Finite Group Databasebeta

← #26183149

Subgroups — #26183149

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#20256952C211yesmaximal, characteristic
2#20252054C211yesmaximal, characteristic, complemented
3#20241849C212yesmaximal, complemented
4#20241844C211yesmaximal, characteristic, complemented
5#20241823C212yesmaximal, complemented
6#20241814C211yesmaximal, characteristic, complemented
7#20241797C212yesmaximal, complemented
8#20241792C211yesmaximal, characteristic, complemented
9#20241771C212yesmaximal, complemented
10C2×#15688852C211yesmaximal, characteristic, complemented
11C8×#41×C47C211yesmaximal, characteristic, complemented
12#15689846C411yescharacteristic
13#15689838C411yescharacteristic
14#15689833C411yescharacteristic
15#15689823—21no—
16#15689814—21no—
17#15689813C412yes—
18#15689802—22no—
19#15689778C411yescharacteristic
20#15689769C412yes—
21#15689680C2211yescharacteristic
22#15689657C2212yes—
23#15689651—22no—
24#15689630—21no—
25C4×#10625350C2212yes—
26C2×C4×D188C2211yescharacteristic
27#117×D47—21no—
28#15689184—22no—
29#15689180C2212yescomplemented
30#15689176C2212yescomplemented
31#15689147C2211yescharacteristic
32C2×C4⋊(C47⋊C8)C2211yescharacteristic
33#15689090—21no—
34#15689079—21no—
35#15688855C2212yescomplemented
36#15688855C2212yescomplemented
37#15688855C2212yescomplemented
38#15688855C2212yescomplemented
39#15688852C2211yescharacteristic
40#15688852C2211yescharacteristic
41#15688852C2211yescharacteristic
42#15688852C2211yescharacteristic
43#15688852C2212yes—
44#15688852C2212yes—
45C4×#41×C47C2211yescharacteristic
46C8×Q8×C47—21no—
47C8×D4×C47—21no—
48C8×D4×C47—22no—
49C47×#431C2211yescharacteristic
50C47×#431C2212yescomplemented
51C2×C4×C8×C47C2211yescharacteristic
52C2×C4×C8×C47C2212yescomplemented
53C2×#10625350C2×C412yes—
54C22×D188C2×C411yescharacteristic
55C22×C4×D47—21no—
56C2×#10625329—21no—
57C22×C47⋊C8—21no—
58#13094024C2×C411yescharacteristic
59#13094024C2×C411yescharacteristic
60#13094019—22no—
61#13094018—22no—
62#13094017—22no—
63#13094016—21no—
64#13094016C2×C412yes—
65#13094014—22no—
66#13094012C2×C412yes—
67#13094010—22no—
68#13094009—22no—
69#13094007—22no—
70#13094006—22no—
71#13094005—22no—
72#13094003—21no—
73#13094003C2×C412yes—
74#13094003C2×C412yes—
75#13094002—22no—
76#13094000—21no—
77#13094000—21no—
78#13093998C2×C412yes—
79#13093997—22no—
80#13093995—21no—
81#13093995—21no—
82#13093992—22no—
83C4×Dic94—21no—
84C4×Dic94C2×C412yes—
85C4×Dic94C2312yes—
86C4×D188—21no—
87C4×D188—21no—
88C4×D188C2×C412yes—
89C4×D188C2×C411yescharacteristic
90C4×D188C2×C411yescharacteristic
91C4×D188C2311yescharacteristic
92C4×D188C2311yescharacteristic
93C4×C22⋊D47—22no—
94C4×C22⋊D47—22no—
95C4×C22⋊D47—22no—
96#13093983—21no—
97#13093983C2×C412yes—
98#13093982—21no—
99#13093982—21no—
100#13093981—22no—
101#13093981—22no—
102#13093980—22no—
103C2×C4⋊Dic47C2×C411yescharacteristic
104C4⋊C4×D47—21no—
105C4⋊C4×D47—21no—
106C2×(C2×D47)⋊C4—21no—
107C22⋊C4×D47—22no—
108C42×D47—22no—
109C4⋊(C47⋊C8)—21no—
110C4⋊(C47⋊C8)—21no—
111C4⋊(C47⋊C8)C2311yescharacteristic
112C4⋊(C47⋊C8)C2311yescharacteristic
113C4⋊(C47⋊C8)C2312yes—
114C22⋊(C47⋊C8)—22no—
115C22⋊(C47⋊C8)—22no—
116#13093935—22no—
117C4×C47⋊C8—22no—
118C2×#41×C47D411yescharacteristic
119C22×C8×C47—21no—
120C22×C8×C47—22no—
121C4×Q8×C47C2×C411yescharacteristic
122C4×Q8×C47D411yescharacteristic
123C4×D4×C47C2×C412yes—
124C4×D4×C47C2×C411yescharacteristic
125C4×D4×C47D412yes—
126C4×D4×C47D411yescharacteristic
127#117×C47D411yescharacteristic
128#117×C47D412yes—
129C2×C42×C47C2311yescharacteristic
130C2×C42×C47C2312yes—
131C4⋊C8×C47—21no—
132C4⋊C8×C47—22no—
133C22⋊C8×C47—21no—
134C22⋊C8×C47—22no—
135C4×C8×C47—21no—
136C4×C8×C47—22no—
137C4×C8×C47C2311yescharacteristic
138C4×C8×C47C2311yescharacteristic
139C4×C8×C47C2312yes—
140C23×D47—21no—
141#41×C47—22no—
142#41×C47—22no—
143C2×Q8×C47C22⋊C411yescharacteristic
144C2×D4×C47C22⋊C412yes—
145C2×D4×C47C22⋊C411yescharacteristic
146C22×C4×C47C22×C411yescharacteristic
147C22×C4×C47C22×C412yes—
148C22×C4×C47C2×D412yes—
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D412yes—
151C22×C4×C47C2×D411yescharacteristic
152C2×C22⋊D47—22no—
153C2×C22⋊D47—22no—
154C22×Dic47—21no—
155#10625350—24no—
156#10625350—42no—
157C2×D188—22no—
158C2×D188—21no—
159C2×D188—41no—
160C2×D188C22×C411yescharacteristic
161C2×D188C22×C411yescharacteristic
162C2×C4×D47—22no—
163C2×C4×D47—21no—
164C2×C4×D47—21no—
165C2×C4×D47—21no—
166C2×C4×D47—21no—
167C2×C4×D47—22no—
168C2×C4×D47—42no—
169C2×C4×D47—42no—
170C2×Dic94—21no—
171C2×Dic94C22×C412yes—
172C2×C8×C47—21no—
173C2×C8×C47—22no—
174C2×C8×C47—21no—
175C2×C8×C47—42no—
176C2×C8×C47—44no—
177C4⋊C4×C47C22⋊C411yescharacteristic
178C4⋊C4×C47C22⋊C412yes—
179C4⋊C4×C47C22⋊C411yescharacteristic
180C4⋊C4×C47C22⋊C412yes—
181C22⋊C4×C47C22⋊C412yes—
182C22⋊C4×C47C22⋊C412yes—
183C22⋊C4×C47C22⋊C412yes—
184C42×C47C22×C411yescharacteristic
185C42×C47C22×C412yes—
186C42×C47C2×D412yes—
187C42×C47C2×D411yescharacteristic
188C42×C47C2×D412yes—
189C42×C47C2×D411yescharacteristic
190C42×C47C2411yescharacteristic
191C22⋊Dic47—22no—
192C22⋊Dic47—22no—
193C22⋊Dic47—22no—
194(C2×D47)⋊C4—22no—
195(C2×D47)⋊C4—22no—
196(C2×D47)⋊C4—21no—
197(C2×D47)⋊C4—21no—
198(C2×D47)⋊C4—22no—
199(C2×D47)⋊C4—22no—
200(C2×D47)⋊C4—21no—
201(C2×D47)⋊C4—21no—
202C4⋊Dic47—21no—
203C4⋊Dic47—21no—
204C4⋊Dic47—21no—
205C4⋊Dic47—21no—
206C4⋊Dic47C22×C411yescharacteristic
207C4⋊Dic47C22×C411yescharacteristic
208C4⋊Dic47C22×C412yes—
209Dic47⋊C4—22no—
210Dic47⋊C4—22no—
211Dic47⋊C4—22no—
212Dic47⋊C4—22no—
213Dic47⋊C4—22no—
214Dic47⋊C4—22no—
215C4×Dic47—22no—
216C4×Dic47—22no—
217C4×Dic47—22no—
218#10625329—42no—
219C2×C47⋊C8—21no—
220C2×C47⋊C8—22no—
221C2×C47⋊C8—21no—
222C2×C47⋊C8—42no—
223C23×C47C2×C22⋊C412yes—
224C23×C47C2×C22⋊C411yescharacteristic
225C22×D47—21no—
226C22×D47—21no—
227C22×D47—42no—
228C22×D47—42no—
229C22×D47—81no—
230Q8×C47—22no—
231D4×C47—24no—
232D4×C47—22no—
233C2×C4×C47—21no—
234C2×C4×C47—22no—
235C2×C4×C47—21no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—41no—
240C2×C4×C47—42no—
241C2×C4×C47C2×C22⋊C412yes—
242C2×C4×C47C2×C22⋊C412yes—
243C2×C4×C47C2×C22⋊C411yescharacteristic
244C2×C4×C47C2×C22⋊C412yes—
245C2×C4×C47C2×C22⋊C411yescharacteristic
246C2×C4×C47C2×C22⋊C411yescharacteristic
247C2×C4×C47#13512yes—
248C2×C4×C47#13512yes—
249C2×C4×C47C23×C411yescharacteristic
250C2×C4×C47C22×D411yescharacteristic
251C2×C4×C47C22×D411yescharacteristic
252C22⋊D47—44no—
253C22⋊D47—82no—
254C2×Dic47—21no—
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—21no—
258C2×Dic47—42no—
259D188—21no—
260D188—21no—
261D188—21no—
262D188—42no—
263D188#13112yes—
264C4×D47—44no—
265C4×D47—41no—
266C4×D47—41no—
267C4×D47—41no—
268C4×D47—41no—
269C4×D47—42no—
270C4×D47—42no—
271C4×D47—82no—
272Dic94—22no—
273Dic94—22no—
274Dic94—81no—
275C8×C47—22no—
276C8×C47—42no—
277C8×C47—41no—
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#41612yes—
288C22×C47C22×C22⋊C411yescharacteristic
289C2×D47—22no—
290C2×D47—41no—
291C2×D47—41no—
292C2×D47—41no—
293C2×D47—41no—
294C2×D47—41no—
295C2×D47—84no—
296C4×C47—22no—
297C4×C47—21no—
298C4×C47—22no—
299C4×C47—21no—
300C4×C47—21no—
301C4×C47—22no—
302C4×C47#41612yes—
303C4×C47C2×#13111yescharacteristic
304C4×C47#56811yescharacteristic
305C4×C47C2×#13511yescharacteristic
306C4×C47C2×#13511yescharacteristic
307Dic47—42no—
308Dic47—41no—
309Dic47—41no—
310Dic47—42no—
311#3475—471no—
312C8×#41D4711yescharacteristic, complemented
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#288011yescharacteristic
325C2×C47C2×#41611yescharacteristic
326C2×C47#290111yescharacteristic
327D47—42no—
328D47—81no—
329#556—471no—
330#555—471no—
331#554—471no—
332#553—941no—
333#552—472no—
334#552—941no—
335#551—942no—
336#547—471no—
337#546—472no—
338#533—471no—
339#529—472no—
340#529—942no—
341#525—941no—
342C4×#41—472no—
343C4×#41C2×D4711yescharacteristic
344C2×C4×D4—471no—
345C2×#117—941no—
346C8×Q8—21no—
347C8×D4—21no—
348C8×D4—22no—
349#433—942no—
350#432—472no—
351#431—472no—
352#431C2×D4711yescharacteristic
353#431C2×D4712yescomplemented
354#423—471no—
355C2×C4⋊C8—471no—
356#408—941no—
357#405—941no—
358C2×C4×C8C2×D4711yescharacteristic
359C2×C4×C8C2×D4712yescomplemented
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×#41—472no—
372C2×#41D18811yescharacteristic
373C22×D4—471no—
374C23×C4—941no—
375C2×M16—941no—
376C22×C8—21no—
377C22×C8—22no—
378C22×C8—941no—
379C4⋊Q8—471no—
380C4⋊Q8—471no—
381#126—942no—
382#126—942no—
383#126—942no—
384#125—472no—
385#125—942no—
386#125—941no—
387#124—472no—
388#124—942no—
389#123—942no—
390#123—942no—
391#123—942no—
392#123—942no—
393C22⋊Q8—472no—
394C22⋊Q8—472no—
395C22⋊Q8—941no—
396C22⋊Q8—941no—
397C22⋊Q8—941no—
398C22⋊Q8—942no—
399#121—472no—
400#121—941no—
401#121—941no—
402#121—942no—
403#120—942no—
404C4×Q8—472no—
405C4×Q8—472no—
406C4×Q8—941no—
407C4×Q8C4×D4711yescharacteristic
408C4×Q8C22⋊D4711yescharacteristic
409C4×D4—471no—
410C4×D4—471no—
411C4×D4—472no—
412C4×D4—471no—
413C4×D4—471no—
414C4×D4—942no—
415C4×D4—942no—
416C4×D4—942no—
417C4×D4—941no—
418C4×D4—941no—
419C4×D4C4×D4712yes—
420C4×D4C4×D4711yescharacteristic
421C4×D4C22⋊D4712yes—
422C4×D4C22⋊D4711yescharacteristic
423#117—472no—
424#117—941no—
425#117—941no—
426#117—941no—
427#117—942no—
428#117—942no—
429#117—942no—
430#117D18811yescharacteristic
431#117D18812yes—
432C2×C4⋊C4—471no—
433C2×C4⋊C4—941no—
434C2×C4⋊C4—941no—
435C2×C22⋊C4—941no—
436C2×C22⋊C4—942no—
437C2×C42—942no—
438C2×C42C22×D4711yescharacteristic
439C2×C42C22×D4712yes—
440C4⋊C8—21no—
441C4⋊C8—22no—
442C4⋊C8—471no—
443C4⋊C8—471no—
444C4⋊C8—472no—
445C4⋊C8—941no—
446C4⋊C8—941no—
447C22⋊C8—21no—
448C22⋊C8—22no—
449C22⋊C8—942no—
450C22⋊C8—942no—
451#97—942no—
452C4×C8—21no—
453C4×C8—22no—
454C4×C8—942no—
455C4×C8C22×D4711yescharacteristic
456C4×C8C22×D4711yescharacteristic
457C4×C8C22×D4712yes—
458C24—941no—
459#41—22no—
460#41—22no—
461#41—944no—
462#41—1882no—
463C2×Q8—472no—
464C2×Q8—941no—
465C2×Q8(C2×D47)⋊C411yescharacteristic
466C2×D4—471no—
467C2×D4—471no—
468C2×D4—942no—
469C2×D4—942no—
470C2×D4—942no—
471C2×D4—941no—
472C2×D4—1881no—
473C2×D4(C2×D47)⋊C412yes—
474C2×D4(C2×D47)⋊C411yescharacteristic
475C22×C4—941no—
476C22×C4—942no—
477C22×C4—942no—
478C22×C4—941no—
479C22×C4—941no—
480C22×C4—941no—
481C22×C4—941no—
482C22×C4—1882no—
483C22×C4—1882no—
484C22×C4C2×C4×D4711yescharacteristic
485C22×C4C2×C4×D4712yes—
486C22×C4C2×D18812yes—
487C22×C4C2×D18811yescharacteristic
488C22×C4C2×C22⋊D4711yescharacteristic
489C22×C4C2×C22⋊D4712yes—
490M16—1882no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—22no—
494C2×C8—42no—
495C2×C8—44no—
496C2×C8—941no—
497C2×C8—942no—
498C2×C8—941no—
499C2×C8—1882no—
500C4⋊C4—471no—
501C4⋊C4—471no—
502C4⋊C4—472no—
503C4⋊C4—942no—
504C4⋊C4—942no—
505C4⋊C4—942no—
506C4⋊C4—942no—
507C4⋊C4—942no—
508C4⋊C4—941no—
509C4⋊C4—941no—
510C4⋊C4—941no—
511C4⋊C4—941no—
512C4⋊C4—942no—
513C4⋊C4(C2×D47)⋊C411yescharacteristic
514C4⋊C4(C2×D47)⋊C411yescharacteristic
515C4⋊C4(C2×D47)⋊C412yes—
516C4⋊C4(C2×D47)⋊C412yes—
517C22⋊C4—942no—
518C22⋊C4—942no—
519C22⋊C4—942no—
520C22⋊C4—942no—
521C22⋊C4—942no—
522C22⋊C4—941no—
523C22⋊C4—941no—
524C22⋊C4—942no—
525C22⋊C4—942no—
526C22⋊C4—941no—
527C22⋊C4—941no—
528C22⋊C4(C2×D47)⋊C412yes—
529C22⋊C4(C2×D47)⋊C412yes—
530C22⋊C4(C2×D47)⋊C412yes—
531C42—942no—
532C42—942no—
533C42—942no—
534C42C2×C4×D4711yescharacteristic
535C42C2×C4×D4712yes—
536C42C2×D18812yes—
537C42C2×D18811yescharacteristic
538C42C2×C22⋊D4712yes—
539C42C2×C22⋊D4711yescharacteristic
540C42C23×D4711yescharacteristic
541C23—941no—
542C23—941no—
543C23—3761no—
544C23—1882no—
545C23—1882no—
546C23C2×(C2×D47)⋊C412yes—
547C23C2×(C2×D47)⋊C411yescharacteristic
548Q8—22no—
549Q8—942no—
550Q8—942no—
551Q8—3761no—
552D4—24no—
553D4—22no—
554D4—472no—
555D4—941no—
556D4—941no—
557D4—941no—
558D4—3762no—
559D4—1882no—
560D4—1884no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—941no—
570C2×C4—942no—
571C2×C4—942no—
572C2×C4—941no—
573C2×C4—3762no—
574C2×C4—1884no—
575C2×C4—1882no—
576C2×C4—1882no—
577C2×C4—1882no—
578C2×C4—1881no—
579C2×C4—1881no—
580C2×C4—1881no—
581C2×C4—1881no—
582C2×C4C2×(C2×D47)⋊C412yes—
583C2×C4C2×(C2×D47)⋊C412yes—
584C2×C4C2×(C2×D47)⋊C411yescharacteristic
585C2×C4C2×(C2×D47)⋊C412yes—
586C2×C4C2×(C2×D47)⋊C411yescharacteristic
587C2×C4C2×(C2×D47)⋊C411yescharacteristic
588C2×C4#1309404612yes—
589C2×C4#1309404612yes—
590C2×C4C22×C4×D4711yescharacteristic
591C2×C4C22×C22⋊D4711yescharacteristic
592C2×C4C22×D18811yescharacteristic
593C8—22no—
594C8—41no—
595C8—42no—
596C8—1881no—
597C8—1881no—
598C8—1882no—
599C22—21no—
600C22—21no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—942no—
605C22—3764no—
606C22—1881no—
607C22—1881no—
608C22—1881no—
609C22—1881no—
610C22—1881no—
611C22#1568912012yes—
612C22C22×(C2×D47)⋊C411yescharacteristic
613C4—21no—
614C4—22no—
615C4—21no—
616C4—22no—
617C4—21no—
618C4—22no—
619C4—1882no—
620C4—1882no—
621C4—1881no—
622C4—1881no—
623C4#1568912012yes—
624C4#131×D4711yescharacteristic
625C4#1568988811yescharacteristic
626C4C2×#1309404611yescharacteristic
627C4C2×#1309404611yescharacteristic
628C2—22no—
629C2—42no—
630C2—3761no—
631C2—1882no—
632C2#2025183511yescharacteristic
633C2C2×#1568912011yescharacteristic
634C2#2025193611yescharacteristic
subgroupG/H
centerC2×C4C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC8×#41×C47C2
socleC22×C47C22×C22⋊C4
prime radical#20256952C2
cosocleC42×C47C24
hypercenterC8×#41D47
hypocenterC47#17508
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
2. #20252054C2semidirect
3. #20241849C2semidirect
4. #20241844C2semidirect
5. #20241823C2semidirect
6. #20241814C2semidirect
7. #20241797C2semidirect
8. #20241792C2semidirect
9. #20241771C2semidirect
10. C2×#15688852C2semidirect
11. C8×#41×C47C2semidirect
29. #15689180C22semidirect
30. #15689176C22semidirect
35. #15688855C22semidirect
36. #15688855C22semidirect
37. #15688855C22semidirect
38. #15688855C22semidirect
50. C47×#431C22semidirect
52. C2×C4×C8×C47C22semidirect
312. C8×#41D47semidirect
353. #431C2×D47semidirect
359. C2×C4×C8C2×D47semidirect
370. C47#17508semidirect

Maximal subgroups

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

subgroupindex
#202569522
#202520542
#202418492
#202418442
#202418232
#202418142
#202417972
#202417922
#202417712
C2×#156888522
C8×#41×C472
#1750847