Finite Group Databasebeta

← #26183864

Subgroups — #26183864

Order 12032.

938 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252477C211yesmaximal, characteristic, complemented
2#20246039C211yesmaximal, characteristic, complemented
3#20241965C212yesmaximal, complemented
4#20241948C212yesmaximal, complemented
5#20241935C212yesmaximal, complemented
6#20241926C212yesmaximal, complemented
7#20241913C211yesmaximal, characteristic, complemented
8#20241896C211yesmaximal, characteristic, complemented
9#20241883C211yesmaximal, characteristic, complemented
10#20241874C211yesmaximal, characteristic, complemented
11C47×#3522C211yesmaximal, characteristic
12#15689254—21no—
13#15689254—22no—
14Q8×C47⋊C8—21no—
15#15689180C2212yes—
16#15689180C2211yescharacteristic
17#15689176C2212yes—
18#15689176C2211yescharacteristic
19#15689030—21no—
20#15689022—21no—
21#15689009—22no—
22#15688991C2212yes—
23#15688989C2212yes—
24C4×C4⋊Dic47C2211yescharacteristic
25#15688969C2211yescharacteristic
26#15688860C2211yescharacteristic
27#15688860C2212yes—
28#15688860C2211yescharacteristic
29#15688857C2212yes—
30#15688857C2212yes—
31#15688857C2211yescharacteristic
32#15688857C2212yes—
33#15688857C2212yes—
34#15688857C2211yescharacteristic
35#15688857C2212yes—
36C47×#559C411yescharacteristic, complemented
37C47×#559C412yescomplemented
38Q82×C47C411yescharacteristic, complemented
39C47×#554C412yescomplemented
40C47×#554C411yescharacteristic, complemented
41C47×#550C411yescharacteristic, complemented
42C47×#548—21no—
43C47×#547—22no—
44C47×#547—21no—
45C47×#542—21no—
46C47×#542—22no—
47C47×#532C2212yes—
48C47×#532C2211yescharacteristic
49C47×#529C2212yes—
50C47×#529C2211yescharacteristic
51C47×#520—21no—
52C2×C4×Q8×C47C2211yescharacteristic
53C2×C4⋊Dic47—21no—
54C2×C4⋊Dic47—22no—
55C2×C4×Dic47—21no—
56C2×C4×Dic47—21no—
57C4⋊(C47⋊C8)—21no—
58C4⋊(C47⋊C8)—22no—
59C4⋊(C47⋊C8)—22no—
60C4⋊(C47⋊C8)—21no—
61C22⋊(C47⋊C8)C2311yescharacteristic
62C22⋊(C47⋊C8)C2311yescharacteristic
63C22⋊(C47⋊C8)C2312yes—
64#13093935—22no—
65#13093935—21no—
66C4×C47⋊C8—21no—
67C4×C47⋊C8—22no—
68C47⋊#95—21no—
69C47⋊#95—22no—
70C47⋊#95—22no—
71C47⋊#95—22no—
72C47⋊#95C2311yescharacteristic
73C47⋊#95C2311yescharacteristic
74C47⋊#95C2312yes—
75C2×#41×C47—21no—
76C2×#41×C47—21no—
77C2×#41×C47—22no—
78C22×Q8×C47D411yescharacteristic
79C4⋊Q8×C47—22no—
80C4⋊Q8×C47—21no—
81C4⋊Q8×C47C2×C412yes—
82C4⋊Q8×C47C2×C411yescharacteristic
83#127×C47C2×C412yes—
84#127×C47C2×C411yescharacteristic
85#126×C47—22no—
86#126×C47—21no—
87#125×C47C2×C412yes—
88#125×C47C2×C411yescharacteristic
89#124×C47—21no—
90#124×C47—22no—
91#124×C47—22no—
92#124×C47—21no—
93#124×C47C2×C412yes—
94#124×C47C2×C411yescharacteristic
95#123×C47—21no—
96#123×C47—22no—
97#123×C47—22no—
98#123×C47—22no—
99#123×C47—22no—
100C22⋊Q8×C47—21no—
101C22⋊Q8×C47—22no—
102C22⋊Q8×C47—22no—
103C22⋊Q8×C47—22no—
104C22⋊Q8×C47—22no—
105C22⋊Q8×C47C2311yescharacteristic
106#121×C47—21no—
107#121×C47—22no—
108#121×C47—22no—
109#121×C47—21no—
110#121×C47C2312yes—
111#121×C47C2311yescharacteristic
112C4×Q8×C47—21no—
113C4×Q8×C47C2×C412yes—
114C4×Q8×C47C2×C411yescharacteristic
115C4×Q8×C47C2×C411yescharacteristic
116C4×Q8×C47C2×C411yescharacteristic
117C4×Q8×C47C2×C412yes—
118C4×Q8×C47D411yescharacteristic
119C4×Q8×C47D411yescharacteristic
120C4×Q8×C47D412yes—
121C4×D4×C47—21no—
122C4×D4×C47—22no—
123C4×D4×C47—22no—
124C4×D4×C47—21no—
125C4×D4×C47—21no—
126C4×D4×C47—22no—
127C4×D4×C47C2×C412yes—
128C4×D4×C47C2×C411yescharacteristic
129C4×D4×C47C2×C412yes—
130C4×D4×C47C2×C412yes—
131C4×D4×C47C2×C412yes—
132#117×C47—21no—
133#117×C47—22no—
134C2×C4⋊C4×C47—22no—
135C2×C4⋊C4×C47—21no—
136C2×C4⋊C4×C47D412yes—
137C2×C4⋊C4×C47D411yescharacteristic
138C2×C42×C47C2311yescharacteristic
139C2×C42×C47C2312yes—
140#41×C47—44no—
141#41×C47—81no—
142#41×C47—82no—
143#41×C47—81no—
144#41×C47—82no—
145C2×Q8×C47—22no—
146C2×Q8×C47—21no—
147C2×Q8×C47—42no—
148C2×Q8×C47C22⋊C411yescharacteristic
149C2×Q8×C47C22⋊C411yescharacteristic
150C2×Q8×C47C22⋊C412yes—
151C2×Q8×C47C22×C411yescharacteristic
152C2×D4×C47—21no—
153C2×D4×C47—22no—
154C2×D4×C47—22no—
155C2×D4×C47—21no—
156C2×D4×C47—22no—
157C2×D4×C47—21no—
158C2×D4×C47C22×C412yes—
159C2×D4×C47C22×C411yescharacteristic
160C22×C4×C47—22no—
161C22×C4×C47—21no—
162C22×C4×C47—22no—
163C22×C4×C47—22no—
164C22×C4×C47—22no—
165C22×C4×C47C2×D411yescharacteristic
166C22×C4×C47C2×D412yes—
167C22×C4×C47C2×D411yescharacteristic
168C22×C4×C47C2×D412yes—
169C22×C4×C47C2411yescharacteristic
170C22×Dic47—21no—
171C22×Dic47—21no—
172C22×Dic47—22no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47—22no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—22no—
179C4⋊C4×C47—22no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47C22⋊C412yes—
183C4⋊C4×C47C22⋊C411yescharacteristic
184C4⋊C4×C47C22⋊C412yes—
185C4⋊C4×C47C22⋊C411yescharacteristic
186C4⋊C4×C47C22⋊C412yes—
187C4⋊C4×C47C22⋊C412yes—
188C4⋊C4×C47C22⋊C412yes—
189C4⋊C4×C47C22×C411yescharacteristic
190C4⋊C4×C47C22×C411yescharacteristic
191C4⋊C4×C47C22×C412yes—
192C22⋊C4×C47—21no—
193C22⋊C4×C47—21no—
194C22⋊C4×C47—22no—
195C22⋊C4×C47—21no—
196C22⋊C4×C47—22no—
197C22⋊C4×C47—22no—
198C22⋊C4×C47—21no—
199C22⋊C4×C47—22no—
200C22⋊C4×C47—22no—
201C22⋊C4×C47—22no—
202C22⋊C4×C47—21no—
203C22⋊C4×C47—22no—
204C42×C47—22no—
205C42×C47—21no—
206C42×C47C22×C411yescharacteristic
207C42×C47C22×C412yes—
208C42×C47C22×C412yes—
209C42×C47C22×C411yescharacteristic
210C42×C47C2×D411yescharacteristic
211C42×C47C2×D412yes—
212C42×C47C2×D412yes—
213C42×C47C2×D411yescharacteristic
214C4⋊Dic47—42no—
215C4⋊Dic47—82no—
216C4×Dic47—42no—
217C4×Dic47—81no—
218C2×C47⋊C8—21no—
219C2×C47⋊C8—21no—
220C2×C47⋊C8—22no—
221C23×C47—22no—
222C23×C47—21no—
223C23×C47C22×D411yescharacteristic
224Q8×C47—22no—
225Q8×C47—22no—
226Q8×C47—42no—
227Q8×C47—41no—
228Q8×C47—81no—
229Q8×C47—82no—
230D4×C47—42no—
231D4×C47—41no—
232D4×C47—44no—
233D4×C47—42no—
234D4×C47—41no—
235D4×C47—41no—
236D4×C47—42no—
237D4×C47—42no—
238D4×C47—81no—
239D4×C47—82no—
240C2×C4×C47—21no—
241C2×C4×C47—22no—
242C2×C4×C47—21no—
243C2×C4×C47—22no—
244C2×C4×C47—22no—
245C2×C4×C47—21no—
246C2×C4×C47—22no—
247C2×C4×C47—21no—
248C2×C4×C47—21no—
249C2×C4×C47—42no—
250C2×C4×C47—41no—
251C2×C4×C47—44no—
252C2×C4×C47—42no—
253C2×C4×C47—82no—
254C2×C4×C47—82no—
255C2×C4×C47—82no—
256C2×C4×C47C2×C22⋊C412yes—
257C2×C4×C47C2×C22⋊C411yescharacteristic
258C2×C4×C47C2×C22⋊C412yes—
259C2×C4×C47C2×C22⋊C411yescharacteristic
260C2×C4×C47C2×C22⋊C412yes—
261C2×C4×C47C2×C22⋊C412yes—
262C2×C4×C47C2×C22⋊C411yescharacteristic
263C2×C4×C47C2×C22⋊C411yescharacteristic
264C2×C4×C47C23×C411yescharacteristic
265C2×C4×C47C22×D411yescharacteristic
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—42no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—42no—
272C2×Dic47—42no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C47⋊C8—41no—
276C47⋊C8—41no—
277C47⋊C8—42no—
278#17602—471nomaximal, Sylow
279C22×C47—21no—
280C22×C47—21no—
281C22×C47—42no—
282C22×C47—41no—
283C22×C47—42no—
284C22×C47—41no—
285C22×C47—42no—
286C22×C47—41no—
287C22×C47C22×C22⋊C411yescharacteristic
288C22×C47#57611yescharacteristic
289C22×C47#57711yescharacteristic
290C4×C47—21no—
291C4×C47—21no—
292C4×C47—21no—
293C4×C47—22no—
294C4×C47—21no—
295C4×C47—22no—
296C4×C47—21no—
297C4×C47—22no—
298C4×C47—22no—
299C4×C47—41no—
300C4×C47—42no—
301C4×C47—41no—
302C4×C47—41no—
303C4×C47#42112yes—
304Dic47—81no—
305Dic47—82no—
306Dic47—81no—
307#3522D4711yescharacteristic
308#2984—471no—
309#2320—471no—
310#1521—472no—
311#1517—472no—
312#1514—472no—
313#1512—472no—
314#1509—471no—
315#1505—471no—
316#1502—471no—
317#1500—471no—
318C2×C47—21no—
319C2×C47—21no—
320C2×C47—82no—
321C2×C47—81no—
322C2×C47#288411yescharacteristic
323C2×C47#289711yescharacteristic
324C2×C47C2×#42111yescharacteristic
325#559Dic4712yescomplemented
326#559Dic4711yescharacteristic, complemented
327Q82Dic4711yescharacteristic, complemented
328#554Dic4711yescharacteristic, complemented
329#554Dic4712yescomplemented
330#550Dic4711yescharacteristic, complemented
331#548—21no—
332#547—22no—
333#547—21no—
334#542—21no—
335#542—22no—
336#532C2×D4711yescharacteristic
337#532C2×D4712yes—
338#529C2×D4711yescharacteristic
339#529C2×D4712yes—
340#520—21no—
341C2×C4×Q8C2×D4711yescharacteristic
342#446—941no—
343#446—942no—
344C8×Q8—941no—
345#432—471no—
346#432—472no—
347#431—471no—
348#431—472no—
349#391—941no—
350#389—941no—
351#387—942no—
352#383—472no—
353#382—472no—
354C4×C4⋊C4—471no—
355#376—471no—
356#328—471no—
357#328—471no—
358#328—472no—
359#327—471no—
360#327—471no—
361#327—472no—
362#327—472no—
363#327—472no—
364#327—472no—
365#327—472no—
366C47#1760211yescharacteristic, complemented, Sylow
367C2×#41—21no—
368C2×#41—21no—
369C2×#41—22no—
370C22×Q8C22⋊D4711yescharacteristic
371C4⋊Q8—22no—
372C4⋊Q8—21no—
373C4⋊Q8C2×Dic4711yescharacteristic
374C4⋊Q8C2×Dic4712yes—
375#127C2×Dic4711yescharacteristic
376#127C2×Dic4712yes—
377#126—22no—
378#126—21no—
379#125C2×Dic4711yescharacteristic
380#125C2×Dic4712yes—
381#124—21no—
382#124—22no—
383#124—22no—
384#124—21no—
385#124C2×Dic4711yescharacteristic
386#124C2×Dic4712yes—
387#123—21no—
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392C22⋊Q8—21no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8C22×D4711yescharacteristic
398#121—21no—
399#121—22no—
400#121—22no—
401#121—21no—
402#121C22×D4711yescharacteristic
403#121C22×D4712yes—
404C4×Q8—21no—
405C4×Q8C2×Dic4711yescharacteristic
406C4×Q8C2×Dic4712yes—
407C4×Q8C2×Dic4711yescharacteristic
408C4×Q8C2×Dic4711yescharacteristic
409C4×Q8C2×Dic4712yes—
410C4×Q8C22⋊D4711yescharacteristic
411C4×Q8C22⋊D4711yescharacteristic
412C4×Q8C22⋊D4712yes—
413C4×D4—21no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—21no—
417C4×D4—21no—
418C4×D4—22no—
419C4×D4C2×Dic4711yescharacteristic
420C4×D4C2×Dic4712yes—
421C4×D4C2×Dic4712yes—
422C4×D4C2×Dic4712yes—
423C4×D4C2×Dic4712yes—
424#117—21no—
425#117—22no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—942no—
429C2×C4⋊C4—941no—
430C2×C4⋊C4C22⋊D4712yes—
431C2×C4⋊C4C22⋊D4711yescharacteristic
432C2×C42—941no—
433C2×C42—941no—
434C2×C42C22×D4711yescharacteristic
435C2×C42C22×D4712yes—
436C4⋊C8—942no—
437C4⋊C8—941no—
438C4⋊C8—941no—
439C4⋊C8—942no—
440C22⋊C8—471no—
441C22⋊C8—471no—
442C22⋊C8—472no—
443#97—942no—
444#97—941no—
445C4×C8—941no—
446C4×C8—942no—
447#95—471no—
448#95—471no—
449#95—472no—
450#95—941no—
451#95—942no—
452#95—942no—
453#95—942no—
454#41—44no—
455#41—81no—
456#41—82no—
457#41—81no—
458#41—82no—
459C2×Q8—22no—
460C2×Q8—21no—
461C2×Q8—42no—
462C2×Q8C22⋊Dic4711yescharacteristic
463C2×Q8C22⋊Dic4711yescharacteristic
464C2×Q8C22⋊Dic4712yes—
465C2×Q8C22×Dic4711yescharacteristic
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4C22×Dic4711yescharacteristic
473C2×D4C22×Dic4712yes—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—941no—
480C22×C4—941no—
481C22×C4—942no—
482C22×C4C2×C22⋊D4711yescharacteristic
483C22×C4C2×C22⋊D4712yes—
484C22×C4C2×C22⋊D4711yescharacteristic
485C22×C4C2×C22⋊D4712yes—
486C22×C4C23×D4711yescharacteristic
487C2×C8—941no—
488C2×C8—941no—
489C2×C8—942no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—3762no—
500C4⋊C4—1882no—
501C4⋊C4C22⋊Dic4712yes—
502C4⋊C4C22⋊Dic4711yescharacteristic
503C4⋊C4C22⋊Dic4712yes—
504C4⋊C4C22⋊Dic4711yescharacteristic
505C4⋊C4C22⋊Dic4712yes—
506C4⋊C4C22⋊Dic4712yes—
507C4⋊C4C22⋊Dic4712yes—
508C4⋊C4C22×Dic4711yescharacteristic
509C4⋊C4C22×Dic4712yes—
510C4⋊C4C22×Dic4711yescharacteristic
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—22no—
514C22⋊C4—21no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—21no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—21no—
522C22⋊C4—22no—
523C42—22no—
524C42—21no—
525C42—3761no—
526C42—1882no—
527C42C22×Dic4711yescharacteristic
528C42C22×Dic4712yes—
529C42C22×Dic4712yes—
530C42C22×Dic4711yescharacteristic
531C42C2×C22⋊D4711yescharacteristic
532C42C2×C22⋊D4712yes—
533C42C2×C22⋊D4712yes—
534C42C2×C22⋊D4711yescharacteristic
535C23—22no—
536C23—21no—
537C23C22×C22⋊D4711yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—42no—
541Q8—41no—
542Q8—81no—
543Q8—82no—
544D4—41no—
545D4—42no—
546D4—44no—
547D4—42no—
548D4—41no—
549D4—41no—
550D4—42no—
551D4—42no—
552D4—81no—
553D4—82no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—1881no—
571C2×C4—1881no—
572C2×C4—1882no—
573C2×C4—1881no—
574C2×C4—1881no—
575C2×C4—1882no—
576C2×C4—1882no—
577C2×C4—1881no—
578C2×C4—1881no—
579C2×C4C2×C22⋊Dic4712yes—
580C2×C4C2×C22⋊Dic4711yescharacteristic
581C2×C4C2×C22⋊Dic4712yes—
582C2×C4C2×C22⋊Dic4711yescharacteristic
583C2×C4C2×C22⋊Dic4712yes—
584C2×C4C2×C22⋊Dic4712yes—
585C2×C4C2×C22⋊Dic4711yescharacteristic
586C2×C4C2×C22⋊Dic4711yescharacteristic
587C2×C4C23×Dic4711yescharacteristic
588C2×C4C22×C22⋊D4711yescharacteristic
589C8—1881no—
590C8—1881no—
591C8—1882no—
592C22—21no—
593C22—21no—
594C22—42no—
595C22—41no—
596C22—42no—
597C22—41no—
598C22—42no—
599C22—41no—
600C22C22×C22⋊Dic4711yescharacteristic
601C22#1568992611yescharacteristic
602C22#1568993411yescharacteristic
603C4—21no—
604C4—22no—
605C4—21no—
606C4—22no—
607C4—21no—
608C4—22no—
609C4—22no—
610C4—21no—
611C4—21no—
612C4—42no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—3761no—
617C4—3762no—
618C4—3761no—
619C4#1568914012yes—
620C2—21no—
621C2—21no—
622C2—82no—
623C2—81no—
624C2#2025185211yescharacteristic
625C2#2025191411yescharacteristic
626C2C2×#1568914011yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
Fitting, prime radicalC47×#3522C2
socleC22×C47C22×C22⋊C4
cosocleC22×C4×C47C24
hypercenter#3522D47
hypocenterC47#17602
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3522

Complements

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

NH ≅ G/Ntype
1. #20252477C2semidirect
2. #20246039C2semidirect
3. #20241965C2semidirect
4. #20241948C2semidirect
5. #20241935C2semidirect
6. #20241926C2semidirect
7. #20241913C2semidirect
8. #20241896C2semidirect
9. #20241883C2semidirect
10. #20241874C2semidirect
36. C47×#559C4semidirect
37. C47×#559C4semidirect
38. Q82×C47C4semidirect
39. C47×#554C4semidirect
40. C47×#554C4semidirect
41. C47×#550C4semidirect
325. #559Dic47semidirect
326. #559Dic47semidirect
327. Q82Dic47semidirect
328. #554Dic47semidirect
329. #554Dic47semidirect
330. #550Dic47semidirect
366. C47#17602semidirect

Maximal subgroups

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

subgroupindex
#202524772
#202460392
#202419652
#202419482
#202419352
#202419262
#202419132
#202418962
#202418832
#202418742
C47×#35222
#1760247