Finite Group Databasebeta

← C2×#20257473 #27124489

Subgroups — C2×#20257473 #27124489

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#20257473C2132yesmaximal, direct product factor
2C2×#15689858C213yesmaximal, complemented
3C2×Q8×Dic94C211yesmaximal, characteristic, complemented
4C2×#15689833C213yesmaximal, complemented
5C2×#15689795C211yesmaximal, characteristic, complemented
6C2×#15689784C212yesmaximal, complemented
7C2×#15689774C216yesmaximal, complemented
8C2×#15689745C216yesmaximal, complemented
9C2×#15689673C213yesmaximal, complemented
10C2×#15689657C213yesmaximal, complemented
11C2×#15689605C212yesmaximal, complemented
12C22×C4×Q8×C47C211yesmaximal, characteristic, complemented
13C22×#10625354C2212yes—
14C22×#10625350C2216yes—
15#15689858C22148yescomplemented
16Q8×Dic94C22116yescomplemented
17#15689833C22148yescomplemented
18#15689795C22116yescomplemented
19#15689784C22132yescomplemented
20#15689774C22196yescomplemented
21#15689745C22196yescomplemented
22#15689673C22148yescomplemented
23C2×#13094024C2213yes—
24C2×#13094023C2216yes—
25C2×#13094021C2213yes—
26#15689657C22148yescomplemented
27C2×#13094018C2216yes—
28C2×#13094016C2213yes—
29C2×#13094013C2216yes—
30C2×#13094012C2213yes—
31C2×#13094012C2216yes—
32C2×#13094009C22112yes—
33C2×#13094007C2216yes—
34C2×#13094003C2211yescharacteristic
35C2×#13094002C22112yes—
36C2×#13094000C2216yes—
37C2×#13093999C2216yes—
38C2×#13093998C2216yes—
39C2×#13093998C2213yes—
40#15689605C22132yescomplemented
41C2×Q8×Dic47C2212yes—
42C2×C4×Dic94C2213yes—
43C2×C4×D188C2216yes—
44C2×C4×D188C2213yes—
45C2×C4×D188C2216yes—
46C2×#13093986C2216yes—
47C2×C4×C22⋊D47C2216yes—
48C2×#13093981C2216yes—
49C2×C4⋊C4×D47C2216yes—
50C23×Q8×C47C2211yescharacteristic
51C2×C4×Q8×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C2×C4×Q8×C47C2213yes—
54C2×C4×Q8×C47C2213yes—
55C2×C4×Q8×C47C22116yescomplemented
56C22×C4⋊C4×C47C2213yes—
57C22×C42×C47C2213yes—
58C2×#10625354—216no—
59C2×#10625354C23116yes—
60C2×#10625350—424no—
61C2×#10625350C23148yes—
62C22×Dic94C2311yescharacteristic
63C22×Dic94C2316yes—
64C22×D188C2316yes—
65C22×D188C2313yes—
66C22×D188C2316yes—
67C22×C22⋊D47C2316yes—
68C22×C4×D47C2316yes—
69C22×C4×D47C23112yes—
70#13094024C23124yes—
71#13094023C23148yes—
72#13094021C23124yes—
73#13094018C23148yes—
74#13094016C23124yes—
75#13094013C23148yes—
76#13094012C23148yes—
77#13094012C23124yes—
78#13094009C23196yes—
79#13094007C23148yes—
80#13094003C2318yes—
81#13094002C23196yes—
82#13094000C23148yes—
83#13093999C23148yes—
84#13093998C23148yes—
85#13093998C23124yes—
86Q8×Dic47C23116yes—
87C4×Dic94C23124yes—
88C4×D188C23148yes—
89C4×D188C23124yes—
90C4×D188C23148yes—
91#13093986C23148yes—
92C4×C22⋊D47C23148yes—
93#13093981C23148yes—
94C2×C4⋊Dic47C2316yes—
95C2×C4⋊Dic47C2311yescharacteristic
96C2×C4⋊Dic47C2313yes—
97C2×Dic47⋊C4C2316yes—
98C2×Dic47⋊C4C2312yes—
99C2×Dic47⋊C4C2316yes—
100C2×Dic47⋊C4C2316yes—
101C4⋊C4×D47C23148yes—
102C2×C22⋊Dic47C2312yes—
103C2×(C2×D47)⋊C4C2316yes—
104C2×(C2×D47)⋊C4C2316yes—
105C2×(C2×D47)⋊C4C2316yes—
106C2×(C2×D47)⋊C4C23112yes—
107C2×(C2×D47)⋊C4C2316yes—
108C2×C4×Dic47C2316yes—
109C22×Q8×C47—28no—
110C22×Q8×C47C2318yes—
111C22×Q8×C47C2313yes—
112C22×Q8×C47C2311yescharacteristic
113C23×C4×C47C2311yescharacteristic
114C23×C4×C47C2313yes—
115C23×C4×C47C2313yes—
116C4×Q8×C47C23124yes—
117C4×Q8×C47C2318yes—
118C4×Q8×C47C23124yes—
119C4×Q8×C47C2318yes—
120C2×C4⋊C4×C47C23124yes—
121C2×C4⋊C4×C47C2313yes—
122C2×C4⋊C4×C47C2313yes—
123C2×C4⋊C4×C47C2316yes—
124C2×C42×C47C2313yes—
125C2×C42×C47C2313yes—
126C2×C42×C47C2313yes—
127C2×C42×C47C2313yes—
128C2×C42×C47C23124yes—
129C24×C47C2411yescharacteristic
130C23×D47C2416yes—
131C2×Q8×C47—232no—
132C2×Q8×C47—212no—
133C2×Q8×C47#4118yescomplemented
134C2×Q8×C47C24112yes—
135C2×Q8×C47C2414yes—
136C22×C4×C47—212no—
137C22×C4×C47—212no—
138C22×C4×C47—24no—
139C22×C4×C47C2411yescharacteristic
140C22×C4×C47C2414yes—
141C22×C4×C47C2413yes—
142C22×C4×C47C2413yes—
143C22×C4×C47C24112yes—
144C22×C4×C47C2413yes—
145C22×C4×C47C24112yes—
146C22×C4×C47C2413yes—
147C22×C4×C47C2411yescharacteristic
148C2×C22⋊D47—224no—
149C2×C22⋊D47C24124yes—
150C22×Dic47C2412yes—
151C22×Dic47C2416yes—
152C22×Dic47C2412yes—
153#10625354—264no—
154#10625350—496no—
155C2×D188—224no—
156C2×D188—212no—
157C2×D188—412no—
158C2×D188C24124yes—
159C2×D188C24112yes—
160C2×D188C24124yes—
161C2×C4×D47—224no—
162C2×C4×D47—424no—
163C2×C4×D47C24124yes—
164C2×C4×D47C24148yes—
165C2×Dic94—24no—
166C2×Dic94—412no—
167C2×Dic94C2414yes—
168C2×Dic94C24124yes—
169C4⋊C4×C47C24112yes—
170C4⋊C4×C47C24112yes—
171C4⋊C4×C47C24124yes—
172C42×C47C24112yes—
173C42×C47C24112yes—
174C42×C47C24112yes—
175C42×C47C24112yes—
176C22⋊Dic47C2418yes—
177(C2×D47)⋊C4C24124yes—
178(C2×D47)⋊C4C24124yes—
179(C2×D47)⋊C4C24124yes—
180(C2×D47)⋊C4C24148yes—
181(C2×D47)⋊C4C24124yes—
182C4⋊Dic47C24124yes—
183C4⋊Dic47C2414yes—
184C4⋊Dic47C24112yes—
185Dic47⋊C4C24124yes—
186Dic47⋊C4C2418yes—
187Dic47⋊C4C24124yes—
188Dic47⋊C4C24124yes—
189C4×Dic47C24124yes—
190C23×C47—22no—
191C23×C47—22no—
192C23×C47H(5,𝔽2)14yes—
193C23×C47C2512yes—
194C23×C47C2511yescharacteristic
195C22×D47—212no—
196C22×D47—212no—
197C22×D47—212no—
198C22×D47C25112yes—
199Q8×C47—224no—
200Q8×C47C2×#41116yescomplemented
201C2×C4×C47—26no—
202C2×C4×C47—26no—
203C2×C4×C47—26no—
204C2×C4×C47—224no—
205C2×C4×C47—22no—
206C2×C4×C47—224no—
207C2×C4×C47—28no—
208C2×C4×C47C2×#41112yes—
209C2×C4×C472−1+414yes—
210C2×C4×C47C2512yes—
211C2×C4×C47C2516yes—
212C2×C4×C47C2516yes—
213C2×C4×C47C2516yes—
214C2×C4×C47C2516yes—
215C2×C4×C47C2512yes—
216C22⋊D47—248no—
217C2×Dic47—24no—
218C2×Dic47—212no—
219C2×Dic47—24no—
220C2×Dic47C2514yes—
221C2×Dic47C25112yes—
222C2×Dic47C2514yes—
223D188—248no—
224D188—224no—
225D188—424no—
226C4×D47—248no—
227C4×D47—448no—
228Dic94—28no—
229Dic94—424no—
230C2×#3522—471nomaximal, Sylow
231C22×C47—22no—
232C22×C47—22no—
233C22×C47—28no—
234C22×C47C22×#4112yes—
235C22×C47C2×H(5,𝔽2)14yes—
236C22×C47C2×H(5,𝔽2)12yes—
237C22×C47C2×2−1+412yes—
238C22×C47C2611yescharacteristic
239C2×D47—212no—
240C2×D47—212no—
241C2×D47—212no—
242C2×D47—424no—
243C4×C47—26no—
244C4×C47—26no—
245C4×C47—26no—
246C4×C47—22no—
247C4×C47C22×#41112yes—
248C4×C47C2×2−1+414yes—
249Dic47—24no—
250Dic47—212no—
251Dic47—24no—
252#3522—4732no—
253C2×#559—473no—
254C2×Q82—471no—
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×Q8D4711yescharacteristic, complemented
264C2×C47—24no—
265C2×C47#352214yescomplemented
266C2×C47C23×#4111yescharacteristic
267C2×C47C22×H(5,𝔽2)11yescharacteristic
268C2×C47C22×2−1+411yescharacteristic
269D47—412no—
270C22×#41—472no—
271C22×#41—476no—
272C23×Q8C2×D4711yescharacteristic
273#559—4748no—
274Q82—4716no—
275#554—4748no—
276#550—4716no—
277#548—4732no—
278#547—4796no—
279#542—4796no—
280#532—4748no—
281C2×C4⋊Q8—476no—
282C2×C4⋊Q8—473no—
283C2×#127—473no—
284#529—4748no—
285C2×#126—476no—
286C2×#125—473no—
287C2×#124—476no—
288C2×#124—473no—
289C2×#124—476no—
290C2×#123—476no—
291C2×#123—4712no—
292C2×C22⋊Q8—476no—
293C2×C22⋊Q8—471no—
294C2×C22⋊Q8—4712no—
295C2×#121—473no—
296C2×#121—476no—
297C2×#121—476no—
298#520—4732no—
299C2×C4×Q8—472no—
300C2×C4×Q8—473no—
301C2×C4×Q8C2×D4711yescharacteristic
302C2×C4×Q8C2×D4713yes—
303C2×C4×Q8C2×D4711yescharacteristic
304C2×C4×Q8C2×D4713yes—
305C2×C4×Q8C2×D47116yescomplemented
306C2×C4×D4—476no—
307C2×C4×D4—476no—
308C2×C4×D4—476no—
309C2×C4×D4—473no—
310C2×C4×D4—476no—
311C2×#117—476no—
312C22×C4⋊C4—476no—
313C22×C4⋊C4C2×D4713yes—
314C22×C42C2×D4713yes—
315C47C2×#352211yescharacteristic, complemented, Sylow
316C2×#41—4716no—
317C2×#41—4748no—
318C2×#41—9416no—
319C2×#41—18824no—
320C22×Q8—28no—
321C22×Q8—471no—
322C22×Q8—476no—
323C22×Q8C22×D4713yes—
324C22×Q8C22×D4718yes—
325C22×Q8C22×D4711yescharacteristic
326C22×D4—476no—
327C22×D4—473no—
328C22×D4—476no—
329C22×D4—476no—
330C23×C4—476no—
331C23×C4—4712no—
332C23×C4C22×D4713yes—
333C23×C4C22×D4713yes—
334C23×C4C22×D4711yescharacteristic
335C4⋊Q8—4748no—
336C4⋊Q8—4724no—
337#127—4724no—
338#126—4748no—
339#125—4724no—
340#124—4748no—
341#124—4724no—
342#124—4748no—
343#123—4748no—
344#123—4796no—
345C22⋊Q8—4748no—
346C22⋊Q8—478no—
347C22⋊Q8—4796no—
348#121—4724no—
349#121—4748no—
350#121—4748no—
351C4×Q8—4716no—
352C4×Q8—4724no—
353C4×Q8C22×D4718yes—
354C4×Q8C22×D47124yes—
355C4×Q8C22×D4718yes—
356C4×Q8C22×D47124yes—
357C4×D4—4748no—
358C4×D4—4748no—
359C4×D4—4748no—
360C4×D4—4724no—
361C4×D4—4748no—
362#117—4748no—
363C2×C4⋊C4—476no—
364C2×C4⋊C4—476no—
365C2×C4⋊C4—472no—
366C2×C4⋊C4—471no—
367C2×C4⋊C4—473no—
368C2×C4⋊C4—476no—
369C2×C4⋊C4—476no—
370C2×C4⋊C4—4748no—
371C2×C4⋊C4C22×D4713yes—
372C2×C4⋊C4C22×D4713yes—
373C2×C4⋊C4C22×D4716yes—
374C2×C4⋊C4C22×D47124yes—
375C2×C22⋊C4—472no—
376C2×C22⋊C4—476no—
377C2×C22⋊C4—476no—
378C2×C22⋊C4—476no—
379C2×C22⋊C4—4712no—
380C2×C22⋊C4—476no—
381C2×C42—476no—
382C2×C42C22×D4713yes—
383C2×C42C22×D4713yes—
384C2×C42C22×D4713yes—
385C2×C42C22×D4713yes—
386C2×C42C22×D47124yes—
387C24—476no—
388C24C23×D4711yescharacteristic
389#41—9464no—
390#41—18896no—
391C2×Q8—232no—
392C2×Q8—212no—
393C2×Q8—474no—
394C2×Q8—4724no—
395C2×Q8—944no—
396C2×Q8—18812no—
397C2×Q8#1062535018yescomplemented
398C2×Q8C23×D47112yes—
399C2×Q8C23×D4714yes—
400C2×D4—4724no—
401C2×D4—4712no—
402C2×D4—4724no—
403C2×D4—4724no—
404C2×D4—9424no—
405C2×D4—9424no—
406C2×D4—9412no—
407C2×D4—18812no—
408C22×C4—212no—
409C22×C4—212no—
410C22×C4—24no—
411C22×C4—472no—
412C22×C4—476no—
413C22×C4—472no—
414C22×C4—4724no—
415C22×C4—4748no—
416C22×C4—9424no—
417C22×C4—18824no—
418C22×C4C23×D4714yes—
419C22×C4C23×D47112yes—
420C22×C4C23×D47112yes—
421C22×C4C23×D4711yescharacteristic
422C22×C4C23×D4713yes—
423C22×C4C23×D4713yes—
424C22×C4C23×D4713yes—
425C22×C4C23×D4713yes—
426C22×C4C23×D4711yescharacteristic
427C4⋊C4—4724no—
428C4⋊C4—478no—
429C4⋊C4—4724no—
430C4⋊C4—4724no—
431C4⋊C4—4724no—
432C4⋊C4—474no—
433C4⋊C4—4712no—
434C4⋊C4C23×D47112yes—
435C4⋊C4C23×D47112yes—
436C4⋊C4C23×D47124yes—
437C22⋊C4—478no—
438C22⋊C4—4724no—
439C22⋊C4—4724no—
440C22⋊C4—4724no—
441C22⋊C4—4748no—
442C22⋊C4—4724no—
443C42—4724no—
444C42C23×D47112yes—
445C42C23×D47112yes—
446C42C23×D47112yes—
447C42C23×D47112yes—
448C23—22no—
449C23—22no—
450C23—4712no—
451C23—9412no—
452C23—9412no—
453C23—9412no—
454C23#1309407014yes—
455C23C24×D4712yes—
456C23C24×D4711yescharacteristic
457Q8—224no—
458Q8—948no—
459Q8—18824no—
460Q8C2×#10625350116yescomplemented
461D4—9448no—
462D4—9448no—
463D4—9424no—
464D4—18824no—
465C2×C4—26no—
466C2×C4—26no—
467C2×C4—26no—
468C2×C4—224no—
469C2×C4—22no—
470C2×C4—224no—
471C2×C4—28no—
472C2×C4—474no—
473C2×C4—4712no—
474C2×C4—474no—
475C2×C4—944no—
476C2×C4—9412no—
477C2×C4—944no—
478C2×C4—9448no—
479C2×C4—18848no—
480C2×C4C2×#10625350112yes—
481C2×C4#1309407214yes—
482C2×C4C24×D4712yes—
483C2×C4C24×D4716yes—
484C2×C4C24×D4716yes—
485C2×C4C24×D4716yes—
486C2×C4C24×D4716yes—
487C2×C4C24×D4712yes—
488C22—22no—
489C22—22no—
490C22—28no—
491C22—9412no—
492C22—9412no—
493C22—9412no—
494C22—18824no—
495C22C22×#1062535012yes—
496C22C2×#1309407014yes—
497C22C2×#1309407012yes—
498C22C2×#1309407212yes—
499C22C25×D4711yescharacteristic
500C4—26no—
501C4—26no—
502C4—26no—
503C4—22no—
504C4—944no—
505C4—9412no—
506C4—944no—
507C4C22×#10625350112yes—
508C4C2×#1309407214yes—
509C2—24no—
510C2—18812no—
511C2#2025747314yesdirect product factor
512C2C23×#1062535011yescharacteristic
513C2C22×#1309407011yescharacteristic
514C2C22×#1309407211yescharacteristic
subgroupG/H
centerC23C24×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC22×C4×Q8×C47C2
socleC23×C47C25
hypercenterC22×C4×Q8D47
hypocenterC47C2×#3522
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #20257473C2direct (N × H)
2. C2×#15689858C2semidirect
3. C2×Q8×Dic94C2semidirect
4. C2×#15689833C2semidirect
5. C2×#15689795C2semidirect
6. C2×#15689784C2semidirect
7. C2×#15689774C2semidirect
8. C2×#15689745C2semidirect
9. C2×#15689673C2semidirect
10. C2×#15689657C2semidirect
11. C2×#15689605C2semidirect
12. C22×C4×Q8×C47C2semidirect
15. #15689858C22semidirect
16. Q8×Dic94C22semidirect
17. #15689833C22semidirect
18. #15689795C22semidirect
19. #15689784C22semidirect
20. #15689774C22semidirect
21. #15689745C22semidirect
22. #15689673C22semidirect
26. #15689657C22semidirect
40. #15689605C22semidirect
55. C2×C4×Q8×C47C22semidirect
133. C2×Q8×C47#41semidirect
200. Q8×C47C2×#41semidirect
263. C22×C4×Q8D47semidirect
265. C2×C47#3522semidirect
305. C2×C4×Q8C2×D47semidirect
315. C47C2×#3522semidirect
397. C2×Q8#10625350semidirect
460. Q8C2×#10625350semidirect
511. C2#20257473direct (N × H)

Maximal subgroups

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

subgroupindex
#202574732
C2×#156898582
C2×Q8×Dic942
C2×#156898332
C2×#156897952
C2×#156897842
C2×#156897742
C2×#156897452
C2×#156896732
C2×#156896572
C2×#156896052
C22×C4×Q8×C472
C2×#352247