Finite Group Databasebeta

← #26183974

Subgroups — #26183974

Order 12032.

930 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252046C211yesmaximal, characteristic, complemented
2#20246281C211yesmaximal, characteristic, complemented
3#20241965C212yesmaximal, complemented
4#20241960C211yesmaximal, characteristic, complemented
5#20241939C212yesmaximal, complemented
6#20241931C211yesmaximal, characteristic
7#20241913C212yesmaximal, complemented
8#20241908C211yesmaximal, characteristic, complemented
9#20241887C212yesmaximal, complemented
10#20241879C211yesmaximal, characteristic
11C47×#3495C211yesmaximal, characteristic
12#15689184—22no—
13#15689180C2212yes—
14#15689176C2212yes—
15#15689150—21no—
16#15689087C2211yescharacteristic
17C2×C22⋊(C47⋊C8)C2211yescharacteristic
18#15689079—21no—
19#15689022—21no—
20#15689016—22no—
21#15688991C2212yes—
22#15688983C2211yescharacteristic
23C4⋊C4×Dic47—21no—
24C4×C4⋊Dic47C2212yes—
25C4×C22⋊Dic47C2211yescharacteristic
26#15688860C2212yes—
27#15688860C2211yescharacteristic
28#15688860C2212yes—
29#15688860C2212yes—
30#15688860C2212yes—
31#15688860C2211yescharacteristic
32#15688860C2212yes—
33#15688857C2212yes—
34#15688857C2211yescharacteristic
35#15688857C2211yescharacteristic
36C47×#556—21no—
37C47×#554—21no—
38C47×#552—22no—
39C47×#552—21no—
40C47×#552—21no—
41C47×#551—22no—
42C47×#544C411yescharacteristic, complemented
43C47×#542C412yescomplemented
44C47×#541C412yescomplemented
45C47×#536C411yescharacteristic, complemented
46C47×#533C2212yes—
47C47×#529C2212yes—
48C47×#525C2211yescharacteristic
49C2×C22⋊Q8×C47C2211yescharacteristic
50C47×#520C411yescharacteristic, complemented
51C47×#519C411yescharacteristic, complemented
52C2×#117×C47C2211yescharacteristic
53C2×#10625329—21no—
54C22×C47⋊C8—21no—
55C2×C4⋊Dic47—22no—
56C2×C4⋊Dic47—21no—
57C2×C22⋊Dic47—21no—
58C2×C22⋊Dic47—21no—
59C2×C4×Dic47—21no—
60C2×C4×Dic47—22no—
61C4⋊(C47⋊C8)—22no—
62C4⋊(C47⋊C8)—22no—
63C22⋊(C47⋊C8)—21no—
64C22⋊(C47⋊C8)—21no—
65C22⋊(C47⋊C8)C2311yescharacteristic
66C22⋊(C47⋊C8)C2311yescharacteristic
67C22⋊(C47⋊C8)C2312yes—
68#13093935—22no—
69C4×C47⋊C8—22no—
70C47⋊#95—22no—
71C47⋊#95—22no—
72C47⋊#95C2311yescharacteristic
73C47⋊#95C2311yescharacteristic
74C47⋊#95C2312yes—
75C2×#41×C47C2×C411yescharacteristic
76C22×Q8×C47C2×C411yescharacteristic
77C23×C4×C47C2311yescharacteristic
78C4⋊Q8×C47—21no—
79C4⋊Q8×C47C2×C412yes—
80C4⋊Q8×C47C2×C412yes—
81#126×C47—22no—
82#126×C47—22no—
83#126×C47—22no—
84#126×C47—22no—
85#125×C47—22no—
86#125×C47—22no—
87#125×C47—21no—
88#125×C47—22no—
89#125×C47C2×C412yes—
90#124×C47—22no—
91#124×C47C2×C412yes—
92#123×C47—21no—
93#123×C47—22no—
94#123×C47—22no—
95#123×C47—22no—
96C22⋊Q8×C47—21no—
97C22⋊Q8×C47—21no—
98C22⋊Q8×C47—22no—
99C22⋊Q8×C47—21no—
100C22⋊Q8×C47—21no—
101C22⋊Q8×C47—22no—
102C22⋊Q8×C47—22no—
103C22⋊Q8×C47C2×C412yes—
104C22⋊Q8×C47C2×C411yescharacteristic
105C22⋊Q8×C47C2312yes—
106C22⋊Q8×C47C2311yescharacteristic
107#121×C47C2×C411yescharacteristic
108#121×C47C2311yescharacteristic
109#120×C47—21no—
110C4×Q8×C47—21no—
111C4×Q8×C47—21no—
112C4×Q8×C47—22no—
113C4×Q8×C47C2×C412yes—
114C4×Q8×C47C2×C412yes—
115C4×Q8×C47C2×C412yes—
116C4×D4×C47—21no—
117C4×D4×C47—21no—
118C4×D4×C47—22no—
119C4×D4×C47—22no—
120C4×D4×C47C2×C412yes—
121#117×C47—22no—
122#117×C47—21no—
123#117×C47—22no—
124#117×C47—22no—
125#117×C47C2×C411yescharacteristic
126#117×C47C2×C411yescharacteristic
127#117×C47C2×C411yescharacteristic
128#117×C47C2×C412yes—
129#117×C47D411yescharacteristic
130#117×C47D412yes—
131#117×C47D411yescharacteristic
132C2×C4⋊C4×C47—21no—
133C2×C4⋊C4×C47—21no—
134C2×C4⋊C4×C47C2×C411yescharacteristic
135C2×C4⋊C4×C47D411yescharacteristic
136C2×C4⋊C4×C47D411yescharacteristic
137C2×C22⋊C4×C47—21no—
138C2×C22⋊C4×C47D412yes—
139C2×C42×C47C2312yes—
140C24×C47C2×D411yescharacteristic
141#41×C47—42no—
142C2×Q8×C47—21no—
143C2×Q8×C47—22no—
144C2×Q8×C47—42no—
145C2×Q8×C47C22×C412yes—
146C2×Q8×C47C22×C411yescharacteristic
147C2×D4×C47—21no—
148C2×D4×C47—21no—
149C2×D4×C47—21no—
150C2×D4×C47C22×C411yescharacteristic
151C22×C4×C47—21no—
152C22×C4×C47—21no—
153C22×C4×C47—21no—
154C22×C4×C47—22no—
155C22×C4×C47—42no—
156C22×C4×C47C22×C411yescharacteristic
157C22×C4×C47C22×C411yescharacteristic
158C22×C4×C47C2×D412yes—
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47C2×D411yescharacteristic
161C22×C4×C47C2×D412yes—
162C22×C4×C47C2×D411yescharacteristic
163C22×C4×C47C2411yescharacteristic
164C22×Dic47—21no—
165C22×Dic47—21no—
166C22×Dic47—22no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—22no—
169C4⋊C4×C47—22no—
170C4⋊C4×C47—22no—
171C4⋊C4×C47—22no—
172C4⋊C4×C47—22no—
173C4⋊C4×C47—22no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47—22no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—22no—
178C4⋊C4×C47—22no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47C22⋊C411yescharacteristic
181C4⋊C4×C47C22⋊C411yescharacteristic
182C4⋊C4×C47C22⋊C412yes—
183C4⋊C4×C47C22⋊C411yescharacteristic
184C4⋊C4×C47C22⋊C411yescharacteristic
185C4⋊C4×C47C22⋊C412yes—
186C4⋊C4×C47C22×C412yes—
187C4⋊C4×C47C22×C411yescharacteristic
188C4⋊C4×C47C22×C411yescharacteristic
189C22⋊C4×C47—22no—
190C22⋊C4×C47—21no—
191C22⋊C4×C47—21no—
192C22⋊C4×C47—22no—
193C22⋊C4×C47—21no—
194C22⋊C4×C47—22no—
195C22⋊C4×C47—22no—
196C22⋊C4×C47—21no—
197C22⋊C4×C47—22no—
198C22⋊C4×C47—21no—
199C22⋊C4×C47C22⋊C412yes—
200C22⋊C4×C47C22⋊C412yes—
201C22⋊C4×C47C22⋊C412yes—
202C22⋊C4×C47C22⋊C412yes—
203C42×C47—22no—
204C42×C47—22no—
205C42×C47—22no—
206C42×C47—21no—
207C42×C47C22×C412yes—
208C42×C47C22×C412yes—
209C42×C47C2×D412yes—
210C42×C47C2×D412yes—
211C22⋊Dic47—42no—
212C22⋊Dic47—81no—
213C4⋊Dic47—44no—
214C4⋊Dic47—42no—
215C4×Dic47—42no—
216C4×Dic47—82no—
217#10625329—42no—
218C2×C47⋊C8—21no—
219C2×C47⋊C8—22no—
220C2×C47⋊C8—21no—
221C2×C47⋊C8—42no—
222C23×C47—21no—
223C23×C47—21no—
224C23×C47—21no—
225C23×C47—41no—
226C23×C47—41no—
227C23×C47C2×C22⋊C411yescharacteristic
228C23×C47C2×C22⋊C411yescharacteristic
229C23×C47C22×D411yescharacteristic
230Q8×C47—42no—
231Q8×C47—41no—
232Q8×C47—42no—
233Q8×C47—82no—
234D4×C47—41no—
235D4×C47—42no—
236D4×C47—41no—
237D4×C47—41no—
238D4×C47—81no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—22no—
244C2×C4×C47—21no—
245C2×C4×C47—21no—
246C2×C4×C47—22no—
247C2×C4×C47—21no—
248C2×C4×C47—22no—
249C2×C4×C47—42no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—42no—
254C2×C4×C47—42no—
255C2×C4×C47—82no—
256C2×C4×C47—81no—
257C2×C4×C47C2×C22⋊C412yes—
258C2×C4×C47C2×C22⋊C412yes—
259C2×C4×C47C2×C22⋊C412yes—
260C2×C4×C47C2×C22⋊C412yes—
261C2×C4×C47C2×C22⋊C411yescharacteristic
262C2×C4×C47C2×C22⋊C411yescharacteristic
263C2×C4×C47C23×C411yescharacteristic
264C2×C4×C47C22×D411yescharacteristic
265C2×Dic47—22no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—42no—
270C2×Dic47—42no—
271C2×Dic47—41no—
272C2×Dic47—42no—
273C2×Dic47—41no—
274C47⋊C8—41no—
275C47⋊C8—41no—
276C47⋊C8—42no—
277#17612—471nomaximal, Sylow
278C22×C47—21no—
279C22×C47—21no—
280C22×C47—21no—
281C22×C47—21no—
282C22×C47—21no—
283C22×C47—21no—
284C22×C47—21no—
285C22×C47—21no—
286C22×C47—41no—
287C22×C47—41no—
288C22×C47—41no—
289C22×C47—81no—
290C22×C47—81no—
291C22×C47C22×C22⋊C411yescharacteristic
292C22×C47#57711yescharacteristic
293C22×C47#57811yescharacteristic
294C4×C47—21no—
295C4×C47—21no—
296C4×C47—21no—
297C4×C47—22no—
298C4×C47—22no—
299C4×C47—22no—
300C4×C47—22no—
301C4×C47—41no—
302C4×C47—42no—
303C4×C47—41no—
304C4×C47—42no—
305C4×C47—41no—
306C4×C47#41012yes—
307Dic47—42no—
308Dic47—82no—
309Dic47—81no—
310#3495D4711yescharacteristic
311#2924—471no—
312#2347—471no—
313#1521—472no—
314#1520—471no—
315#1515—472no—
316#1513—471no—
317#1509—472no—
318#1508—471no—
319#1503—472no—
320#1501—471no—
321C2×C47—21no—
322C2×C47—21no—
323C2×C47—41no—
324C2×C47—41no—
325C2×C47—81no—
326C2×C47C2×#41011yescharacteristic
327C2×C47#289811yescharacteristic
328C2×C47#290111yescharacteristic
329#556—21no—
330#554—21no—
331#552—22no—
332#552—21no—
333#552—21no—
334#551—22no—
335#544Dic4711yescharacteristic, complemented
336#542Dic4712yescomplemented
337#541Dic4712yescomplemented
338#536Dic4711yescharacteristic, complemented
339#533C2×D4712yes—
340#529C2×D4712yes—
341#525C2×D4711yescharacteristic
342C2×C22⋊Q8C2×D4711yescharacteristic
343#520Dic4711yescharacteristic, complemented
344#519Dic4711yescharacteristic, complemented
345C2×#117C2×D4711yescharacteristic
346#433—942no—
347#432—472no—
348#431—472no—
349#424—941no—
350#407—471no—
351C2×C22⋊C8—471no—
352#405—941no—
353#389—941no—
354#388—942no—
355#383—472no—
356#380—471no—
357C4×C4⋊C4—472no—
358C4×C4⋊C4—941no—
359C4×C22⋊C4—471no—
360#328—471no—
361#328—471no—
362#328—472no—
363#328—472no—
364#328—472no—
365#328—472no—
366#328—472no—
367#327—471no—
368#327—471no—
369#327—472no—
370C47#1761211yescharacteristic, complemented, Sylow
371C2×#41C2×Dic4711yescharacteristic
372C22×Q8C2×Dic4711yescharacteristic
373C23×C4C22×D4711yescharacteristic
374C2×M16—941no—
375C22×C8—941no—
376C4⋊Q8—21no—
377C4⋊Q8C2×Dic4712yes—
378C4⋊Q8C2×Dic4712yes—
379#126—22no—
380#126—22no—
381#126—22no—
382#126—22no—
383#125—22no—
384#125—22no—
385#125—21no—
386#125—22no—
387#125C2×Dic4712yes—
388#124—22no—
389#124C2×Dic4712yes—
390#123—21no—
391#123—22no—
392#123—22no—
393#123—22no—
394C22⋊Q8—21no—
395C22⋊Q8—21no—
396C22⋊Q8—22no—
397C22⋊Q8—21no—
398C22⋊Q8—21no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401C22⋊Q8C2×Dic4712yes—
402C22⋊Q8C2×Dic4711yescharacteristic
403C22⋊Q8C22×D4712yes—
404C22⋊Q8C22×D4711yescharacteristic
405#121C2×Dic4711yescharacteristic
406#121C22×D4711yescharacteristic
407#120—21no—
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8—22no—
411C4×Q8C2×Dic4712yes—
412C4×Q8C2×Dic4712yes—
413C4×Q8C2×Dic4712yes—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4C2×Dic4712yes—
419#117—22no—
420#117—21no—
421#117—22no—
422#117—22no—
423#117C2×Dic4711yescharacteristic
424#117C2×Dic4711yescharacteristic
425#117C2×Dic4711yescharacteristic
426#117C2×Dic4712yes—
427#117C22⋊D4711yescharacteristic
428#117C22⋊D4712yes—
429#117C22⋊D4711yescharacteristic
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—942no—
433C2×C4⋊C4—941no—
434C2×C4⋊C4C2×Dic4711yescharacteristic
435C2×C4⋊C4C22⋊D4711yescharacteristic
436C2×C4⋊C4C22⋊D4711yescharacteristic
437C2×C22⋊C4—21no—
438C2×C22⋊C4—941no—
439C2×C22⋊C4—941no—
440C2×C22⋊C4C22⋊D4712yes—
441C2×C42—942no—
442C2×C42—941no—
443C2×C42C22×D4712yes—
444C4⋊C8—942no—
445C4⋊C8—942no—
446C22⋊C8—471no—
447C22⋊C8—471no—
448C22⋊C8—472no—
449C22⋊C8—941no—
450C22⋊C8—941no—
451#97—942no—
452C4×C8—942no—
453#95—471no—
454#95—471no—
455#95—472no—
456#95—942no—
457#95—942no—
458C24C2×C22⋊D4711yescharacteristic
459#41—42no—
460C2×Q8—21no—
461C2×Q8—22no—
462C2×Q8—42no—
463C2×Q8C22×Dic4712yes—
464C2×Q8C22×Dic4711yescharacteristic
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4C22×Dic4711yescharacteristic
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—42no—
474C22×C4—941no—
475C22×C4—941no—
476C22×C4—942no—
477C22×C4C22×Dic4711yescharacteristic
478C22×C4C22×Dic4711yescharacteristic
479C22×C4C2×C22⋊D4712yes—
480C22×C4C2×C22⋊D4711yescharacteristic
481C22×C4C2×C22⋊D4711yescharacteristic
482C22×C4C2×C22⋊D4712yes—
483C22×C4C2×C22⋊D4711yescharacteristic
484C22×C4C23×D4711yescharacteristic
485M16—1882no—
486C2×C8—941no—
487C2×C8—942no—
488C2×C8—941no—
489C2×C8—1882no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—1882no—
504C4⋊C4—1884no—
505C4⋊C4C22⋊Dic4711yescharacteristic
506C4⋊C4C22⋊Dic4711yescharacteristic
507C4⋊C4C22⋊Dic4712yes—
508C4⋊C4C22⋊Dic4711yescharacteristic
509C4⋊C4C22⋊Dic4711yescharacteristic
510C4⋊C4C22⋊Dic4712yes—
511C4⋊C4C22×Dic4712yes—
512C4⋊C4C22×Dic4711yescharacteristic
513C4⋊C4C22×Dic4711yescharacteristic
514C22⋊C4—22no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—22no—
518C22⋊C4—21no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—21no—
522C22⋊C4—22no—
523C22⋊C4—21no—
524C22⋊C4—3761no—
525C22⋊C4—1882no—
526C22⋊C4C22⋊Dic4712yes—
527C22⋊C4C22⋊Dic4712yes—
528C22⋊C4C22⋊Dic4712yes—
529C22⋊C4C22⋊Dic4712yes—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—21no—
534C42—3762no—
535C42—1882no—
536C42C22×Dic4712yes—
537C42C22×Dic4712yes—
538C42C2×C22⋊D4712yes—
539C42C2×C22⋊D4712yes—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—41no—
544C23—41no—
545C23C2×C22⋊Dic4711yescharacteristic
546C23C2×C22⋊Dic4711yescharacteristic
547C23C22×C22⋊D4711yescharacteristic
548Q8—42no—
549Q8—41no—
550Q8—42no—
551Q8—82no—
552D4—41no—
553D4—42no—
554D4—41no—
555D4—41no—
556D4—81no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—942no—
576C2×C4—1881no—
577C2×C4—1881no—
578C2×C4—1882no—
579C2×C4—1882no—
580C2×C4—1882no—
581C2×C4—1881no—
582C2×C4—1881no—
583C2×C4—1881no—
584C2×C4C2×C22⋊Dic4712yes—
585C2×C4C2×C22⋊Dic4712yes—
586C2×C4C2×C22⋊Dic4712yes—
587C2×C4C2×C22⋊Dic4712yes—
588C2×C4C2×C22⋊Dic4711yescharacteristic
589C2×C4C2×C22⋊Dic4711yescharacteristic
590C2×C4C23×Dic4711yescharacteristic
591C2×C4C22×C22⋊D4711yescharacteristic
592C8—1881no—
593C8—1881no—
594C8—1882no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—81no—
607C22—81no—
608C22C22×C22⋊Dic4711yescharacteristic
609C22#1568993111yescharacteristic
610C22#1568993811yescharacteristic
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—41no—
619C4—42no—
620C4—41no—
621C4—41no—
622C4—42no—
623C4—3762no—
624C4—3761no—
625C4—1882no—
626C4#1568909712yes—
627C2—21no—
628C2—21no—
629C2—41no—
630C2—41no—
631C2—81no—
632C2C2×#1568909711yescharacteristic
633C2#2025192111yescharacteristic
634C2#2025193511yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#3495C2
socleC22×C47C22×C22⋊C4
prime radicalC47×#525C22
cosocleC22×C4×C47C24
hypercenter#3495D47
hypocenterC47#17612
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×#117 ◁ #3495

Complements

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

NH ≅ G/Ntype
1. #20252046C2semidirect
2. #20246281C2semidirect
3. #20241965C2semidirect
4. #20241960C2semidirect
5. #20241939C2semidirect
7. #20241913C2semidirect
8. #20241908C2semidirect
9. #20241887C2semidirect
42. C47×#544C4semidirect
43. C47×#542C4semidirect
44. C47×#541C4semidirect
45. C47×#536C4semidirect
50. C47×#520C4semidirect
51. C47×#519C4semidirect
335. #544Dic47semidirect
336. #542Dic47semidirect
337. #541Dic47semidirect
338. #536Dic47semidirect
343. #520Dic47semidirect
344. #519Dic47semidirect
370. C47#17612semidirect

Maximal subgroups

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

subgroupindex
#202520462
#202462812
#202419652
#202419602
#202419392
#202419312
#202419132
#202419082
#202418872
#202418792
C47×#34952
#1761247