Finite Group Databasebeta

← #26183146

Subgroups — #26183146

Order 12032.

980 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#41×C47⋊C8C211yesmaximal, characteristic, complemented
2#20252054C211yesmaximal, characteristic, complemented
3#20241848C212yesmaximal, complemented
4#20241843C211yesmaximal, characteristic, complemented
5#20241822C212yesmaximal, complemented
6#20241813C211yesmaximal, characteristic, complemented
7#20241796C212yesmaximal, complemented
8#20241791C211yesmaximal, characteristic, complemented
9#20241770C212yesmaximal, complemented
10C2×D4⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
11C47×#3475C211yesmaximal, characteristic
12Q8×C47⋊C8—21no—
13D4×C47⋊C8—22no—
14D4×C47⋊C8—21no—
15#15689184—22no—
16#15689180C2212yescomplemented
17#15689176C2211yescharacteristic
18#15689176C2212yescomplemented
19#15689176C2212yescomplemented
20#15689147C2211yescharacteristic
21C2×C4⋊(C47⋊C8)C2211yescharacteristic
22#15689090—21no—
23#15689079—21no—
24C2×C4×C47⋊C8C2211yescharacteristic
25C2×C4×C47⋊C8C2212yescomplemented
26Q8⋊(C47⋊C8)C2212yescomplemented
27Q8⋊(C47⋊C8)C2212yescomplemented
28Q8⋊(C47⋊C8)C2212yescomplemented
29Q8⋊(C47⋊C8)C2212yescomplemented
30D4⋊(C47⋊C8)C2211yescharacteristic
31D4⋊(C47⋊C8)C2211yescharacteristic
32D4⋊(C47⋊C8)C2211yescharacteristic
33D4⋊(C47⋊C8)C2211yescharacteristic
34D4⋊(C47⋊C8)C2212yes—
35D4⋊(C47⋊C8)C2212yes—
36C47×#556C411yescharacteristic
37C47×#555C411yescharacteristic
38C47×#554C411yescharacteristic
39C47×#553—21no—
40C47×#552—21no—
41C47×#552C412yes—
42C47×#551—22no—
43C47×#547C411yescharacteristic
44C47×#546C412yes—
45C47×#533C2211yescharacteristic
46C47×#529—22no—
47C47×#529C2212yes—
48C47×#525—21no—
49C4×#41×C47C2212yes—
50C4×#41×C47C2211yescharacteristic
51C2×C4×D4×C47C2211yescharacteristic
52C2×#117×C47—21no—
53C2×#10625329—21no—
54C22×C47⋊C8—21no—
55C22×C47⋊C8—22no—
56C22×C47⋊C8—21no—
57C4⋊(C47⋊C8)—21no—
58C4⋊(C47⋊C8)—22no—
59C4⋊(C47⋊C8)—21no—
60C4⋊(C47⋊C8)—21no—
61C4⋊(C47⋊C8)C2311yescharacteristic
62C4⋊(C47⋊C8)C2311yescharacteristic
63C4⋊(C47⋊C8)C2312yes—
64C22⋊(C47⋊C8)—22no—
65C22⋊(C47⋊C8)—22no—
66C22⋊(C47⋊C8)—21no—
67C22⋊(C47⋊C8)—22no—
68#13093935—22no—
69C4×C47⋊C8—21no—
70C4×C47⋊C8—22no—
71C4×C47⋊C8—22no—
72C4×C47⋊C8C2311yescharacteristic
73C4×C47⋊C8C2311yescharacteristic
74C4×C47⋊C8C2312yes—
75C2×#41×C47C2×C412yes—
76C2×#41×C47D411yescharacteristic
77C22×D4×C47C2×C411yescharacteristic
78C23×C4×C47—21no—
79C4⋊Q8×C47C2×C411yescharacteristic
80C4⋊Q8×C47C2×C411yescharacteristic
81#126×C47—22no—
82#126×C47—22no—
83#126×C47—22no—
84#125×C47—22no—
85#125×C47—21no—
86#125×C47C2×C412yes—
87#124×C47—22no—
88#124×C47C2×C412yes—
89#123×C47—22no—
90#123×C47—22no—
91#123×C47—22no—
92#123×C47—22no—
93C22⋊Q8×C47—21no—
94C22⋊Q8×C47—21no—
95C22⋊Q8×C47—21no—
96C22⋊Q8×C47—22no—
97C22⋊Q8×C47C2×C412yes—
98C22⋊Q8×C47C2×C412yes—
99#121×C47—21no—
100#121×C47—21no—
101#121×C47—22no—
102#121×C47C2×C412yes—
103#120×C47—22no—
104C4×Q8×C47—21no—
105C4×Q8×C47C2×C412yes—
106C4×Q8×C47C2×C411yescharacteristic
107C4×Q8×C47D411yescharacteristic
108C4×Q8×C47C2312yes—
109C4×D4×C47—22no—
110C4×D4×C47—22no—
111C4×D4×C47—22no—
112C4×D4×C47—21no—
113C4×D4×C47—21no—
114C4×D4×C47C2×C411yescharacteristic
115C4×D4×C47C2×C411yescharacteristic
116C4×D4×C47C2×C412yes—
117C4×D4×C47C2×C412yes—
118C4×D4×C47C2×C411yescharacteristic
119C4×D4×C47D412yes—
120C4×D4×C47D411yescharacteristic
121C4×D4×C47C2311yescharacteristic
122C4×D4×C47C2311yescharacteristic
123#117×C47—21no—
124#117×C47—21no—
125#117×C47—21no—
126#117×C47—22no—
127#117×C47—22no—
128#117×C47—22no—
129#117×C47C2×C412yes—
130#117×C47D411yescharacteristic
131#117×C47D412yes—
132C2×C4⋊C4×C47—21no—
133C2×C4⋊C4×C47—21no—
134C2×C4⋊C4×C47C2×C411yescharacteristic
135C2×C22⋊C4×C47—21no—
136C2×C22⋊C4×C47—22no—
137C2×C42×C47—22no—
138C2×C42×C47C2311yescharacteristic
139C2×C42×C47C2312yes—
140C24×C47—21no—
141#41×C47—22no—
142#41×C47—22no—
143#41×C47—24no—
144#41×C47—42no—
145C2×Q8×C47—21no—
146C2×Q8×C47C22⋊C411yescharacteristic
147C2×Q8×C47C22×C412yes—
148C2×D4×C47—22no—
149C2×D4×C47—22no—
150C2×D4×C47—22no—
151C2×D4×C47—21no—
152C2×D4×C47—41no—
153C2×D4×C47C22⋊C412yes—
154C2×D4×C47C22⋊C411yescharacteristic
155C2×D4×C47C22×C411yescharacteristic
156C2×D4×C47C22×C411yescharacteristic
157C22×C4×C47—21no—
158C22×C4×C47—22no—
159C22×C4×C47—21no—
160C22×C4×C47—21no—
161C22×C4×C47—21no—
162C22×C4×C47—21no—
163C22×C4×C47—22no—
164C22×C4×C47—42no—
165C22×C4×C47—42no—
166C22×C4×C47C22×C411yescharacteristic
167C22×C4×C47C22×C412yes—
168C22×C4×C47C2×D412yes—
169C22×C4×C47C2×D411yescharacteristic
170C22×C4×C47C2×D412yes—
171C22×C4×C47C2×D411yescharacteristic
172C4⋊C4×C47—21no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47—22no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—22no—
178C4⋊C4×C47—22no—
179C4⋊C4×C47—22no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—22no—
182C4⋊C4×C47C22⋊C411yescharacteristic
183C4⋊C4×C47C22⋊C411yescharacteristic
184C4⋊C4×C47C22⋊C412yes—
185C4⋊C4×C47C22⋊C412yes—
186C4⋊C4×C47C22×C411yescharacteristic
187C4⋊C4×C47C22×C411yescharacteristic
188C4⋊C4×C47C22×C412yes—
189C22⋊C4×C47—22no—
190C22⋊C4×C47—22no—
191C22⋊C4×C47—22no—
192C22⋊C4×C47—22no—
193C22⋊C4×C47—22no—
194C22⋊C4×C47—21no—
195C22⋊C4×C47—21no—
196C22⋊C4×C47—22no—
197C22⋊C4×C47—22no—
198C22⋊C4×C47—21no—
199C22⋊C4×C47—21no—
200C22⋊C4×C47C22⋊C412yes—
201C22⋊C4×C47C22⋊C412yes—
202C22⋊C4×C47C22⋊C412yes—
203C42×C47—22no—
204C42×C47—22no—
205C42×C47—22no—
206C42×C47C22×C411yescharacteristic
207C42×C47C22×C412yes—
208C42×C47C2×D412yes—
209C42×C47C2×D412yes—
210C42×C47C2×D411yescharacteristic
211C42×C47C2×D411yescharacteristic
212C42×C47C2411yescharacteristic
213#10625329—42no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—21no—
217C2×C47⋊C8—22no—
218C2×C47⋊C8—22no—
219C2×C47⋊C8—21no—
220C2×C47⋊C8—42no—
221C2×C47⋊C8—44no—
222C2×C47⋊C8—42no—
223C23×C47—21no—
224C23×C47—21no—
225C23×C47—42no—
226C23×C47—42no—
227C23×C47—81no—
228C23×C47C2×C22⋊C412yes—
229C23×C47C2×C22⋊C411yescharacteristic
230Q8×C47—22no—
231Q8×C47—22no—
232Q8×C47—22no—
233Q8×C47—81no—
234D4×C47—24no—
235D4×C47—21no—
236D4×C47—21no—
237D4×C47—21no—
238D4×C47—22no—
239D4×C47—42no—
240D4×C47—44no—
241D4×C47—82no—
242D4×C47#13112yes—
243C2×C4×C47—21no—
244C2×C4×C47—22no—
245C2×C4×C47—21no—
246C2×C4×C47—22no—
247C2×C4×C47—22no—
248C2×C4×C47—22no—
249C2×C4×C47—21no—
250C2×C4×C47—22no—
251C2×C4×C47—22no—
252C2×C4×C47—21no—
253C2×C4×C47—42no—
254C2×C4×C47—41no—
255C2×C4×C47—44no—
256C2×C4×C47—42no—
257C2×C4×C47—41no—
258C2×C4×C47—41no—
259C2×C4×C47—41no—
260C2×C4×C47—41no—
261C2×C4×C47—42no—
262C2×C4×C47—42no—
263C2×C4×C47—82no—
264C2×C4×C47C2×C22⋊C412yes—
265C2×C4×C47C2×C22⋊C412yes—
266C2×C4×C47C2×C22⋊C411yescharacteristic
267C2×C4×C47C2×C22⋊C412yes—
268C2×C4×C47C2×C22⋊C411yescharacteristic
269C2×C4×C47C2×C22⋊C411yescharacteristic
270C2×C4×C47#13512yes—
271C2×C4×C47#13512yes—
272C2×C4×C47C23×C411yescharacteristic
273C2×C4×C47C22×D411yescharacteristic
274C2×C4×C47C22×D411yescharacteristic
275C47⋊C8—22no—
276C47⋊C8—41no—
277C47⋊C8—42no—
278C47⋊C8—41no—
279C47⋊C8—41no—
280C47⋊C8—42no—
281#17508—471nomaximal, Sylow
282C22×C47—21no—
283C22×C47—21no—
284C22×C47—22no—
285C22×C47—22no—
286C22×C47—22no—
287C22×C47—22no—
288C22×C47—41no—
289C22×C47—41no—
290C22×C47—41no—
291C22×C47—41no—
292C22×C47—41no—
293C22×C47—84no—
294C22×C47#41612yes—
295C22×C47C22×C22⋊C411yescharacteristic
296C4×C47—22no—
297C4×C47—21no—
298C4×C47—22no—
299C4×C47—21no—
300C4×C47—22no—
301C4×C47—21no—
302C4×C47—42no—
303C4×C47—41no—
304C4×C47—41no—
305C4×C47—42no—
306C4×C47#41612yes—
307C4×C47C2×#13111yescharacteristic
308C4×C47#56811yescharacteristic
309C4×C47C2×#13511yescharacteristic
310C4×C47C2×#13511yescharacteristic
311#3475D4711yescharacteristic
312C8×#41—471no—
313#2925—471no—
314#1494—472no—
315#1493—471no—
316#1488—472no—
317#1486—471no—
318#1482—472no—
319#1481—471no—
320#1476—472no—
321C2×D4⋊C8—471no—
322C2×C47—22no—
323C2×C47—42no—
324C2×C47—42no—
325C2×C47—81no—
326C2×C47#288011yescharacteristic
327C2×C47C2×#41611yescharacteristic
328C2×C47#290111yescharacteristic
329#556Dic4711yescharacteristic
330#555Dic4711yescharacteristic
331#554Dic4711yescharacteristic
332#553—21no—
333#552—21no—
334#552Dic4712yes—
335#551—22no—
336#547Dic4711yescharacteristic
337#546Dic4712yes—
338#533C2×D4711yescharacteristic
339#529—22no—
340#529C2×D4712yes—
341#525—21no—
342C4×#41C2×D4712yes—
343C4×#41C2×D4711yescharacteristic
344C2×C4×D4C2×D4711yescharacteristic
345C2×#117—21no—
346C8×Q8—941no—
347C8×D4—942no—
348C8×D4—941no—
349#433—942no—
350#432—472no—
351#431—471no—
352#431—472no—
353#431—472no—
354#423—471no—
355C2×C4⋊C8—471no—
356#408—941no—
357#405—941no—
358C2×C4×C8—471no—
359C2×C4×C8—472no—
360Q8⋊C8—472no—
361Q8⋊C8—472no—
362Q8⋊C8—472no—
363Q8⋊C8—472no—
364D4⋊C8—471no—
365D4⋊C8—471no—
366D4⋊C8—471no—
367D4⋊C8—471no—
368D4⋊C8—472no—
369D4⋊C8—472no—
370C47#1750811yescharacteristic, complemented, Sylow
371C2×#41C2×Dic4712yes—
372C2×#41C22⋊D4711yescharacteristic
373C22×D4C2×Dic4711yescharacteristic
374C23×C4—21no—
375C2×M16—941no—
376C22×C8—941no—
377C22×C8—942no—
378C22×C8—941no—
379C4⋊Q8C2×Dic4711yescharacteristic
380C4⋊Q8C2×Dic4711yescharacteristic
381#126—22no—
382#126—22no—
383#126—22no—
384#125—22no—
385#125—21no—
386#125C2×Dic4712yes—
387#124—22no—
388#124C2×Dic4712yes—
389#123—22no—
390#123—22no—
391#123—22no—
392#123—22no—
393C22⋊Q8—21no—
394C22⋊Q8—21no—
395C22⋊Q8—21no—
396C22⋊Q8—22no—
397C22⋊Q8C2×Dic4712yes—
398C22⋊Q8C2×Dic4712yes—
399#121—21no—
400#121—21no—
401#121—22no—
402#121C2×Dic4712yes—
403#120—22no—
404C4×Q8—21no—
405C4×Q8C2×Dic4712yes—
406C4×Q8C2×Dic4711yescharacteristic
407C4×Q8C22⋊D4711yescharacteristic
408C4×Q8C22×D4712yes—
409C4×D4—22no—
410C4×D4—22no—
411C4×D4—22no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4C2×Dic4711yescharacteristic
415C4×D4C2×Dic4711yescharacteristic
416C4×D4C2×Dic4712yes—
417C4×D4C2×Dic4712yes—
418C4×D4C2×Dic4711yescharacteristic
419C4×D4C22⋊D4712yes—
420C4×D4C22⋊D4711yescharacteristic
421C4×D4C22×D4711yescharacteristic
422C4×D4C22×D4711yescharacteristic
423#117—21no—
424#117—21no—
425#117—21no—
426#117—22no—
427#117—22no—
428#117—22no—
429#117C2×Dic4712yes—
430#117C22⋊D4711yescharacteristic
431#117C22⋊D4712yes—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4C2×Dic4711yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—22no—
437C2×C42—22no—
438C2×C42C22×D4711yescharacteristic
439C2×C42C22×D4712yes—
440C4⋊C8—471no—
441C4⋊C8—471no—
442C4⋊C8—472no—
443C4⋊C8—941no—
444C4⋊C8—941no—
445C4⋊C8—942no—
446C4⋊C8—941no—
447C22⋊C8—942no—
448C22⋊C8—942no—
449C22⋊C8—941no—
450C22⋊C8—942no—
451#97—942no—
452C4×C8—471no—
453C4×C8—471no—
454C4×C8—472no—
455C4×C8—942no—
456C4×C8—941no—
457C4×C8—942no—
458C24—21no—
459#41—22no—
460#41—22no—
461#41—24no—
462#41—42no—
463C2×Q8—21no—
464C2×Q8C22⋊Dic4711yescharacteristic
465C2×Q8C22×Dic4712yes—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—41no—
471C2×D4C22⋊Dic4712yes—
472C2×D4C22⋊Dic4711yescharacteristic
473C2×D4C22×Dic4711yescharacteristic
474C2×D4C22×Dic4711yescharacteristic
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4C22×Dic4711yescharacteristic
485C22×C4C22×Dic4712yes—
486C22×C4C2×C22⋊D4712yes—
487C22×C4C2×C22⋊D4711yescharacteristic
488C22×C4C2×C22⋊D4712yes—
489C22×C4C2×C22⋊D4711yescharacteristic
490M16—1882no—
491C2×C8—941no—
492C2×C8—941no—
493C2×C8—942no—
494C2×C8—941no—
495C2×C8—942no—
496C2×C8—941no—
497C2×C8—1884no—
498C2×C8—1882no—
499C2×C8—1882no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4C22⋊Dic4711yescharacteristic
511C4⋊C4C22⋊Dic4711yescharacteristic
512C4⋊C4C22⋊Dic4712yes—
513C4⋊C4C22⋊Dic4712yes—
514C4⋊C4C22×Dic4711yescharacteristic
515C4⋊C4C22×Dic4711yescharacteristic
516C4⋊C4C22×Dic4712yes—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—21no—
527C22⋊C4—21no—
528C22⋊C4C22⋊Dic4712yes—
529C22⋊C4C22⋊Dic4712yes—
530C22⋊C4C22⋊Dic4712yes—
531C42—22no—
532C42—22no—
533C42—22no—
534C42C22×Dic4711yescharacteristic
535C42C22×Dic4712yes—
536C42C2×C22⋊D4712yes—
537C42C2×C22⋊D4711yescharacteristic
538C42C2×C22⋊D4712yes—
539C42C2×C22⋊D4711yescharacteristic
540C42C23×D4711yescharacteristic
541C23—21no—
542C23—21no—
543C23—42no—
544C23—42no—
545C23—81no—
546C23C2×C22⋊Dic4712yes—
547C23C2×C22⋊Dic4711yescharacteristic
548Q8—22no—
549Q8—22no—
550Q8—22no—
551Q8—81no—
552D4—24no—
553D4—21no—
554D4—22no—
555D4—21no—
556D4—21no—
557D4—42no—
558D4—44no—
559D4—82no—
560D4#1309403112yes—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—82no—
582C2×C4C2×C22⋊Dic4712yes—
583C2×C4C2×C22⋊Dic4712yes—
584C2×C4C2×C22⋊Dic4711yescharacteristic
585C2×C4C2×C22⋊Dic4712yes—
586C2×C4C2×C22⋊Dic4711yescharacteristic
587C2×C4C2×C22⋊Dic4711yescharacteristic
588C2×C4#1309404412yes—
589C2×C4#1309404412yes—
590C2×C4C23×Dic4711yescharacteristic
591C2×C4C22×C22⋊D4711yescharacteristic
592C2×C4C22×C22⋊D4711yescharacteristic
593C8—942no—
594C8—1881no—
595C8—1882no—
596C8—1881no—
597C8—1881no—
598C8—1882no—
599C22—21no—
600C22—21no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—84no—
611C22#1568911912yes—
612C22C22×C22⋊Dic4711yescharacteristic
613C4—21no—
614C4—22no—
615C4—21no—
616C4—22no—
617C4—21no—
618C4—22no—
619C4—42no—
620C4—42no—
621C4—41no—
622C4—41no—
623C4#1568911912yes—
624C4C2×#1309403111yescharacteristic
625C4#1568989011yescharacteristic
626C4C2×#1309404411yescharacteristic
627C4C2×#1309404411yescharacteristic
628C2—22no—
629C2—42no—
630C2—42no—
631C2—81no—
632C2#2025183711yescharacteristic
633C2C2×#1568911911yescharacteristic
634C2#2025193511yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
Fitting, prime radicalC47×#3475C2
socleC22×C47C22×C22⋊C4
cosocleC42×C47C24
hypercenter#3475D47
hypocenterC47#17508
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #3475

Complements

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

NH ≅ G/Ntype
1. #41×C47⋊C8C2semidirect
2. #20252054C2semidirect
3. #20241848C2semidirect
4. #20241843C2semidirect
5. #20241822C2semidirect
6. #20241813C2semidirect
7. #20241796C2semidirect
8. #20241791C2semidirect
9. #20241770C2semidirect
10. C2×D4⋊(C47⋊C8)C2semidirect
16. #15689180C22semidirect
18. #15689176C22semidirect
19. #15689176C22semidirect
25. C2×C4×C47⋊C8C22semidirect
26. Q8⋊(C47⋊C8)C22semidirect
27. Q8⋊(C47⋊C8)C22semidirect
28. Q8⋊(C47⋊C8)C22semidirect
29. Q8⋊(C47⋊C8)C22semidirect
370. C47#17508semidirect

Maximal subgroups

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

subgroupindex
#41×C47⋊C82
#202520542
#202418482
#202418432
#202418222
#202418132
#202417962
#202417912
#202417702
C2×D4⋊(C47⋊C8)2
C47×#34752
#1750847