256C4×C47—28no—
257C4×C47—28no—
258C4×C47C2×2−1+418yes—
259Dic47—22no—
260Dic47—28no—
261Dic47—24no—
262Dic47—28no—
263#3544—4732no—
264C2×#563—474no—
265C2×Q82D4711yescharacteristic, complemented
266C2×#555—471no—
267C2×#554—472no—
268C2×#552—474no—
269C2×#547—474no—
270C2×#544—474no—
271C2×#542—474no—
272C2×#536—474no—
273C22×C4⋊Q8—471no—
274C2×#529—472no—
275C2×C47#354414yescomplemented
276C2×C47C22×H(5,𝔽2)11yescharacteristic
277C2×C47C22×2−1+412yes—
278D47—24no—
279D47—48no—
280C22×#41—474no—
281C23×Q8—472no—
282#563—4764no—
283Q82C2×D47116yescomplemented
284#555—4716no—
285#554—4732no—
286#552—4764no—
287#547—4764no—
288#544—4764no—
289#542—4764no—
290#536—4764no—
291C2×C4⋊Q8—472no—
292C2×C4⋊Q8—474no—
293C2×C4⋊Q8—4716no—
294C2×C4⋊Q8—474no—
295C2×C4⋊Q8—471no—
296C2×C4⋊Q8C2×D4711yescharacteristic
297C2×C4⋊Q8C2×D4714yes—
298C2×C4⋊Q8C2×D4714yes—
299#529—4732no—
300C2×#126—474no—
301C2×#126—478no—
302C2×#125—472no—
303C2×#125—478no—
304C2×#125—474no—
305C2×#124—472no—
306C2×#124—474no—
307C2×#124—474no—
308C2×#123—474no—
309C2×#123—478no—
310C2×#123—478no—
311C2×C22⋊Q8—472no—
312C2×C22⋊Q8—474no—
313C2×C22⋊Q8—478no—
314C2×C22⋊Q8—478no—
315C2×C22⋊Q8—478no—
316C2×#121—472no—
317C2×#121—474no—
318C2×C4×Q8C2×D4712yes—
319C2×C4×Q8C2×D4714yes—
320C2×C4×D4—474no—
321C2×C4×D4—472no—
322C2×C4×D4—474no—
323C2×#117—474no—
324C2×#117—474no—
325C2×#117—474no—
326C22×C4⋊C4—474no—
327C22×C4⋊C4—474no—
328C22×C42—471no—
329C47C2×#354411yescharacteristic, complemented, Sylow
330C2×#41—4732no—
331C2×#41—18816no—
332C22×Q8—4716no—
333C22×Q8—472no—
334C22×Q8—474no—
335C22×Q8—1888no—
336C22×Q8C22×D4712yes—
337C22×Q8C22×D4714yes—
338C22×Q8C22×D4712yes—
339C22×D4—472no—
340C22×D4—474no—
341C22×D4—474no—
342C23×C4—474no—
343C23×C4—472no—
344C23×C4—474no—
345C23×C4—471no—
346C23×C4—478no—
347C4⋊Q8—4716no—
348C4⋊Q8—4732no—
349C4⋊Q8—4732no—
350C4⋊Q8—478no—
351C4⋊Q8C22×D4718yes—
352C4⋊Q8C22×D47132yes—
353C4⋊Q8C22×D47132yes—
354#126—4732no—
355#126—4764no—
356#125—4716no—
357#125—4764no—
358#125—4732no—
359#124—4716no—
360#124—4732no—
361#124—4732no—
362#123—4732no—
363#123—4764no—
364#123—4764no—
365C22⋊Q8—4716no—
366C22⋊Q8—4732no—
367C22⋊Q8—4764no—
368C22⋊Q8—4764no—
369C22⋊Q8—4764no—
370#121—4716no—
371#121—4732no—
372C4×Q8C22×D47116yes—
373C4×Q8C22×D47132yes—
374C4×D4—4732no—
375C4×D4—4716no—
376C4×D4—4732no—
377#117—4732no—
378#117—4732no—
379#117—4732no—
380C2×C4⋊C4—474no—
381C2×C4⋊C4—478no—
382C2×C4⋊C4—478no—
383C2×C4⋊C4—472no—
384C2×C4⋊C4—478no—
385C2×C4⋊C4—474no—
386C2×C4⋊C4—474no—
387C2×C4⋊C4—474no—
388C2×C4⋊C4—4732no—
389C2×C4⋊C4—4732no—
390C2×C4⋊C4C22×D4712yes—
391C2×C4⋊C4C22×D4714yes—
392C2×C4⋊C4C22×D4714yes—
393C2×C4⋊C4C22×D4718yes—
394C2×C22⋊C4—474no—
395C2×C22⋊C4—474no—
396C2×C22⋊C4—478no—
397C2×C22⋊C4—474no—
398C2×C22⋊C4—478no—
399C2×C22⋊C4—474no—
400C2×C22⋊C4—474no—
401C2×C42—474no—
402C2×C42—472no—
403C2×C42—471no—
404C2×C42—478no—
405C2×C42C22×D4714yes—
406C2×C42C22×D4714yes—
407C2×C42C22×D4711yescharacteristic
408C24—471no—
409C24—474no—
410#41—18864no—
411C2×Q8—28no—
412C2×Q8—44no—
413C2×Q8—48no—
414C2×Q8—478no—
415C2×Q8—4716no—
416C2×Q8—1888no—
417C2×Q8—1884no—
418C2×Q8—18832no—
419C2×Q8C23×D4718yes—
420C2×Q8C23×D47116yes—
421C2×Q8C23×D4718yes—
422C2×D4—478no—
423C2×D4—4716no—
424C2×D4—4716no—
425C2×D4—9416no—
426C2×D4—948no—
427C2×D4—1888no—
428C22×C4—474no—
429C22×C4—472no—
430C22×C4—471no—
431C22×C4—474no—
432C22×C4—478no—
433C22×C4—474no—
434C22×C4—4716no—
435C22×C4—4716no—
436C22×C4—4732no—
437C22×C4—944no—
438C22×C4—9416no—
439C22×C4—948no—
440C22×C4—9416no—
441C22×C4—18816no—
442C22×C4C23×D4711yescharacteristic
443C22×C4C23×D4714yes—
444C22×C4C23×D4714yes—
445C22×C4C23×D4714yes—
446C22×C4C23×D4712yes—
447C4⋊C4—4732no—
448C4⋊C4—4732no—
449C4⋊C4—4732no—
450C4⋊C4—4716no—
451C4⋊C4—4716no—
452C4⋊C4—4716no—
453C4⋊C4—478no—
454C4⋊C4—4716no—
455C4⋊C4C23×D4718yes—
456C4⋊C4C23×D47116yes—
457C4⋊C4C23×D47116yes—
458C4⋊C4C23×D47132yes—
459C22⋊C4—4716no—
460C22⋊C4—4716no—
461C22⋊C4—4732no—
462C22⋊C4—4716no—
463C22⋊C4—4732no—
464C22⋊C4—4716no—
465C22⋊C4—4716no—
466C42—478no—
467C42—474no—
468C42—4716no—
469C42C23×D4714yes—
470C42C23×D47116yes—
471C42C23×D47116yes—
472C23—474no—
473C23—472no—
474C23—478no—
475C23—944no—
476C23—948no—
477C23—948no—
478C23—948no—
479C23C24×D4711yescharacteristic
480Q8—216no—
481Q8—48no—
482Q8—416no—
483Q8—18816no—
484Q8—1888no—
485D4—9432no—
486D4—9416no—
487D4—18816no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—474no—
493C2×C4—478no—
494C2×C4—472no—
495C2×C4—478no—
496C2×C4—948no—
497C2×C4—9432no—
498C2×C4—948no—
499C2×C4—942no—
500C2×C4—948no—
501C2×C4—944no—
502C2×C4—9416no—
503C2×C4—9432no—
504C2×C4—18832no—
505C2×C4#1309407218yes—
506C2×C4C24×D4712yes—
507C2×C4C24×D4718yes—
508C2×C4C24×D4718yes—
509C2×C4C24×D4718yes—
510C2×C4C24×D4714yes—
511C22—474no—
512C22—944no—
513C22—948no—
514C22—948no—
515C22—948no—
516C22—948no—
517C22—18816no—
518C22H(5,𝔽2)×D4712yes—
519C22C2×#1309407214yes—
520C22C25×D4711yescharacteristic
521C4—28no—
522C4—22no—
523C4—28no—
524C4—28no—
525C4—948no—
526C4—942no—
527C4—944no—
528C4—948no—
529C4C2×#1309407218yes—
530C2—944no—
531C2—1888no—
532C2#2025773314yesdirect product factor
533C2C2×H(5,𝔽2)×D4711yescharacteristic
534C2C22×#1309407212yes—
subgroupG/H
centerC23C24×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC2×Q82×C47C2
socleC23×C47C25
hypercenterC2×Q82D47
hypocenterC47C2×#3544
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
1. #20257733C2direct (N × H)
2. C2×#15689872C2semidirect
3. C2×#15689839C2semidirect
4. C2×#15689833C2semidirect
5. C2×#15689814C2semidirect
6. C2×#15689776C2semidirect
7. C2×#15689750C2semidirect
8. C2×#15689745C2semidirect
9. C2×#15689698C2semidirect
10. C2×C4⋊Q8×D47C2semidirect
11. C2×#15689660C2semidirect
12. C2×Q82×C47C2semidirect
15. #15689872C22semidirect
16. #15689839C22semidirect
17. #15689833C22semidirect
18. #15689814C22semidirect
19. #15689776C22semidirect
20. #15689750C22semidirect
21. #15689745C22semidirect
22. #15689698C22semidirect
27. C4⋊Q8×D47C22semidirect
28. #15689660C22semidirect
56. Q82×C47C22semidirect
265. C2×Q82D47semidirect
275. C2×C47#3544semidirect
283. Q82C2×D47semidirect
329. C47C2×#3544semidirect
532. C2#20257733direct (N × H)

Maximal subgroups

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

subgroupindex
#202577332
C2×#156898722
C2×#156898392
C2×#156898332
C2×#156898142
C2×#156897762
C2×#156897502
C2×#156897452
C2×#156896982
C2×C4⋊Q8×D472
C2×#156896602
C2×Q82×C472
C2×#354447