Finite Group Databasebeta

← C2×#20257010 #27123376

Subgroups — C2×#20257010 #27123376

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#20257010C2132yesmaximal, direct product factor
2C2×#15689852C211yesmaximal, characteristic, complemented
3C2×#15689843C213yesmaximal, complemented
4C2×#15689812C216yesmaximal, complemented
5C2×Q8×D188C211yesmaximal, characteristic, complemented
6C2×#15689780C212yesmaximal, complemented
7C2×#15689778C213yesmaximal, complemented
8C2×#15689680C213yesmaximal, complemented
9C2×#15689655C216yesmaximal, complemented
10C2×C4×#10625350C213yesmaximal, complemented
11C2×C4×Q8×D47C212yesmaximal, complemented
12C2×C4×#41×C47C211yesmaximal, characteristic, complemented
13C22×#10625350C2213yes—
14C22×Q8×D47C2212yes—
15#15689852C22116yescomplemented
16#15689843C22148yescomplemented
17#15689812C22196yescomplemented
18Q8×D188C22116yescomplemented
19#15689780C22132yescomplemented
20#15689778C22148yescomplemented
21#15689680C22148yescomplemented
22C2×#13094024C2213yes—
23C2×#13094024C2213yes—
24#15689655C22196yescomplemented
25C2×#13094017C22112yes—
26C2×#13094016C2216yes—
27C2×#13094015C2216yes—
28C2×#13094011C2216yes—
29C2×#13094006C22112yes—
30C2×#13094004C2216yes—
31C2×#13094003C2216yes—
32C2×#13094003C2216yes—
33C2×#13093995C2216yes—
34C4×#10625350C22148yescomplemented
35C2×Q8×Dic47C2212yes—
36C2×C4×Dic94C2216yes—
37C2×C4×Dic94C2213yes—
38C2×C4×Dic94C2213yes—
39C2×#13093989C2216yes—
40C4×Q8×D47C22132yescomplemented
41C2×C4×D188C2211yescharacteristic
42C2×C4×D188C2213yes—
43C2×C4×C22⋊D47C2216yes—
44C2×C4×C22⋊D47C2216yes—
45C2×#13093983C2213yes—
46C2×#13093983C2216yes—
47C2×#13093981C2216yes—
48C2×C4⋊C4×D47C2216yes—
49C2×C42×D47C2216yes—
50C22×#41×C47C2211yescharacteristic
51C4×#41×C47C22116yescomplemented
52C2×C4×Q8×C47C2211yescharacteristic
53C2×C4×Q8×C47C2211yescharacteristic
54C2×C4×D4×C47C2213yes—
55C2×C4×D4×C47C2213yes—
56C2×#117×C47C2213yes—
57C22×C42×C47C2213yes—
58C2×#10625350—224no—
59C2×#10625350C23124yes—
60C22×Dic94C2316yes—
61C22×Dic94C2313yes—
62C2×Q8×D47—216no—
63C2×Q8×D47C23116yes—
64C22×D188C2311yescharacteristic
65C22×C22⋊D47C2316yes—
66C22×C4×D47C2312yes—
67C22×C4×D47C2316yes—
68C22×C4×D47C2316yes—
69#13094024C23124yes—
70#13094024C23124yes—
71#13094017C23196yes—
72#13094016C23148yes—
73#13094015C23148yes—
74#13094011C23148yes—
75#13094006C23196yes—
76#13094004C23148yes—
77#13094003C23148yes—
78#13094003C23148yes—
79#13093995C23148yes—
80Q8×Dic47C23116yes—
81C4×Dic94C23148yes—
82C4×Dic94C23124yes—
83C4×Dic94C23124yes—
84#13093989C23148yes—
85C4×D188C23124yes—
86C4×D188C2318yes—
87C4×C22⋊D47C23148yes—
88C4×C22⋊D47C23148yes—
89#13093983C23124yes—
90#13093983C23148yes—
91#13093981C23148yes—
92C2×C4⋊Dic47C2316yes—
93C2×C4⋊Dic47C2311yescharacteristic
94C2×C4⋊Dic47C2313yes—
95C2×C4⋊Dic47C2316yes—
96C2×C4⋊Dic47C2316yes—
97C2×Dic47⋊C4C23112yes—
98C2×Dic47⋊C4C2316yes—
99C2×Dic47⋊C4C2316yes—
100C4⋊C4×D47C23148yes—
101C2×C22⋊Dic47C2316yes—
102C2×C22⋊Dic47C23112yes—
103C2×(C2×D47)⋊C4C2312yes—
104C2×(C2×D47)⋊C4C2316yes—
105C2×C4×Dic47C2316yes—
106C2×C4×Dic47C2316yes—
107C2×C4×Dic47C2316yes—
108C42×D47C23148yes—
109C2×#41×C47—28no—
110C2×#41×C47C2318yes—
111C22×Q8×C47C2311yescharacteristic
112C22×D4×C47C2313yes—
113C23×C4×C47C2313yes—
114C23×C4×C47C2313yes—
115C23×C4×C47C2313yes—
116C4×Q8×C47C2318yes—
117C4×Q8×C47C2318yes—
118C4×D4×C47C23124yes—
119C4×D4×C47C23124yes—
120#117×C47C23124yes—
121C2×C4⋊C4×C47C2313yes—
122C2×C4⋊C4×C47C2313yes—
123C2×C22⋊C4×C47C2316yes—
124C2×C42×C47C2311yescharacteristic
125C2×C42×C47C2313yes—
126C2×C42×C47C2313yes—
127C2×C42×C47C2313yes—
128C2×C42×C47C23124yes—
129C24×C47C2413yes—
130C23×D47C2412yes—
131#41×C47—232no—
132C2×Q8×C47#4118yescomplemented
133C2×Q8×C47C2414yes—
134C2×D4×C47—212no—
135C2×D4×C47C24112yes—
136C22×C4×C47—212no—
137C22×C4×C47—212no—
138C22×C4×C47—212no—
139C22×C4×C47C2411yescharacteristic
140C22×C4×C47C2413yes—
141C22×C4×C47C2413yes—
142C22×C4×C47C24112yes—
143C22×C4×C47C24112yes—
144C22×C4×C47C24112yes—
145C22×C4×C47C2411yescharacteristic
146C22×C4×C47C2413yes—
147C22×C4×C47C2411yescharacteristic
148C2×C22⋊D47—224no—
149C2×C22⋊D47C24124yes—
150C22×Dic47C2412yes—
151C22×Dic47C2416yes—
152C22×Dic47C2416yes—
153Q8×D47—264no—
154#10625350—296no—
155C2×D188#4118yescomplemented
156C2×D188C2414yes—
157C2×C4×D47—224no—
158C2×C4×D47—224no—
159C2×C4×D47—28no—
160C2×C4×D47C2418yes—
161C2×C4×D47C24124yes—
162C2×C4×D47C24124yes—
163C2×Dic94—224no—
164C2×Dic94—212no—
165C2×Dic94C24124yes—
166C2×Dic94C24112yes—
167C4⋊C4×C47C24112yes—
168C4⋊C4×C47C24112yes—
169C22⋊C4×C47C24124yes—
170C42×C47C2414yes—
171C42×C47C24112yes—
172C42×C47C24112yes—
173C42×C47C24112yes—
174C22⋊Dic47C24124yes—
175C22⋊Dic47C24148yes—
176(C2×D47)⋊C4C2418yes—
177(C2×D47)⋊C4C24124yes—
178C4⋊Dic47C24124yes—
179C4⋊Dic47C2414yes—
180C4⋊Dic47C24112yes—
181C4⋊Dic47C24124yes—
182C4⋊Dic47C24124yes—
183Dic47⋊C4C24148yes—
184Dic47⋊C4C24124yes—
185Dic47⋊C4C24124yes—
186C4×Dic47C24124yes—
187C4×Dic47C24124yes—
188C4×Dic47C24124yes—
189C23×C47—26no—
190C23×C47—26no—
191C23×C47—26no—
192C23×C47C2516yes—
193C23×C47C2511yescharacteristic
194C22×D47—24no—
195C22×D47—24no—
196C22×D47C2×#4118yes—
197C22×D47C2514yes—
198Q8×C47C2×#41116yescomplemented
199D4×C47—224no—
200C2×C4×C47—26no—
201C2×C4×C47—22no—
202C2×C4×C47—26no—
203C2×C4×C47—224no—
204C2×C4×C47—224no—
205C2×C4×C47—224no—
206C2×C4×C47C2×#4114yes—
207C2×C4×C47C2×#41112yes—
208C2×C4×C472−1+414yes—
209C2×C4×C47C2516yes—
210C2×C4×C47C2512yes—
211C2×C4×C47C2516yes—
212C2×C4×C47C2516yes—
213C2×C4×C47C2512yes—
214C2×C4×C47C2512yes—
215C22⋊D47—248no—
216C2×Dic47—212no—
217C2×Dic47—212no—
218C2×Dic47—24no—
219C2×Dic47C2514yes—
220C2×Dic47C25112yes—
221C2×Dic47C25112yes—
222D188C2×#41116yescomplemented
223C4×D47—248no—
224C4×D47—248no—
225C4×D47—216no—
226Dic94—224no—
227Dic94—248no—
228C2×#3482—471nomaximal, Sylow
229C22×C47—26no—
230C22×C47—26no—
231C22×C47—26no—
232C22×C47—412no—
233C22×C47C22×#4112yes—
234C22×C47C22×#4112yes—
235C22×C47C2×2−1+412yes—
236C22×C47C2611yescharacteristic
237C2×D47—24no—
238C2×D47—24no—
239C2×D47—216no—
240C2×D47C22×#4118yes—
241C4×C47—26no—
242C4×C47—22no—
243C4×C47—26no—
244C4×C47C22×#4114yes—
245C4×C47C22×#41112yes—
246C4×C47C2×2−1+414yes—
247Dic47—212no—
248Dic47—212no—
249Dic47—24no—
250#3482—4732no—
251C2×#557—471no—
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—473no—
260C2×C4×#41D4711yescharacteristic, complemented
261C22×C4×Q8—472no—
262C2×C47—46no—
263C2×C47#348214yescomplemented
264C2×C47C23×#4111yescharacteristic
265C2×C47C23×#4111yescharacteristic
266C2×C47C22×2−1+411yescharacteristic
267D47—28no—
268C22×#41—473no—
269C22×#41C2×D4711yescharacteristic
270C23×Q8—472no—
271#557—4716no—
272#556—4748no—
273#552—4796no—
274D4×Q8—4716no—
275#548—4732no—
276#547—4748no—
277#533—4748no—
278C2×C4⋊Q8—473no—
279C2×C4⋊Q8—473no—
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⋊Q8—476no—
289C2×#121—476no—
290C4×#41—4748no—
291C4×#41C2×D47116yescomplemented
292C2×C4×Q8—472no—
293C2×C4×Q8—4732no—
294C2×C4×Q8—476no—
295C2×C4×Q8—476no—
296C2×C4×Q8—473no—
297C2×C4×Q8—473no—
298C2×C4×Q8C2×D4711yescharacteristic
299C2×C4×Q8C2×D4711yescharacteristic
300C2×C4×D4—476no—
301C2×C4×D4—476no—
302C2×C4×D4—471no—
303C2×C4×D4—473no—
304C2×C4×D4C2×D4713yes—
305C2×C4×D4C2×D4713yes—
306C2×#117—473no—
307C2×#117—476no—
308C2×#117—476no—
309C2×#117C2×D4713yes—
310C22×C4⋊C4—476no—
311C22×C42—476no—
312C22×C42C2×D4713yes—
313C47C2×#348211yescharacteristic, complemented, Sylow
314C2×#41—28no—
315C2×#41—4724no—
316C2×#41—9424no—
317C2×#41C22×D4718yes—
318C22×Q8—4716no—
319C22×Q8—476no—
320C22×Q8—473no—
321C22×Q8—9416no—
322C22×Q8C22×D4711yescharacteristic
323C22×D4—476no—
324C22×D4—471no—
325C22×D4C22×D4713yes—
326C23×C4—476no—
327C23×C4—476no—
328C23×C4—472no—
329C23×C4C22×D4713yes—
330C23×C4C22×D4713yes—
331C23×C4C22×D4713yes—
332C4⋊Q8—4724no—
333C4⋊Q8—4724no—
334#126—4796no—
335#125—4748no—
336#125—4748no—
337#124—4748no—
338#123—4796no—
339C22⋊Q8—4748no—
340C22⋊Q8—4748no—
341C22⋊Q8—4748no—
342#121—4748no—
343C4×Q8—4716no—
344C4×Q8—4748no—
345C4×Q8—4748no—
346C4×Q8—4724no—
347C4×Q8—4724no—
348C4×Q8C22×D4718yes—
349C4×Q8C22×D4718yes—
350C4×D4—4748no—
351C4×D4—4748no—
352C4×D4—478no—
353C4×D4—4724no—
354C4×D4C22×D47124yes—
355C4×D4C22×D47124yes—
356#117—4724no—
357#117—4748no—
358#117—4748no—
359#117C22×D47124yes—
360C2×C4⋊C4—476no—
361C2×C4⋊C4—4712no—
362C2×C4⋊C4—471no—
363C2×C4⋊C4—476no—
364C2×C4⋊C4—473no—
365C2×C4⋊C4—476no—
366C2×C4⋊C4—476no—
367C2×C4⋊C4—476no—
368C2×C4⋊C4—4748no—
369C2×C4⋊C4C22×D4713yes—
370C2×C4⋊C4C22×D4713yes—
371C2×C22⋊C4—476no—
372C2×C22⋊C4—4712no—
373C2×C22⋊C4—472no—
374C2×C22⋊C4—476no—
375C2×C22⋊C4C22×D4716yes—
376C2×C42—476no—
377C2×C42—476no—
378C2×C42—476no—
379C2×C42—4748no—
380C2×C42C22×D4713yes—
381C2×C42C22×D4713yes—
382C2×C42C22×D47124yes—
383C2×C42C22×D4711yescharacteristic
384C2×C42C22×D4713yes—
385C24—472no—
386C24C23×D4713yes—
387#41—232no—
388#41—9496no—
389C2×Q8—4724no—
390C2×Q8—4712no—
391C2×Q8—9464no—
392C2×Q8—9412no—
393C2×Q8—9424no—
394C2×Q8#1062535018yescomplemented
395C2×Q8C23×D4714yes—
396C2×D4—212no—
397C2×D4—4724no—
398C2×D4—478no—
399C2×D4—474no—
400C2×D4—9424no—
401C2×D4C23×D47112yes—
402C22×C4—212no—
403C22×C4—212no—
404C22×C4—212no—
405C22×C4—472no—
406C22×C4—476no—
407C22×C4—476no—
408C22×C4—478no—
409C22×C4—4724no—
410C22×C4—4724no—
411C22×C4—9424no—
412C22×C4—9424no—
413C22×C4—948no—
414C22×C4C23×D4713yes—
415C22×C4C23×D4711yescharacteristic
416C22×C4C23×D4713yes—
417C22×C4C23×D47112yes—
418C22×C4C23×D47112yes—
419C22×C4C23×D47112yes—
420C22×C4C23×D4711yescharacteristic
421C22×C4C23×D4713yes—
422C22×C4C23×D4711yescharacteristic
423C4⋊C4—4748no—
424C4⋊C4—4724no—
425C4⋊C4—4724no—
426C4⋊C4—4724no—
427C4⋊C4—474no—
428C4⋊C4—4712no—
429C4⋊C4—4724no—
430C4⋊C4—4724no—
431C4⋊C4C23×D47112yes—
432C4⋊C4C23×D47112yes—
433C22⋊C4—4724no—
434C22⋊C4—4748no—
435C22⋊C4—478no—
436C22⋊C4—4724no—
437C22⋊C4C23×D47124yes—
438C42—4724no—
439C42—4724no—
440C42—4724no—
441C42C23×D47112yes—
442C42C23×D4714yes—
443C42C23×D47112yes—
444C42C23×D47112yes—
445C23—26no—
446C23—26no—
447C23—26no—
448C23—478no—
449C23—474no—
450C23—944no—
451C23—944no—
452C23C24×D4716yes—
453C23C24×D4711yescharacteristic
454Q8—9424no—
455Q8—9448no—
456Q8C2×#10625350116yescomplemented
457D4—224no—
458D4—4716no—
459D4—9448no—
460C2×C4—26no—
461C2×C4—22no—
462C2×C4—26no—
463C2×C4—224no—
464C2×C4—224no—
465C2×C4—224no—
466C2×C4—474no—
467C2×C4—4712no—
468C2×C4—4712no—
469C2×C4—9448no—
470C2×C4—9412no—
471C2×C4—9412no—
472C2×C4—944no—
473C2×C4—9448no—
474C2×C4—9416no—
475C2×C4#41×D4714yes—
476C2×C4C2×#10625350112yes—
477C2×C4#1309407114yes—
478C2×C4C24×D4716yes—
479C2×C4C24×D4712yes—
480C2×C4C24×D4716yes—
481C2×C4C24×D4712yes—
482C2×C4C24×D4716yes—
483C2×C4C24×D4712yes—
484C22—26no—
485C22—26no—
486C22—26no—
487C22—412no—
488C22—478no—
489C22—944no—
490C22—944no—
491C22—9416no—
492C22C2×#41×D4712yes—
493C22C22×#1062535012yes—
494C22C2×#1309407112yes—
495C22C25×D4711yescharacteristic
496C4—26no—
497C4—22no—
498C4—26no—
499C4—9412no—
500C4—9412no—
501C4—944no—
502C4C2×#41×D4714yes—
503C4C22×#10625350112yes—
504C4C2×#1309407114yes—
505C2—46no—
506C2—948no—
507C2#2025701014yesdirect product factor
508C2C22×#41×D4711yescharacteristic
509C2C23×#1062535011yescharacteristic
510C2C22×#1309407111yescharacteristic
subgroupG/H
centerC22×C4C23×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC2×C4×#41×C47C2
socleC23×C47C25
hypercenterC2×C4×#41D47
hypocenterC47C2×#3482
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. #20257010C2direct (N × H)
2. C2×#15689852C2semidirect
3. C2×#15689843C2semidirect
4. C2×#15689812C2semidirect
5. C2×Q8×D188C2semidirect
6. C2×#15689780C2semidirect
7. C2×#15689778C2semidirect
8. C2×#15689680C2semidirect
9. C2×#15689655C2semidirect
10. C2×C4×#10625350C2semidirect
11. C2×C4×Q8×D47C2semidirect
12. C2×C4×#41×C47C2semidirect
15. #15689852C22semidirect
16. #15689843C22semidirect
17. #15689812C22semidirect
18. Q8×D188C22semidirect
19. #15689780C22semidirect
20. #15689778C22semidirect
21. #15689680C22semidirect
24. #15689655C22semidirect
34. C4×#10625350C22semidirect
40. C4×Q8×D47C22semidirect
51. C4×#41×C47C22semidirect
132. C2×Q8×C47#41semidirect
155. C2×D188#41semidirect
198. Q8×C47C2×#41semidirect
222. D188C2×#41semidirect
260. C2×C4×#41D47semidirect
263. C2×C47#3482semidirect
291. C4×#41C2×D47semidirect
313. C47C2×#3482semidirect
394. C2×Q8#10625350semidirect
456. Q8C2×#10625350semidirect
507. C2#20257010direct (N × H)

Maximal subgroups

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

subgroupindex
#202570102
C2×#156898522
C2×#156898432
C2×#156898122
C2×Q8×D1882
C2×#156897802
C2×#156897782
C2×#156896802
C2×#156896552
C2×C4×#106253502
C2×C4×Q8×D472
C2×C4×#41×C472
C2×#348247