Finite Group Databasebeta

← C2×#20257009 #27123367

Subgroups — C2×#20257009 #27123367

Order 12032.

7706 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257009C2132yesmaximal, direct product factor
2C2×#15689852C211yesmaximal, characteristic, complemented
3C2×#15689847C213yesmaximal, complemented
4C2×#15689815C216yesmaximal, complemented
5C2×#15689784C212yesmaximal, complemented
6C2×#15689777C213yesmaximal, complemented
7C2×#15689678C213yesmaximal, complemented
8C2×#15689659C216yesmaximal, complemented
9C2×C4×#10625354C211yesmaximal, characteristic, complemented
10C2×C4×#10625352C213yesmaximal, complemented
11C22×Q8×Dic47C212yesmaximal, complemented
12C2×D4×Q8×C47C211yesmaximal, characteristic, complemented
13C22×#10625354C2211yescharacteristic
14C22×#10625352C2213yes—
15#15689852C22116yescomplemented
16#15689847C22148yescomplemented
17#15689815C22196yescomplemented
18#15689784C22132yescomplemented
19#15689777C22148yescomplemented
20#15689678C22148yescomplemented
21C2×#13094024C2213yes—
22#15689659C22196yescomplemented
23C2×#13094019C22112yes—
24C2×#13094015C2216yes—
25C2×#13094015C2216yes—
26C2×#13094013C2216yes—
27C2×#13094008C22112yes—
28C2×#13094004C2216yes—
29C2×#13094000C2216yes—
30C2×#13093998C2216yes—
31C4×#10625354C22116yescomplemented
32C4×#10625352C22148yescomplemented
33C2×Q8×Dic47C2211yescharacteristic
34C2×Q8×Dic47C2212yes—
35C2×Q8×Dic47C2216yes—
36C2×Q8×Dic47C2216yes—
37C2×Q8×Dic47C22132yescomplemented
38C2×C4×Dic94C2213yes—
39C2×#13093989C2213yes—
40C2×C4×D188C2213yes—
41C2×D4×Dic47C2211yescharacteristic
42C2×#13093986C2213yes—
43C2×C4×C22⋊D47C2216yes—
44C2×(C22⋊D47)⋊C4C2216yes—
45C2×#13093982C2213yes—
46C2×#13093982C2213yes—
47C2×#13093982C2216yes—
48C2×#13093980C2216yes—
49C22×C4⋊Dic47C2216yes—
50C22×C4×Dic47C2216yes—
51C2×C42×D47C2213yes—
52C23×Q8×C47C2212yes—
53D4×Q8×C47C22116yescomplemented
54C2×C4⋊Q8×C47C2213yes—
55C2×C22⋊Q8×C47C2216yes—
56C2×C4×Q8×C47C2211yescharacteristic
57C2×C4×D4×C47C2213yes—
58C2×#10625354—28no—
59C2×#10625354C2318yes—
60C2×#10625352—224no—
61C2×#10625352C23124yes—
62C22×Dic94C2313yes—
63C22×D188C2313yes—
64C22×C22⋊D47C2316yes—
65C23×Dic47C2312yes—
66C23×Dic47C2316yes—
67C22×C4×D47C2313yes—
68C22×C4×D47C2313yes—
69C22×C4×D47C2313yes—
70#13094024C23124yes—
71#13094019C23196yes—
72#13094015C23148yes—
73#13094015C23148yes—
74#13094013C23148yes—
75#13094008C23196yes—
76#13094004C23148yes—
77#13094000C23148yes—
78#13093998C23148yes—
79Q8×Dic47C23116yes—
80Q8×Dic47C23148yes—
81Q8×Dic47C2318yes—
82Q8×Dic47C23148yes—
83C4×Dic94C23124yes—
84#13093989C23124yes—
85C4×D188C23124yes—
86D4×Dic47C2318yes—
87#13093986C23124yes—
88C4×C22⋊D47C23148yes—
89(C22⋊D47)⋊C4C23148yes—
90#13093982C23124yes—
91#13093982C23124yes—
92#13093982C23148yes—
93#13093980C23148yes—
94C2×C4⋊Dic47C23148yes—
95C2×C4⋊Dic47C2313yes—
96C2×C4⋊Dic47C2316yes—
97C2×C4⋊Dic47C23112yes—
98C2×C4⋊Dic47C2311yescharacteristic
99C2×C4⋊Dic47C2316yes—
100C2×Dic47⋊C4C2316yes—
101C2×Dic47⋊C4C2316yes—
102C2×C22⋊Dic47C2312yes—
103C2×(C2×D47)⋊C4C2316yes—
104C2×(C2×D47)⋊C4C2316yes—
105C2×(C2×D47)⋊C4C23112yes—
106C2×C4×Dic47C2311yescharacteristic
107C2×C4×Dic47C2313yes—
108C2×C4×Dic47C2313yes—
109C2×C4×Dic47C2316yes—
110C2×C4×Dic47C2316yes—
111C2×C4×Dic47C2316yes—
112C2×C4×Dic47C23148yes—
113C42×D47C23124yes—
114C22×Q8×C47—216no—
115C22×Q8×C47C23116yes—
116C22×Q8×C47C2316yes—
117C22×Q8×C47C2311yescharacteristic
118C22×D4×C47C2311yescharacteristic
119C23×C4×C47C2316yes—
120C4⋊Q8×C47C23124yes—
121C22⋊Q8×C47C23148yes—
122C4×Q8×C47C2318yes—
123C4×D4×C47C23124yes—
124C2×C4⋊C4×C47C2313yes—
125C2×C4⋊C4×C47C2313yes—
126C2×C4⋊C4×C47C2316yes—
127C2×C22⋊C4×C47C2316yes—
128C2×C42×C47C2313yes—
129C24×C47C2412yes—
130C23×D47C2413yes—
131C2×Q8×C47—264no—
132C2×Q8×C47—224no—
133C2×Q8×C47#4118yescomplemented
134C2×Q8×C47C24124yes—
135C2×Q8×C47C2414yes—
136C2×D4×C47#4118yescomplemented
137C2×D4×C47C2414yes—
138C22×C4×C47—224no—
139C22×C4×C47C2413yes—
140C22×C4×C47C2416yes—
141C22×C4×C47C24124yes—
142C22×C4×C47C2411yescharacteristic
143C22×C4×C47C2413yes—
144C2×C22⋊D47—224no—
145C2×C22⋊D47C24124yes—
146C22×Dic47—224no—
147C22×Dic47—28no—
148C22×Dic47C2412yes—
149C22×Dic47C2418yes—
150C22×Dic47C2413yes—
151C22×Dic47C2411yescharacteristic
152C22×Dic47C2416yes—
153C22×Dic47C24124yes—
154C22×Dic47C2411yescharacteristic
155#10625354—232no—
156#10625352—296no—
157C2×D188—212no—
158C2×D188C24112yes—
159C2×C4×D47—212no—
160C2×C4×D47—212no—
161C2×C4×D47—212no—
162C2×C4×D47C24112yes—
163C2×C4×D47C24112yes—
164C2×C4×D47C24112yes—
165C2×Dic94—212no—
166C2×Dic94C24112yes—
167C4⋊C4×C47C24112yes—
168C4⋊C4×C47C24112yes—
169C4⋊C4×C47C24124yes—
170C22⋊C4×C47C24124yes—
171C42×C47C24112yes—
172C22⋊Dic47C2418yes—
173(C2×D47)⋊C4C24124yes—
174(C2×D47)⋊C4C24124yes—
175(C2×D47)⋊C4C24148yes—
176C4⋊Dic47C24112yes—
177C4⋊Dic47C24124yes—
178C4⋊Dic47C24148yes—
179C4⋊Dic47C2414yes—
180C4⋊Dic47C24124yes—
181Dic47⋊C4C24124yes—
182Dic47⋊C4C24124yes—
183C4×Dic47C2414yes—
184C4×Dic47C24112yes—
185C4×Dic47C24112yes—
186C4×Dic47C24124yes—
187C4×Dic47C24124yes—
188C4×Dic47C24124yes—
189C23×C47—24no—
190C23×C47—24no—
191C23×C47C2×#4118yes—
192C23×C47C2514yes—
193C23×C47C2511yescharacteristic
194C22×D47—26no—
195C22×D47—26no—
196C22×D47—26no—
197C22×D47C2516yes—
198Q8×C47—248no—
199Q8×C47C2×#41116yescomplemented
200D4×C47C2×#41116yescomplemented
201C2×C4×C47—212no—
202C2×C4×C47—26no—
203C2×C4×C47—248no—
204C2×C4×C47C2×#41112yes—
205C2×C4×C47C2×#4114yes—
206C2×C4×C47C25112yes—
207C2×C4×C47C2516yes—
208C2×C4×C47C2516yes—
209C2×C4×C47C2512yes—
210C22⋊D47—248no—
211C2×Dic47—212no—
212C2×Dic47—22no—
213C2×Dic47—26no—
214C2×Dic47—248no—
215C2×Dic47—24no—
216C2×Dic47—216no—
217C2×Dic472−1+414yes—
218C2×Dic47C2514yes—
219C2×Dic47C2512yes—
220C2×Dic47C2516yes—
221C2×Dic47C25112yes—
222C2×Dic47C2512yes—
223D188—224no—
224C4×D47—224no—
225C4×D47—224no—
226C4×D47—224no—
227Dic94—224no—
228C2×#3482—471nomaximal, Sylow
229C22×C47—24no—
230C22×C47—24no—
231C22×C47—216no—
232C22×C47C22×#4112yes—
233C22×C47C22×#4118yes—
234C22×C47C22×#4112yes—
235C22×C47C2×2−1+412yes—
236C22×C47C2611yescharacteristic
237C2×D47—26no—
238C2×D47—26no—
239C2×D47—26no—
240C2×D47—412no—
241C4×C47—212no—
242C4×C47—26no—
243C4×C47C22×#41112yes—
244C4×C47C22×#4114yes—
245Dic47—212no—
246Dic47—22no—
247Dic47—26no—
248Dic47—24no—
249Dic47C2×2−1+414yes—
250#3482—4732no—
251C2×#557—471no—
252C2×#556—473no—
253C2×#552—476no—
254C2×D4×Q8D4711yescharacteristic, complemented
255C2×#548—472no—
256C2×#547—473no—
257C2×#533—473no—
258C2×#529—476no—
259C2×C4×#41—471no—
260C2×C4×#41—473no—
261C22×C4×Q8—472no—
262C2×C47—28no—
263C2×C47#348214yescomplemented
264C2×C47C23×#4111yescharacteristic
265C2×C47C23×#4111yescharacteristic
266C2×C47C22×2−1+411yescharacteristic
267D47—46no—
268C22×#41—471no—
269C22×#41—473no—
270C23×Q8C2×D4712yes—
271#557—4716no—
272#556—4748no—
273#552—4796no—
274D4×Q8C2×D47116yescomplemented
275#548—4732no—
276#547—4748no—
277#533—4748no—
278C2×C4⋊Q8—473no—
279C2×C4⋊Q8C2×D4713yes—
280#529—4796no—
281C2×#126—4712no—
282C2×#125—476no—
283C2×#125—476no—
284C2×#124—476no—
285C2×#123—4712no—
286C2×C22⋊Q8—476no—
287C2×C22⋊Q8—476no—
288C2×C22⋊Q8C2×D4716yes—
289C2×#121—476no—
290C4×#41—4716no—
291C4×#41—4748no—
292C2×C4×Q8—471no—
293C2×C4×Q8—476no—
294C2×C4×Q8—472no—
295C2×C4×Q8—476no—
296C2×C4×Q8—4732no—
297C2×C4×Q8—473no—
298C2×C4×Q8—473no—
299C2×C4×Q8C2×D4711yescharacteristic
300C2×C4×D4—471no—
301C2×C4×D4—476no—
302C2×C4×D4—473no—
303C2×C4×D4—476no—
304C2×C4×D4—473no—
305C2×C4×D4C2×D4713yes—
306C2×#117—473no—
307C2×#117—473no—
308C2×#117—476no—
309C2×#117—476no—
310C22×C4⋊C4—476no—
311C22×C42—476no—
312C22×C42—473no—
313C47C2×#348211yescharacteristic, complemented, Sylow
314C2×#41—478no—
315C2×#41—4724no—
316C2×#41—948no—
317C2×#41—9424no—
318C22×Q8—216no—
319C22×Q8—473no—
320C22×Q8C22×D4716yes—
321C22×Q8C22×D47116yes—
322C22×Q8C22×D4711yescharacteristic
323C22×D4—476no—
324C22×D4—473no—
325C22×D4C22×D4711yescharacteristic
326C23×C4—476no—
327C23×C4—472no—
328C23×C4—473no—
329C23×C4—473no—
330C23×C4—473no—
331C23×C4C22×D4716yes—
332C4⋊Q8—4724no—
333C4⋊Q8C22×D47124yes—
334#126—4796no—
335#125—4748no—
336#125—4748no—
337#124—4748no—
338#123—4796no—
339C22⋊Q8—4748no—
340C22⋊Q8—4748no—
341C22⋊Q8C22×D47148yes—
342#121—4748no—
343C4×Q8—478no—
344C4×Q8—4748no—
345C4×Q8—4716no—
346C4×Q8—4748no—
347C4×Q8—4724no—
348C4×Q8—4724no—
349C4×Q8C22×D4718yes—
350C4×D4—478no—
351C4×D4—4748no—
352C4×D4—4724no—
353C4×D4—4748no—
354C4×D4—4724no—
355C4×D4C22×D47124yes—
356#117—4724no—
357#117—4724no—
358#117—4748no—
359#117—4748no—
360C2×C4⋊C4—473no—
361C2×C4⋊C4—476no—
362C2×C4⋊C4—4712no—
363C2×C4⋊C4—4748no—
364C2×C4⋊C4—471no—
365C2×C4⋊C4—476no—
366C2×C4⋊C4—476no—
367C2×C4⋊C4—476no—
368C2×C4⋊C4C22×D4713yes—
369C2×C4⋊C4C22×D4713yes—
370C2×C4⋊C4C22×D4716yes—
371C2×C22⋊C4—472no—
372C2×C22⋊C4—476no—
373C2×C22⋊C4—476no—
374C2×C22⋊C4—4712no—
375C2×C22⋊C4C22×D4716yes—
376C2×C42—473no—
377C2×C42—476no—
378C2×C42—476no—
379C2×C42—476no—
380C2×C42—473no—
381C2×C42—471no—
382C2×C42—4748no—
383C2×C42—4724no—
384C2×C42C22×D4713yes—
385C24—473no—
386C24C23×D4712yes—
387#41—9432no—
388#41—9496no—
389C2×Q8—264no—
390C2×Q8—224no—
391C2×Q8—4712no—
392C2×Q8—9412no—
393C2×Q8#1062535218yescomplemented
394C2×Q8C23×D47124yes—
395C2×Q8C23×D4714yes—
396C2×D4—4724no—
397C2×D4—4712no—
398C2×D4—9424no—
399C2×D4—9412no—
400C2×D4#1062535418yescomplemented
401C2×D4C23×D4714yes—
402C22×C4—224no—
403C22×C4—471no—
404C22×C4—478no—
405C22×C4—4724no—
406C22×C4—472no—
407C22×C4—471no—
408C22×C4—473no—
409C22×C4—476no—
410C22×C4—4712no—
411C22×C4—4712no—
412C22×C4—4712no—
413C22×C4—9412no—
414C22×C4—948no—
415C22×C4—9424no—
416C22×C4—9412no—
417C22×C4—9412no—
418C22×C4C23×D47124yes—
419C22×C4C23×D4716yes—
420C22×C4C23×D4713yes—
421C22×C4C23×D4713yes—
422C22×C4C23×D4711yescharacteristic
423C4⋊C4—4724no—
424C4⋊C4—4724no—
425C4⋊C4—4712no—
426C4⋊C4—4724no—
427C4⋊C4—4748no—
428C4⋊C4—474no—
429C4⋊C4—4724no—
430C4⋊C4C23×D47112yes—
431C4⋊C4C23×D47112yes—
432C4⋊C4C23×D47124yes—
433C22⋊C4—478no—
434C22⋊C4—4724no—
435C22⋊C4—4724no—
436C22⋊C4—4748no—
437C22⋊C4C23×D47124yes—
438C42—4712no—
439C42—474no—
440C42—4712no—
441C42—4724no—
442C42—4724no—
443C42—4724no—
444C42C23×D47112yes—
445C23—24no—
446C23—24no—
447C23—476no—
448C23—946no—
449C23—946no—
450C23—946no—
451C23C2×#1062535418yes—
452C23C24×D4714yes—
453C23C24×D4711yescharacteristic
454Q8—248no—
455Q8—9424no—
456Q8C2×#10625352116yescomplemented
457D4—9448no—
458D4—9424no—
459D4C2×#10625354116yescomplemented
460C2×C4—212no—
461C2×C4—26no—
462C2×C4—248no—
463C2×C4—474no—
464C2×C4—472no—
465C2×C4—474no—
466C2×C4—472no—
467C2×C4—476no—
468C2×C4—4712no—
469C2×C4—9424no—
470C2×C4—9416no—
471C2×C4—942no—
472C2×C4—946no—
473C2×C4—9412no—
474C2×C4—9448no—
475C2×C4—944no—
476C2×C4—9424no—
477C2×C4—9424no—
478C2×C4C2×#10625352112yes—
479C2×C4C2×#1062535414yes—
480C2×C4C24×D47112yes—
481C2×C4C24×D4716yes—
482C2×C4C24×D4716yes—
483C2×C4C24×D4712yes—
484C22—24no—
485C22—24no—
486C22—216no—
487C22—946no—
488C22—946no—
489C22—946no—
490C22—18812no—
491C22C22×#1062535212yes—
492C22C22×#1062535418yes—
493C22C22×#1062535412yes—
494C222−1+4×D4712yes—
495C22C25×D4711yescharacteristic
496C4—212no—
497C4—26no—
498C4—474no—
499C4—942no—
500C4—946no—
501C4—9412no—
502C4—944no—
503C4C22×#10625352112yes—
504C4C22×#1062535414yes—
505C2—28no—
506C2—1886no—
507C2#2025700914yesdirect product factor
508C2C23×#1062535211yescharacteristic
509C2C23×#1062535411yescharacteristic
510C2C2×2−1+4×D4711yescharacteristic
subgroupG/H
centerC23C24×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC2×D4×Q8×C47C2
socleC23×C47C25
hypercenterC2×D4×Q8D47
hypocenterC47C2×#3482
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. #20257009C2direct (N × H)
2. C2×#15689852C2semidirect
3. C2×#15689847C2semidirect
4. C2×#15689815C2semidirect
5. C2×#15689784C2semidirect
6. C2×#15689777C2semidirect
7. C2×#15689678C2semidirect
8. C2×#15689659C2semidirect
9. C2×C4×#10625354C2semidirect
10. C2×C4×#10625352C2semidirect
11. C22×Q8×Dic47C2semidirect
12. C2×D4×Q8×C47C2semidirect
15. #15689852C22semidirect
16. #15689847C22semidirect
17. #15689815C22semidirect
18. #15689784C22semidirect
19. #15689777C22semidirect
20. #15689678C22semidirect
22. #15689659C22semidirect
31. C4×#10625354C22semidirect
32. C4×#10625352C22semidirect
37. C2×Q8×Dic47C22semidirect
53. D4×Q8×C47C22semidirect
133. C2×Q8×C47#41semidirect
136. C2×D4×C47#41semidirect
199. Q8×C47C2×#41semidirect
200. D4×C47C2×#41semidirect
254. C2×D4×Q8D47semidirect
263. C2×C47#3482semidirect
274. D4×Q8C2×D47semidirect
313. C47C2×#3482semidirect
393. C2×Q8#10625352semidirect
400. C2×D4#10625354semidirect
456. Q8C2×#10625352semidirect
459. D4C2×#10625354semidirect
507. C2#20257009direct (N × H)

Maximal subgroups

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

subgroupindex
#202570092
C2×#156898522
C2×#156898472
C2×#156898152
C2×#156897842
C2×#156897772
C2×#156896782
C2×#156896592
C2×C4×#106253542
C2×C4×#106253522
C22×Q8×Dic472
C2×D4×Q8×C472
C2×#348247