Finite Group Databasebeta

← #26182856

Subgroups — #26182856

Order 12032.

984 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#20252499C211yesmaximal, characteristic, complemented
2#20252355C211yesmaximal, characteristic, complemented
3#20241805C212yesmaximal, complemented
4#20241801C212yesmaximal, complemented
5#20241796C212yesmaximal, complemented
6#20241787C212yesmaximal, complemented
7#20241779C211yesmaximal, characteristic, complemented
8#20241775C211yesmaximal, characteristic, complemented
9#20241770C211yesmaximal, characteristic, complemented
10C2×D4⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
11C47×#3460C211yesmaximal, characteristic
12#15689254—21no—
13Q8×C47⋊C8—21no—
14#15689201—21no—
15#15689195—22no—
16D4×C47⋊C8—22no—
17D4×C47⋊C8—21no—
18#15689176C2211yescharacteristic
19#15689176C2211yescharacteristic
20#15689176C2212yescomplemented
21#15689176C2212yescomplemented
22C2×C4⋊(C47⋊C8)C2211yescharacteristic
23C2×C4⋊(C47⋊C8)C2212yescomplemented
24C4×#10625329C2212yescomplemented
25C2×C4×C47⋊C8C2211yescharacteristic
26Q8⋊(C47⋊C8)C2211yescharacteristic, complemented
27Q8⋊(C47⋊C8)C2211yescharacteristic, complemented
28Q8⋊(C47⋊C8)C2212yescomplemented
29D4⋊(C47⋊C8)C2212yes—
30D4⋊(C47⋊C8)C2212yes—
31D4⋊(C47⋊C8)C2212yes—
32D4⋊(C47⋊C8)C2212yes—
33D4⋊(C47⋊C8)C2211yescharacteristic, complemented
34D4⋊(C47⋊C8)C2211yescharacteristic, complemented
35D4⋊(C47⋊C8)C2212yescomplemented
36C47×#544C411yescharacteristic
37C47×#542—21no—
38C47×#542C412yes—
39C47×#540—21no—
40C47×#540C412yes—
41C47×#538—22no—
42C47×#537C411yescharacteristic
43C47×#535C411yescharacteristic
44C47×#535C411yescharacteristic
45C47×#532—21no—
46C47×#532C2212yes—
47C47×#529—21no—
48C47×#529C2212yes—
49C47×#525—22no—
50C4×#41×C47C2211yescharacteristic
51C4×#41×C47C2211yescharacteristic
52C2×C4×D4×C47C2211yescharacteristic
53C2×#10625329—22no—
54C22×C47⋊C8—21no—
55C22×C47⋊C8—21no—
56C22×C47⋊C8—22no—
57C4⋊(C47⋊C8)—21no—
58C4⋊(C47⋊C8)—22no—
59C4⋊(C47⋊C8)—21no—
60C4⋊(C47⋊C8)—22no—
61C4⋊(C47⋊C8)C2312yes—
62C4⋊(C47⋊C8)C2311yescharacteristic
63C4⋊(C47⋊C8)C2311yescharacteristic
64C22⋊(C47⋊C8)—21no—
65C22⋊(C47⋊C8)—21no—
66C22⋊(C47⋊C8)—22no—
67C22⋊(C47⋊C8)—22no—
68#13093935—22no—
69C4×C47⋊C8—21no—
70C4×C47⋊C8—22no—
71C4×C47⋊C8—21no—
72C4×C47⋊C8C2311yescharacteristic
73C4×C47⋊C8C2311yescharacteristic
74C4×C47⋊C8C2312yes—
75C2×#41×C47—22no—
76C2×#41×C47C2×C411yescharacteristic
77C2×#41×C47C2×C411yescharacteristic
78C22×D4×C47C2×C411yescharacteristic
79C23×C4×C47—21no—
80C4⋊Q8×C47—21no—
81C4⋊Q8×C47C2×C412yes—
82#127×C47—21no—
83#127×C47C2×C412yes—
84#126×C47—22no—
85#126×C47—22no—
86#125×C47—21no—
87#125×C47C2×C412yes—
88#124×C47—21no—
89#124×C47—22no—
90#124×C47—22no—
91#124×C47C2×C412yes—
92#123×C47—22no—
93#123×C47—22no—
94#123×C47—22no—
95#123×C47—22no—
96C22⋊Q8×C47—21no—
97C22⋊Q8×C47—22no—
98C22⋊Q8×C47—22no—
99C22⋊Q8×C47C2×C412yes—
100#121×C47—22no—
101#121×C47—21no—
102#121×C47—22no—
103#121×C47—21no—
104#121×C47—21no—
105#121×C47—22no—
106#121×C47—22no—
107#121×C47C2×C412yes—
108#121×C47C2×C412yes—
109#121×C47C2×C412yes—
110#120×C47—22no—
111#120×C47—22no—
112C4×Q8×C47C2×C411yescharacteristic
113C4×Q8×C47D411yescharacteristic
114C4×Q8×C47D411yescharacteristic
115C4×Q8×C47C2311yescharacteristic
116C4×D4×C47—21no—
117C4×D4×C47—21no—
118C4×D4×C47—22no—
119C4×D4×C47—21no—
120C4×D4×C47—21no—
121C4×D4×C47—22no—
122C4×D4×C47C2×C412yes—
123C4×D4×C47C2×C411yescharacteristic
124C4×D4×C47D412yes—
125C4×D4×C47D411yescharacteristic
126C4×D4×C47D412yes—
127C4×D4×C47D411yescharacteristic
128C4×D4×C47C2312yes—
129C4×D4×C47C2311yescharacteristic
130#117×C47—21no—
131#117×C47—22no—
132#117×C47C2×C411yescharacteristic
133#117×C47C2×C411yescharacteristic
134#117×C47C2×C412yes—
135C2×C4⋊C4×C47C2×C411yescharacteristic
136C2×C22⋊C4×C47—21no—
137C2×C42×C47—21no—
138C2×C42×C47C2311yescharacteristic
139C2×C42×C47C2312yes—
140C24×C47—21no—
141#41×C47—24no—
142#41×C47—24no—
143#41×C47—82no—
144#41×C47—82no—
145C2×Q8×C47—22no—
146C2×Q8×C47C22⋊C411yescharacteristic
147C2×Q8×C47C22×C411yescharacteristic
148C2×D4×C47—21no—
149C2×D4×C47—22no—
150C2×D4×C47—22no—
151C2×D4×C47—22no—
152C2×D4×C47—21no—
153C2×D4×C47—22no—
154C2×D4×C47—22no—
155C2×D4×C47—24no—
156C2×D4×C47C22⋊C412yes—
157C2×D4×C47C22⋊C411yescharacteristic
158C2×D4×C47C22×C412yes—
159C2×D4×C47C22×C411yescharacteristic
160C22×C4×C47—21no—
161C22×C4×C47—21no—
162C22×C4×C47—21no—
163C22×C4×C47—22no—
164C22×C4×C47—21no—
165C22×C4×C47—22no—
166C22×C4×C47—81no—
167C22×C4×C47—81no—
168C22×C4×C47C22×C412yes—
169C22×C4×C47C22×C411yescharacteristic
170C22×C4×C47C2×D411yescharacteristic
171C22×C4×C47C2×D412yes—
172C22×C4×C47C2×D411yescharacteristic
173C22×C4×C47C2×D412yes—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—22no—
177C4⋊C4×C47—22no—
178C4⋊C4×C47—22no—
179C4⋊C4×C47—22no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47C22⋊C411yescharacteristic
182C4⋊C4×C47C22⋊C411yescharacteristic
183C4⋊C4×C47C22⋊C412yes—
184C4⋊C4×C47C22⋊C412yes—
185C4⋊C4×C47C22×C412yes—
186C4⋊C4×C47C22×C411yescharacteristic
187C4⋊C4×C47C22×C411yescharacteristic
188C22⋊C4×C47—21no—
189C22⋊C4×C47—22no—
190C22⋊C4×C47—22no—
191C22⋊C4×C47—21no—
192C22⋊C4×C47—22no—
193C22⋊C4×C47—22no—
194C22⋊C4×C47—22no—
195C22⋊C4×C47—22no—
196C22⋊C4×C47—22no—
197C22⋊C4×C47—22no—
198C22⋊C4×C47—21no—
199C22⋊C4×C47C22⋊C412yes—
200C22⋊C4×C47C22⋊C412yes—
201C22⋊C4×C47C22⋊C412yes—
202C42×C47—21no—
203C42×C47—21no—
204C42×C47—21no—
205C42×C47C22×C411yescharacteristic
206C42×C47C22×C412yes—
207C42×C47C2×D411yescharacteristic
208C42×C47C2×D412yes—
209C42×C47C2×D411yescharacteristic
210C42×C47C2×D412yes—
211C42×C47C2411yescharacteristic
212#10625329—44no—
213C2×C47⋊C8—21no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—21no—
217C2×C47⋊C8—22no—
218C2×C47⋊C8—22no—
219C2×C47⋊C8—42no—
220C2×C47⋊C8—81no—
221C2×C47⋊C8—82no—
222C23×C47—22no—
223C23×C47—21no—
224C23×C47—42no—
225C23×C47—81no—
226C23×C47—81no—
227C23×C47C2×C22⋊C412yes—
228C23×C47C2×C22⋊C411yescharacteristic
229Q8×C47—22no—
230Q8×C47—22no—
231Q8×C47—82no—
232D4×C47—24no—
233D4×C47—24no—
234D4×C47—22no—
235D4×C47—22no—
236D4×C47—42no—
237D4×C47—42no—
238D4×C47—82no—
239D4×C47—82no—
240D4×C47—82no—
241D4×C47—82no—
242D4×C47—82no—
243C2×C4×C47—24no—
244C2×C4×C47—21no—
245C2×C4×C47—22no—
246C2×C4×C47—21no—
247C2×C4×C47—22no—
248C2×C4×C47—21no—
249C2×C4×C47—21no—
250C2×C4×C47—21no—
251C2×C4×C47—22no—
252C2×C4×C47—41no—
253C2×C4×C47—42no—
254C2×C4×C47—42no—
255C2×C4×C47—41no—
256C2×C4×C47—42no—
257C2×C4×C47—41no—
258C2×C4×C47—42no—
259C2×C4×C47—81no—
260C2×C4×C47—82no—
261C2×C4×C47—81no—
262C2×C4×C47C2×C22⋊C411yescharacteristic
263C2×C4×C47C2×C22⋊C412yes—
264C2×C4×C47C2×C22⋊C411yescharacteristic
265C2×C4×C47C2×C22⋊C412yes—
266C2×C4×C47C2×C22⋊C412yes—
267C2×C4×C47C2×C22⋊C411yescharacteristic
268C2×C4×C47#13114yes—
269C2×C4×C47C23×C411yescharacteristic
270C2×C4×C47C22×D411yescharacteristic
271C2×C4×C47C22×D411yescharacteristic
272C47⋊C8—22no—
273C47⋊C8—41no—
274C47⋊C8—42no—
275C47⋊C8—41no—
276C47⋊C8—42no—
277C47⋊C8—41no—
278#17468—471nomaximal, Sylow
279C22×C47—21no—
280C22×C47—21no—
281C22×C47—22no—
282C22×C47—22no—
283C22×C47—22no—
284C22×C47—41no—
285C22×C47—42no—
286C22×C47—42no—
287C22×C47—41no—
288C22×C47—41no—
289C22×C47—42no—
290C22×C47—84no—
291C22×C47#40812yes—
292C22×C47C22×C22⋊C411yescharacteristic
293C4×C47—22no—
294C4×C47—21no—
295C4×C47—22no—
296C4×C47—21no—
297C4×C47—21no—
298C4×C47—22no—
299C4×C47—41no—
300C4×C47—42no—
301C4×C47—41no—
302C4×C47—41no—
303C4×C47#40812yes—
304C4×C47C2×#13112yes—
305C4×C47#57611yescharacteristic
306C4×C47#57711yescharacteristic
307#3460D4711yescharacteristic
308#2988—471no—
309#2965—471no—
310#1484—472no—
311#1483—472no—
312#1482—472no—
313#1480—472no—
314#1478—471no—
315#1477—471no—
316#1476—471no—
317C2×D4⋊C8—471no—
318C2×C47—22no—
319C2×C47—42no—
320C2×C47—81no—
321C2×C47—82no—
322C2×C47C2×#40811yescharacteristic
323C2×C47#289711yescharacteristic
324C2×C47#290111yescharacteristic
325#544Dic4711yescharacteristic
326#542—21no—
327#542Dic4712yes—
328#540—21no—
329#540Dic4712yes—
330#538—22no—
331#537Dic4711yescharacteristic
332#535Dic4711yescharacteristic
333#535Dic4711yescharacteristic
334#532—21no—
335#532C2×D4712yes—
336#529—21no—
337#529C2×D4712yes—
338#525—22no—
339C4×#41C2×D4711yescharacteristic
340C4×#41C2×D4711yescharacteristic
341C2×C4×D4C2×D4711yescharacteristic
342#446—941no—
343C8×Q8—941no—
344#436—941no—
345#435—942no—
346C8×D4—942no—
347C8×D4—941no—
348#431—471no—
349#431—472no—
350#431—471no—
351#431—472no—
352C2×C4⋊C8—471no—
353C2×C4⋊C8—472no—
354C4×M16—472no—
355C2×C4×C8—471no—
356Q8⋊C8—471no—
357Q8⋊C8—471no—
358Q8⋊C8—472no—
359D4⋊C8—472no—
360D4⋊C8—472no—
361D4⋊C8—472no—
362D4⋊C8—472no—
363D4⋊C8—471no—
364D4⋊C8—471no—
365D4⋊C8—472no—
366C47#1746811yescharacteristic, complemented, Sylow
367C2×#41—22no—
368C2×#41C2×Dic4711yescharacteristic
369C2×#41C2×Dic4711yescharacteristic
370C22×D4C2×Dic4711yescharacteristic
371C23×C4—21no—
372C2×M16—942no—
373C22×C8—941no—
374C22×C8—942no—
375C22×C8—941no—
376C4⋊Q8—21no—
377C4⋊Q8C2×Dic4712yes—
378#127—21no—
379#127C2×Dic4712yes—
380#126—22no—
381#126—22no—
382#125—21no—
383#125C2×Dic4712yes—
384#124—21no—
385#124—22no—
386#124—22no—
387#124C2×Dic4712yes—
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392C22⋊Q8—21no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8C2×Dic4712yes—
396#121—22no—
397#121—21no—
398#121—22no—
399#121—21no—
400#121—21no—
401#121—22no—
402#121—22no—
403#121C2×Dic4712yes—
404#121C2×Dic4712yes—
405#121C2×Dic4712yes—
406#120—22no—
407#120—22no—
408C4×Q8C2×Dic4711yescharacteristic
409C4×Q8C22⋊D4711yescharacteristic
410C4×Q8C22⋊D4711yescharacteristic
411C4×Q8C22×D4711yescharacteristic
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—22no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—22no—
418C4×D4C2×Dic4712yes—
419C4×D4C2×Dic4711yescharacteristic
420C4×D4C22⋊D4712yes—
421C4×D4C22⋊D4711yescharacteristic
422C4×D4C22⋊D4712yes—
423C4×D4C22⋊D4711yescharacteristic
424C4×D4C22×D4712yes—
425C4×D4C22×D4711yescharacteristic
426#117—21no—
427#117—22no—
428#117C2×Dic4711yescharacteristic
429#117C2×Dic4711yescharacteristic
430#117C2×Dic4712yes—
431C2×C4⋊C4C2×Dic4711yescharacteristic
432C2×C22⋊C4—21no—
433C2×C42—21no—
434C2×C42C22×D4711yescharacteristic
435C2×C42C22×D4712yes—
436C4⋊C8—472no—
437C4⋊C8—471no—
438C4⋊C8—471no—
439C4⋊C8—941no—
440C4⋊C8—942no—
441C4⋊C8—942no—
442C4⋊C8—941no—
443C22⋊C8—941no—
444C22⋊C8—942no—
445C22⋊C8—941no—
446C22⋊C8—942no—
447#97—942no—
448C4×C8—471no—
449C4×C8—471no—
450C4×C8—472no—
451C4×C8—941no—
452C4×C8—941no—
453C4×C8—942no—
454C24—21no—
455#41—24no—
456#41—24no—
457#41—82no—
458#41—82no—
459C2×Q8—22no—
460C2×Q8C22⋊Dic4711yescharacteristic
461C2×Q8C22×Dic4711yescharacteristic
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—24no—
470C2×D4C22⋊Dic4712yes—
471C2×D4C22⋊Dic4711yescharacteristic
472C2×D4C22×Dic4712yes—
473C2×D4C22×Dic4711yescharacteristic
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—81no—
481C22×C4—81no—
482C22×C4C22×Dic4712yes—
483C22×C4C22×Dic4711yescharacteristic
484C22×C4C2×C22⋊D4711yescharacteristic
485C22×C4C2×C22⋊D4712yes—
486C22×C4C2×C22⋊D4711yescharacteristic
487C22×C4C2×C22⋊D4712yes—
488M16—1884no—
489C2×C8—941no—
490C2×C8—941no—
491C2×C8—942no—
492C2×C8—941no—
493C2×C8—941no—
494C2×C8—942no—
495C2×C8—3761no—
496C2×C8—3762no—
497C2×C8—1882no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4C22⋊Dic4711yescharacteristic
506C4⋊C4C22⋊Dic4711yescharacteristic
507C4⋊C4C22⋊Dic4712yes—
508C4⋊C4C22⋊Dic4712yes—
509C4⋊C4C22×Dic4712yes—
510C4⋊C4C22×Dic4711yescharacteristic
511C4⋊C4C22×Dic4711yescharacteristic
512C22⋊C4—21no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—21no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—21no—
523C22⋊C4C22⋊Dic4712yes—
524C22⋊C4C22⋊Dic4712yes—
525C22⋊C4C22⋊Dic4712yes—
526C42—21no—
527C42—21no—
528C42—21no—
529C42C22×Dic4711yescharacteristic
530C42C22×Dic4712yes—
531C42C2×C22⋊D4711yescharacteristic
532C42C2×C22⋊D4712yes—
533C42C2×C22⋊D4711yescharacteristic
534C42C2×C22⋊D4712yes—
535C42C23×D4711yescharacteristic
536C23—22no—
537C23—21no—
538C23—42no—
539C23—81no—
540C23—81no—
541C23C2×C22⋊Dic4712yes—
542C23C2×C22⋊Dic4711yescharacteristic
543Q8—22no—
544Q8—22no—
545Q8—82no—
546D4—24no—
547D4—22no—
548D4—24no—
549D4—22no—
550D4—42no—
551D4—42no—
552D4—82no—
553D4—82no—
554D4—82no—
555D4—82no—
556D4—82no—
557C2×C4—24no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—81no—
576C2×C4C2×C22⋊Dic4711yescharacteristic
577C2×C4C2×C22⋊Dic4712yes—
578C2×C4C2×C22⋊Dic4711yescharacteristic
579C2×C4C2×C22⋊Dic4712yes—
580C2×C4C2×C22⋊Dic4712yes—
581C2×C4C2×C22⋊Dic4711yescharacteristic
582C2×C4#1309403114yes—
583C2×C4C23×Dic4711yescharacteristic
584C2×C4C22×C22⋊D4711yescharacteristic
585C2×C4C22×C22⋊D4711yescharacteristic
586C8—942no—
587C8—1881no—
588C8—1882no—
589C8—1882no—
590C8—1881no—
591C8—1881no—
592C22—21no—
593C22—21no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—42no—
598C22—41no—
599C22—41no—
600C22—42no—
601C22—41no—
602C22—42no—
603C22—84no—
604C22#1568909012yes—
605C22C22×C22⋊Dic4711yescharacteristic
606C4—21no—
607C4—22no—
608C4—21no—
609C4—21no—
610C4—22no—
611C4—22no—
612C4—42no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4#1568909012yes—
617C4C2×#1309403112yes—
618C4#1568992611yescharacteristic
619C4#1568993411yescharacteristic
620C2—22no—
621C2—42no—
622C2—82no—
623C2—81no—
624C2C2×#1568909011yescharacteristic
625C2#2025191411yescharacteristic
626C2#2025193511yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
Fitting, prime radicalC47×#3460C2
socleC22×C47C22×C22⋊C4
cosocleC42×C47C24
hypercenter#3460D47
hypocenterC47#17468
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #3460

Complements

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

NH ≅ G/Ntype
1. #20252499C2semidirect
2. #20252355C2semidirect
3. #20241805C2semidirect
4. #20241801C2semidirect
5. #20241796C2semidirect
6. #20241787C2semidirect
7. #20241779C2semidirect
8. #20241775C2semidirect
9. #20241770C2semidirect
10. C2×D4⋊(C47⋊C8)C2semidirect
20. #15689176C22semidirect
21. #15689176C22semidirect
23. C2×C4⋊(C47⋊C8)C22semidirect
24. C4×#10625329C22semidirect
26. Q8⋊(C47⋊C8)C22semidirect
27. Q8⋊(C47⋊C8)C22semidirect
28. Q8⋊(C47⋊C8)C22semidirect
33. D4⋊(C47⋊C8)C22semidirect
34. D4⋊(C47⋊C8)C22semidirect
35. D4⋊(C47⋊C8)C22semidirect
366. C47#17468semidirect

Maximal subgroups

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

subgroupindex
#202524992
#202523552
#202418052
#202418012
#202417962
#202417872
#202417792
#202417752
#202417702
C2×D4⋊(C47⋊C8)2
C47×#34602
#1746847