Finite Group Databasebeta

← #26220655

Subgroups — #26220655

Order 12032.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689224C211yesmaximal, characteristic
2#20246033C211yesmaximal, characteristic, complemented
3#20246025C211yesmaximal, characteristic
4#20244271C211yesmaximal, characteristic
5#20244124C211yesmaximal, characteristic, complemented
6#20244121C211yesmaximal, characteristic, complemented
7#20244088C211yesmaximal, characteristic
8#20244081C211yesmaximal, characteristic, complemented
9#20244068C211yesmaximal, characteristic
10#20244061C211yesmaximal, characteristic, complemented
11#20243288C211yesmaximal, characteristic, complemented
12#103×Dic47C211yesmaximal, characteristic, complemented
13#102×Dic47C211yesmaximal, characteristic
14#20243074C211yesmaximal, characteristic, complemented
15C47×#1968C211yesmaximal, characteristic
16C2×Q8×Dic47C2211yescharacteristic
17C2×D4×Dic47C2211yescharacteristic
18#15689224C418yescomplemented
19C2×#13093958C2211yescharacteristic
20C2×#13093958C2211yescharacteristic
21C2×#13093948C2211yescharacteristic
22C2×#13093948C2211yescharacteristic
23C2×#13093945C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25C2×#13093935C2211yescharacteristic
26C2×#13093934C2211yescharacteristic
27C2×#13093934C2211yescharacteristic
28C2×C8×Dic47C2211yescharacteristic
29C2×C8×Dic47C2211yescharacteristic
30#15689030C2211yescharacteristic
31#15689026C2211yescharacteristic
32#15689021C2211yescharacteristic
33#15689021C2211yescharacteristic
34#15689019C2211yescharacteristic
35#15689019C2211yescharacteristic
36#15689017—21no—
37#15689011—21no—
38#15689008—21no—
39#15689006—21no—
40#15688990—21no—
41#15688987—21no—
42#15688984—21no—
43#15688981—21no—
44C4⋊C4×Dic47C2211yescharacteristic
45C4⋊C4×Dic47C2211yescharacteristic
46#15688969C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C22×SD16×C47C2211yescharacteristic
54C2×#106×C47C2211yescharacteristic
55C2×#103×C47C2211yescharacteristic
56C2×#102×C47C2211yescharacteristic
57C2×C4×C8×C47C2211yescharacteristic
58C47×#391C2211yescharacteristic
59C47×#390C2211yescharacteristic
60C23×Dic47—21no—
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63Q8×Dic47—22no—
64Q8×Dic47C2×C414yes—
65D4×Dic47—22no—
66D4×Dic47C2×C414yes—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47—21no—
78C2×C22⋊Dic47—21no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47C2311yescharacteristic
83C2×C4×Dic47C2311yescharacteristic
84#13093958—22no—
85#13093958C2×C414yes—
86#13093948—22no—
87#13093948C2×C414yes—
88#13093945—22no—
89#13093945C2×C414yes—
90#13093935—22no—
91#13093934—22no—
92#13093934C2×C414yes—
93C8×Dic47—22no—
94C8×Dic47—22no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95D411yescharacteristic
104C47⋊#95D411yescharacteristic
105C47⋊#95D411yescharacteristic
106C47⋊#95D411yescharacteristic
107C22×Q8×C47C2311yescharacteristic
108C22×D4×C47C2311yescharacteristic
109C2×SD16×C47—22no—
110C2×SD16×C47—22no—
111C2×SD16×C47C2×C414yes—
112C22×C8×C47C2311yescharacteristic
113C22×C8×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C22⋊C4×C47—21no—
117C2×C22⋊C4×C47—21no—
118C2×C42×C47C2311yescharacteristic
119#106×C47—22no—
120#103×C47—22no—
121#102×C47—22no—
122C4×C8×C47—22no—
123#95×C47—21no—
124#95×C47—21no—
125C24×C47—21no—
126C2×Q8×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47—21no—
131C2×Q8×C47C22×C412yes—
132C2×D4×C47—21no—
133C2×D4×C47—21no—
134C2×D4×C47—21no—
135C2×D4×C47—21no—
136C2×D4×C47—21no—
137C2×D4×C47C22×C412yes—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2411yescharacteristic
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—24no—
147C22×Dic47—21no—
148C22×Dic47—21no—
149C22×Dic47—81no—
150C22×Dic47—81no—
151C22×Dic47C2×D411yescharacteristic
152C22×Dic47C2×D411yescharacteristic
153C22×Dic47C2×D411yescharacteristic
154C22×Dic47C2×D411yescharacteristic
155SD16×C47—28no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47C22×C412yes—
159C2×C8×C47#4112yes—
160C2×C8×C47#4112yes—
161C2×C8×C47#4112yes—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C22⋊C4×C47—81no—
167C22⋊C4×C47—81no—
168C42×C47—21no—
169C42×C47—21no—
170C22⋊Dic47—24no—
171C22⋊Dic47—81no—
172C22⋊Dic47—81no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—24no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47C22×C412yes—
181C4⋊Dic47C22×C412yes—
182Dic47⋊C4—81no—
183Dic47⋊C4—81no—
184Dic47⋊C4—81no—
185Dic47⋊C4—81no—
186C4×Dic47—21no—
187C4×Dic47—21no—
188C4×Dic47—24no—
189C4×Dic47—81no—
190C4×Dic47—81no—
191C4×Dic47C22×C412yes—
192C4×Dic47#4112yes—
193C2×C47⋊C8—21no—
194C2×C47⋊C8—21no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8C22×C412yes—
198C2×C47⋊C8#4112yes—
199C23×C47—22no—
200C23×C47—41no—
201C23×C47—41no—
202C23×C47—41no—
203C23×C47—41no—
204C23×C47—41no—
205C23×C47—41no—
206C23×C47C22×D411yescharacteristic
207Q8×C47—22no—
208Q8×C47—22no—
209Q8×C47—41no—
210Q8×C47—41no—
211D4×C47—22no—
212D4×C47—22no—
213D4×C47—41no—
214D4×C47—41no—
215C2×C4×C47—22no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47C23×C411yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×C4×C47C2×#4111yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×Dic47—21no—
237C2×Dic47—21no—
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—84no—
259C2×Dic47C4×D412yes—
260C2×Dic47C4×D412yes—
261C8×C47—22no—
262C8×C47#11714yes—
263C47⋊C8—22no—
264C47⋊C8—22no—
265#21291—471nomaximal, Sylow
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—42no—
269C22×C47—81no—
270C22×C47—81no—
271C22×C47—81no—
272C22×C47—81no—
273C22×C47C2×C4×D411yescharacteristic
274C22×C47#53211yescharacteristic
275C22×C47#53211yescharacteristic
276C22×C47#57611yescharacteristic
277C22×C47#57711yescharacteristic
278C22×C47#57711yescharacteristic
279C22×C47#57811yescharacteristic
280C4×C47—41no—
281C4×C47—41no—
282C4×C47—42no—
283C4×C47—81no—
284C4×C47—81no—
285C4×C47C2×#11711yescharacteristic
286C4×C47C4×#4111yescharacteristic
287C4×C47#52911yescharacteristic
288C4×C47#52911yescharacteristic
289Dic47—22no—
290Dic47—22no—
291Dic47—41no—
292Dic47—41no—
293Dic47—42no—
294Dic47—42no—
295Dic47—81no—
296Dic47—81no—
297C2×#440—471no—
298#2320—471no—
299#2319—471no—
300#1973—471no—
301#1968D4711yescharacteristic
302#1943—471no—
303#1942—471no—
304#1936—471no—
305#1935—471no—
306#1934—471no—
307#1933—471no—
308#1776—471no—
309C4×#103—471no—
310C4×#102—471no—
311#1726—471no—
312C2×C47—82no—
313C2×C47#229311yescharacteristic
314C2×C47#294611yescharacteristic
315C2×C47#294711yescharacteristic
316C2×C47#317411yescharacteristic
317C2×C47#317511yescharacteristic
318C2×C47#317811yescharacteristic
319C2×C47#317911yescharacteristic
320C22×SD16C2×D4711yescharacteristic
321C2×C4×Q8—471no—
322C2×C4×D4—471no—
323#440—478no—
324C2×#107—471no—
325C2×#107—471no—
326C2×#106C2×D4711yescharacteristic
327C2×#103—471no—
328C2×#103—471no—
329C2×#103C2×D4711yescharacteristic
330C2×#102—471no—
331C2×#102—471no—
332C2×#102C2×D4711yescharacteristic
333C2×#97—471no—
334C2×#97—471no—
335C2×#97—471no—
336C2×C4×C8—471no—
337C2×C4×C8—471no—
338C2×C4×C8C2×D4711yescharacteristic
339#391—471no—
340#391C2×D4711yescharacteristic
341#390—471no—
342#390C2×D4711yescharacteristic
343#389—471no—
344#389—471no—
345#389—471no—
346#389—471no—
347#388—941no—
348#388—941no—
349#387—941no—
350#387—941no—
351#383—941no—
352#382—941no—
353#381—941no—
354#380—941no—
355C4×C4⋊C4—471no—
356C4×C4⋊C4—471no—
357#376—471no—
358#340—471no—
359#340—471no—
360#340—471no—
361#340—471no—
362#340—471no—
363#340—471no—
364C47#2129111yescharacteristic, complemented, Sylow
365C22×Q8C22×D4711yescharacteristic
366C22×D4C22×D4711yescharacteristic
367C23×C4—941no—
368C2×SD16—22no—
369C2×SD16—22no—
370C2×SD16C4×D4714yes—
371C22×C8—471no—
372C22×C8—471no—
373C22×C8C22×D4711yescharacteristic
374C22×C8C22×D4711yescharacteristic
375C4×Q8—474no—
376C4×Q8—942no—
377C4×D4—474no—
378C4×D4—942no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—471no—
381C2×C4⋊C4—471no—
382C2×C4⋊C4—471no—
383C2×C4⋊C4—941no—
384C2×C4⋊C4—941no—
385C2×C4⋊C4—941no—
386C2×C4⋊C4—941no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4C22×D4711yescharacteristic
389C2×C4⋊C4C22×D4711yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—941no—
393C2×C22⋊C4—941no—
394C2×C22⋊C4—941no—
395C2×C42—471no—
396C2×C42—471no—
397C2×C42—941no—
398C2×C42—941no—
399C2×C42—941no—
400C2×C42C22×D4711yescharacteristic
401#107—474no—
402#107—942no—
403#106—22no—
404#103—22no—
405#103—474no—
406#103—942no—
407#102—22no—
408#102—474no—
409#102—942no—
410#97—474no—
411#97—942no—
412#97—942no—
413C4×C8—22no—
414C4×C8—942no—
415C4×C8—942no—
416#95—21no—
417#95—21no—
418#95—471no—
419#95—471no—
420#95—471no—
421#95—471no—
422#95—941no—
423#95—941no—
424#95—941no—
425#95—941no—
426#95—941no—
427#95—941no—
428#95—941no—
429#95—941no—
430C24—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8C2×C4×D4712yes—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×C4×D4712yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—471no—
447C22×C4—471no—
448C22×C4—471no—
449C22×C4—471no—
450C22×C4—944no—
451C22×C4—941no—
452C22×C4—941no—
453C22×C4—941no—
454C22×C4—941no—
455C22×C4—3761no—
456C22×C4—3761no—
457C22×C4D4×D4711yescharacteristic
458C22×C4D4×D4711yescharacteristic
459C22×C4C23×D4711yescharacteristic
460SD16—28no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—472no—
464C2×C8—472no—
465C2×C8—941no—
466C2×C8—941no—
467C2×C8—941no—
468C2×C8—941no—
469C2×C8C2×C4×D4712yes—
470C2×C8#1062535212yes—
471C2×C8#1062535212yes—
472C2×C8#1062535212yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—472no—
478C4⋊C4—472no—
479C4⋊C4—944no—
480C4⋊C4—941no—
481C4⋊C4—941no—
482C4⋊C4—941no—
483C4⋊C4—941no—
484C4⋊C4—941no—
485C4⋊C4—941no—
486C4⋊C4—3761no—
487C4⋊C4—3761no—
488C4⋊C4—3761no—
489C4⋊C4—3761no—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—944no—
493C22⋊C4—3761no—
494C22⋊C4—3761no—
495C42—21no—
496C42—21no—
497C42—472no—
498C42—472no—
499C42—944no—
500C42—941no—
501C42—941no—
502C42—3761no—
503C42—3761no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D4711yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—41no—
515Q8—41no—
516D4—22no—
517D4—22no—
518D4—41no—
519D4—41no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—472no—
536C2×C4—472no—
537C2×C4—941no—
538C2×C4—941no—
539C2×C4—941no—
540C2×C4—941no—
541C2×C4—942no—
542C2×C4—942no—
543C2×C4—941no—
544C2×C4—941no—
545C2×C4—941no—
546C2×C4—941no—
547C2×C4—941no—
548C2×C4—941no—
549C2×C4—3764no—
550C2×C4—1881no—
551C2×C4—1881no—
552C2×C4—1881no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4—1881no—
556C2×C4—1881no—
557C2×C4—1881no—
558C2×C4—1881no—
559C2×C4—1881no—
560C2×C4C22×C4×D4711yescharacteristic
561C2×C4#41×D4711yescharacteristic
562C2×C4#41×D4711yescharacteristic
563C2×C4C2×#1062535211yescharacteristic
564C2×C4C2×#1062535211yescharacteristic
565C2×C4C2×#1062535211yescharacteristic
566C8—22no—
567C8—942no—
568C8—942no—
569C8#1309398014yes—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C4×D4×D4711yescharacteristic
578C22#1568966811yescharacteristic
579C22#1568967111yescharacteristic
580C22#1568992511yescharacteristic
581C22#1568992911yescharacteristic
582C22#1568992911yescharacteristic
583C22#1568993711yescharacteristic
584C4—42no—
585C4—41no—
586C4—41no—
587C4—81no—
588C4—81no—
589C4—942no—
590C4—942no—
591C4—3761no—
592C4—3761no—
593C4—1882no—
594C4—1882no—
595C4—1881no—
596C4—1881no—
597C4C2×#1309398011yescharacteristic
598C4C4×#1062535211yescharacteristic
599C4#1568965211yescharacteristic
600C4#1568965411yescharacteristic
601C2—82no—
602C2#2024585011yescharacteristic
603C2#2025221511yescharacteristic
604C2#2025222411yescharacteristic
605C2#2025401311yescharacteristic
606C2#2025402411yescharacteristic
607C2#2025405111yescharacteristic
608C2#2025406611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1968C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#1968D47
hypocenterC47#21291
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
2. #20246033C2semidirect
5. #20244124C2semidirect
6. #20244121C2semidirect
8. #20244081C2semidirect
10. #20244061C2semidirect
11. #20243288C2semidirect
12. #103×Dic47C2semidirect
14. #20243074C2semidirect
18. #15689224C4semidirect
364. C47#21291semidirect

Maximal subgroups

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

subgroupindex
C2×#156892242
#202460332
#202460252
#202442712
#202441242
#202441212
#202440882
#202440812
#202440682
#202440612
#202432882
#103×Dic472
#102×Dic472
#202430742
C47×#19682
#2129147