Finite Group Databasebeta

← C2×#20256983 #27123341

Subgroups — C2×#20256983 #27123341

Order 12032.

7802 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256983C2132yesmaximal, direct product factor
2C2×#15689858C213yesmaximal, complemented
3C2×#15689852C211yesmaximal, characteristic, complemented
4C2×#15689846C213yesmaximal, complemented
5C2×#15689821C216yesmaximal, complemented
6C2×#15689795C211yesmaximal, characteristic, complemented
7C2×#15689784C212yesmaximal, complemented
8C2×#15689673C213yesmaximal, complemented
9C2×#15689660C216yesmaximal, complemented
10C2×C4×#10625354C212yesmaximal, complemented
11C2×C4×#10625350C213yesmaximal, complemented
12C22×C4×Q8×C47C211yesmaximal, characteristic, complemented
13C22×#10625354C2212yes—
14C22×#10625350C2213yes—
15#15689858C22148yescomplemented
16#15689852C22116yescomplemented
17#15689846C22148yescomplemented
18#15689821C22196yescomplemented
19#15689795C22116yescomplemented
20#15689784C22132yescomplemented
21#15689673C22148yescomplemented
22C2×#13094024C2213yes—
23C2×#13094021C2213yes—
24#15689660C22196yescomplemented
25C2×#13094019C22112yes—
26C2×#13094015C2216yes—
27C2×#13094013C2216yes—
28C2×#13094012C2216yes—
29C2×#13094009C22112yes—
30C2×#13094000C2216yes—
31C2×#13093999C2216yes—
32C2×#13093998C2216yes—
33C2×#13093998C2216yes—
34C4×#10625354C22132yescomplemented
35C4×#10625350C22148yescomplemented
36C2×Q8×Dic47C2212yes—
37C2×C4×Dic94C2213yes—
38C2×C4×Dic94C2213yes—
39C2×C4×D188C2213yes—
40C2×C4×D188C2213yes—
41C2×C4×D188C2216yes—
42C2×C4×D188C2216yes—
43C2×#13093986C2216yes—
44C2×C4×C22⋊D47C2216yes—
45C2×C4×C22⋊D47C2216yes—
46C2×#13093983C2211yescharacteristic
47C2×#13093982C2216yes—
48C2×#13093981C2216yes—
49C2×C42×D47C2216yes—
50C23×Q8×C47C2211yescharacteristic
51C2×C4×Q8×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C2×C4×Q8×C47C2213yes—
54C2×C4×Q8×C47C2213yes—
55C2×C4×Q8×C47C22116yescomplemented
56C22×C4⋊C4×C47C2213yes—
57C22×C42×C47C2213yes—
58C2×#10625354—216no—
59C2×#10625354C23116yes—
60C2×#10625350—224no—
61C2×#10625350C23124yes—
62C22×Dic94C2313yes—
63C22×D188C2316yes—
64C22×D188C2313yes—
65C22×D188C2316yes—
66C22×C22⋊D47C2316yes—
67C22×C4×D47C2316yes—
68C22×C4×D47C2316yes—
69C22×C4×D47C2316yes—
70#13094024C23124yes—
71#13094021C23124yes—
72#13094019C23196yes—
73#13094015C23148yes—
74#13094013C23148yes—
75#13094012C23148yes—
76#13094009C23196yes—
77#13094000C23148yes—
78#13093999C23148yes—
79#13093998C23148yes—
80#13093998C23148yes—
81Q8×Dic47C23116yes—
82C4×Dic94C23124yes—
83C4×Dic94C23124yes—
84C4×D188C23124yes—
85C4×D188C23124yes—
86C4×D188C23148yes—
87C4×D188C23148yes—
88#13093986C23148yes—
89C4×C22⋊D47C23148yes—
90C4×C22⋊D47C23148yes—
91#13093983C2318yes—
92#13093982C23148yes—
93#13093981C23148yes—
94C2×C4⋊Dic47C2316yes—
95C2×C4⋊Dic47C2311yescharacteristic
96C2×C4⋊Dic47C2311yescharacteristic
97C2×C4⋊Dic47C2316yes—
98C2×Dic47⋊C4C2316yes—
99C2×Dic47⋊C4C2316yes—
100C2×C22⋊Dic47C2312yes—
101C2×(C2×D47)⋊C4C2316yes—
102C2×(C2×D47)⋊C4C2316yes—
103C2×(C2×D47)⋊C4C23112yes—
104C2×(C2×D47)⋊C4C23112yes—
105C2×C4×Dic47C2312yes—
106C2×C4×Dic47C2316yes—
107C2×C4×Dic47C2316yes—
108C42×D47C23148yes—
109C22×Q8×C47—28no—
110C22×Q8×C47C2318yes—
111C22×Q8×C47C2313yes—
112C22×Q8×C47C2311yescharacteristic
113C23×C4×C47C2313yes—
114C23×C4×C47C2313yes—
115C23×C4×C47C2311yescharacteristic
116C4×Q8×C47C2318yes—
117C4×Q8×C47C23124yes—
118C4×Q8×C47C2318yes—
119C4×Q8×C47C23124yes—
120C2×C4⋊C4×C47C2313yes—
121C2×C4⋊C4×C47C2313yes—
122C2×C4⋊C4×C47C2316yes—
123C2×C4⋊C4×C47C23124yes—
124C2×C42×C47C2313yes—
125C2×C42×C47C2313yes—
126C2×C42×C47C2313yes—
127C2×C42×C47C2313yes—
128C2×C42×C47C23124yes—
129C24×C47C2411yescharacteristic
130C23×D47C2416yes—
131C2×Q8×C47—232no—
132C2×Q8×C47—212no—
133C2×Q8×C47#4118yescomplemented
134C2×Q8×C47C24112yes—
135C2×Q8×C47C2414yes—
136C22×C4×C47—212no—
137C22×C4×C47—46no—
138C22×C4×C47#4118yescomplemented
139C22×C4×C47C2413yes—
140C22×C4×C47C2413yes—
141C22×C4×C47C2413yes—
142C22×C4×C47C24112yes—
143C22×C4×C47C24112yes—
144C22×C4×C47C2411yescharacteristic
145C22×C4×C47C2413yes—
146C22×C4×C47C2411yescharacteristic
147C22×C4×C47C2414yes—
148C2×C22⋊D47—224no—
149C2×C22⋊D47C24124yes—
150C22×Dic47C2412yes—
151C22×Dic47C2412yes—
152C22×Dic47C2416yes—
153#10625354—264no—
154#10625350—296no—
155C2×D188—212no—
156C2×D188—224no—
157C2×D188—412no—
158C2×D188C24124yes—
159C2×D188C24112yes—
160C2×D188C24124yes—
161C2×C4×D47—224no—
162C2×C4×D47—224no—
163C2×C4×D47—412no—
164C2×C4×D47C24124yes—
165C2×C4×D47C24124yes—
166C2×C4×D47C24124yes—
167C2×Dic94—212no—
168C2×Dic94C24112yes—
169C4⋊C4×C47C24112yes—
170C4⋊C4×C47C24112yes—
171C4⋊C4×C47C24124yes—
172C42×C47C24112yes—
173C42×C47C24112yes—
174C42×C47C24112yes—
175C42×C47C24112yes—
176C22⋊Dic47C2418yes—
177(C2×D47)⋊C4C24124yes—
178(C2×D47)⋊C4C24124yes—
179(C2×D47)⋊C4C24148yes—
180(C2×D47)⋊C4C24148yes—
181C4⋊Dic47C24124yes—
182C4⋊Dic47C2414yes—
183C4⋊Dic47C2414yes—
184C4⋊Dic47C24124yes—
185Dic47⋊C4C24124yes—
186Dic47⋊C4C24124yes—
187C4×Dic47C2418yes—
188C4×Dic47C24124yes—
189C4×Dic47C24124yes—
190C23×C47—22no—
191C23×C47—22no—
192C23×C47C2×#4114yes—
193C23×C47C2512yes—
194C23×C47C2511yescharacteristic
195C22×D47—212no—
196C22×D47—212no—
197C22×D47—212no—
198C22×D47C25112yes—
199Q8×C47—224no—
200Q8×C47C2×#41116yescomplemented
201C2×C4×C47—26no—
202C2×C4×C47—26no—
203C2×C4×C47—26no—
204C2×C4×C47—224no—
205C2×C4×C47—412no—
206C2×C4×C47C2×#41112yes—
207C2×C4×C47C2×#4114yes—
208C2×C4×C47C2×#4114yes—
209C2×C4×C47C2×#41116yescomplemented
210C2×C4×C47C2516yes—
211C2×C4×C47C2516yes—
212C2×C4×C47C2516yes—
213C2×C4×C47C2516yes—
214C2×C4×C47C2512yes—
215C2×C4×C47C2512yes—
216C22⋊D47—248no—
217C2×Dic47—212no—
218C2×Dic47—24no—
219C2×Dic47—24no—
220C2×Dic47C2514yes—
221C2×Dic47C2514yes—
222C2×Dic47C25112yes—
223D188—224no—
224D188—248no—
225D188—424no—
226C4×D47—248no—
227C4×D47—248no—
228C4×D47—424no—
229Dic94—224no—
230C2×#3480—471nomaximal, Sylow
231C22×C47—22no—
232C22×C47—22no—
233C22×C47—28no—
234C22×C47C22×#4112yes—
235C22×C47C22×#4114yes—
236C22×C47C22×#4112yes—
237C22×C47#58512yes—
238C22×C47C2611yescharacteristic
239C2×D47—212no—
240C2×D47—212no—
241C2×D47—212no—
242C2×D47—424no—
243C4×C47—26no—
244C4×C47—26no—
245C4×C47—26no—
246C4×C47C22×#41112yes—
247C4×C47C22×#4114yes—
248C4×C47C22×#4114yes—
249Dic47—212no—
250Dic47—24no—
251Dic47—24no—
252#3480—4732no—
253C2×#559—473no—
254C2×#557—471no—
255C2×#556—473no—
256C2×#553—476no—
257C2×#550—471no—
258C2×#548—472no—
259C2×#532—473no—
260C2×#529—476no—
261C2×C4×#41—473no—
262C2×C4×#41—472no—
263C22×C4×Q8D4711yescharacteristic, complemented
264C2×C47—24no—
265C2×C47#348014yescomplemented
266C2×C47C23×#4111yescharacteristic
267C2×C47C23×#4111yescharacteristic
268C2×C47C2×#58511yescharacteristic
269D47—412no—
270C22×#41—472no—
271C22×#41—473no—
272C23×Q8C2×D4711yescharacteristic
273#559—4748no—
274#557—4716no—
275#556—4748no—
276#553—4796no—
277#550—4716no—
278#548—4732no—
279#532—4748no—
280C2×C4⋊Q8—473no—
281C2×#127—473no—
282#529—4796no—
283C2×#126—4712no—
284C2×#125—476no—
285C2×#124—476no—
286C2×#124—476no—
287C2×#123—4712no—
288C2×C22⋊Q8—476no—
289C2×#121—476no—
290C2×#121—476no—
291C2×#121—476no—
292C4×#41—4732no—
293C4×#41—4748no—
294C2×C4×Q8—472no—
295C2×C4×Q8—473no—
296C2×C4×Q8—473no—
297C2×C4×Q8C2×D4711yescharacteristic
298C2×C4×Q8C2×D4713yes—
299C2×C4×Q8C2×D4711yescharacteristic
300C2×C4×Q8C2×D4713yes—
301C2×C4×Q8C2×D47116yescomplemented
302C2×C4×D4—476no—
303C2×C4×D4—476no—
304C2×C4×D4—476no—
305C2×C4×D4—476no—
306C2×C4×D4—473no—
307C2×C4×D4—473no—
308C2×C4×D4—476no—
309C2×#117—471no—
310C2×#117—476no—
311C2×#117—476no—
312C22×C4⋊C4C2×D4713yes—
313C22×C42—476no—
314C22×C42C2×D4713yes—
315C47C2×#348011yescharacteristic, complemented, Sylow
316C2×#41—4716no—
317C2×#41—4724no—
318C2×#41—9416no—
319C2×#41—9424no—
320C22×Q8—28no—
321C22×Q8—473no—
322C22×Q8C22×D4713yes—
323C22×Q8C22×D4718yes—
324C22×Q8C22×D4711yescharacteristic
325C22×D4—476no—
326C22×D4—476no—
327C22×D4—473no—
328C22×D4—476no—
329C23×C4—476no—
330C23×C4—476no—
331C23×C4—476no—
332C23×C4C22×D4713yes—
333C23×C4C22×D4713yes—
334C23×C4C22×D4711yescharacteristic
335C4⋊Q8—4724no—
336#127—4724no—
337#126—4796no—
338#125—4748no—
339#124—4748no—
340#124—4748no—
341#123—4796no—
342C22⋊Q8—4748no—
343#121—4748no—
344#121—4748no—
345#121—4748no—
346C4×Q8—4716no—
347C4×Q8—4724no—
348C4×Q8—4724no—
349C4×Q8C22×D4718yes—
350C4×Q8C22×D47124yes—
351C4×Q8C22×D4718yes—
352C4×Q8C22×D47124yes—
353C4×D4—4748no—
354C4×D4—4748no—
355C4×D4—4748no—
356C4×D4—4748no—
357C4×D4—4724no—
358C4×D4—4724no—
359C4×D4—4748no—
360#117—478no—
361#117—4748no—
362#117—4748no—
363C2×C4⋊C4—471no—
364C2×C4⋊C4—471no—
365C2×C4⋊C4—476no—
366C2×C4⋊C4—476no—
367C2×C4⋊C4—476no—
368C2×C4⋊C4—476no—
369C2×C4⋊C4C22×D4713yes—
370C2×C4⋊C4C22×D4713yes—
371C2×C4⋊C4C22×D4716yes—
372C2×C4⋊C4C22×D47124yes—
373C2×C22⋊C4—472no—
374C2×C22⋊C4—476no—
375C2×C22⋊C4—4712no—
376C2×C22⋊C4—476no—
377C2×C22⋊C4—4712no—
378C2×C42—476no—
379C2×C42—476no—
380C2×C42—472no—
381C2×C42—4748no—
382C2×C42C22×D4713yes—
383C2×C42C22×D4713yes—
384C2×C42C22×D47124yes—
385C2×C42C22×D4713yes—
386C2×C42C22×D4713yes—
387C24—476no—
388C24C23×D4711yescharacteristic
389#41—9464no—
390#41—9496no—
391C2×Q8—232no—
392C2×Q8—212no—
393C2×Q8—4712no—
394C2×Q8—9412no—
395C2×Q8#1062535018yescomplemented
396C2×Q8C23×D47112yes—
397C2×Q8C23×D4714yes—
398C2×D4—4724no—
399C2×D4—4724no—
400C2×D4—4712no—
401C2×D4—4724no—
402C2×D4—9424no—
403C2×D4—9424no—
404C2×D4—9412no—
405C2×D4—18812no—
406C22×C4—212no—
407C22×C4—46no—
408C22×C4—472no—
409C22×C4—472no—
410C22×C4—476no—
411C22×C4—4724no—
412C22×C4—4724no—
413C22×C4—4724no—
414C22×C4—9424no—
415C22×C4—9424no—
416C22×C4—18812no—
417C22×C4#1062535418yescomplemented
418C22×C4C23×D4713yes—
419C22×C4C23×D4713yes—
420C22×C4C23×D4713yes—
421C22×C4C23×D47112yes—
422C22×C4C23×D47112yes—
423C22×C4C23×D4713yes—
424C22×C4C23×D4711yescharacteristic
425C22×C4C23×D4714yes—
426C22×C4C23×D4711yescharacteristic
427C4⋊C4—4724no—
428C4⋊C4—4724no—
429C4⋊C4—474no—
430C4⋊C4—474no—
431C4⋊C4—4724no—
432C4⋊C4—4724no—
433C4⋊C4C23×D47112yes—
434C4⋊C4C23×D47112yes—
435C4⋊C4C23×D47124yes—
436C22⋊C4—478no—
437C22⋊C4—4724no—
438C22⋊C4—4748no—
439C22⋊C4—4724no—
440C22⋊C4—4748no—
441C42—478no—
442C42—4724no—
443C42—4724no—
444C42C23×D47112yes—
445C42C23×D47112yes—
446C42C23×D47112yes—
447C42C23×D47112yes—
448C23—22no—
449C23—22no—
450C23—4712no—
451C23—9412no—
452C23—9412no—
453C23—9412no—
454C23C2×#1062535414yes—
455C23C24×D4712yes—
456C23C24×D4711yescharacteristic
457Q8—224no—
458Q8—9424no—
459Q8C2×#10625350116yescomplemented
460D4—9448no—
461D4—9448no—
462D4—9424no—
463D4—18824no—
464C2×C4—26no—
465C2×C4—26no—
466C2×C4—26no—
467C2×C4—224no—
468C2×C4—412no—
469C2×C4—474no—
470C2×C4—474no—
471C2×C4—4712no—
472C2×C4—944no—
473C2×C4—9412no—
474C2×C4—944no—
475C2×C4—9448no—
476C2×C4—9448no—
477C2×C4—18824no—
478C2×C4C2×#10625350112yes—
479C2×C4C2×#1062535414yes—
480C2×C4C2×#1062535414yes—
481C2×C4C2×#10625354116yescomplemented
482C2×C4C24×D4716yes—
483C2×C4C24×D4716yes—
484C2×C4C24×D4716yes—
485C2×C4C24×D4716yes—
486C2×C4C24×D4712yes—
487C2×C4C24×D4712yes—
488C22—22no—
489C22—22no—
490C22—28no—
491C22—9412no—
492C22—9412no—
493C22—9412no—
494C22—18824no—
495C22C22×#1062535012yes—
496C22C22×#1062535414yes—
497C22C22×#1062535412yes—
498C22#1568995912yes—
499C22C25×D4711yescharacteristic
500C4—26no—
501C4—26no—
502C4—26no—
503C4—944no—
504C4—9412no—
505C4—944no—
506C4C22×#10625350112yes—
507C4C22×#1062535414yes—
508C4C22×#1062535414yes—
509C2—24no—
510C2—18812no—
511C2#2025698314yesdirect product factor
512C2C23×#1062535011yescharacteristic
513C2C23×#1062535411yescharacteristic
514C2C2×#1568995911yescharacteristic
subgroupG/H
centerC22×C4C23×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC22×C4×Q8×C47C2
socleC23×C47C25
hypercenterC22×C4×Q8D47
hypocenterC47C2×#3480
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #20256983C2direct (N × H)
2. C2×#15689858C2semidirect
3. C2×#15689852C2semidirect
4. C2×#15689846C2semidirect
5. C2×#15689821C2semidirect
6. C2×#15689795C2semidirect
7. C2×#15689784C2semidirect
8. C2×#15689673C2semidirect
9. C2×#15689660C2semidirect
10. C2×C4×#10625354C2semidirect
11. C2×C4×#10625350C2semidirect
12. C22×C4×Q8×C47C2semidirect
15. #15689858C22semidirect
16. #15689852C22semidirect
17. #15689846C22semidirect
18. #15689821C22semidirect
19. #15689795C22semidirect
20. #15689784C22semidirect
21. #15689673C22semidirect
24. #15689660C22semidirect
34. C4×#10625354C22semidirect
35. C4×#10625350C22semidirect
55. C2×C4×Q8×C47C22semidirect
133. C2×Q8×C47#41semidirect
138. C22×C4×C47#41semidirect
200. Q8×C47C2×#41semidirect
209. C2×C4×C47C2×#41semidirect
263. C22×C4×Q8D47semidirect
265. C2×C47#3480semidirect
301. C2×C4×Q8C2×D47semidirect
315. C47C2×#3480semidirect
395. C2×Q8#10625350semidirect
417. C22×C4#10625354semidirect
459. Q8C2×#10625350semidirect
481. C2×C4C2×#10625354semidirect
511. C2#20256983direct (N × H)

Maximal subgroups

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

subgroupindex
#202569832
C2×#156898582
C2×#156898522
C2×#156898462
C2×#156898212
C2×#156897952
C2×#156897842
C2×#156896732
C2×#156896602
C2×C4×#106253542
C2×C4×#106253502
C22×C4×Q8×C472
C2×#348047