Finite Group Databasebeta

← C2×#20257474 #27124490

Subgroups — C2×#20257474 #27124490

Order 12032.

7758 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257474C2132yesmaximal, direct product factor
2C2×#15689858C213yesmaximal, complemented
3C2×#15689834C213yesmaximal, complemented
4C2×#15689795C211yesmaximal, characteristic, complemented
5C2×#15689785C212yesmaximal, complemented
6C2×#15689776C216yesmaximal, complemented
7C2×#15689747C216yesmaximal, complemented
8C2×#15689674C213yesmaximal, complemented
9C2×#15689660C213yesmaximal, complemented
10C2×#15689605C212yesmaximal, complemented
11C2×C4×Q8×D47C211yesmaximal, characteristic, complemented
12C2×Q82×C47C211yesmaximal, characteristic, complemented
13C22×#10625354C2212yes—
14C22×#10625354C2216yes—
15C22×Q8×D47C2211yescharacteristic
16#15689858C22148yescomplemented
17#15689834C22148yescomplemented
18#15689795C22116yescomplemented
19#15689785C22132yescomplemented
20#15689776C22196yescomplemented
21#15689747C22196yescomplemented
22#15689674C22148yescomplemented
23C2×#13094021C2213yes—
24#15689660C22148yescomplemented
25C2×#13094019C2216yes—
26C2×#13094015C2213yes—
27C2×#13094013C2216yes—
28C2×#13094013C2213yes—
29C2×#13094012C2216yes—
30C2×#13094009C22112yes—
31C2×#13094009C2216yes—
32C2×#13094004C2216yes—
33C2×#13094000C22112yes—
34C2×#13094000C2211yescharacteristic
35C2×#13093999C2216yes—
36C2×#13093999C2216yes—
37C2×#13093999C2213yes—
38#15689605C22132yescomplemented
39C2×Q8×Dic47C2211yescharacteristic
40C2×C4×Dic94C2213yes—
41C2×#13093989C2213yes—
42C4×Q8×D47C22116yescomplemented
43C2×C4×D188C2216yes—
44C2×C4×D188C2216yes—
45C2×C4×D188C2213yes—
46C2×#13093986C2216yes—
47C2×#13093986C2216yes—
48C2×#13093981C2216yes—
49C2×C4⋊C4×D47C2213yes—
50C2×C4⋊C4×D47C2216yes—
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—216no—
59C2×#10625354—424no—
60C2×#10625354C23116yes—
61C2×#10625354C23148yes—
62C22×Dic94C2313yes—
63C2×Q8×D47—28no—
64C2×Q8×D47C2318yes—
65C22×D188C2313yes—
66C22×D188C2316yes—
67C22×D188C2316yes—
68C22×D188C2316yes—
69C22×C4×D47C2311yescharacteristic
70C22×C4×D47C2313yes—
71C22×C4×D47C2313yes—
72C22×C4×D47C2316yes—
73C22×C4×D47C23112yes—
74#13094021C23124yes—
75#13094019C23148yes—
76#13094015C23124yes—
77#13094013C23148yes—
78#13094013C23124yes—
79#13094012C23148yes—
80#13094009C23196yes—
81#13094009C23148yes—
82#13094004C23148yes—
83#13094000C23196yes—
84#13094000C2318yes—
85#13093999C23148yes—
86#13093999C23148yes—
87#13093999C23124yes—
88Q8×Dic47C2318yes—
89C4×Dic94C23124yes—
90#13093989C23124yes—
91C4×D188C23148yes—
92C4×D188C23148yes—
93C4×D188C23124yes—
94#13093986C23148yes—
95#13093986C23148yes—
96#13093981C23148yes—
97C2×C4⋊Dic47C2313yes—
98C2×C4⋊Dic47C2311yescharacteristic
99C2×C4⋊Dic47C2316yes—
100C2×Dic47⋊C4C2316yes—
101C2×Dic47⋊C4C2316yes—
102C2×Dic47⋊C4C2312yes—
103C4⋊C4×D47C23124yes—
104C4⋊C4×D47C23148yes—
105C2×(C2×D47)⋊C4C2316yes—
106C2×(C2×D47)⋊C4C2316yes—
107C2×(C2×D47)⋊C4C23112yes—
108C2×(C2×D47)⋊C4C2312yes—
109C2×(C2×D47)⋊C4C2316yes—
110C2×(C2×D47)⋊C4C2316yes—
111C2×C4×Dic47C2313yes—
112C2×C4×Dic47C2313yes—
113C2×C4×Dic47C2313yes—
114C42×D47C23124yes—
115C22×Q8×C47C2311yescharacteristic
116C22×Q8×C47C2316yes—
117C22×Q8×C47C2311yescharacteristic
118C4⋊Q8×C47C23148yes—
119C4⋊Q8×C47C23124yes—
120C4×Q8×C47C23116yes—
121C4×Q8×C47C2318yes—
122C4×Q8×C47C23124yes—
123C2×C4⋊C4×C47C2316yes—
124C2×C4⋊C4×C47C2313yes—
125C2×C4⋊C4×C47C2313yes—
126C2×C4⋊C4×C47C2316yes—
127C2×C42×C47C2313yes—
128C2×C42×C47C2316yes—
129C23×D47C2416yes—
130C23×D47C2411yescharacteristic
131C2×Q8×C47—24no—
132C2×Q8×C47—412no—
133C2×Q8×C47#4118yescomplemented
134C2×Q8×C47C2414yes—
135C2×Q8×C47C24124yes—
136C2×Q8×C47C2414yes—
137C22×C4×C47C2412yes—
138C22×C4×C47C2413yes—
139C22×C4×C47C2416yes—
140C22×C4×C47C2413yes—
141C22×C4×C47C2411yescharacteristic
142C22×Dic47C2411yescharacteristic
143C22×Dic47C2413yes—
144C22×Dic47C2413yes—
145C22×Dic47C2412yes—
146#10625354—264no—
147#10625354—496no—
148Q8×D47—232no—
149C2×D188—224no—
150C2×D188—224no—
151C2×D188—212no—
152C2×D188—412no—
153C2×D188C24112yes—
154C2×D188C24124yes—
155C2×D188C24124yes—
156C2×D188C24124yes—
157C2×C4×D47—212no—
158C2×C4×D47—212no—
159C2×C4×D47—224no—
160C2×C4×D47—24no—
161C2×C4×D47—424no—
162C2×C4×D47C2414yes—
163C2×C4×D47C24112yes—
164C2×C4×D47C24112yes—
165C2×C4×D47C24124yes—
166C2×C4×D47C24148yes—
167C2×Dic94—212no—
168C2×Dic94C24112yes—
169C4⋊C4×C47C24124yes—
170C4⋊C4×C47C24112yes—
171C4⋊C4×C47C24112yes—
172C4⋊C4×C47C24124yes—
173C42×C47C24112yes—
174C42×C47C24124yes—
175(C2×D47)⋊C4C24124yes—
176(C2×D47)⋊C4C24124yes—
177(C2×D47)⋊C4C24148yes—
178(C2×D47)⋊C4C2418yes—
179(C2×D47)⋊C4C24124yes—
180(C2×D47)⋊C4C24124yes—
181C4⋊Dic47C24112yes—
182C4⋊Dic47C2414yes—
183C4⋊Dic47C24124yes—
184Dic47⋊C4C24124yes—
185Dic47⋊C4C24124yes—
186Dic47⋊C4C2418yes—
187C4×Dic47C24112yes—
188C4×Dic47C24112yes—
189C4×Dic47C24112yes—
190C23×C47C2511yescharacteristic
191C22×D47—22no—
192C22×D47—22no—
193C22×D47—212no—
194C22×D47—212no—
195C22×D47—212no—
196C22×D47H(5,𝔽2)14yes—
197C22×D47C25112yes—
198C22×D47C2512yes—
199Q8×C47—28no—
200Q8×C47—424no—
201Q8×C47C2×#41116yescomplemented
202C2×C4×C47—26no—
203C2×C4×C47—212no—
204C2×C4×C47—24no—
205C2×C4×C47C2×#41112yes—
206C2×C4×C472−1+414yes—
207C2×C4×C47C2516yes—
208C2×C4×C47C25112yes—
209C2×C4×C47C2514yes—
210C2×C4×C47C2516yes—
211C2×C4×C47C2512yes—
212C2×Dic47—26no—
213C2×Dic47—24no—
214C2×Dic47—26no—
215C2×Dic47—22no—
216C2×Dic47C2512yes—
217C2×Dic47C2516yes—
218C2×Dic47C2516yes—
219C2×Dic47C2514yes—
220D188—248no—
221D188—248no—
222D188—224no—
223D188—424no—
224C4×D47—224no—
225C4×D47—224no—
226C4×D47—248no—
227C4×D47—28no—
228C4×D47—448no—
229Dic94—224no—
230C2×#3522—471nomaximal, Sylow
231C22×C47C22×#4112yes—
232C22×C47C2×H(5,𝔽2)12yes—
233C22×C47C2×2−1+412yes—
234C22×C47C2611yescharacteristic
235C2×D47—22no—
236C2×D47—22no—
237C2×D47—28no—
238C2×D47—212no—
239C2×D47—212no—
240C2×D47—212no—
241C2×D47—424no—
242C2×D47C2×H(5,𝔽2)14yes—
243C4×C47—26no—
244C4×C47—212no—
245C4×C47—24no—
246C4×C47C22×#41112yes—
247C4×C47C2×2−1+414yes—
248Dic47—26no—
249Dic47—24no—
250Dic47—26no—
251Dic47—22no—
252#3522—4732no—
253C2×#559—473no—
254C2×Q82D4711yescharacteristic, complemented
255C2×#554—473no—
256C2×#550—471no—
257C2×#548—472no—
258C2×#547—476no—
259C2×#542—476no—
260C2×#532—473no—
261C2×#529—473no—
262C2×#520—472no—
263C22×C4×Q8—471no—
264C2×C47#352214yescomplemented
265C2×C47C23×#4111yescharacteristic
266C2×C47C22×H(5,𝔽2)11yescharacteristic
267C2×C47C22×2−1+411yescharacteristic
268D47—24no—
269D47—412no—
270C22×#41—472no—
271C22×#41—476no—
272C23×Q8—471no—
273#559—4748no—
274Q82C2×D47116yescomplemented
275#554—4748no—
276#550—4716no—
277#548—4732no—
278#547—4796no—
279#542—4796no—
280#532—4748no—
281C2×C4⋊Q8C2×D4713yes—
282C2×C4⋊Q8C2×D4716yes—
283C2×#127—473no—
284#529—4748no—
285C2×#126—476no—
286C2×#125—473no—
287C2×#124—473no—
288C2×#124—476no—
289C2×#124—476no—
290C2×#123—476no—
291C2×#123—4712no—
292C2×C22⋊Q8—471no—
293C2×C22⋊Q8—4712no—
294C2×C22⋊Q8—476no—
295C2×#121—473no—
296C2×#121—476no—
297C2×#121—476no—
298#520—4732no—
299C2×C4×Q8—471no—
300C2×C4×Q8—4716no—
301C2×C4×Q8—473no—
302C2×C4×Q8—473no—
303C2×C4×Q8C2×D4711yescharacteristic
304C2×C4×Q8C2×D4712yes—
305C2×C4×Q8C2×D4713yes—
306C2×C4×D4—476no—
307C2×C4×D4—476no—
308C2×C4×D4—476no—
309C2×C4×D4—476no—
310C2×C4×D4—473no—
311C2×#117—476no—
312C22×C4⋊C4—473no—
313C22×C4⋊C4—476no—
314C22×C42—473no—
315C47C2×#352211yescharacteristic, complemented, Sylow
316C2×#41—4716no—
317C2×#41—4748no—
318C2×#41—9416no—
319C2×#41—18824no—
320C22×Q8—478no—
321C22×Q8—473no—
322C22×Q8—948no—
323C22×Q8C22×D4711yescharacteristic
324C22×Q8C22×D4716yes—
325C22×Q8C22×D4711yescharacteristic
326C22×D4—473no—
327C22×D4—476no—
328C22×D4—476no—
329C22×D4—476no—
330C23×C4—473no—
331C23×C4—476no—
332C23×C4—473no—
333C23×C4—471no—
334C23×C4—4712no—
335C4⋊Q8C22×D47124yes—
336C4⋊Q8C22×D47148yes—
337#127—4724no—
338#126—4748no—
339#125—4724no—
340#124—4724no—
341#124—4748no—
342#124—4748no—
343#123—4748no—
344#123—4796no—
345C22⋊Q8—478no—
346C22⋊Q8—4796no—
347C22⋊Q8—4748no—
348#121—4724no—
349#121—4748no—
350#121—4748no—
351C4×Q8—478no—
352C4×Q8—4724no—
353C4×Q8—4724no—
354C4×Q8C22×D4718yes—
355C4×Q8C22×D47116yes—
356C4×Q8C22×D47124yes—
357C4×D4—4748no—
358C4×D4—4748no—
359C4×D4—4748no—
360C4×D4—4748no—
361C4×D4—4724no—
362#117—4748no—
363C2×C4⋊C4—473no—
364C2×C4⋊C4—476no—
365C2×C4⋊C4—476no—
366C2×C4⋊C4—472no—
367C2×C4⋊C4—471no—
368C2×C4⋊C4—476no—
369C2×C4⋊C4—4724no—
370C2×C4⋊C4—4748no—
371C2×C4⋊C4C22×D4716yes—
372C2×C4⋊C4C22×D4713yes—
373C2×C4⋊C4C22×D4713yes—
374C2×C4⋊C4C22×D4716yes—
375C2×C22⋊C4—476no—
376C2×C22⋊C4—476no—
377C2×C22⋊C4—472no—
378C2×C22⋊C4—476no—
379C2×C22⋊C4—4712no—
380C2×C22⋊C4—476no—
381C2×C42—473no—
382C2×C42—473no—
383C2×C42—473no—
384C2×C42—4724no—
385C2×C42C22×D4716yes—
386C2×C42C22×D4713yes—
387C24—471no—
388C24—476no—
389#41—9464no—
390#41—18896no—
391C2×Q8—24no—
392C2×Q8—412no—
393C2×Q8—4712no—
394C2×Q8—9432no—
395C2×Q8—9412no—
396C2×Q8#1062535418yescomplemented
397C2×Q8C23×D4714yes—
398C2×Q8C23×D47124yes—
399C2×Q8C23×D4714yes—
400C2×D4—4712no—
401C2×D4—4724no—
402C2×D4—4724no—
403C2×D4—4724no—
404C2×D4—9424no—
405C2×D4—9424no—
406C2×D4—9412no—
407C2×D4—18812no—
408C22×C4—471no—
409C22×C4—473no—
410C22×C4—473no—
411C22×C4—472no—
412C22×C4—474no—
413C22×C4—4712no—
414C22×C4—4712no—
415C22×C4—4724no—
416C22×C4—4748no—
417C22×C4—9412no—
418C22×C4—9412no—
419C22×C4—9424no—
420C22×C4—944no—
421C22×C4—18824no—
422C22×C4C23×D4713yes—
423C22×C4C23×D4712yes—
424C22×C4C23×D4716yes—
425C22×C4C23×D4713yes—
426C22×C4C23×D4711yescharacteristic
427C4⋊C4—4724no—
428C4⋊C4—4724no—
429C4⋊C4—478no—
430C4⋊C4—4712no—
431C4⋊C4—474no—
432C4⋊C4—4724no—
433C4⋊C4C23×D47124yes—
434C4⋊C4C23×D47112yes—
435C4⋊C4C23×D47112yes—
436C4⋊C4C23×D47124yes—
437C22⋊C4—4724no—
438C22⋊C4—4724no—
439C22⋊C4—478no—
440C22⋊C4—4724no—
441C22⋊C4—4748no—
442C22⋊C4—4724no—
443C42—4712no—
444C42—4712no—
445C42—4712no—
446C42C23×D47112yes—
447C42C23×D47124yes—
448C23—474no—
449C23—472no—
450C23—4712no—
451C23—942no—
452C23—942no—
453C23—9412no—
454C23—9412no—
455C23—9412no—
456C23C24×D4711yescharacteristic
457Q8—28no—
458Q8—424no—
459Q8—9424no—
460Q8C2×#10625354116yescomplemented
461D4—9448no—
462D4—9448no—
463D4—9424no—
464D4—18824no—
465C2×C4—26no—
466C2×C4—212no—
467C2×C4—24no—
468C2×C4—472no—
469C2×C4—476no—
470C2×C4—476no—
471C2×C4—474no—
472C2×C4—9424no—
473C2×C4—944no—
474C2×C4—946no—
475C2×C4—946no—
476C2×C4—942no—
477C2×C4—9424no—
478C2×C4—9448no—
479C2×C4—948no—
480C2×C4—18848no—
481C2×C4C2×#10625354112yes—
482C2×C4#1309407214yes—
483C2×C4C24×D4716yes—
484C2×C4C24×D4714yes—
485C2×C4C24×D47112yes—
486C2×C4C24×D4716yes—
487C2×C4C24×D4712yes—
488C22—474no—
489C22—942no—
490C22—942no—
491C22—948no—
492C22—9412no—
493C22—9412no—
494C22—9412no—
495C22—18824no—
496C22C22×#1062535412yes—
497C22H(5,𝔽2)×D4712yes—
498C22C2×#1309407212yes—
499C22C25×D4711yescharacteristic
500C4—26no—
501C4—212no—
502C4—24no—
503C4—944no—
504C4—946no—
505C4—946no—
506C4—942no—
507C4C22×#10625354112yes—
508C4C2×#1309407214yes—
509C2—944no—
510C2—18812no—
511C2#2025747414yesdirect product factor
512C2C23×#1062535411yescharacteristic
513C2C2×H(5,𝔽2)×D4711yescharacteristic
514C2C22×#1309407211yescharacteristic
subgroupG/H
centerC23C24×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC2×Q82×C47C2
socleC23×C47C25
hypercenterC2×Q82D47
hypocenterC47C2×#3522
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
1. #20257474C2direct (N × H)
2. C2×#15689858C2semidirect
3. C2×#15689834C2semidirect
4. C2×#15689795C2semidirect
5. C2×#15689785C2semidirect
6. C2×#15689776C2semidirect
7. C2×#15689747C2semidirect
8. C2×#15689674C2semidirect
9. C2×#15689660C2semidirect
10. C2×#15689605C2semidirect
11. C2×C4×Q8×D47C2semidirect
12. C2×Q82×C47C2semidirect
16. #15689858C22semidirect
17. #15689834C22semidirect
18. #15689795C22semidirect
19. #15689785C22semidirect
20. #15689776C22semidirect
21. #15689747C22semidirect
22. #15689674C22semidirect
24. #15689660C22semidirect
38. #15689605C22semidirect
42. C4×Q8×D47C22semidirect
52. Q82×C47C22semidirect
133. C2×Q8×C47#41semidirect
201. Q8×C47C2×#41semidirect
254. C2×Q82D47semidirect
264. C2×C47#3522semidirect
274. Q82C2×D47semidirect
315. C47C2×#3522semidirect
396. C2×Q8#10625354semidirect
460. Q8C2×#10625354semidirect
511. C2#20257474direct (N × H)

Maximal subgroups

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

subgroupindex
#202574742
C2×#156898582
C2×#156898342
C2×#156897952
C2×#156897852
C2×#156897762
C2×#156897472
C2×#156896742
C2×#156896602
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