Finite Group Databasebeta

← C2×#20257406 #27124544

Subgroups — C2×#20257406 #27124544

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#20257406C2132yesmaximal, direct product factor
2C2×#15689847C216yesmaximal, complemented
3C2×#15689846C213yesmaximal, complemented
4C2×#15689839C213yesmaximal, complemented
5C2×Q8×D188C211yesmaximal, characteristic, complemented
6C2×#15689785C212yesmaximal, complemented
7C2×#15689747C216yesmaximal, complemented
8C2×#15689678C213yesmaximal, complemented
9C2×#15689660C213yesmaximal, complemented
10C2×#15689599C212yesmaximal, complemented
11C2×C4×#10625354C211yesmaximal, characteristic, complemented
12C2×Q82×C47C211yesmaximal, characteristic, complemented
13C22×#10625354C2211yescharacteristic
14C22×Q8×D47C2212yes—
15#15689847C22196yescomplemented
16#15689846C22148yescomplemented
17#15689839C22148yescomplemented
18Q8×D188C22116yescomplemented
19#15689785C22132yescomplemented
20#15689747C22196yescomplemented
21#15689678C22148yescomplemented
22C2×#13094024C2213yes—
23#15689660C22148yescomplemented
24C2×#13094019C22112yes—
25C2×#13094019C2216yes—
26C2×#13094015C2213yes—
27C2×#13094013C22112yes—
28C2×#13094013C2211yescharacteristic
29C2×#13094012C2216yes—
30C2×#13094009C2216yes—
31C2×#13094004C2216yes—
32C2×#13094004C2216yes—
33C2×#13094000C2216yes—
34C2×#13094000C2213yes—
35C2×#13093999C2216yes—
36C2×#13093999C2213yes—
37#15689599C22132yescomplemented
38C4×#10625354C22116yescomplemented
39C2×Q8×Dic47C2211yescharacteristic
40C2×Q8×Dic47C2216yes—
41C2×C4×Dic94C2216yes—
42C2×#13093989C2216yes—
43C2×C4×D188C2213yes—
44C2×C4×D188C2216yes—
45C2×C4×D188C2213yes—
46C2×#13093986C2213yes—
47C2×#13093982C2213yes—
48C2×#13093982C2216yes—
49C2×#13093982C2216yes—
50C2×#13093981C2216yes—
51C2×C42×D47C2213yes—
52Q82×C47C22116yescomplemented
53C2×C4⋊Q8×C47C2216yes—
54C2×C4⋊Q8×C47C2213yes—
55C2×C4×Q8×C47C2211yescharacteristic
56C2×C4×Q8×C47C2212yes—
57C2×C4×Q8×C47C2213yes—
58C2×#10625354—28no—
59C2×#10625354C2318yes—
60C22×Dic94C2316yes—
61C2×Q8×D47—216no—
62C2×Q8×D47C23116yes—
63C22×D188C2313yes—
64C22×D188C2311yescharacteristic
65C22×D188C2316yes—
66C22×C4×D47C2313yes—
67C22×C4×D47C2313yes—
68C22×C4×D47C2316yes—
69C22×C4×D47C2313yes—
70#13094024C23124yes—
71#13094019C23196yes—
72#13094019C23148yes—
73#13094015C23124yes—
74#13094013C23196yes—
75#13094013C2318yes—
76#13094012C23148yes—
77#13094009C23148yes—
78#13094004C23148yes—
79#13094004C23148yes—
80#13094000C23148yes—
81#13094000C23124yes—
82#13093999C23148yes—
83#13093999C23124yes—
84Q8×Dic47C23148yes—
85Q8×Dic47C2318yes—
86C4×Dic94C23148yes—
87#13093989C23148yes—
88C4×D188C23148yes—
89C4×D188C23124yes—
90C4×D188C23124yes—
91#13093986C23124yes—
92#13093982C23148yes—
93#13093982C23148yes—
94#13093982C23124yes—
95#13093981C23148yes—
96C2×C4⋊Dic47C2313yes—
97C2×C4⋊Dic47C2313yes—
98C2×C4⋊Dic47C2316yes—
99C2×C4⋊Dic47C2316yes—
100C2×Dic47⋊C4C2316yes—
101C2×Dic47⋊C4C2316yes—
102C2×(C2×D47)⋊C4C2312yes—
103C2×(C2×D47)⋊C4C2316yes—
104C2×(C2×D47)⋊C4C2316yes—
105C2×(C2×D47)⋊C4C23112yes—
106C2×(C2×D47)⋊C4C2312yes—
107C2×(C2×D47)⋊C4C2316yes—
108C2×(C2×D47)⋊C4C2316yes—
109C2×C4×Dic47C2311yescharacteristic
110C2×C4×Dic47C2313yes—
111C2×C4×Dic47C2313yes—
112C2×C4×Dic47C23112yes—
113C42×D47C23124yes—
114C22×Q8×C47C2311yescharacteristic
115C22×Q8×C47C2316yes—
116C22×Q8×C47C2311yescharacteristic
117C4⋊Q8×C47C23148yes—
118C4⋊Q8×C47C23124yes—
119C4×Q8×C47C23116yes—
120C4×Q8×C47C2318yes—
121C4×Q8×C47C23124yes—
122C2×C4⋊C4×C47C2313yes—
123C2×C4⋊C4×C47C2313yes—
124C2×C4⋊C4×C47C2316yes—
125C2×C4⋊C4×C47C2316yes—
126C2×C42×C47C2313yes—
127C2×C42×C47C2316yes—
128C23×D47C2412yes—
129C23×D47C2413yes—
130C2×Q8×C47—24no—
131C2×Q8×C47—412no—
132C2×Q8×C47#4118yes—
133C2×Q8×C47C2414yes—
134C2×Q8×C47C24124yes—
135C2×Q8×C47C2414yes—
136C22×C4×C47C2412yes—
137C22×C4×C47C2416yes—
138C22×C4×C47C2413yes—
139C22×C4×C47C2413yes—
140C22×C4×C47C2411yescharacteristic
141C22×Dic47C2413yes—
142C22×Dic47C2411yescharacteristic
143C22×Dic47C2411yescharacteristic
144C22×Dic47C2416yes—
145#10625354—232no—
146Q8×D47—264no—
147C2×D188—212no—
148C2×D188—24no—
149C2×D188—412no—
150C2×D188C24112yes—
151C2×D188C2414yes—
152C2×D188C24124yes—
153C2×C4×D47—224no—
154C2×C4×D47—212no—
155C2×C4×D47—212no—
156C2×C4×D47—46no—
157C2×C4×D47C24112yes—
158C2×C4×D47C24112yes—
159C2×C4×D47C24124yes—
160C2×C4×D47C24112yes—
161C2×Dic94—224no—
162C2×Dic94C24124yes—
163C4⋊C4×C47C24112yes—
164C4⋊C4×C47C24112yes—
165C4⋊C4×C47C24124yes—
166C4⋊C4×C47C24124yes—
167C42×C47C24112yes—
168C42×C47C24124yes—
169(C2×D47)⋊C4C2418yes—
170(C2×D47)⋊C4C24124yes—
171(C2×D47)⋊C4C24124yes—
172(C2×D47)⋊C4C24148yes—
173(C2×D47)⋊C4C2418yes—
174(C2×D47)⋊C4C24124yes—
175(C2×D47)⋊C4C24124yes—
176C4⋊Dic47C24112yes—
177C4⋊Dic47C24112yes—
178C4⋊Dic47C24124yes—
179C4⋊Dic47C24124yes—
180Dic47⋊C4C24124yes—
181Dic47⋊C4C24124yes—
182C4×Dic47C2414yes—
183C4×Dic47C24112yes—
184C4×Dic47C24112yes—
185C4×Dic47C24148yes—
186C23×C47C2511yescharacteristic
187C22×D47—24no—
188C22×D47—26no—
189C22×D47—26no—
190C22×D47—24no—
191C22×D47—26no—
192C22×D47—24no—
193C22×D47C2514yes—
194C22×D47C2516yes—
195Q8×C47—28no—
196Q8×C47—424no—
197Q8×C47C2×#41116yes—
198C2×C4×C47—24no—
199C2×C4×C47—26no—
200C2×C4×C47—212no—
201C2×C4×C47C2×#41112yes—
202C2×C4×C472−1+414yes—
203C2×C4×C47C25112yes—
204C2×C4×C47C2516yes—
205C2×C4×C47C2514yes—
206C2×C4×C47C2516yes—
207C2×C4×C47C2512yes—
208C2×Dic47—212no—
209C2×Dic47—22no—
210C2×Dic47—26no—
211C2×Dic47—22no—
212C2×Dic47C2516yes—
213C2×Dic47C2512yes—
214C2×Dic47C2512yes—
215C2×Dic47C25112yes—
216D188—224no—
217D188—28no—
218D188—424no—
219C4×D47—248no—
220C4×D47—224no—
221C4×D47—224no—
222C4×D47—412no—
223Dic94—248no—
224C2×#3518—471nomaximal, Sylow
225C22×C47C22×#4112yes—
226C22×C47C2×2−1+412yes—
227C22×C47#58512yes—
228C22×C47C2611yescharacteristic
229C2×D47—24no—
230C2×D47—26no—
231C2×D47—26no—
232C2×D47—24no—
233C2×D47—26no—
234C2×D47—24no—
235C2×D47—48no—
236C2×D47—412no—
237C4×C47—24no—
238C4×C47—26no—
239C4×C47—212no—
240C4×C47C22×#41112yes—
241C4×C47C2×2−1+414yes—
242Dic47—212no—
243Dic47—22no—
244Dic47—26no—
245Dic47—22no—
246#3518—4732no—
247C2×Q82D4711yescharacteristic, complemented
248C2×#556—476no—
249C2×#556—473no—
250C2×#555—473no—
251C2×D4×Q8—471no—
252C2×#548—472no—
253C2×#542—476no—
254C2×#533—473no—
255C2×#529—473no—
256C2×#519—472no—
257C2×C4×#41—471no—
258C2×C47#351814yescomplemented
259C2×C47C23×#4111yescharacteristic
260C2×C47C22×2−1+411yescharacteristic
261C2×C47C2×#58511yescharacteristic
262D47—44no—
263D47—46no—
264C22×#41—471no—
265C23×Q8—472no—
266Q82C2×D47116yescomplemented
267#556—4796no—
268#556—4748no—
269#555—4748no—
270D4×Q8—4716no—
271#548—4732no—
272#542—4796no—
273#533—4748no—
274C2×C4⋊Q8—473no—
275C2×C4⋊Q8C2×D4713yes—
276C2×C4⋊Q8C2×D4716yes—
277#529—4748no—
278C2×#126—476no—
279C2×#126—4712no—
280C2×#125—473no—
281C2×#124—471no—
282C2×#124—4712no—
283C2×#124—476no—
284C2×#123—476no—
285C2×C22⋊Q8—473no—
286C2×C22⋊Q8—476no—
287C2×C22⋊Q8—476no—
288C2×C22⋊Q8—476no—
289C2×#121—473no—
290C2×#121—476no—
291#519—4732no—
292C4×#41—4716no—
293C2×C4×Q8—471no—
294C2×C4×Q8—476no—
295C2×C4×Q8—476no—
296C2×C4×Q8—476no—
297C2×C4×Q8C2×D4711yescharacteristic
298C2×C4×Q8C2×D4712yes—
299C2×C4×Q8C2×D4713yes—
300C2×C4×D4—473no—
301C2×C4×D4—473no—
302C2×C4×D4—476no—
303C2×C4×D4—473no—
304C2×#117—473no—
305C2×#117—476no—
306C2×#117—476no—
307C2×#117—476no—
308C22×C42—473no—
309C47C2×#351811yescharacteristic, complemented, Sylow
310C2×#41—478no—
311C2×#41—948no—
312C22×Q8—4716no—
313C22×Q8—476no—
314C22×Q8—9416no—
315C22×Q8C22×D4711yescharacteristic
316C22×Q8C22×D4716yes—
317C22×Q8C22×D4711yescharacteristic
318C22×D4—471no—
319C22×D4—473no—
320C22×D4—476no—
321C23×C4—476no—
322C23×C4—473no—
323C23×C4—473no—
324C23×C4—473no—
325C4⋊Q8—4724no—
326C4⋊Q8C22×D47124yes—
327C4⋊Q8C22×D47148yes—
328#126—4748no—
329#126—4796no—
330#125—4724no—
331#124—478no—
332#124—4796no—
333#124—4748no—
334#123—4748no—
335C22⋊Q8—4724no—
336C22⋊Q8—4748no—
337C22⋊Q8—4748no—
338C22⋊Q8—4748no—
339#121—4724no—
340#121—4748no—
341C4×Q8—478no—
342C4×Q8—4748no—
343C4×Q8—4748no—
344C4×Q8—4748no—
345C4×Q8C22×D4718yes—
346C4×Q8C22×D47116yes—
347C4×Q8C22×D47124yes—
348C4×D4—4724no—
349C4×D4—4724no—
350C4×D4—4748no—
351C4×D4—4724no—
352#117—4724no—
353#117—4748no—
354#117—4748no—
355#117—4748no—
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⋊C4C22×D4713yes—
363C2×C4⋊C4C22×D4713yes—
364C2×C4⋊C4C22×D4716yes—
365C2×C4⋊C4C22×D4716yes—
366C2×C22⋊C4—476no—
367C2×C22⋊C4—472no—
368C2×C22⋊C4—476no—
369C2×C22⋊C4—4712no—
370C2×C22⋊C4—472no—
371C2×C22⋊C4—476no—
372C2×C22⋊C4—476no—
373C2×C42—473no—
374C2×C42—4712no—
375C2×C42—473no—
376C2×C42—471no—
377C2×C42—4724no—
378C2×C42C22×D4716yes—
379C2×C42C22×D4713yes—
380C24—472no—
381C24—473no—
382#41—9432no—
383C2×Q8—24no—
384C2×Q8—412no—
385C2×Q8—4724no—
386C2×Q8—9464no—
387C2×Q8—9424no—
388C2×Q8#1062535418yes—
389C2×Q8C23×D4714yes—
390C2×Q8C23×D47124yes—
391C2×Q8C23×D4714yes—
392C2×D4—474no—
393C2×D4—4712no—
394C2×D4—4724no—
395C2×D4—9412no—
396C2×D4—944no—
397C2×D4—18812no—
398C22×C4—473no—
399C22×C4—471no—
400C22×C4—471no—
401C22×C4—476no—
402C22×C4—4712no—
403C22×C4—4712no—
404C22×C4—4724no—
405C22×C4—4712no—
406C22×C4—9424no—
407C22×C4—9412no—
408C22×C4—9412no—
409C22×C4—1886no—
410C22×C4C23×D4716yes—
411C22×C4C23×D4713yes—
412C22×C4C23×D4712yes—
413C22×C4C23×D4713yes—
414C22×C4C23×D4711yescharacteristic
415C4⋊C4—4724no—
416C4⋊C4—4724no—
417C4⋊C4—4712no—
418C4⋊C4—4712no—
419C4⋊C4—4724no—
420C4⋊C4—4724no—
421C4⋊C4C23×D47112yes—
422C4⋊C4C23×D47112yes—
423C4⋊C4C23×D47124yes—
424C4⋊C4C23×D47124yes—
425C22⋊C4—4724no—
426C22⋊C4—478no—
427C22⋊C4—4724no—
428C22⋊C4—4748no—
429C22⋊C4—478no—
430C22⋊C4—4724no—
431C22⋊C4—4724no—
432C42—474no—
433C42—4712no—
434C42—4712no—
435C42—4748no—
436C42C23×D47112yes—
437C42C23×D47124yes—
438C23—474no—
439C23—476no—
440C23—944no—
441C23—946no—
442C23—946no—
443C23—944no—
444C23—946no—
445C23—944no—
446C23C24×D4711yescharacteristic
447Q8—28no—
448Q8—424no—
449Q8—9448no—
450Q8C2×#10625354116yes—
451D4—9424no—
452D4—948no—
453D4—18824no—
454C2×C4—24no—
455C2×C4—26no—
456C2×C4—212no—
457C2×C4—476no—
458C2×C4—472no—
459C2×C4—472no—
460C2×C4—4712no—
461C2×C4—942no—
462C2×C4—9412no—
463C2×C4—946no—
464C2×C4—942no—
465C2×C4—9448no—
466C2×C4—9424no—
467C2×C4—9424no—
468C2×C4—18812no—
469C2×C4C2×#10625354112yes—
470C2×C4#1309407214yes—
471C2×C4C24×D47112yes—
472C2×C4C24×D4716yes—
473C2×C4C24×D4714yes—
474C2×C4C24×D4716yes—
475C2×C4C24×D4712yes—
476C22—944no—
477C22—946no—
478C22—946no—
479C22—944no—
480C22—946no—
481C22—944no—
482C22—1888no—
483C22—18812no—
484C22C22×#1062535412yes—
485C22C2×#1309407212yes—
486C22#1568995812yes—
487C22C25×D4711yescharacteristic
488C4—24no—
489C4—26no—
490C4—212no—
491C4—942no—
492C4—9412no—
493C4—946no—
494C4—942no—
495C4C22×#10625354112yes—
496C4C2×#1309407214yes—
497C2—1884no—
498C2—1886no—
499C2#2025740614yesdirect product factor
500C2C23×#1062535411yescharacteristic
501C2C22×#1309407211yescharacteristic
502C2C2×#1568995811yescharacteristic
subgroupG/H
centerC23C24×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC2×Q82×C47C2
socleC23×C47C25
hypercenterC2×Q82D47
hypocenterC47C2×#3518
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
1. #20257406C2direct (N × H)
2. C2×#15689847C2semidirect
3. C2×#15689846C2semidirect
4. C2×#15689839C2semidirect
5. C2×Q8×D188C2semidirect
6. C2×#15689785C2semidirect
7. C2×#15689747C2semidirect
8. C2×#15689678C2semidirect
9. C2×#15689660C2semidirect
10. C2×#15689599C2semidirect
11. C2×C4×#10625354C2semidirect
12. C2×Q82×C47C2semidirect
15. #15689847C22semidirect
16. #15689846C22semidirect
17. #15689839C22semidirect
18. Q8×D188C22semidirect
19. #15689785C22semidirect
20. #15689747C22semidirect
21. #15689678C22semidirect
23. #15689660C22semidirect
37. #15689599C22semidirect
38. C4×#10625354C22semidirect
52. Q82×C47C22semidirect
247. C2×Q82D47semidirect
258. C2×C47#3518semidirect
266. Q82C2×D47semidirect
309. C47C2×#3518semidirect
499. C2#20257406direct (N × H)

Maximal subgroups

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

subgroupindex
#202574062
C2×#156898472
C2×#156898462
C2×#156898392
C2×Q8×D1882
C2×#156897852
C2×#156897472
C2×#156896782
C2×#156896602
C2×#156895992
C2×C4×#106253542
C2×Q82×C472
C2×#351847