Finite Group Databasebeta

← #26184462

Subgroups — #26184462

Order 12032.

876 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252408C211yesmaximal, characteristic, complemented
2#20252039C211yesmaximal, characteristic, complemented
3#20242019C212yesmaximal, complemented
4#20242015C212yesmaximal, complemented
5#20242003C211yesmaximal, characteristic, complemented
6#20241995C211yesmaximal, characteristic, complemented
7#20241995C212yesmaximal, complemented
8#20241990C212yesmaximal, complemented
9#20241983C211yesmaximal, characteristic, complemented
10#20241976C211yesmaximal, characteristic, complemented
11C47×#3515C211yesmaximal, characteristic
12Q8×C47⋊C8—21no—
13#15689201—21no—
14#15689201—22no—
15#15689184—22no—
16#15689180C2212yes—
17#15689180C2211yescharacteristic
18#15689180C2212yes—
19#15689180C2212yes—
20#15689150—21no—
21#15689090—21no—
22C4×#10625329C2211yescharacteristic
23C4×#10625329C2211yescharacteristic
24C4×#10625329C2212yes—
25C2×#13093935C2211yescharacteristic
26#15688865C2211yescharacteristic
27#15688865C2211yescharacteristic
28#15688865C2212yes—
29#15688865C2212yes—
30#15688865C2212yes—
31#15688865C2212yes—
32#15688865C2212yes—
33#15688863C2211yescharacteristic
34#15688863C2211yescharacteristic
35#15688863C2212yes—
36C47×#557C411yescharacteristic
37C47×#554C411yescharacteristic
38C47×#554C411yescharacteristic
39C47×#553C412yes—
40C47×#552—22no—
41C47×#552—21no—
42C47×#552C412yes—
43C47×#547—21no—
44C47×#547C411yescharacteristic
45C47×#541—21no—
46C47×#539—22no—
47C47×#533C2211yescharacteristic
48C47×#529C2211yescharacteristic
49C47×#529C2212yes—
50C2×#125×C47C2212yes—
51C47×#520—21no—
52C4×#41×C47C2211yescharacteristic
53C2×#10625329—21no—
54C2×#10625329—22no—
55C2×#10625329—21no—
56C22×C47⋊C8—21no—
57C4⋊(C47⋊C8)—21no—
58C4⋊(C47⋊C8)—22no—
59C4⋊(C47⋊C8)—22no—
60C4⋊(C47⋊C8)—22no—
61C22⋊(C47⋊C8)—22no—
62C22⋊(C47⋊C8)—22no—
63C22⋊(C47⋊C8)—22no—
64C22⋊(C47⋊C8)—21no—
65#13093935—21no—
66#13093935C2311yescharacteristic
67#13093935C2311yescharacteristic
68#13093935C2311yescharacteristic
69#13093935C2311yescharacteristic
70#13093935C2312yes—
71#13093935C2312yes—
72C4×C47⋊C8—21no—
73C4×C47⋊C8—22no—
74C4×C47⋊C8—21no—
75C2×#41×C47—21no—
76C2×#41×C47D411yescharacteristic
77C4⋊Q8×C47—21no—
78C4⋊Q8×C47C2×C411yescharacteristic
79C4⋊Q8×C47C2×C411yescharacteristic
80#126×C47—22no—
81#126×C47—22no—
82#126×C47—22no—
83#126×C47—21no—
84#125×C47—22no—
85#125×C47—22no—
86#125×C47—21no—
87#125×C47—22no—
88#125×C47—22no—
89#125×C47C2×C412yes—
90#125×C47C2×C412yes—
91#125×C47C2311yescharacteristic
92#125×C47C2312yes—
93#124×C47C2311yescharacteristic
94#123×C47—22no—
95#123×C47—21no—
96#123×C47—22no—
97#123×C47—22no—
98#123×C47—22no—
99C22⋊Q8×C47—21no—
100C22⋊Q8×C47—21no—
101C22⋊Q8×C47—21no—
102C22⋊Q8×C47—22no—
103C22⋊Q8×C47—21no—
104C22⋊Q8×C47—22no—
105C22⋊Q8×C47C2×C411yescharacteristic
106C22⋊Q8×C47C2×C412yes—
107#121×C47—22no—
108#121×C47C2×C412yes—
109#121×C47C2×C411yescharacteristic
110C4×Q8×C47—21no—
111C4×Q8×C47—21no—
112C4×Q8×C47—21no—
113C4×Q8×C47C2×C412yes—
114C4×Q8×C47C2×C411yescharacteristic
115C4×Q8×C47C2×C411yescharacteristic
116C4×Q8×C47D411yescharacteristic
117C4×D4×C47—21no—
118C4×D4×C47—22no—
119C4×D4×C47—21no—
120C4×D4×C47—22no—
121C4×D4×C47C2×C411yescharacteristic
122C4×D4×C47C2×C412yes—
123C4×D4×C47C2×C411yescharacteristic
124C4×D4×C47C2×C412yes—
125C4×D4×C47D411yescharacteristic
126C4×D4×C47D412yes—
127#117×C47—22no—
128#117×C47—21no—
129#117×C47—22no—
130#117×C47C2×C412yes—
131#117×C47D411yescharacteristic
132#117×C47D412yes—
133C2×C4⋊C4×C47—22no—
134C2×C4⋊C4×C47—22no—
135C2×C4⋊C4×C47—21no—
136C2×C4⋊C4×C47C2×C412yes—
137C2×C4⋊C4×C47C2×C412yes—
138C2×C42×C47C2312yes—
139C2×C42×C47C2311yescharacteristic
140#41×C47—42no—
141#41×C47—44no—
142C2×Q8×C47—21no—
143C2×Q8×C47—21no—
144C2×Q8×C47C22⋊C411yescharacteristic
145C2×Q8×C47C22×C411yescharacteristic
146C2×D4×C47—22no—
147C2×D4×C47—21no—
148C2×D4×C47C22⋊C411yescharacteristic
149C2×D4×C47C22⋊C412yes—
150C2×D4×C47C22×C411yescharacteristic
151C22×C4×C47—22no—
152C22×C4×C47—22no—
153C22×C4×C47—21no—
154C22×C4×C47—22no—
155C22×C4×C47C22×C412yes—
156C22×C4×C47C22×C411yescharacteristic
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2×D412yes—
159C22×C4×C47C2×D412yes—
160C22×C4×C47C2×D411yescharacteristic
161C4⋊C4×C47—22no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47—22no—
165C4⋊C4×C47—22no—
166C4⋊C4×C47—22no—
167C4⋊C4×C47—22no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—22no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47—22no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—22no—
176C4⋊C4×C47—22no—
177C4⋊C4×C47C22⋊C411yescharacteristic
178C4⋊C4×C47C22⋊C411yescharacteristic
179C4⋊C4×C47C22⋊C412yes—
180C4⋊C4×C47C22⋊C412yes—
181C4⋊C4×C47C22×C411yescharacteristic
182C4⋊C4×C47C22×C411yescharacteristic
183C4⋊C4×C47C22×C412yes—
184C4⋊C4×C47C22×C412yes—
185C22⋊C4×C47—22no—
186C22⋊C4×C47—22no—
187C22⋊C4×C47—21no—
188C22⋊C4×C47—21no—
189C22⋊C4×C47—22no—
190C22⋊C4×C47—22no—
191C22⋊C4×C47—21no—
192C22⋊C4×C47—21no—
193C22⋊C4×C47—21no—
194C22⋊C4×C47C22⋊C412yes—
195C22⋊C4×C47C22⋊C412yes—
196C22⋊C4×C47C22⋊C412yes—
197C42×C47—22no—
198C42×C47—22no—
199C42×C47—21no—
200C42×C47C22×C412yes—
201C42×C47C22×C411yescharacteristic
202C42×C47C2×D412yes—
203C42×C47C2×D411yescharacteristic
204C42×C47C2×D411yescharacteristic
205C42×C47C2×D412yes—
206C42×C47C2411yescharacteristic
207#10625329—44no—
208#10625329—42no—
209#10625329—42no—
210C2×C47⋊C8—21no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—22no—
213C2×C47⋊C8—22no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—42no—
217C23×C47—21no—
218C23×C47C2×C22⋊C411yescharacteristic
219C23×C47C2×C22⋊C412yes—
220Q8×C47—22no—
221Q8×C47—41no—
222Q8×C47—42no—
223Q8×C47—81no—
224D4×C47—22no—
225D4×C47—42no—
226D4×C47—41no—
227D4×C47—44no—
228D4×C47—81no—
229C2×C4×C47—22no—
230C2×C4×C47—21no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—42no—
236C2×C4×C47—41no—
237C2×C4×C47—42no—
238C2×C4×C47—41no—
239C2×C4×C47—42no—
240C2×C4×C47—44no—
241C2×C4×C47—42no—
242C2×C4×C47—82no—
243C2×C4×C47—82no—
244C2×C4×C47C2×C22⋊C411yescharacteristic
245C2×C4×C47C2×C22⋊C412yes—
246C2×C4×C47C2×C22⋊C411yescharacteristic
247C2×C4×C47C2×C22⋊C412yes—
248C2×C4×C47C2×C22⋊C412yes—
249C2×C4×C47C2×C22⋊C411yescharacteristic
250C2×C4×C47C23×C411yescharacteristic
251C2×C4×C47C22×D411yescharacteristic
252C2×C4×C47C22×D411yescharacteristic
253C47⋊C8—22no—
254C47⋊C8—41no—
255C47⋊C8—41no—
256C47⋊C8—42no—
257C47⋊C8—41no—
258C47⋊C8—42no—
259#17682—471nomaximal, Sylow
260C22×C47—21no—
261C22×C47—22no—
262C22×C47—21no—
263C22×C47—22no—
264C22×C47—22no—
265C22×C47—21no—
266C22×C47—41no—
267C22×C47—41no—
268C22×C47—41no—
269C22×C47C22×C22⋊C411yescharacteristic
270C4×C47—21no—
271C4×C47—21no—
272C4×C47—22no—
273C4×C47—21no—
274C4×C47—22no—
275C4×C47—21no—
276C4×C47—22no—
277C4×C47—21no—
278C4×C47—41no—
279C4×C47—41no—
280C4×C47—42no—
281C4×C47—41no—
282C4×C47—42no—
283C4×C47#42112yes—
284#3515D4711yescharacteristic
285#2971—471no—
286#2923—471no—
287#1535—472no—
288#1534—472no—
289#1531—471no—
290#1529—471no—
291#1529—472no—
292#1528—472no—
293#1526—471no—
294#1524—471no—
295C2×C47—41no—
296C2×C47—42no—
297C2×C47—81no—
298C2×C47#288911yescharacteristic
299C2×C47C2×#42111yescharacteristic
300C2×C47#290111yescharacteristic
301#557Dic4711yescharacteristic
302#554Dic4711yescharacteristic
303#554Dic4711yescharacteristic
304#553Dic4712yes—
305#552—22no—
306#552—21no—
307#552Dic4712yes—
308#547—21no—
309#547Dic4711yescharacteristic
310#541—21no—
311#539—22no—
312#533C2×D4711yescharacteristic
313#529C2×D4711yescharacteristic
314#529C2×D4712yes—
315C2×#125C2×D4712yes—
316#520—21no—
317C4×#41C2×D4711yescharacteristic
318C8×Q8—941no—
319#436—941no—
320#436—942no—
321#433—942no—
322#432—472no—
323#432—472no—
324#432—471no—
325#432—472no—
326#424—941no—
327#408—941no—
328C4×M16—471no—
329C4×M16—471no—
330C4×M16—472no—
331C2×#97—471no—
332#330—471no—
333#330—471no—
334#330—472no—
335#330—472no—
336#330—472no—
337#330—472no—
338#330—472no—
339#329—471no—
340#329—471no—
341#329—472no—
342C47#1768211yescharacteristic, complemented, Sylow
343C2×#41—21no—
344C2×#41C22⋊D4711yescharacteristic
345C2×M16—941no—
346C2×M16—942no—
347C2×M16—941no—
348C22×C8—941no—
349C4⋊Q8—21no—
350C4⋊Q8C2×Dic4711yescharacteristic
351C4⋊Q8C2×Dic4711yescharacteristic
352#126—22no—
353#126—22no—
354#126—22no—
355#126—21no—
356#125—22no—
357#125—22no—
358#125—21no—
359#125—22no—
360#125—22no—
361#125C2×Dic4712yes—
362#125C2×Dic4712yes—
363#125C22×D4711yescharacteristic
364#125C22×D4712yes—
365#124C22×D4711yescharacteristic
366#123—22no—
367#123—21no—
368#123—22no—
369#123—22no—
370#123—22no—
371C22⋊Q8—21no—
372C22⋊Q8—21no—
373C22⋊Q8—21no—
374C22⋊Q8—22no—
375C22⋊Q8—21no—
376C22⋊Q8—22no—
377C22⋊Q8C2×Dic4711yescharacteristic
378C22⋊Q8C2×Dic4712yes—
379#121—22no—
380#121C2×Dic4712yes—
381#121C2×Dic4711yescharacteristic
382C4×Q8—21no—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8C2×Dic4712yes—
386C4×Q8C2×Dic4711yescharacteristic
387C4×Q8C2×Dic4711yescharacteristic
388C4×Q8C22⋊D4711yescharacteristic
389C4×D4—21no—
390C4×D4—22no—
391C4×D4—21no—
392C4×D4—22no—
393C4×D4C2×Dic4711yescharacteristic
394C4×D4C2×Dic4712yes—
395C4×D4C2×Dic4711yescharacteristic
396C4×D4C2×Dic4712yes—
397C4×D4C22⋊D4711yescharacteristic
398C4×D4C22⋊D4712yes—
399#117—22no—
400#117—21no—
401#117—22no—
402#117C2×Dic4712yes—
403#117C22⋊D4711yescharacteristic
404#117C22⋊D4712yes—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4C2×Dic4712yes—
409C2×C4⋊C4C2×Dic4712yes—
410C2×C42C22×D4712yes—
411C2×C42C22×D4711yescharacteristic
412C4⋊C8—942no—
413C4⋊C8—942no—
414C4⋊C8—941no—
415C4⋊C8—942no—
416C22⋊C8—942no—
417C22⋊C8—942no—
418C22⋊C8—942no—
419C22⋊C8—941no—
420#97—471no—
421#97—471no—
422#97—471no—
423#97—471no—
424#97—472no—
425#97—472no—
426#97—941no—
427C4×C8—941no—
428C4×C8—941no—
429C4×C8—942no—
430#41—42no—
431#41—44no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8C22⋊Dic4711yescharacteristic
435C2×Q8C22×Dic4711yescharacteristic
436C2×D4—22no—
437C2×D4—21no—
438C2×D4C22⋊Dic4711yescharacteristic
439C2×D4C22⋊Dic4712yes—
440C2×D4C22×Dic4711yescharacteristic
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4C22×Dic4712yes—
446C22×C4C22×Dic4711yescharacteristic
447C22×C4C2×C22⋊D4711yescharacteristic
448C22×C4C2×C22⋊D4712yes—
449C22×C4C2×C22⋊D4712yes—
450C22×C4C2×C22⋊D4711yescharacteristic
451M16—1884no—
452M16—1882no—
453M16—1882no—
454C2×C8—941no—
455C2×C8—942no—
456C2×C8—941no—
457C2×C8—942no—
458C2×C8—941no—
459C2×C8—941no—
460C2×C8—1882no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4C22⋊Dic4711yescharacteristic
478C4⋊C4C22⋊Dic4711yescharacteristic
479C4⋊C4C22⋊Dic4712yes—
480C4⋊C4C22⋊Dic4712yes—
481C4⋊C4C22×Dic4711yescharacteristic
482C4⋊C4C22×Dic4711yescharacteristic
483C4⋊C4C22×Dic4712yes—
484C4⋊C4C22×Dic4712yes—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4C22⋊Dic4712yes—
495C22⋊C4C22⋊Dic4712yes—
496C22⋊C4C22⋊Dic4712yes—
497C42—22no—
498C42—22no—
499C42—21no—
500C42C22×Dic4712yes—
501C42C22×Dic4711yescharacteristic
502C42C2×C22⋊D4711yescharacteristic
503C42C2×C22⋊D4711yescharacteristic
504C42C2×C22⋊D4712yes—
505C42C2×C22⋊D4712yes—
506C42C23×D4711yescharacteristic
507C23—21no—
508C23C2×C22⋊Dic4711yescharacteristic
509C23C2×C22⋊Dic4712yes—
510Q8—22no—
511Q8—41no—
512Q8—42no—
513Q8—81no—
514D4—22no—
515D4—42no—
516D4—41no—
517D4—44no—
518D4—81no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4C2×C22⋊Dic4711yescharacteristic
535C2×C4C2×C22⋊Dic4712yes—
536C2×C4C2×C22⋊Dic4711yescharacteristic
537C2×C4C2×C22⋊Dic4712yes—
538C2×C4C2×C22⋊Dic4712yes—
539C2×C4C2×C22⋊Dic4711yescharacteristic
540C2×C4C23×Dic4711yescharacteristic
541C2×C4C22×C22⋊D4711yescharacteristic
542C2×C4C22×C22⋊D4711yescharacteristic
543C8—942no—
544C8—1881no—
545C8—1881no—
546C8—1882no—
547C8—1881no—
548C8—1882no—
549C22—21no—
550C22—22no—
551C22—22no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22C22×C22⋊Dic4711yescharacteristic
559C4—21no—
560C4—22no—
561C4—21no—
562C4—22no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—22no—
567C4—41no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—41no—
572C4#1568914012yes—
573C2—41no—
574C2—42no—
575C2—81no—
576C2#2025187111yescharacteristic
577C2C2×#1568914011yescharacteristic
578C2#2025193511yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
Fitting, prime radicalC47×#3515C2
socleC22×C47C22×C22⋊C4
cosocleC42×C47C24
hypercenter#3515D47
hypocenterC47#17682
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3515

Complements

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

NH ≅ G/Ntype
1. #20252408C2semidirect
2. #20252039C2semidirect
3. #20242019C2semidirect
4. #20242015C2semidirect
5. #20242003C2semidirect
6. #20241995C2semidirect
7. #20241995C2semidirect
8. #20241990C2semidirect
9. #20241983C2semidirect
10. #20241976C2semidirect
342. C47#17682semidirect

Maximal subgroups

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

subgroupindex
#202524082
#202520392
#202420192
#202420152
#202420032
#202419952
#202419902
#202419832
#202419762
C47×#35152
#1768247