Finite Group Databasebeta

← C2×#20257405 #27124548

Subgroups — C2×#20257405 #27124548

Order 12032.

7118 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257405C2132yesmaximal, direct product factor
2C2×Q8×Dic94C211yesmaximal, characteristic, complemented
3C2×#15689843C213yesmaximal, complemented
4C2×#15689841C216yesmaximal, complemented
5C2×#15689838C213yesmaximal, complemented
6C2×Q8×D188C211yesmaximal, characteristic, complemented
7C2×#15689780C212yesmaximal, complemented
8C2×#15689738C216yesmaximal, complemented
9C2×#15689680C213yesmaximal, complemented
10C2×#15689657C213yesmaximal, complemented
11C2×#15689599C212yesmaximal, complemented
12C2×C4×#41×C47C211yesmaximal, characteristic, complemented
13C22×Q8×D47C2212yes—
14Q8×Dic94C22116yescomplemented
15#15689843C22148yescomplemented
16#15689841C22196yescomplemented
17#15689838C22148yescomplemented
18Q8×D188C22116yescomplemented
19#15689780C22132yescomplemented
20#15689738C22196yescomplemented
21#15689680C22148yescomplemented
22C2×#13094024C2213yes—
23C2×#13094024C2213yes—
24C2×#13094023C2216yes—
25#15689657C22148yescomplemented
26C2×#13094018C2216yes—
27C2×#13094017C22112yes—
28C2×#13094016C2213yes—
29C2×#13094012C2211yescharacteristic
30C2×#13094011C2216yes—
31C2×#13094010C22112yes—
32C2×#13094007C2216yes—
33C2×#13094004C2216yes—
34C2×#13094003C2216yes—
35C2×#13094003C2213yes—
36C2×#13094003C2216yes—
37C2×#13093998C2213yes—
38C2×#13093995C2216yes—
39#15689599C22132yescomplemented
40C2×Q8×Dic47C2212yes—
41C2×C4×Dic94C2216yes—
42C2×C4×Dic94C2213yes—
43C2×C4×Dic94C2216yes—
44C2×#13093989C2216yes—
45C2×C4×D188C2213yes—
46C2×C4×C22⋊D47C2216yes—
47C2×#13093983C2216yes—
48C2×#13093982C2216yes—
49C2×#13093981C2216yes—
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—
58C22×Dic94C2316yes—
59C22×Dic94C2311yescharacteristic
60C22×Dic94C2316yes—
61C2×Q8×D47—216no—
62C2×Q8×D47C23116yes—
63C22×D188C2311yescharacteristic
64C22×C22⋊D47C2316yes—
65C22×C4×D47C2316yes—
66#13094024C23124yes—
67#13094024C23124yes—
68#13094023C23148yes—
69#13094018C23148yes—
70#13094017C23196yes—
71#13094016C23124yes—
72#13094012C2318yes—
73#13094011C23148yes—
74#13094010C23196yes—
75#13094007C23148yes—
76#13094004C23148yes—
77#13094003C23148yes—
78#13094003C23148yes—
79#13094003C23124yes—
80#13093998C23124yes—
81#13093995C23148yes—
82Q8×Dic47C23116yes—
83C4×Dic94C23148yes—
84C4×Dic94C23148yes—
85C4×Dic94C23124yes—
86#13093989C23148yes—
87C4×D188C23124yes—
88C4×C22⋊D47C23148yes—
89#13093983C23148yes—
90#13093982C23148yes—
91#13093981C23148yes—
92C2×C4⋊Dic47C2316yes—
93C2×C4⋊Dic47C2313yes—
94C2×C4⋊Dic47C2313yes—
95C2×C4⋊Dic47C2316yes—
96C2×Dic47⋊C4C2316yes—
97C2×Dic47⋊C4C2316yes—
98C2×Dic47⋊C4C2316yes—
99C2×Dic47⋊C4C2316yes—
100C2×C22⋊Dic47C23112yes—
101C2×C22⋊Dic47C2316yes—
102C2×(C2×D47)⋊C4C2312yes—
103C2×(C2×D47)⋊C4C2316yes—
104C2×(C2×D47)⋊C4C2312yes—
105C2×(C2×D47)⋊C4C2316yes—
106C2×C4×Dic47C2316yes—
107C2×C4×Dic47C23112yes—
108C2×#41×C47—28no—
109C2×#41×C47C2318yes—
110C22×Q8×C47C2311yescharacteristic
111C22×D4×C47C2313yes—
112C23×C4×C47C2313yes—
113C23×C4×C47C2313yes—
114C23×C4×C47C2313yes—
115C4×Q8×C47C2318yes—
116C4×Q8×C47C2318yes—
117C4×D4×C47C23124yes—
118C4×D4×C47C23124yes—
119#117×C47C23124yes—
120C2×C4⋊C4×C47C2313yes—
121C2×C4⋊C4×C47C2313yes—
122C2×C22⋊C4×C47C2316yes—
123C2×C42×C47C2311yescharacteristic
124C2×C42×C47C2313yes—
125C2×C42×C47C2313yes—
126C2×C42×C47C2313yes—
127C2×C42×C47C23124yes—
128C24×C47C2413yes—
129C23×D47C2412yes—
130#41×C47—232no—
131C2×Q8×C47#4118yes—
132C2×Q8×C47C2414yes—
133C2×D4×C47—212no—
134C2×D4×C47C24112yes—
135C22×C4×C47—212no—
136C22×C4×C47—212no—
137C22×C4×C47—46no—
138C22×C4×C47C2411yescharacteristic
139C22×C4×C47C2411yescharacteristic
140C22×C4×C47C2413yes—
141C22×C4×C47C2413yes—
142C22×C4×C47C24112yes—
143C22×C4×C47C24112yes—
144C22×C4×C47C24112yes—
145C22×C4×C47C2413yes—
146C22×C4×C47C2411yescharacteristic
147C2×C22⋊D47—412no—
148C2×C22⋊D47C24124yes—
149C22×Dic47C2412yes—
150C22×Dic47C2416yes—
151C22×Dic47C2416yes—
152Q8×D47—264no—
153C2×D188—24no—
154C2×D188C2414yes—
155C2×C4×D47—224no—
156C2×C4×D47C24124yes—
157C2×Dic94—224no—
158C2×Dic94—24no—
159C2×Dic94—412no—
160C2×Dic94C24124yes—
161C2×Dic94C2414yes—
162C2×Dic94C24124yes—
163C4⋊C4×C47C24112yes—
164C4⋊C4×C47C24112yes—
165C22⋊C4×C47C24124yes—
166C42×C47C2414yes—
167C42×C47C24112yes—
168C42×C47C24112yes—
169C42×C47C24112yes—
170C22⋊Dic47C24148yes—
171C22⋊Dic47C24124yes—
172(C2×D47)⋊C4C2418yes—
173(C2×D47)⋊C4C24124yes—
174(C2×D47)⋊C4C2418yes—
175(C2×D47)⋊C4C24124yes—
176C4⋊Dic47C24124yes—
177C4⋊Dic47C24112yes—
178C4⋊Dic47C24112yes—
179C4⋊Dic47C24124yes—
180Dic47⋊C4C24124yes—
181Dic47⋊C4C24124yes—
182Dic47⋊C4C24124yes—
183Dic47⋊C4C24124yes—
184C4×Dic47C24124yes—
185C4×Dic47C24148yes—
186C23×C47—26no—
187C23×C47—26no—
188C23×C47—26no—
189C23×C47C2516yes—
190C23×C47C2511yescharacteristic
191C22×D47—24no—
192C22×D47—24no—
193C22×D47—24no—
194C22×D47C2514yes—
195Q8×C47C2×#41116yes—
196D4×C47—224no—
197C2×C4×C47—22no—
198C2×C4×C47—26no—
199C2×C4×C47—26no—
200C2×C4×C47—22no—
201C2×C4×C47—224no—
202C2×C4×C47—224no—
203C2×C4×C47—412no—
204C2×C4×C47C2×#41112yes—
205C2×C4×C472−1+414yes—
206C2×C4×C47C2516yes—
207C2×C4×C47C2512yes—
208C2×C4×C47C2512yes—
209C2×C4×C47C2516yes—
210C2×C4×C47C2516yes—
211C2×C4×C47C2512yes—
212C22⋊D47—424no—
213C2×Dic47—212no—
214C2×Dic47—24no—
215C2×Dic47—212no—
216C2×Dic47C2514yes—
217C2×Dic47C25112yes—
218C2×Dic47C25112yes—
219D188—28no—
220C4×D47—248no—
221Dic94—248no—
222Dic94—28no—
223Dic94—424no—
224C2×#3518—471nomaximal, Sylow
225C22×C47—26no—
226C22×C47—26no—
227C22×C47—26no—
228C22×C47—412no—
229C22×C47C22×#4112yes—
230C22×C47C2×2−1+412yes—
231C22×C47#58512yes—
232C22×C47C2611yescharacteristic
233C2×D47—24no—
234C2×D47—24no—
235C2×D47—24no—
236C2×D47—48no—
237C4×C47—22no—
238C4×C47—26no—
239C4×C47—26no—
240C4×C47—22no—
241C4×C47C22×#41112yes—
242C4×C47C2×2−1+414yes—
243Dic47—212no—
244Dic47—24no—
245Dic47—212no—
246#3518—4732no—
247C2×Q82—471no—
248C2×#556—473no—
249C2×#556—476no—
250C2×#555—473no—
251C2×D4×Q8—471no—
252C2×#548—472no—
253C2×#542—476no—
254C2×#533—473no—
255C2×#529—473no—
256C2×#519—472no—
257C2×C4×#41D4711yescharacteristic, complemented
258C2×C47—46no—
259C2×C47#351814yescomplemented
260C2×C47C23×#4111yescharacteristic
261C2×C47C22×2−1+411yescharacteristic
262C2×C47C2×#58511yescharacteristic
263D47—44no—
264C22×#41C2×D4711yescharacteristic
265C23×Q8—472no—
266Q82—4716no—
267#556—4748no—
268#556—4796no—
269#555—4748no—
270D4×Q8—4716no—
271#548—4732no—
272#542—4796no—
273#533—4748no—
274C2×C4⋊Q8—476no—
275C2×C4⋊Q8—473no—
276C2×C4⋊Q8—473no—
277#529—4748no—
278C2×#126—4712no—
279C2×#126—476no—
280C2×#125—473no—
281C2×#124—476no—
282C2×#124—4712no—
283C2×#124—471no—
284C2×#123—476no—
285C2×C22⋊Q8—473no—
286C2×C22⋊Q8—476no—
287C2×C22⋊Q8—476no—
288C2×C22⋊Q8—476no—
289C2×#121—476no—
290C2×#121—473no—
291#519—4732no—
292C4×#41C2×D47116yescomplemented
293C2×C4×Q8—472no—
294C2×C4×Q8—476no—
295C2×C4×Q8—476no—
296C2×C4×Q8—473no—
297C2×C4×Q8—476no—
298C2×C4×Q8C2×D4711yescharacteristic
299C2×C4×Q8C2×D4711yescharacteristic
300C2×C4×D4—476no—
301C2×C4×D4—473no—
302C2×C4×D4C2×D4713yes—
303C2×C4×D4C2×D4713yes—
304C2×#117—476no—
305C2×#117—476no—
306C2×#117—476no—
307C2×#117C2×D4713yes—
308C22×C42C2×D4713yes—
309C47C2×#351811yescharacteristic, complemented, Sylow
310C2×#41—28no—
311C2×#41C22×D4718yes—
312C22×Q8—4716no—
313C22×Q8—471no—
314C22×Q8—476no—
315C22×Q8—476no—
316C22×Q8—9416no—
317C22×Q8C22×D4711yescharacteristic
318C22×D4—476no—
319C22×D4—471no—
320C22×D4C22×D4713yes—
321C23×C4—476no—
322C23×C4C22×D4713yes—
323C23×C4C22×D4713yes—
324C23×C4C22×D4713yes—
325C4⋊Q8—4748no—
326C4⋊Q8—4724no—
327C4⋊Q8—4724no—
328#126—4796no—
329#126—4748no—
330#125—4724no—
331#124—4748no—
332#124—4796no—
333#124—478no—
334#123—4748no—
335C22⋊Q8—4724no—
336C22⋊Q8—4748no—
337C22⋊Q8—4748no—
338C22⋊Q8—4748no—
339#121—4748no—
340#121—4724no—
341C4×Q8—4716no—
342C4×Q8—4748no—
343C4×Q8—4748no—
344C4×Q8—4724no—
345C4×Q8—4748no—
346C4×Q8C22×D4718yes—
347C4×Q8C22×D4718yes—
348C4×D4—4748no—
349C4×D4—4724no—
350C4×D4C22×D47124yes—
351C4×D4C22×D47124yes—
352#117—4748no—
353#117—4748no—
354#117—4748no—
355#117C22×D47124yes—
356C2×C4⋊C4—473no—
357C2×C4⋊C4—473no—
358C2×C4⋊C4—476no—
359C2×C4⋊C4—476no—
360C2×C4⋊C4—476no—
361C2×C4⋊C4—476no—
362C2×C4⋊C4—476no—
363C2×C4⋊C4—476no—
364C2×C4⋊C4C22×D4713yes—
365C2×C4⋊C4C22×D4713yes—
366C2×C22⋊C4—4712no—
367C2×C22⋊C4—476no—
368C2×C22⋊C4—476no—
369C2×C22⋊C4—472no—
370C2×C22⋊C4—472no—
371C2×C22⋊C4—476no—
372C2×C22⋊C4C22×D4716yes—
373C2×C42—4712no—
374C2×C42—476no—
375C2×C42C22×D4713yes—
376C2×C42C22×D4713yes—
377C2×C42C22×D4713yes—
378C2×C42C22×D47124yes—
379C2×C42C22×D4711yescharacteristic
380C24—472no—
381C24C23×D4713yes—
382#41—232no—
383C2×Q8—474no—
384C2×Q8—4724no—
385C2×Q8—4724no—
386C2×Q8—9464no—
387C2×Q8—9424no—
388C2×Q8—944no—
389C2×Q8—18812no—
390C2×Q8#1062535018yes—
391C2×Q8C23×D4714yes—
392C2×D4—212no—
393C2×D4—4724no—
394C2×D4—474no—
395C2×D4—944no—
396C2×D4—18812no—
397C2×D4C23×D47112yes—
398C22×C4—212no—
399C22×C4—212no—
400C22×C4—46no—
401C22×C4—472no—
402C22×C4—476no—
403C22×C4—476no—
404C22×C4—4724no—
405C22×C4—9424no—
406C22×C4C23×D47112yes—
407C22×C4C23×D47112yes—
408C22×C4C23×D4713yes—
409C22×C4C23×D4711yescharacteristic
410C22×C4C23×D4711yescharacteristic
411C22×C4C23×D4713yes—
412C22×C4C23×D47112yes—
413C22×C4C23×D4713yes—
414C22×C4C23×D4711yescharacteristic
415C4⋊C4—4724no—
416C4⋊C4—4724no—
417C4⋊C4—4724no—
418C4⋊C4—4724no—
419C4⋊C4—4712no—
420C4⋊C4—4712no—
421C4⋊C4—4724no—
422C4⋊C4—4724no—
423C4⋊C4C23×D47112yes—
424C4⋊C4C23×D47112yes—
425C22⋊C4—4748no—
426C22⋊C4—4724no—
427C22⋊C4—4724no—
428C22⋊C4—478no—
429C22⋊C4—478no—
430C22⋊C4—4724no—
431C22⋊C4C23×D47124yes—
432C42—4748no—
433C42—4724no—
434C42C23×D4714yes—
435C42C23×D47112yes—
436C42C23×D47112yes—
437C42C23×D47112yes—
438C23—26no—
439C23—26no—
440C23—26no—
441C23—474no—
442C23—944no—
443C23—944no—
444C23—944no—
445C23C24×D4716yes—
446C23C24×D4711yescharacteristic
447Q8—9448no—
448Q8—948no—
449Q8—18824no—
450Q8C2×#10625350116yes—
451D4—224no—
452D4—948no—
453D4—18824no—
454C2×C4—22no—
455C2×C4—26no—
456C2×C4—22no—
457C2×C4—26no—
458C2×C4—224no—
459C2×C4—224no—
460C2×C4—412no—
461C2×C4—474no—
462C2×C4—4712no—
463C2×C4—4712no—
464C2×C4—9412no—
465C2×C4—944no—
466C2×C4—9412no—
467C2×C4—9448no—
468C2×C4C2×#10625350112yes—
469C2×C4#1309407114yes—
470C2×C4C24×D4716yes—
471C2×C4C24×D4712yes—
472C2×C4C24×D4712yes—
473C2×C4C24×D4716yes—
474C2×C4C24×D4716yes—
475C2×C4C24×D4712yes—
476C22—26no—
477C22—26no—
478C22—26no—
479C22—412no—
480C22—944no—
481C22—944no—
482C22—944no—
483C22—1888no—
484C22C22×#1062535012yes—
485C22C2×#1309407112yes—
486C22#1568995912yes—
487C22C25×D4711yescharacteristic
488C4—22no—
489C4—26no—
490C4—22no—
491C4—26no—
492C4—9412no—
493C4—944no—
494C4—9412no—
495C4C22×#10625350112yes—
496C4C2×#1309407114yes—
497C2—46no—
498C2—1884no—
499C2#2025740514yesdirect product factor
500C2C23×#1062535011yescharacteristic
501C2C22×#1309407111yescharacteristic
502C2C2×#1568995911yescharacteristic
subgroupG/H
centerC23C24×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC2×C4×#41×C47C2
socleC23×C47C25
hypercenterC2×C4×#41D47
hypocenterC47C2×#3518
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. #20257405C2direct (N × H)
2. C2×Q8×Dic94C2semidirect
3. C2×#15689843C2semidirect
4. C2×#15689841C2semidirect
5. C2×#15689838C2semidirect
6. C2×Q8×D188C2semidirect
7. C2×#15689780C2semidirect
8. C2×#15689738C2semidirect
9. C2×#15689680C2semidirect
10. C2×#15689657C2semidirect
11. C2×#15689599C2semidirect
12. C2×C4×#41×C47C2semidirect
14. Q8×Dic94C22semidirect
15. #15689843C22semidirect
16. #15689841C22semidirect
17. #15689838C22semidirect
18. Q8×D188C22semidirect
19. #15689780C22semidirect
20. #15689738C22semidirect
21. #15689680C22semidirect
25. #15689657C22semidirect
39. #15689599C22semidirect
51. C4×#41×C47C22semidirect
257. C2×C4×#41D47semidirect
259. C2×C47#3518semidirect
292. C4×#41C2×D47semidirect
309. C47C2×#3518semidirect
499. C2#20257405direct (N × H)

Maximal subgroups

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

subgroupindex
#202574052
C2×Q8×Dic942
C2×#156898432
C2×#156898412
C2×#156898382
C2×Q8×D1882
C2×#156897802
C2×#156897382
C2×#156896802
C2×#156896572
C2×#156895992
C2×C4×#41×C472
C2×#351847