Finite Group Databasebeta

← C2×#20257008 #27123377

Subgroups — C2×#20257008 #27123377

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#20257008C2132yesmaximal, direct product factor
2C2×#15689846C213yesmaximal, complemented
3C2×#15689811C216yesmaximal, complemented
4C2×Q8×D188C211yesmaximal, characteristic, complemented
5C2×#15689785C212yesmaximal, complemented
6C2×#15689776C213yesmaximal, complemented
7C2×#15689678C213yesmaximal, complemented
8C2×#15689658C216yesmaximal, complemented
9C2×C4×#10625354C211yesmaximal, characteristic, complemented
10C2×C4×#10625354C213yesmaximal, complemented
11C2×C4×Q8×D47C212yesmaximal, complemented
12C2×C47×#557C211yesmaximal, characteristic, complemented
13C22×#10625354C2213yes—
14C22×#10625354C2211yescharacteristic
15C22×Q8×D47C2212yes—
16#15689846C22148yescomplemented
17#15689811C22196yescomplemented
18Q8×D188C22116yescomplemented
19#15689785C22132yescomplemented
20#15689776C22148yescomplemented
21#15689678C22148yescomplemented
22C2×#13094024C2213yes—
23#15689658C22196yescomplemented
24C2×#13094019C22112yes—
25C2×#13094015C2216yes—
26C2×#13094012C2216yes—
27C2×#13094009C22112yes—
28C2×#13094004C2216yes—
29C2×#13094004C2216yes—
30C2×#13094000C2216yes—
31C2×#13093999C2216yes—
32C4×#10625354C22116yescomplemented
33C4×#10625354C22148yescomplemented
34C2×Q8×Dic47C2211yescharacteristic
35C2×Q8×Dic47C2213yes—
36C2×C4×Dic94C2216yes—
37C2×#13093989C2216yes—
38C4×Q8×D47C22132yescomplemented
39C2×C4×D188C2213yes—
40C2×C4×D188C2213yes—
41C2×C4×D188C2216yes—
42C2×#13093986C2211yescharacteristic
43C2×#13093986C2213yes—
44C2×#13093986C2216yes—
45C2×#13093982C2213yes—
46C2×#13093982C2216yes—
47C2×#13093982C2213yes—
48C2×#13093982C2216yes—
49C2×C4⋊C4×D47C2216yes—
50C2×C42×D47C2216yes—
51C2×C42×D47C2213yes—
52C47×#557C22116yescomplemented
53C2×C4⋊Q8×C47C2213yes—
54C2×#125×C47C2216yes—
55C2×C4×Q8×C47C2212yes—
56C2×C4×Q8×C47C2211yescharacteristic
57C2×C4×Q8×C47C2213yes—
58C2×#10625354—28no—
59C2×#10625354—224no—
60C2×#10625354C23124yes—
61C2×#10625354C2318yes—
62C22×Dic94C2316yes—
63C2×Q8×D47—216no—
64C2×Q8×D47C23116yes—
65C22×D188C2316yes—
66C22×D188C2313yes—
67C22×D188C2311yescharacteristic
68C22×C4×D47C2312yes—
69C22×C4×D47C2316yes—
70C22×C4×D47C2316yes—
71C22×C4×D47C2313yes—
72C22×C4×D47C2313yes—
73C22×C4×D47C2313yes—
74#13094024C23124yes—
75#13094019C23196yes—
76#13094015C23148yes—
77#13094012C23148yes—
78#13094009C23196yes—
79#13094004C23148yes—
80#13094004C23148yes—
81#13094000C23148yes—
82#13093999C23148yes—
83Q8×Dic47C2318yes—
84Q8×Dic47C23124yes—
85C4×Dic94C23148yes—
86#13093989C23148yes—
87C4×D188C23124yes—
88C4×D188C23124yes—
89C4×D188C23148yes—
90#13093986C2318yes—
91#13093986C23124yes—
92#13093986C23148yes—
93#13093982C23124yes—
94#13093982C23148yes—
95#13093982C23124yes—
96#13093982C23148yes—
97C2×C4⋊Dic47C2313yes—
98C2×C4⋊Dic47C2313yes—
99C2×C4⋊Dic47C2316yes—
100C2×C4⋊Dic47C2316yes—
101C2×Dic47⋊C4C23112yes—
102C4⋊C4×D47C23148yes—
103C2×(C2×D47)⋊C4C2312yes—
104C2×(C2×D47)⋊C4C2316yes—
105C2×(C2×D47)⋊C4C2316yes—
106C2×(C2×D47)⋊C4C2316yes—
107C2×(C2×D47)⋊C4C23112yes—
108C2×C4×Dic47C2313yes—
109C2×C4×Dic47C2311yescharacteristic
110C2×C4×Dic47C2313yes—
111C2×C4×Dic47C2316yes—
112C2×C4×Dic47C2316yes—
113C42×D47C23148yes—
114C42×D47C23124yes—
115C22×Q8×C47C2313yes—
116C22×Q8×C47C2311yescharacteristic
117C4⋊Q8×C47C23124yes—
118#125×C47C23148yes—
119C4×Q8×C47C23116yes—
120C4×Q8×C47C2318yes—
121C4×Q8×C47C23124yes—
122C2×C4⋊C4×C47C2311yescharacteristic
123C2×C4⋊C4×C47C2313yes—
124C2×C4⋊C4×C47C2316yes—
125C2×C4⋊C4×C47C2316yes—
126C2×C4⋊C4×C47C2316yes—
127C2×C42×C47C2316yes—
128C2×C42×C47C2313yes—
129C23×D47C2413yes—
130C23×D47C2412yes—
131C2×Q8×C47—212no—
132C2×Q8×C47#4118yescomplemented
133C2×Q8×C47C24112yes—
134C2×Q8×C47C2414yes—
135C22×C4×C47C2412yes—
136C22×C4×C47C2413yes—
137C22×C4×C47C2416yes—
138C22×C4×C47C2413yes—
139C22×C4×C47C2411yescharacteristic
140C22×Dic47C2413yes—
141C22×Dic47C2411yescharacteristic
142C22×Dic47C2416yes—
143C22×Dic47C2411yescharacteristic
144#10625354—232no—
145#10625354—296no—
146Q8×D47—264no—
147C2×D188—224no—
148C2×D188—212no—
149C2×D188#4118yescomplemented
150C2×D188C24124yes—
151C2×D188C24112yes—
152C2×D188C2414yes—
153C2×C4×D47—224no—
154C2×C4×D47—224no—
155C2×C4×D47—212no—
156C2×C4×D47—212no—
157C2×C4×D47—28no—
158C2×C4×D47—212no—
159C2×C4×D47C2418yes—
160C2×C4×D47C24124yes—
161C2×C4×D47C24124yes—
162C2×C4×D47C24112yes—
163C2×C4×D47C24112yes—
164C2×C4×D47C24112yes—
165C2×Dic94—224no—
166C2×Dic94C24124yes—
167C4⋊C4×C47C2414yes—
168C4⋊C4×C47C24112yes—
169C4⋊C4×C47C24124yes—
170C4⋊C4×C47C24124yes—
171C4⋊C4×C47C24124yes—
172C42×C47C24124yes—
173C42×C47C24112yes—
174(C2×D47)⋊C4C2418yes—
175(C2×D47)⋊C4C24124yes—
176(C2×D47)⋊C4C24124yes—
177(C2×D47)⋊C4C24124yes—
178(C2×D47)⋊C4C24148yes—
179C4⋊Dic47C24112yes—
180C4⋊Dic47C24112yes—
181C4⋊Dic47C24124yes—
182C4⋊Dic47C24124yes—
183Dic47⋊C4C24148yes—
184C4×Dic47C24112yes—
185C4×Dic47C2414yes—
186C4×Dic47C24112yes—
187C4×Dic47C24124yes—
188C4×Dic47C24124yes—
189C23×C47C2511yescharacteristic
190C22×D47—24no—
191C22×D47—24no—
192C22×D47—26no—
193C22×D47—26no—
194C22×D47—26no—
195C22×D47C2×#4118yes—
196C22×D47C2516yes—
197C22×D47C2514yes—
198Q8×C47—224no—
199Q8×C47C2×#41116yescomplemented
200C2×C4×C47—212no—
201C2×C4×C47—24no—
202C2×C4×C47—26no—
203C2×C4×C47C2×#4114yes—
204C2×C4×C47C2×#41112yes—
205C2×C4×C47C2514yes—
206C2×C4×C47C25112yes—
207C2×C4×C47C2516yes—
208C2×C4×C47C2512yes—
209C2×C4×C47C2516yes—
210C2×Dic47—212no—
211C2×Dic47—26no—
212C2×Dic47—22no—
213C2×Dic472−1+414yes—
214C2×Dic47C2516yes—
215C2×Dic47C2512yes—
216C2×Dic47C25112yes—
217C2×Dic47C2512yes—
218D188—248no—
219D188—224no—
220D188C2×#41116yescomplemented
221C4×D47—248no—
222C4×D47—248no—
223C4×D47—224no—
224C4×D47—224no—
225C4×D47—216no—
226C4×D47—224no—
227Dic94—248no—
228C2×#3482—471nomaximal, Sylow
229C22×C47C22×#4112yes—
230C22×C47C22×#4112yes—
231C22×C47C2×2−1+412yes—
232C22×C47C2611yescharacteristic
233C2×D47—24no—
234C2×D47—24no—
235C2×D47—216no—
236C2×D47—26no—
237C2×D47—26no—
238C2×D47—26no—
239C2×D47—412no—
240C2×D47C22×#4118yes—
241C4×C47—212no—
242C4×C47—24no—
243C4×C47—26no—
244C4×C47C22×#4114yes—
245C4×C47C22×#41112yes—
246Dic47—212no—
247Dic47—26no—
248Dic47—22no—
249Dic47C2×2−1+414yes—
250#3482—4732no—
251C2×#557D4711yescharacteristic, complemented
252C2×#556—473no—
253C2×#552—476no—
254C2×D4×Q8—471no—
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#348214yescomplemented
263C2×C47C23×#4111yescharacteristic
264C2×C47C23×#4111yescharacteristic
265C2×C47C22×2−1+411yescharacteristic
266D47—28no—
267D47—46no—
268C22×#41—473no—
269C22×#41—471no—
270C23×Q8—472no—
271#557C2×D47116yescomplemented
272#556—4748no—
273#552—4796no—
274D4×Q8—4716no—
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×#125C2×D4716yes—
284C2×#124—476no—
285C2×#123—4712no—
286C2×C22⋊Q8—476no—
287C2×C22⋊Q8—476no—
288C2×C22⋊Q8—476no—
289C2×#121—476no—
290C4×#41—4748no—
291C4×#41—4716no—
292C2×C4×Q8—471no—
293C2×C4×Q8—473no—
294C2×C4×Q8—4732no—
295C2×C4×Q8—476no—
296C2×C4×Q8—476no—
297C2×C4×Q8C2×D4712yes—
298C2×C4×Q8C2×D4711yescharacteristic
299C2×C4×Q8C2×D4713yes—
300C2×C4×D4—471no—
301C2×C4×D4—476no—
302C2×C4×D4—473no—
303C2×C4×D4—473no—
304C2×C4×D4—473no—
305C2×C4×D4—476no—
306C2×#117—473no—
307C2×#117—476no—
308C2×#117—473no—
309C2×#117—476no—
310C22×C4⋊C4—476no—
311C22×C42—476no—
312C22×C42—473no—
313C47C2×#348211yescharacteristic, complemented, Sylow
314C2×#41—4724no—
315C2×#41—478no—
316C2×#41—9424no—
317C2×#41—948no—
318C22×Q8—4716no—
319C22×Q8—476no—
320C22×Q8—9416no—
321C22×Q8C22×D4713yes—
322C22×Q8C22×D4711yescharacteristic
323C22×D4—471no—
324C22×D4—476no—
325C22×D4—473no—
326C23×C4—476no—
327C23×C4—476no—
328C23×C4—473no—
329C23×C4—473no—
330C23×C4—473no—
331C23×C4—472no—
332C4⋊Q8—4724no—
333C4⋊Q8C22×D47124yes—
334#126—4796no—
335#125—4748no—
336#125C22×D47148yes—
337#124—4748no—
338#123—4796no—
339C22⋊Q8—4748no—
340C22⋊Q8—4748no—
341C22⋊Q8—4748no—
342#121—4748no—
343C4×Q8—478no—
344C4×Q8—4724no—
345C4×Q8—4748no—
346C4×Q8—4748no—
347C4×Q8C22×D47116yes—
348C4×Q8C22×D4718yes—
349C4×Q8C22×D47124yes—
350C4×D4—478no—
351C4×D4—4724no—
352C4×D4—4748no—
353C4×D4—4724no—
354C4×D4—4724no—
355C4×D4—4748no—
356#117—4724no—
357#117—4748no—
358#117—4724no—
359#117—4748no—
360C2×C4⋊C4—473no—
361C2×C4⋊C4—4712no—
362C2×C4⋊C4—476no—
363C2×C4⋊C4—473no—
364C2×C4⋊C4—476no—
365C2×C4⋊C4—4748no—
366C2×C4⋊C4C22×D4716yes—
367C2×C4⋊C4C22×D4711yescharacteristic
368C2×C4⋊C4C22×D4713yes—
369C2×C4⋊C4C22×D4716yes—
370C2×C4⋊C4C22×D4716yes—
371C2×C22⋊C4—472no—
372C2×C22⋊C4—476no—
373C2×C22⋊C4—476no—
374C2×C22⋊C4—476no—
375C2×C22⋊C4—4712no—
376C2×C42—473no—
377C2×C42—476no—
378C2×C42—476no—
379C2×C42—471no—
380C2×C42—473no—
381C2×C42—4748no—
382C2×C42—4724no—
383C2×C42C22×D4713yes—
384C2×C42C22×D4716yes—
385C24—472no—
386C24—473no—
387#41—9496no—
388#41—9432no—
389C2×Q8—212no—
390C2×Q8—4724no—
391C2×Q8—9464no—
392C2×Q8—9424no—
393C2×Q8#1062535418yescomplemented
394C2×Q8C23×D47112yes—
395C2×Q8C23×D4714yes—
396C2×D4—478no—
397C2×D4—474no—
398C2×D4—4724no—
399C2×D4—4712no—
400C2×D4—9424no—
401C2×D4—9412no—
402C22×C4—471no—
403C22×C4—473no—
404C22×C4—471no—
405C22×C4—476no—
406C22×C4—478no—
407C22×C4—4724no—
408C22×C4—4724no—
409C22×C4—4712no—
410C22×C4—4712no—
411C22×C4—4712no—
412C22×C4—948no—
413C22×C4—9412no—
414C22×C4—9424no—
415C22×C4—9412no—
416C22×C4—9424no—
417C22×C4—9412no—
418C22×C4C23×D4712yes—
419C22×C4C23×D4713yes—
420C22×C4C23×D4716yes—
421C22×C4C23×D4711yescharacteristic
422C22×C4C23×D4713yes—
423C4⋊C4—4748no—
424C4⋊C4—4712no—
425C4⋊C4—4724no—
426C4⋊C4—4712no—
427C4⋊C4—4724no—
428C4⋊C4C23×D47124yes—
429C4⋊C4C23×D4714yes—
430C4⋊C4C23×D47112yes—
431C4⋊C4C23×D47124yes—
432C4⋊C4C23×D47124yes—
433C22⋊C4—478no—
434C22⋊C4—4724no—
435C22⋊C4—4724no—
436C22⋊C4—4724no—
437C22⋊C4—4748no—
438C42—474no—
439C42—4712no—
440C42—4712no—
441C42—4724no—
442C42—4724no—
443C42C23×D47112yes—
444C42C23×D47124yes—
445C23—478no—
446C23—474no—
447C23—476no—
448C23—944no—
449C23—944no—
450C23—946no—
451C23—946no—
452C23—946no—
453C23C24×D4711yescharacteristic
454Q8—224no—
455Q8—9448no—
456Q8C2×#10625354116yescomplemented
457D4—4716no—
458D4—9448no—
459D4—9424no—
460C2×C4—212no—
461C2×C4—24no—
462C2×C4—26no—
463C2×C4—474no—
464C2×C4—472no—
465C2×C4—476no—
466C2×C4—472no—
467C2×C4—4712no—
468C2×C4—9416no—
469C2×C4—9424no—
470C2×C4—9448no—
471C2×C4—946no—
472C2×C4—942no—
473C2×C4—9412no—
474C2×C4—9424no—
475C2×C4—9448no—
476C2×C4—9424no—
477C2×C4#41×D4714yes—
478C2×C4C2×#10625354112yes—
479C2×C4C24×D4714yes—
480C2×C4C24×D4716yes—
481C2×C4C24×D47112yes—
482C2×C4C24×D4712yes—
483C2×C4C24×D4716yes—
484C22—478no—
485C22—944no—
486C22—944no—
487C22—9416no—
488C22—946no—
489C22—946no—
490C22—946no—
491C22—18812no—
492C22C2×#41×D4712yes—
493C22C22×#1062535412yes—
494C222−1+4×D4712yes—
495C22C25×D4711yescharacteristic
496C4—212no—
497C4—24no—
498C4—26no—
499C4—474no—
500C4—946no—
501C4—942no—
502C4—9412no—
503C4C2×#41×D4714yes—
504C4C22×#10625354112yes—
505C2—948no—
506C2—1886no—
507C2#2025700814yesdirect product factor
508C2C22×#41×D4711yescharacteristic
509C2C23×#1062535411yescharacteristic
510C2C2×2−1+4×D4711yescharacteristic
subgroupG/H
centerC23C24×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC2×C47×#557C2
socleC23×C47C25
hypercenterC2×#557D47
hypocenterC47C2×#3482
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. #20257008C2direct (N × H)
2. C2×#15689846C2semidirect
3. C2×#15689811C2semidirect
4. C2×Q8×D188C2semidirect
5. C2×#15689785C2semidirect
6. C2×#15689776C2semidirect
7. C2×#15689678C2semidirect
8. C2×#15689658C2semidirect
9. C2×C4×#10625354C2semidirect
10. C2×C4×#10625354C2semidirect
11. C2×C4×Q8×D47C2semidirect
12. C2×C47×#557C2semidirect
16. #15689846C22semidirect
17. #15689811C22semidirect
18. Q8×D188C22semidirect
19. #15689785C22semidirect
20. #15689776C22semidirect
21. #15689678C22semidirect
23. #15689658C22semidirect
32. C4×#10625354C22semidirect
33. C4×#10625354C22semidirect
38. C4×Q8×D47C22semidirect
52. C47×#557C22semidirect
132. C2×Q8×C47#41semidirect
149. C2×D188#41semidirect
199. Q8×C47C2×#41semidirect
220. D188C2×#41semidirect
251. C2×#557D47semidirect
262. C2×C47#3482semidirect
271. #557C2×D47semidirect
313. C47C2×#3482semidirect
393. C2×Q8#10625354semidirect
456. Q8C2×#10625354semidirect
507. C2#20257008direct (N × H)

Maximal subgroups

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

subgroupindex
#202570082
C2×#156898462
C2×#156898112
C2×Q8×D1882
C2×#156897852
C2×#156897762
C2×#156896782
C2×#156896582
C2×C4×#106253542
C2×C4×Q8×D472
C2×C47×#5572
C2×#348247