Finite Group Databasebeta

← C2×#20257485 #27124492

Subgroups — C2×#20257485 #27124492

Order 12032.

7022 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257485C2132yesmaximal, direct product factor
2C2×#15689852C213yesmaximal, complemented
3C2×#15689847C213yesmaximal, complemented
4C2×#15689811C216yesmaximal, complemented
5C2×Q8×D188C211yesmaximal, characteristic, complemented
6C2×Q8×D188C212yesmaximal, complemented
7C2×#15689736C216yesmaximal, complemented
8C2×#15689678C213yesmaximal, complemented
9C2×#15689660C213yesmaximal
10C2×#15689599C212yesmaximal, complemented
11C2×C4×Q8×D47C211yesmaximal, characteristic, complemented
12C2×C47×#557C211yesmaximal, characteristic, complemented
13C22×Q8×D47C2211yescharacteristic
14C22×Q8×D47C2212yes—
15#15689852C22148yescomplemented
16#15689847C22148yescomplemented
17#15689811C22196yescomplemented
18Q8×D188C22116yescomplemented
19Q8×D188C22132yescomplemented
20#15689736C22196yescomplemented
21#15689678C22148yescomplemented
22C2×#13094024C2213yes—
23C2×#13094024C2216yes—
24#15689660C22148yes—
25C2×#13094019C22112yes—
26C2×#13094019C2216yes—
27C2×#13094015C22112yes—
28C2×#13094015C2213yes—
29C2×#13094015C2216yes—
30C2×#13094013C2213yes—
31C2×#13094009C2216yes—
32C2×#13094004C2216yes—
33C2×#13094004C2216yes—
34C2×#13094004C2216yes—
35C2×#13094000C2213yes—
36C2×#13093999C2211yescharacteristic
37#15689599C22132yescomplemented
38C2×Q8×Dic47C2211yescharacteristic
39C2×Q8×Dic47C2216yes—
40C2×C4×Dic94C2213yes—
41C2×C4×Dic94C2216yes—
42C2×#13093989C2213yes—
43C2×#13093989C2216yes—
44C4×Q8×D47C22116yescomplemented
45C2×C4×D188C2216yes—
46C2×C4×D188C2213yes—
47C2×#13093982C2216yes—
48C2×#13093982C2216yes—
49C2×#13093981C2216yes—
50C2×C4⋊C4×D47C2213yes—
51C2×C42×D47C2213yes—
52C47×#557C22116yescomplemented
53C2×C4⋊Q8×C47C2213yes—
54C2×#125×C47C2216yes—
55C2×C4×Q8×C47C2211yescharacteristic
56C2×C4×Q8×C47C2212yes—
57C2×C4×Q8×C47C2213yes—
58C22×Dic94C2313yes—
59C22×Dic94C2316yes—
60C2×Q8×D47—28no—
61C2×Q8×D47—216no—
62C2×Q8×D47C2318yes—
63C2×Q8×D47C23116yes—
64C22×D188C2312yes—
65C22×D188C2311yescharacteristic
66C22×C4×D47C2311yescharacteristic
67C22×C4×D47C2313yes—
68C22×C4×D47C2313yes—
69C22×C4×D47C2316yes—
70#13094024C23148yes—
71#13094024C23124yes—
72#13094019C23148yes—
73#13094019C23196yes—
74#13094015C23148yes—
75#13094015C23196yes—
76#13094015C23124yes—
77#13094013C23124yes—
78#13094009C23148yes—
79#13094004C23148yes—
80#13094004C23148yes—
81#13094004C23148yes—
82#13094000C23124yes—
83#13093999C2318yes—
84Q8×Dic47C2318yes—
85Q8×Dic47C23148yes—
86C4×Dic94C23148yes—
87C4×Dic94C23124yes—
88#13093989C23124yes—
89#13093989C23148yes—
90C4×D188C23148yes—
91C4×D188C23124yes—
92#13093982C23148yes—
93#13093982C23148yes—
94#13093981C23148yes—
95C2×C4⋊Dic47C2313yes—
96C2×C4⋊Dic47C2316yes—
97C2×C4⋊Dic47C2313yes—
98C2×C4⋊Dic47C2316yes—
99C2×C4⋊Dic47C23112yes—
100C2×Dic47⋊C4C2316yes—
101C2×Dic47⋊C4C2316yes—
102C2×Dic47⋊C4C2316yes—
103C4⋊C4×D47C23124yes—
104C2×(C2×D47)⋊C4C2312yes—
105C2×(C2×D47)⋊C4C2316yes—
106C2×(C2×D47)⋊C4C2316yes—
107C2×(C2×D47)⋊C4C2316yes—
108C2×(C2×D47)⋊C4C2316yes—
109C2×C4×Dic47C2313yes—
110C2×C4×Dic47C2313yes—
111C2×C4×Dic47C2313yes—
112C2×C4×Dic47C23112yes—
113C42×D47C23124yes—
114C22×Q8×C47C2313yes—
115C22×Q8×C47C2311yescharacteristic
116C4⋊Q8×C47C23124yes—
117#125×C47C23148yes—
118C4×Q8×C47C2318yes—
119C4×Q8×C47C23116yes—
120C4×Q8×C47C23124yes—
121C2×C4⋊C4×C47C2311yescharacteristic
122C2×C4⋊C4×C47C2313yes—
123C2×C4⋊C4×C47C2316yes—
124C2×C4⋊C4×C47C2316yes—
125C2×C4⋊C4×C47C2316yes—
126C2×C42×C47C2316yes—
127C2×C42×C47C2313yes—
128C23×D47C2412yes—
129C23×D47C2411yescharacteristic
130C2×Q8×C47—212no—
131C2×Q8×C47#4118yes—
132C2×Q8×C47C24112yes—
133C2×Q8×C47C2414yes—
134C22×C4×C47C2412yes—
135C22×C4×C47C2413yes—
136C22×C4×C47C2416yes—
137C22×C4×C47C2413yes—
138C22×C4×C47C2411yescharacteristic
139C22×Dic47C2411yescharacteristic
140C22×Dic47C2413yes—
141C22×Dic47C2413yes—
142C22×Dic47C2416yes—
143Q8×D47—232no—
144Q8×D47—264no—
145C2×D188—28no—
146C2×D188—24no—
147C2×D188C2418yes—
148C2×D188C2414yes—
149C2×C4×D47—212no—
150C2×C4×D47—212no—
151C2×C4×D47—224no—
152C2×C4×D47—24no—
153C2×C4×D47C2414yes—
154C2×C4×D47C24112yes—
155C2×C4×D47C24112yes—
156C2×C4×D47C24124yes—
157C2×Dic94—212no—
158C2×Dic94—224no—
159C2×Dic94C24112yes—
160C2×Dic94C24124yes—
161C4⋊C4×C47C2414yes—
162C4⋊C4×C47C24112yes—
163C4⋊C4×C47C24124yes—
164C4⋊C4×C47C24124yes—
165C4⋊C4×C47C24124yes—
166C42×C47C24124yes—
167C42×C47C24112yes—
168(C2×D47)⋊C4C2418yes—
169(C2×D47)⋊C4C24124yes—
170(C2×D47)⋊C4C24124yes—
171(C2×D47)⋊C4C24124yes—
172(C2×D47)⋊C4C24124yes—
173C4⋊Dic47C24112yes—
174C4⋊Dic47C24124yes—
175C4⋊Dic47C24112yes—
176C4⋊Dic47C24124yes—
177C4⋊Dic47C24148yes—
178Dic47⋊C4C24124yes—
179Dic47⋊C4C24124yes—
180Dic47⋊C4C24124yes—
181C4×Dic47C24112yes—
182C4×Dic47C24112yes—
183C4×Dic47C24112yes—
184C4×Dic47C24148yes—
185C23×C47C2511yescharacteristic
186C22×D47—22no—
187C22×D47—22no—
188C22×D47—24no—
189C22×D47—24no—
190C22×D47—24no—
191C22×D472−1+414yes—
192C22×D47C2514yes—
193C22×D47C2512yes—
194Q8×C47—224no—
195Q8×C47C2×#41116yes—
196C2×C4×C47—26no—
197C2×C4×C47—24no—
198C2×C4×C47—212no—
199C2×C4×C47C2×#41112yes—
200C2×C4×C472−1+414yes—
201C2×C4×C47C25112yes—
202C2×C4×C47C2516yes—
203C2×C4×C47C2514yes—
204C2×C4×C47C2516yes—
205C2×C4×C47C2512yes—
206C2×Dic47—26no—
207C2×Dic47—212no—
208C2×Dic47—26no—
209C2×Dic47—22no—
210C2×Dic47C2512yes—
211C2×Dic47C2516yes—
212C2×Dic47C2516yes—
213C2×Dic47C25112yes—
214D188—216no—
215D188—28no—
216C4×D47—224no—
217C4×D47—224no—
218C4×D47—248no—
219C4×D47—28no—
220Dic94—224no—
221Dic94—248no—
222C2×#3523—471nomaximal, Sylow
223C22×C47C22×#4112yes—
224C22×C47C2×2−1+412yes—
225C22×C47C2×2−1+412yes—
226C22×C47C2611yescharacteristic
227C2×D47—22no—
228C2×D47—22no—
229C2×D47—28no—
230C2×D47—24no—
231C2×D47—24no—
232C2×D47—24no—
233C2×D47—48no—
234C2×D47C2×2−1+414yes—
235C4×C47—26no—
236C4×C47—24no—
237C4×C47—212no—
238C4×C47C22×#41112yes—
239C4×C47C2×2−1+414yes—
240Dic47—26no—
241Dic47—212no—
242Dic47—26no—
243Dic47—22no—
244#3523—4732no—
245C2×#557—473no—
246C2×#557D4711yescharacteristic, complemented
247C2×#556—473no—
248C2×#552—476no—
249C2×D4×Q8—472no—
250C2×D4×Q8—471no—
251C2×#541—476no—
252C2×#533—473no—
253C2×#529—473no—
254C2×#519—472no—
255C22×C4×Q8—471no—
256C2×C47#352314yescomplemented
257C2×C47C23×#4111yescharacteristic
258C2×C47C22×2−1+411yescharacteristic
259C2×C47C22×2−1+411yescharacteristic
260D47—24no—
261D47—44no—
262C23×Q8—471no—
263C23×Q8—472no—
264#557—4748no—
265#557C2×D47116yescomplemented
266#556—4748no—
267#552—4796no—
268D4×Q8—4732no—
269D4×Q8—4716no—
270#541—4796no—
271#533—4748no—
272C2×C4⋊Q8—473no—
273C2×C4⋊Q8—476no—
274C2×C4⋊Q8C2×D4713yes—
275#529—4748no—
276C2×#126—476no—
277C2×#126—4712no—
278C2×#125—473no—
279C2×#125—476no—
280C2×#125—4712no—
281C2×#125C2×D4716yes—
282C2×#124—473no—
283C2×#123—476no—
284C2×C22⋊Q8—473no—
285C2×C22⋊Q8—476no—
286C2×C22⋊Q8—476no—
287C2×C22⋊Q8—476no—
288C2×#121—471no—
289#519—4732no—
290C2×C4×Q8—471no—
291C2×C4×Q8—476no—
292C2×C4×Q8—4716no—
293C2×C4×Q8—473no—
294C2×C4×Q8—476no—
295C2×C4×Q8—473no—
296C2×C4×Q8—476no—
297C2×C4×Q8C2×D4711yescharacteristic
298C2×C4×Q8C2×D4712yes—
299C2×C4×Q8C2×D4713yes—
300C2×C4×D4—476no—
301C2×C4×D4—473no—
302C2×#117—476no—
303C2×#117—476no—
304C2×#117—476no—
305C22×C4⋊C4—473no—
306C22×C42—473no—
307C47C2×#352311yescharacteristic, complemented, Sylow
308C22×Q8—478no—
309C22×Q8—4716no—
310C22×Q8—473no—
311C22×Q8—476no—
312C22×Q8—948no—
313C22×Q8—9416no—
314C22×Q8C22×D4713yes—
315C22×Q8C22×D4711yescharacteristic
316C22×D4—472no—
317C22×D4—471no—
318C23×C4—473no—
319C23×C4—473no—
320C23×C4—476no—
321C23×C4—471no—
322C4⋊Q8—4724no—
323C4⋊Q8—4748no—
324C4⋊Q8C22×D47124yes—
325#126—4748no—
326#126—4796no—
327#125—4724no—
328#125—4748no—
329#125—4796no—
330#125C22×D47148yes—
331#124—4724no—
332#123—4748no—
333C22⋊Q8—4724no—
334C22⋊Q8—4748no—
335C22⋊Q8—4748no—
336C22⋊Q8—4748no—
337#121—478no—
338C4×Q8—478no—
339C4×Q8—4748no—
340C4×Q8—4724no—
341C4×Q8—4748no—
342C4×Q8—4724no—
343C4×Q8—4748no—
344C4×Q8C22×D4718yes—
345C4×Q8C22×D47116yes—
346C4×Q8C22×D47124yes—
347C4×D4—4748no—
348C4×D4—4724no—
349#117—4748no—
350#117—4748no—
351#117—4748no—
352C2×C4⋊C4—473no—
353C2×C4⋊C4—476no—
354C2×C4⋊C4—476no—
355C2×C4⋊C4—476no—
356C2×C4⋊C4—476no—
357C2×C4⋊C4—473no—
358C2×C4⋊C4—476no—
359C2×C4⋊C4—4712no—
360C2×C4⋊C4—4724no—
361C2×C4⋊C4C22×D4713yes—
362C2×C4⋊C4C22×D4716yes—
363C2×C4⋊C4C22×D4716yes—
364C2×C4⋊C4C22×D4711yescharacteristic
365C2×C4⋊C4C22×D4716yes—
366C2×C22⋊C4—472no—
367C2×C22⋊C4—476no—
368C2×C22⋊C4—476no—
369C2×C22⋊C4—476no—
370C2×C22⋊C4—476no—
371C2×C42—473no—
372C2×C42—4712no—
373C2×C42—473no—
374C2×C42—473no—
375C2×C42—4724no—
376C2×C42C22×D4716yes—
377C2×C42C22×D4713yes—
378C24—471no—
379C24—472no—
380C2×Q8—212no—
381C2×Q8—4712no—
382C2×Q8—4724no—
383C2×Q8—9432no—
384C2×Q8—9464no—
385C2×Q8—9412no—
386C2×Q8—9424no—
387C2×Q8#1062535418yes—
388C2×Q8C23×D47112yes—
389C2×Q8C23×D4714yes—
390C2×D4—478no—
391C2×D4—474no—
392C2×D4—948no—
393C2×D4—944no—
394C22×C4—473no—
395C22×C4—471no—
396C22×C4—476no—
397C22×C4—473no—
398C22×C4—474no—
399C22×C4—4712no—
400C22×C4—4712no—
401C22×C4—4724no—
402C22×C4—9412no—
403C22×C4—9412no—
404C22×C4—9424no—
405C22×C4—944no—
406C22×C4C23×D4716yes—
407C22×C4C23×D4713yes—
408C22×C4C23×D4712yes—
409C22×C4C23×D4713yes—
410C22×C4C23×D4711yescharacteristic
411C4⋊C4—4724no—
412C4⋊C4—4724no—
413C4⋊C4—4724no—
414C4⋊C4—4712no—
415C4⋊C4—4724no—
416C4⋊C4—4712no—
417C4⋊C4—4724no—
418C4⋊C4—4748no—
419C4⋊C4C23×D47112yes—
420C4⋊C4C23×D47124yes—
421C4⋊C4C23×D47124yes—
422C4⋊C4C23×D4714yes—
423C4⋊C4C23×D47124yes—
424C22⋊C4—478no—
425C22⋊C4—4724no—
426C22⋊C4—4724no—
427C22⋊C4—4724no—
428C22⋊C4—4724no—
429C42—4712no—
430C42—4712no—
431C42—4712no—
432C42—4748no—
433C42C23×D47112yes—
434C42C23×D47124yes—
435C23—474no—
436C23—472no—
437C23—474no—
438C23—942no—
439C23—942no—
440C23—944no—
441C23—944no—
442C23—944no—
443C23C24×D4711yescharacteristic
444Q8—224no—
445Q8—9424no—
446Q8—9448no—
447Q8C2×#10625354116yes—
448D4—9416no—
449D4—948no—
450C2×C4—26no—
451C2×C4—24no—
452C2×C4—212no—
453C2×C4—476no—
454C2×C4—472no—
455C2×C4—476no—
456C2×C4—4712no—
457C2×C4—9424no—
458C2×C4—9412no—
459C2×C4—946no—
460C2×C4—946no—
461C2×C4—942no—
462C2×C4—9424no—
463C2×C4—9448no—
464C2×C4—948no—
465C2×C4C2×#10625354112yes—
466C2×C4#1309407114yes—
467C2×C4C24×D47112yes—
468C2×C4C24×D4716yes—
469C2×C4C24×D4714yes—
470C2×C4C24×D4716yes—
471C2×C4C24×D4712yes—
472C22—474no—
473C22—942no—
474C22—942no—
475C22—948no—
476C22—944no—
477C22—944no—
478C22—944no—
479C22—1888no—
480C22C22×#1062535412yes—
481C222−1+4×D4712yes—
482C22C2×#1309407112yes—
483C22C25×D4711yescharacteristic
484C4—26no—
485C4—24no—
486C4—212no—
487C4—946no—
488C4—9412no—
489C4—946no—
490C4—942no—
491C4C22×#10625354112yes—
492C4C2×#1309407114yes—
493C2—944no—
494C2—1884no—
495C2#2025748514yesdirect product factor
496C2C23×#1062535411yescharacteristic
497C2C2×2−1+4×D4711yescharacteristic
498C2C22×#1309407111yescharacteristic
subgroupG/H
centerC23C24×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC2×C47×#557C2
socleC23×C47C25
prime radicalC2×#13093999C22
hypercenterC2×#557D47
hypocenterC47C2×#3523
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. #20257485C2direct (N × H)
2. C2×#15689852C2semidirect
3. C2×#15689847C2semidirect
4. C2×#15689811C2semidirect
5. C2×Q8×D188C2semidirect
6. C2×Q8×D188C2semidirect
7. C2×#15689736C2semidirect
8. C2×#15689678C2semidirect
10. C2×#15689599C2semidirect
11. C2×C4×Q8×D47C2semidirect
12. C2×C47×#557C2semidirect
15. #15689852C22semidirect
16. #15689847C22semidirect
17. #15689811C22semidirect
18. Q8×D188C22semidirect
19. Q8×D188C22semidirect
20. #15689736C22semidirect
21. #15689678C22semidirect
37. #15689599C22semidirect
44. C4×Q8×D47C22semidirect
52. C47×#557C22semidirect
246. C2×#557D47semidirect
256. C2×C47#3523semidirect
265. #557C2×D47semidirect
307. C47C2×#3523semidirect
495. C2#20257485direct (N × H)

Maximal subgroups

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

subgroupindex
#202574852
C2×#156898522
C2×#156898472
C2×#156898112
C2×Q8×D1882
C2×#156897362
C2×#156896782
C2×#156896602
C2×#156895992
C2×C4×Q8×D472
C2×C47×#5572
C2×#352347