Finite Group Databasebeta

← #26186910

Subgroups — #26186910

Order 12032.

622 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252537C211yesmaximal, characteristic, complemented
2#20252464C211yesmaximal, characteristic, complemented
3#20242302C211yesmaximal, characteristic, complemented
4#20242291C211yesmaximal, characteristic, complemented
5#20242279C211yesmaximal, characteristic, complemented
6#20242273C211yesmaximal, characteristic, complemented
7#20242212C211yesmaximal, characteristic, complemented
8SD16⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
9SD16×C47⋊C8C211yesmaximal, characteristic, complemented
10D8×C47⋊C8C211yesmaximal, characteristic, complemented
11#20242155C211yesmaximal, characteristic, complemented
12#20241848C211yesmaximal, characteristic, complemented
13#20241839C211yesmaximal, characteristic, complemented
14M16×C47⋊C8C211yesmaximal, characteristic, complemented
15C47×#3171C211yesmaximal, characteristic
16#15689254C2211yescharacteristic
17Q8×C47⋊C8C2211yescharacteristic
18#15689201—21no—
19#15689201C2211yescharacteristic
20#15689201C2211yescharacteristic
21#15689201C2211yescharacteristic
22#15689195—21no—
23#15689195—21no—
24#15689195—21no—
25D4×C47⋊C8C2211yescharacteristic
26D4×C47⋊C8C2211yescharacteristic
27D4×C47⋊C8C2211yescharacteristic
28#15689184—21no—
29#15689180C2211yescharacteristic
30#15689176C2211yescharacteristic
31#15689150—21no—
32#15689147C2211yescharacteristic
33#15689090—21no—
34#15689090—21no—
35C2×C4×C47⋊C8C2211yescharacteristic
36#15688876C2211yescharacteristic
37#15688876C2211yescharacteristic
38#15688874C2211yescharacteristic
39#15688874C2211yescharacteristic
40Q8⋊(C47⋊C8)C2211yescharacteristic
41Q8⋊(C47⋊C8)C2211yescharacteristic
42D4⋊(C47⋊C8)C2211yescharacteristic
43D4⋊(C47⋊C8)C2211yescharacteristic
44D4⋊(C47⋊C8)C2211yescharacteristic
45D4⋊(C47⋊C8)C2211yescharacteristic
46D4⋊(C47⋊C8)C2211yescharacteristic
47D4⋊(C47⋊C8)C2211yescharacteristic
48#15688844C2211yescharacteristic
49#15688844C2211yescharacteristic
50C8×C47⋊C8C2211yescharacteristic
51C8×C47⋊C8C2211yescharacteristic
52C47×#532C2211yescharacteristic
53C47×#529C2211yescharacteristic
54C47×#501C411yescharacteristic
55C47×#496C411yescharacteristic
56C47×#490C411yescharacteristic
57C47×#489C411yescharacteristic
58C47×#469C411yescharacteristic
59C47×#469C411yescharacteristic
60C47×#468C411yescharacteristic
61C47×#468C411yescharacteristic
62C4×SD16×C47C2211yescharacteristic
63C4×SD16×C47C2211yescharacteristic
64C4×D8×C47C2211yescharacteristic
65C4×D8×C47C2211yescharacteristic
66C4×M16×C47C2211yescharacteristic
67C2×#10625329—21no—
68C2×#10625329—21no—
69C2×#10625329—21no—
70C2×#10625329—21no—
71C22×C47⋊C8—21no—
72C22×C47⋊C8—21no—
73C22×C47⋊C8—21no—
74C22×C47⋊C8—21no—
75C4⋊(C47⋊C8)—21no—
76C4⋊(C47⋊C8)—21no—
77C4⋊(C47⋊C8)—21no—
78C4⋊(C47⋊C8)D411yescharacteristic
79C4⋊(C47⋊C8)D411yescharacteristic
80C4⋊(C47⋊C8)C2311yescharacteristic
81C4⋊(C47⋊C8)C2311yescharacteristic
82C4⋊(C47⋊C8)C2311yescharacteristic
83C4⋊(C47⋊C8)C2311yescharacteristic
84C22⋊(C47⋊C8)—21no—
85C22⋊(C47⋊C8)—21no—
86C22⋊(C47⋊C8)—21no—
87C22⋊(C47⋊C8)—21no—
88C22⋊(C47⋊C8)—21no—
89C22⋊(C47⋊C8)—21no—
90C22⋊(C47⋊C8)—21no—
91C22⋊(C47⋊C8)D411yescharacteristic
92C22⋊(C47⋊C8)D411yescharacteristic
93#13093935—21no—
94#13093935—21no—
95C4×C47⋊C8—21no—
96C4×C47⋊C8—21no—
97C4×C47⋊C8C2311yescharacteristic
98C4×C47⋊C8C2311yescharacteristic
99C4×C47⋊C8C2311yescharacteristic
100C4×C47⋊C8C2311yescharacteristic
101C2×#41×C47—21no—
102C2×SD16×C47C2×C411yescharacteristic
103C2×SD16×C47C2×C411yescharacteristic
104C2×D8×C47C2×C411yescharacteristic
105C2×D8×C47C2×C411yescharacteristic
106C2×M16×C47C2×C411yescharacteristic
107C2×M16×C47C2×C411yescharacteristic
108C4⋊Q8×C47C2×C411yescharacteristic
109#127×C47C2×C411yescharacteristic
110#126×C47—21no—
111#125×C47C2×C411yescharacteristic
112#124×C47—21no—
113#124×C47C2×C411yescharacteristic
114#123×C47—21no—
115C22⋊Q8×C47C2×C411yescharacteristic
116#121×C47—21no—
117#121×C47C2×C411yescharacteristic
118#121×C47C2×C411yescharacteristic
119#121×C47C2×C411yescharacteristic
120C4×Q8×C47C2311yescharacteristic
121C4×D4×C47—21no—
122C4×D4×C47—21no—
123C4×D4×C47C2311yescharacteristic
124C4×D4×C47C2311yescharacteristic
125C4×D4×C47C2311yescharacteristic
126#117×C47—21no—
127C2×C42×C47C2311yescharacteristic
128#107×C47C2×C411yescharacteristic
129#107×C47C2×C411yescharacteristic
130#106×C47C2×C411yescharacteristic
131#106×C47C2×C411yescharacteristic
132#103×C47C2×C411yescharacteristic
133#103×C47C2×C411yescharacteristic
134#102×C47C2×C411yescharacteristic
135#102×C47C2×C411yescharacteristic
136#102×C47C2×C411yescharacteristic
137#102×C47C2×C411yescharacteristic
138#102×C47C2×C411yescharacteristic
139#102×C47C2×C411yescharacteristic
140#97×C47C2×C411yescharacteristic
141#97×C47C2×C411yescharacteristic
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144#41×C47—81no—
145#41×C47—81no—
146C2×Q8×C47—21no—
147C2×Q8×C47C22×C411yescharacteristic
148C2×D4×C47—21no—
149C2×D4×C47—21no—
150C2×D4×C47—21no—
151C2×D4×C47—21no—
152C2×D4×C47—21no—
153C2×D4×C47C22×C411yescharacteristic
154C2×D4×C47C22×C411yescharacteristic
155C2×D4×C47C22×C411yescharacteristic
156C22×C4×C47—21no—
157C22×C4×C47—21no—
158C22×C4×C47—21no—
159C22×C4×C47C22×C411yescharacteristic
160C22×C4×C47C2×D411yescharacteristic
161C22×C4×C47#4111yescharacteristic
162SD16×C47—22no—
163SD16×C47—22no—
164D8×C47—22no—
165D8×C47—22no—
166M16×C47—22no—
167M16×C47—22no—
168C2×C8×C47C22×C411yescharacteristic
169C2×C8×C47C22×C411yescharacteristic
170C2×C8×C47C22×C411yescharacteristic
171C2×C8×C47C22×C411yescharacteristic
172C4⋊C4×C47—21no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47C22×C411yescharacteristic
178C4⋊C4×C47C22×C411yescharacteristic
179C4⋊C4×C47C22×C411yescharacteristic
180C4⋊C4×C47C22×C411yescharacteristic
181C22⋊C4×C47—21no—
182C22⋊C4×C47—21no—
183C22⋊C4×C47—21no—
184C22⋊C4×C47—21no—
185C22⋊C4×C47—21no—
186C22⋊C4×C47—21no—
187C22⋊C4×C47—21no—
188C22⋊C4×C47—21no—
189C22⋊C4×C47—21no—
190C42×C47—21no—
191C42×C47C22×C411yescharacteristic
192C42×C47C2×D411yescharacteristic
193C42×C47#4111yescharacteristic
194C42×C47C2411yescharacteristic
195#10625329—81no—
196#10625329—81no—
197#10625329—81no—
198#10625329—81no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—42no—
206C2×C47⋊C8—81no—
207C2×C47⋊C8—81no—
208C2×C47⋊C8—81no—
209C2×C47⋊C8C2×D411yescharacteristic
210C2×C47⋊C8C2×D411yescharacteristic
211C2×C47⋊C8C2×D411yescharacteristic
212C2×C47⋊C8C2×D411yescharacteristic
213C23×C47—21no—
214C23×C47—21no—
215C23×C47—21no—
216C23×C47C4×D411yescharacteristic
217Q8×C47—21no—
218Q8×C47—21no—
219Q8×C47—81no—
220D4×C47—21no—
221D4×C47—21no—
222D4×C47—21no—
223D4×C47—21no—
224D4×C47—21no—
225D4×C47—21no—
226D4×C47—81no—
227D4×C47—81no—
228D4×C47—81no—
229D4×C47—81no—
230D4×C47—81no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47C4×D411yescharacteristic
248C2×C4×C47C4×D411yescharacteristic
249C2×C4×C47C4×D411yescharacteristic
250C2×C4×C47C23×C411yescharacteristic
251C2×C4×C47C22×D411yescharacteristic
252C2×C4×C47C2×#4111yescharacteristic
253C8×C47—21no—
254C8×C47—21no—
255C8×C47#13112yes—
256C8×C47#13112yes—
257C47⋊C8—22no—
258C47⋊C8—21no—
259C47⋊C8—21no—
260C47⋊C8—21no—
261C47⋊C8—21no—
262C47⋊C8—41no—
263C47⋊C8—41no—
264C47⋊C8—41no—
265C47⋊C8—41no—
266C47⋊C8—41no—
267#17968—471nomaximal, Sylow
268C22×C47—21no—
269C22×C47—21no—
270C22×C47—21no—
271C22×C47—41no—
272C22×C47—41no—
273C22×C47—41no—
274C22×C47—41no—
275C22×C47—41no—
276C22×C47—41no—
277C22×C47—41no—
278C22×C47—41no—
279C22×C47—41no—
280C22×C47C2×C4×D411yescharacteristic
281C4×C47—21no—
282C4×C47—21no—
283C4×C47—21no—
284C4×C47—21no—
285C4×C47—41no—
286C4×C47—41no—
287C4×C47—41no—
288C4×C47—41no—
289C4×C47—41no—
290C4×C47C2×#13111yescharacteristic
291C4×C47C2×#13111yescharacteristic
292C4×C47#57611yescharacteristic
293C4×C47#57711yescharacteristic
294#3171D4711yescharacteristic
295#2993—471no—
296#2981—471no—
297#1594—471no—
298#1592—471no—
299#1590—471no—
300#1589—471no—
301#1580—471no—
302SD16⋊C8—471no—
303C8×SD16—471no—
304C8×D8—471no—
305#1569—471no—
306#1494—471no—
307#1492—471no—
308C8×M16—471no—
309C2×C47—41no—
310C2×C47—81no—
311C2×C47—81no—
312C2×C47—81no—
313C2×C47#293111yescharacteristic
314C2×C47#294611yescharacteristic
315C2×C47#295611yescharacteristic
316#532C2×D4711yescharacteristic
317#529C2×D4711yescharacteristic
318#501Dic4711yescharacteristic
319#496Dic4711yescharacteristic
320#490Dic4711yescharacteristic
321#489Dic4711yescharacteristic
322#469Dic4711yescharacteristic
323#469Dic4711yescharacteristic
324#468Dic4711yescharacteristic
325#468Dic4711yescharacteristic
326#446—471no—
327C8×Q8—471no—
328C4×SD16C2×D4711yescharacteristic
329C4×SD16C2×D4711yescharacteristic
330C4×D8C2×D4711yescharacteristic
331C4×D8C2×D4711yescharacteristic
332#436—471no—
333#436—471no—
334#436—471no—
335#436—941no—
336#435—941no—
337#435—941no—
338#435—941no—
339C8×D4—471no—
340C8×D4—471no—
341C8×D4—471no—
342#433—941no—
343#432—471no—
344#431—471no—
345#424—941no—
346#423—471no—
347#408—941no—
348#408—941no—
349C4×M16C2×D4711yescharacteristic
350C2×C4×C8—471no—
351#335—471no—
352#335—471no—
353#334—471no—
354#334—471no—
355Q8⋊C8—471no—
356Q8⋊C8—471no—
357D4⋊C8—471no—
358D4⋊C8—471no—
359D4⋊C8—471no—
360D4⋊C8—471no—
361D4⋊C8—471no—
362D4⋊C8—471no—
363#322—471no—
364#322—471no—
365C82—471no—
366C82—471no—
367C47#1796811yescharacteristic, complemented, Sylow
368C2×#41—21no—
369C2×SD16C2×Dic4711yescharacteristic
370C2×SD16C2×Dic4711yescharacteristic
371C2×D8C2×Dic4711yescharacteristic
372C2×D8C2×Dic4711yescharacteristic
373C2×M16—941no—
374C2×M16—941no—
375C2×M16—941no—
376C2×M16—941no—
377C2×M16C2×Dic4711yescharacteristic
378C2×M16C2×Dic4711yescharacteristic
379C22×C8—941no—
380C22×C8—941no—
381C22×C8—941no—
382C22×C8—941no—
383C4⋊Q8C2×Dic4711yescharacteristic
384#127C2×Dic4711yescharacteristic
385#126—21no—
386#125C2×Dic4711yescharacteristic
387#124—21no—
388#124C2×Dic4711yescharacteristic
389#123—21no—
390C22⋊Q8C2×Dic4711yescharacteristic
391#121—21no—
392#121C2×Dic4711yescharacteristic
393#121C2×Dic4711yescharacteristic
394#121C2×Dic4711yescharacteristic
395C4×Q8C22×D4711yescharacteristic
396C4×D4—21no—
397C4×D4—21no—
398C4×D4C22×D4711yescharacteristic
399C4×D4C22×D4711yescharacteristic
400C4×D4C22×D4711yescharacteristic
401#117—21no—
402C2×C42C22×D4711yescharacteristic
403#107C2×Dic4711yescharacteristic
404#107C2×Dic4711yescharacteristic
405#106C2×Dic4711yescharacteristic
406#106C2×Dic4711yescharacteristic
407C4⋊C8—471no—
408C4⋊C8—471no—
409C4⋊C8—471no—
410C4⋊C8—471no—
411C4⋊C8—471no—
412C4⋊C8—471no—
413C4⋊C8—941no—
414C4⋊C8—941no—
415C4⋊C8—941no—
416#103C2×Dic4711yescharacteristic
417#103C2×Dic4711yescharacteristic
418#102C2×Dic4711yescharacteristic
419#102C2×Dic4711yescharacteristic
420#102C2×Dic4711yescharacteristic
421#102C2×Dic4711yescharacteristic
422#102C2×Dic4711yescharacteristic
423#102C2×Dic4711yescharacteristic
424C22⋊C8—471no—
425C22⋊C8—471no—
426C22⋊C8—941no—
427C22⋊C8—941no—
428C22⋊C8—941no—
429C22⋊C8—941no—
430C22⋊C8—941no—
431C22⋊C8—941no—
432C22⋊C8—941no—
433#97—941no—
434#97—941no—
435#97C2×Dic4711yescharacteristic
436#97C2×Dic4711yescharacteristic
437C4×C8—471no—
438C4×C8—471no—
439C4×C8—471no—
440C4×C8—471no—
441C4×C8—941no—
442C4×C8—941no—
443C4×C8C22×D4711yescharacteristic
444C4×C8C22×D4711yescharacteristic
445#41—81no—
446#41—81no—
447C2×Q8—21no—
448C2×Q8C22×Dic4711yescharacteristic
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4C22×Dic4711yescharacteristic
455C2×D4C22×Dic4711yescharacteristic
456C2×D4C22×Dic4711yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4D4×D4711yescharacteristic
461C22×C4#1062535211yescharacteristic
462C22×C4C22×Dic4711yescharacteristic
463SD16—22no—
464SD16—22no—
465D8—22no—
466D8—22no—
467M16—22no—
468M16—22no—
469M16—3761no—
470M16—3761no—
471M16—3761no—
472M16—3761no—
473C2×C8—471no—
474C2×C8—471no—
475C2×C8—471no—
476C2×C8—471no—
477C2×C8—941no—
478C2×C8—941no—
479C2×C8—941no—
480C2×C8—941no—
481C2×C8—941no—
482C2×C8—941no—
483C2×C8—3761no—
484C2×C8—3761no—
485C2×C8—3761no—
486C2×C8—1882no—
487C2×C8C22×Dic4711yescharacteristic
488C2×C8C22×Dic4711yescharacteristic
489C2×C8C22×Dic4711yescharacteristic
490C2×C8C22×Dic4711yescharacteristic
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4C22×Dic4711yescharacteristic
497C4⋊C4C22×Dic4711yescharacteristic
498C4⋊C4C22×Dic4711yescharacteristic
499C4⋊C4C22×Dic4711yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C42—21no—
510C42D4×D4711yescharacteristic
511C42#1062535211yescharacteristic
512C42C22×Dic4711yescharacteristic
513C42C23×D4711yescharacteristic
514C23—21no—
515C23—21no—
516C23—21no—
517C23D4×Dic4711yescharacteristic
518Q8—21no—
519Q8—21no—
520Q8—81no—
521D4—21no—
522D4—21no—
523D4—21no—
524D4—21no—
525D4—21no—
526D4—21no—
527D4—81no—
528D4—81no—
529D4—81no—
530D4—81no—
531D4—81no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4D4×Dic4711yescharacteristic
549C2×C4D4×Dic4711yescharacteristic
550C2×C4D4×Dic4711yescharacteristic
551C2×C4C23×Dic4711yescharacteristic
552C2×C4C2×D4×D4711yescharacteristic
553C2×C4C2×#1062535211yescharacteristic
554C8—21no—
555C8—21no—
556C8—942no—
557C8—941no—
558C8—941no—
559C8—941no—
560C8—941no—
561C8—1881no—
562C8—1881no—
563C8—1881no—
564C8—1881no—
565C8—1881no—
566C8#1309403112yes—
567C8#1309403112yes—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22C2×D4×Dic4711yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4C2×#1309403111yescharacteristic
591C4C2×#1309403111yescharacteristic
592C4#1568992711yescharacteristic
593C4#1568993311yescharacteristic
594C2—41no—
595C2—81no—
596C2—81no—
597C2—81no—
598C2#2025210011yescharacteristic
599C2#2025222111yescharacteristic
600C2#2025229811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
Fitting, prime radicalC47×#3171C2
socleC22×C47C2×C4×D4
cosocleC42×C47C24
hypercenter#3171D47
hypocenterC47#17968
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3171

Complements

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

NH ≅ G/Ntype
1. #20252537C2semidirect
2. #20252464C2semidirect
3. #20242302C2semidirect
4. #20242291C2semidirect
5. #20242279C2semidirect
6. #20242273C2semidirect
7. #20242212C2semidirect
8. SD16⋊(C47⋊C8)C2semidirect
9. SD16×C47⋊C8C2semidirect
10. D8×C47⋊C8C2semidirect
11. #20242155C2semidirect
12. #20241848C2semidirect
13. #20241839C2semidirect
14. M16×C47⋊C8C2semidirect
367. C47#17968semidirect

Maximal subgroups

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

subgroupindex
#202525372
#202524642
#202423022
#202422912
#202422792
#202422732
#202422122
SD16⋊(C47⋊C8)2
SD16×C47⋊C82
D8×C47⋊C82
#202421552
#202418482
#202418392
M16×C47⋊C82
C47×#31712
#1796847