Finite Group Databasebeta

← #26250198

Subgroups — #26250198

Order 12032.

1174 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689400C211yesmaximal, characteristic, complemented
2C2×C4×#10625336C211yesmaximal, characteristic, complemented
3#20248301C211yesmaximal, characteristic
4#20244588C212yesmaximal, complemented
5#20244389C212yesmaximal, complemented
6#20244350C211yesmaximal, characteristic, complemented
7#20244183C211yesmaximal, characteristic, complemented
8#20243710C212yesmaximal, complemented
9#20243587C212yesmaximal, complemented
10#20243393C211yesmaximal, characteristic, complemented
11C2×C47×#556C211yesmaximal, characteristic, complemented
12C22×#10625336C2211yescharacteristic
13C2×#13093998C2212yes—
14C2×#13093995—21no—
15C2×C4×D188C2211yescharacteristic
16#15689400—24no—
17C4×#10625336—24no—
18C2×#13093957C2212yes—
19C2×#13093955C2211yescharacteristic
20C2×C4⋊(C47⋊C8)C2211yescharacteristic
21C2×#13093948C2211yescharacteristic
22C2×#13093948C2212yes—
23C2×#13093944C2212yes—
24C2×#13093944C2211yescharacteristic
25C2×C22⋊(C47⋊C8)C2212yes—
26C2×C22⋊(C47⋊C8)C2212yes—
27C2×C4×C47⋊C8C2211yescharacteristic
28#15689039—22no—
29#15688993C2211yescharacteristic
30#15688983C2212yes—
31#15688980—21no—
32#15688885C2212yes—
33#15688885C2212yes—
34#15688885C2212yes—
35C47×#556—24no—
36C47×#556—24no—
37C2×C4⋊Q8×C47C2211yescharacteristic
38C2×#126×C47—22no—
39C2×#124×C47—21no—
40C2×C22⋊Q8×C47C2212yes—
41C2×C4×Q8×C47—21no—
42C2×C4×Q8×C47C2211yescharacteristic
43C2×C4×D4×C47C2211yescharacteristic
44C2×#117×C47C2212yes—
45C22×D188C2311yescharacteristic
46C22×C22⋊D47—22no—
47C22×C4×D47—21no—
48C2×#10625336—22no—
49C2×#10625336—22no—
50C2×#10625336—22no—
51C22×C47⋊C8C2311yescharacteristic
52C22×C47⋊C8C2312yes—
53C22×C47⋊C8C2311yescharacteristic
54#13093998—24no—
55#13093998—42no—
56#13093995—44no—
57C4×D188—22no—
58C4×D188—22no—
59C2×C4⋊Dic47—21no—
60C2×C4⋊Dic47—21no—
61C2×C4⋊Dic47C2311yescharacteristic
62C2×C4⋊Dic47C2312yes—
63C2×C22⋊Dic47—22no—
64C2×(C2×D47)⋊C4—22no—
65C2×(C2×D47)⋊C4—21no—
66#13093957—24no—
67#13093955—22no—
68C4⋊(C47⋊C8)—22no—
69#13093948—22no—
70#13093948—42no—
71#13093944—24no—
72#13093944—22no—
73C22⋊(C47⋊C8)—24no—
74C22⋊(C47⋊C8)—42no—
75C4×C47⋊C8—22no—
76C47⋊#95—22no—
77C22×Q8×C47—21no—
78C22×Q8×C47C2311yescharacteristic
79C22×D4×C47D411yescharacteristic
80C23×C4×C47C2312yes—
81C4⋊Q8×C47—22no—
82C4⋊Q8×C47D414yescomplemented
83#126×C47—216no—
84#124×C47—28no—
85C22⋊Q8×C47—24no—
86C22⋊Q8×C47—24no—
87C4×Q8×C47—28no—
88C4×Q8×C47—22no—
89C4×Q8×C47D414yescomplemented
90C4×D4×C47—22no—
91C4×D4×C47—22no—
92#117×C47—24no—
93#117×C47—24no—
94C2×C4⋊C4×C47—22no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47D411yescharacteristic
98C2×C4⋊C4×C47C2311yescharacteristic
99C2×C4⋊C4×C47C2312yes—
100C2×C22⋊C4×C47—22no—
101C2×C22⋊C4×C47—22no—
102C2×C22⋊C4×C47D412yes—
103C2×C42×C47—22no—
104C2×C42×C47—21no—
105C2×C42×C47C2311yescharacteristic
106C24×C47C2×D412yes—
107C23×D47—21no—
108C2×Q8×C47—24no—
109C2×Q8×C47—21no—
110C2×Q8×C47—21no—
111C2×Q8×C47—21no—
112C2×Q8×C47—21no—
113C2×Q8×C47—21no—
114C2×Q8×C47—42no—
115C2×Q8×C47—42no—
116C2×Q8×C47C2×D412yes—
117C2×D4×C47—22no—
118C2×D4×C47—22no—
119C2×D4×C47—22no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47C2×D412yes—
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2411yescharacteristic
133C2×C22⋊D47—44no—
134C2×C22⋊D47—44no—
135C2×C22⋊D47—82no—
136C22×Dic47—21no—
137C22×Dic47—22no—
138C2×D188—21no—
139C2×D188—21no—
140C2×D188—21no—
141C2×D188—21no—
142C2×D188—21no—
143C2×D188#4112yes—
144C2×C4×D47—42no—
145C2×C4×D47—42no—
146C2×C4×D47—42no—
147C4⋊C4×C47—28no—
148C4⋊C4×C47—24no—
149C4⋊C4×C47—24no—
150C4⋊C4×C47—22no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47C2×D412yes—
154C4⋊C4×C47C2×D414yes—
155C22⋊C4×C47—28no—
156C22⋊C4×C47—28no—
157C22⋊C4×C47—24no—
158C42×C47—28no—
159C42×C47—24no—
160C42×C47C2×D412yes—
161C42×C47C2×D412yes—
162C22⋊Dic47—44no—
163#10625336—28no—
164(C2×D47)⋊C4—44no—
165(C2×D47)⋊C4—42no—
166C4⋊Dic47—21no—
167C4⋊Dic47—21no—
168C4⋊Dic47—22no—
169C4⋊Dic47—22no—
170C4⋊Dic47—42no—
171C4⋊Dic47—81no—
172C2×C47⋊C8—21no—
173C2×C47⋊C8—21no—
174C2×C47⋊C8—21no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—22no—
177C2×C47⋊C8—21no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8—21no—
180C2×C47⋊C8#4112yes—
181C23×C47—22no—
182C23×C47—22no—
183C23×C47—22no—
184C23×C47—22no—
185C23×C47—22no—
186C23×C47—22no—
187C23×C47—22no—
188C23×C47C22×D411yescharacteristic
189C22×D47—22no—
190C22×D47—41no—
191C22×D47—41no—
192C22×D47—41no—
193C22×D47—41no—
194C22×D47—41no—
195C22×D47—41no—
196Q8×C47—22no—
197Q8×C47—22no—
198Q8×C47—41no—
199Q8×C47—41no—
200Q8×C47—48no—
201D4×C47—44no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—22no—
207C2×C4×C47—24no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—41no—
211C2×C4×C47—41no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—42no—
217C2×C4×C47—42no—
218C2×C4×C47—42no—
219C2×C4×C47—42no—
220C2×C4×C47#12012yes—
221C2×C4×C47#12012yes—
222C2×C4×C47#13512yes—
223C2×C4×C47AGL(1,ℤ/8)12yes—
224C2×C4×C47C22×D411yescharacteristic
225C2×C4×C47C22×D411yescharacteristic
226C2×C4×C47C2×#4111yescharacteristic
227C2×C4×C47C2×#4111yescharacteristic
228C2×C4×C47H(5,𝔽2)11yescharacteristic
229C2×C4×C472−1+411yescharacteristic
230C22⋊D47—88no—
231C2×Dic47—41no—
232C2×Dic47—42no—
233C2×Dic47—41no—
234C2×Dic47—42no—
235C2×Dic47—41no—
236C2×Dic47—42no—
237D188—22no—
238D188—22no—
239D188—41no—
240D188—41no—
241C4×D47—84no—
242C47⋊C8—22no—
243C47⋊C8—41no—
244C47⋊C8—42no—
245#24119—471nomaximal, Sylow
246C22×C47—42no—
247C22×C47—42no—
248C22×C47—42no—
249C22×C47—42no—
250C22×C47—42no—
251C22×C47—42no—
252C22×C47—42no—
253C22×C47C2×#12011yescharacteristic
254C22×C47#54611yescharacteristic
255C22×C47#54711yescharacteristic
256C22×C47C2×#13511yescharacteristic
257C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C47#57711yescharacteristic
259C22×C47#57811yescharacteristic
260C2×D47—42no—
261C2×D47—42no—
262C2×D47—42no—
263C2×D47—81no—
264C2×D47—81no—
265C2×D47—81no—
266C2×D47—81no—
267C4×C47—22no—
268C4×C47—42no—
269C4×C47—42no—
270C4×C47—44no—
271C4×C47—42no—
272C4×C47#44914yes—
273C4×C47#52511yescharacteristic
274C4×C47#53711yescharacteristic
275C4×C47#54811yescharacteristic
276C4×C47#55011yescharacteristic
277Dic47—81no—
278Dic47—82no—
279C2×#556D4711yescharacteristic, complemented
280C2×#474—471no—
281C2×C4×SD16—471no—
282#2537—471no—
283#2036—472no—
284#2000—472no—
285#1990—471no—
286#1954—471no—
287#1875—472no—
288#1854—472no—
289#1803—471no—
290C2×C47—82no—
291C2×C47#240711yescharacteristic
292C2×C47C2×#44911yescharacteristic
293C2×C47#301311yescharacteristic
294C2×C47#330211yescharacteristic
295C2×C47#332411yescharacteristic
296C2×C47#334011yescharacteristic
297C2×C47#335311yescharacteristic
298D47—82no—
299C22×SD16—471no—
300#556—24no—
301#556—24no—
302C2×C4⋊Q8C2×D4711yescharacteristic
303C2×#126—22no—
304C2×#124—21no—
305C2×C22⋊Q8C2×D4712yes—
306C2×#121—472no—
307C2×#121—941no—
308C2×C4×Q8—21no—
309C2×C4×Q8C2×D4711yescharacteristic
310C2×C4×D4—471no—
311C2×C4×D4C2×D4711yescharacteristic
312C2×#117C2×D4712yes—
313#474—944no—
314C4×SD16—944no—
315C2×#107—472no—
316C2×#106—471no—
317C2×C4⋊C8—471no—
318C2×#103—471no—
319C2×#103—472no—
320C2×#102—471no—
321C2×#102—472no—
322C2×C22⋊C8—472no—
323C2×C22⋊C8—472no—
324C2×C4×C8—471no—
325#394—942no—
326#384—471no—
327#380—472no—
328#380—941no—
329#340—472no—
330#340—472no—
331#340—472no—
332C47#2411911yescharacteristic, complemented, Sylow
333C22×Q8—21no—
334C22×Q8C22×D4711yescharacteristic
335C22×D4—471no—
336C22×D4—942no—
337C22×D4C22⋊D4711yescharacteristic
338C23×C4—941no—
339C23×C4C22×D4712yes—
340C2×SD16—942no—
341C2×SD16—942no—
342C2×SD16—942no—
343C22×C8—471no—
344C22×C8—471no—
345C22×C8—472no—
346C4⋊Q8—22no—
347C4⋊Q8C22⋊D4714yescomplemented
348#126—216no—
349#124—28no—
350C22⋊Q8—24no—
351C22⋊Q8—24no—
352#121—944no—
353#121—1882no—
354#121—1884no—
355C4×Q8—28no—
356C4×Q8—22no—
357C4×Q8C22⋊D4714yescomplemented
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—942no—
361C4×D4—942no—
362#117—24no—
363#117—24no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—471no—
368C2×C4⋊C4—472no—
369C2×C4⋊C4—941no—
370C2×C4⋊C4—941no—
371C2×C4⋊C4C22⋊D4711yescharacteristic
372C2×C4⋊C4C22×D4711yescharacteristic
373C2×C4⋊C4C22×D4712yes—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—942no—
377C2×C22⋊C4—941no—
378C2×C22⋊C4—942no—
379C2×C22⋊C4C22⋊D4712yes—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42C22×D4711yescharacteristic
383#107—944no—
384#106—942no—
385C4⋊C8—942no—
386#103—942no—
387#103—1882no—
388#102—942no—
389#102—944no—
390C22⋊C8—944no—
391C22⋊C8—1882no—
392C4×C8—942no—
393#95—942no—
394C24—941no—
395C24C2×C22⋊D4712yes—
396C2×Q8—24no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8C2×C22⋊D4712yes—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—472no—
409C2×D4—941no—
410C2×D4—941no—
411C2×D4—941no—
412C2×D4—941no—
413C2×D4—941no—
414C2×D4—3762no—
415C2×D4—1884no—
416C2×D4—1884no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—941no—
427C22×C4—942no—
428C22×C4—1882no—
429C22×C4—1882no—
430C22×C4—1882no—
431C22×C4C2×C22⋊D4712yes—
432C22×C4C2×C22⋊D4711yescharacteristic
433C22×C4C2×C22⋊D4711yescharacteristic
434C22×C4C23×D4711yescharacteristic
435SD16—948no—
436C2×C8—472no—
437C2×C8—941no—
438C2×C8—941no—
439C2×C8—941no—
440C2×C8—942no—
441C2×C8—942no—
442C2×C8—941no—
443C2×C8—942no—
444C2×C8—941no—
445C4⋊C4—28no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—941no—
452C4⋊C4—941no—
453C4⋊C4—942no—
454C4⋊C4—942no—
455C4⋊C4—3761no—
456C4⋊C4—1882no—
457C4⋊C4C2×C22⋊D4712yes—
458C4⋊C4C2×C22⋊D4714yes—
459C22⋊C4—28no—
460C22⋊C4—28no—
461C22⋊C4—24no—
462C22⋊C4—1884no—
463C22⋊C4—1884no—
464C22⋊C4—1882no—
465C42—28no—
466C42—24no—
467C42C2×C22⋊D4712yes—
468C42C2×C22⋊D4712yes—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—942no—
477C23—1881no—
478C23—1881no—
479C23—1881no—
480C23—1881no—
481C23—1881no—
482C23—1881no—
483C23C22×C22⋊D4711yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—48no—
487Q8—41no—
488Q8—41no—
489D4—44no—
490D4—942no—
491D4—942no—
492D4—3768no—
493D4—1881no—
494D4—1881no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—3764no—
514C2×C4—1882no—
515C2×C4—1881no—
516C2×C4—1881no—
517C2×C4—1881no—
518C2×C4—1882no—
519C2×C4—1882no—
520C2×C4#1309399312yes—
521C2×C4#1309399312yes—
522C2×C4#1309404412yes—
523C2×C4#1309404912yes—
524C2×C4C22×C22⋊D4711yescharacteristic
525C2×C4C22×C22⋊D4711yescharacteristic
526C2×C4#41×D4711yescharacteristic
527C2×C4#41×D4711yescharacteristic
528C2×C4#1309407011yescharacteristic
529C2×C4#1309407111yescharacteristic
530C8—942no—
531C8—1882no—
532C8—1881no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—3761no—
541C22—3761no—
542C22—3761no—
543C22—3761no—
544C22—1882no—
545C22—1882no—
546C22—1882no—
547C22C2×#1309399311yescharacteristic
548C22#1568976111yescharacteristic
549C22#1568977211yescharacteristic
550C22C2×#1309404411yescharacteristic
551C22C2×#1309404911yescharacteristic
552C22#1568993111yescharacteristic
553C22#1568993811yescharacteristic
554C4—22no—
555C4—42no—
556C4—44no—
557C4—42no—
558C4—42no—
559C4—3762no—
560C4—3761no—
561C4#1568926814yes—
562C4#1568963011yescharacteristic
563C4#1568970111yescharacteristic
564C4#1568978011yescharacteristic
565C4#1568979211yescharacteristic
566C2—82no—
567C2—3762no—
568C2#2024672711yescharacteristic
569C2C2×#1568926811yescharacteristic
570C2#2025270311yescharacteristic
571C2#2025529911yescharacteristic
572C2#2025561311yescharacteristic
573C2#2025579511yescharacteristic
574C2#2025596611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#556C2
socleC23×C47C22×D4
prime radical#20248301C2
hypercenterC2×#556D47
hypocenterC47#24119
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. C2×#15689400C2semidirect
2. C2×C4×#10625336C2semidirect
4. #20244588C2semidirect
5. #20244389C2semidirect
6. #20244350C2semidirect
7. #20244183C2semidirect
8. #20243710C2semidirect
9. #20243587C2semidirect
10. #20243393C2semidirect
11. C2×C47×#556C2semidirect
82. C4⋊Q8×C47D4semidirect
89. C4×Q8×C47D4semidirect
279. C2×#556D47semidirect
332. C47#24119semidirect
347. C4⋊Q8C22⋊D47semidirect
357. C4×Q8C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156894002
C2×C4×#106253362
#202483012
#202445882
#202443892
#202443502
#202441832
#202437102
#202435872
#202433932
C2×C47×#5562
#2411947