Finite Group Databasebeta

← #26224130

Subgroups — #26224130

Order 12032.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253176C211yesmaximal, characteristic
2#20252328C211yesmaximal, characteristic, complemented
3#20244332C211yesmaximal, characteristic, complemented
4#20244163C211yesmaximal, characteristic, complemented
5#20243987C211yesmaximal, characteristic, complemented
6#20243987C211yesmaximal, characteristic, complemented
7#20243663C211yesmaximal, characteristic
8#20243663C211yesmaximal, characteristic
9#20243408C211yesmaximal, characteristic
10#20243273C211yesmaximal, characteristic, complemented
11#20243268C211yesmaximal, characteristic, complemented
12#20243260C211yesmaximal, characteristic, complemented
13#20243230C211yesmaximal, characteristic
14#20243223C211yesmaximal, characteristic
15C47×#2875C211yesmaximal, characteristic
16#15689680C2211yescharacteristic
17#15689430C411yescharacteristic, complemented
18#15689430C411yescharacteristic, complemented
19#15689423C411yescharacteristic, complemented
20#15689423C411yescharacteristic, complemented
21#15689418C411yescharacteristic, complemented
22#15689413C411yescharacteristic, complemented
23#15689408C411yescharacteristic, complemented
24#15689403C411yescharacteristic, complemented
25#15689251—21no—
26#15689251—21no—
27#15689251—21no—
28#15689251—21no—
29C2×#13093959C2211yescharacteristic
30C2×#13093959C2211yescharacteristic
31#15689159C2211yescharacteristic
32#15689159C2211yescharacteristic
33#15689149C2211yescharacteristic
34#15689149C2211yescharacteristic
35#15689147C2211yescharacteristic
36C2×C4⋊(C47⋊C8)C2211yescharacteristic
37C2×#13093951C2211yescharacteristic
38C2×#13093951C2211yescharacteristic
39#15689119C2211yescharacteristic
40#15689119C2211yescharacteristic
41#15689078C2211yescharacteristic
42#15689078C2211yescharacteristic
43C2×#13093934C2211yescharacteristic
44C2×C8×Dic47C2211yescharacteristic
45C4×C4⋊Dic47C2211yescharacteristic
46#15688887C2211yescharacteristic
47#15688887C2211yescharacteristic
48#15688884C2211yescharacteristic
49#15688884C2211yescharacteristic
50#15688884C2211yescharacteristic
51#15688884C2211yescharacteristic
52#15688882C2211yescharacteristic
53#15688882C2211yescharacteristic
54#15688877C2211yescharacteristic
55#15688877C2211yescharacteristic
56C2×#131×C47C2211yescharacteristic
57C2×#131×C47C2211yescharacteristic
58C4×#41×C47C2211yescharacteristic
59C47×#405—21no—
60C47×#405—21no—
61C47×#405C2211yescharacteristic
62C47×#405C2211yescharacteristic
63C4×M16×C47—21no—
64C4×M16×C47C2211yescharacteristic
65C2×#97×C47—21no—
66C2×#97×C47C2211yescharacteristic
67C2×#10625329C2311yescharacteristic
68C2×#10625329C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70C22×C47⋊C8C2311yescharacteristic
71#13094024C2×C411yescharacteristic
72#13094024C2×C411yescharacteristic
73#13094016C2×C411yescharacteristic
74#13094016C2×C411yescharacteristic
75#13094003—21no—
76#13094003—21no—
77C4×Dic94—21no—
78C4×Dic94—21no—
79#13093983C2311yescharacteristic
80#13093983C2311yescharacteristic
81C2×C4⋊Dic47—21no—
82C2×C4×Dic47C2311yescharacteristic
83C2×C4×Dic47C2311yescharacteristic
84#13093959—22no—
85#13093959—22no—
86#13093957C2×C411yescharacteristic
87#13093957C2×C411yescharacteristic
88#13093957C2×C411yescharacteristic
89#13093957C2×C411yescharacteristic
90#13093955C2×C411yescharacteristic
91#13093955C2×C411yescharacteristic
92#13093955C2×C411yescharacteristic
93#13093955C2×C411yescharacteristic
94C4⋊(C47⋊C8)—21no—
95C4⋊(C47⋊C8)—21no—
96C4⋊(C47⋊C8)C2×C411yescharacteristic
97C4⋊(C47⋊C8)C2×C411yescharacteristic
98C4⋊(C47⋊C8)C2×C411yescharacteristic
99C4⋊(C47⋊C8)C2×C411yescharacteristic
100Dic47⋊C8—21no—
101Dic47⋊C8—21no—
102Dic47⋊C8—21no—
103Dic47⋊C8—21no—
104#13093951—22no—
105#13093951—22no—
106#13093948C2×C411yescharacteristic
107#13093948C2×C411yescharacteristic
108#13093948C2×C411yescharacteristic
109#13093948C2×C411yescharacteristic
110#13093945C2×C411yescharacteristic
111#13093945C2×C411yescharacteristic
112#13093945C2×C411yescharacteristic
113#13093945C2×C411yescharacteristic
114#13093934—21no—
115#13093934—21no—
116#13093934—22no—
117C8×Dic47—21no—
118C8×Dic47—21no—
119C8×Dic47—22no—
120C47⋊#95—21no—
121C2×#41×C47C2311yescharacteristic
122#131×C47—22no—
123#131×C47—22no—
124#131×C47—22no—
125#131×C47Q814yescomplemented
126C2×M16×C47—21no—
127C2×M16×C47—21no—
128C2×M16×C47C2311yescharacteristic
129C2×M16×C47C2311yescharacteristic
130C22×C8×C47—21no—
131C22×C8×C47—21no—
132C22×C8×C47C2311yescharacteristic
133C22×C8×C47C2311yescharacteristic
134C4×Q8×C47C2×C411yescharacteristic
135C4×Q8×C47C2×C411yescharacteristic
136C4×D4×C47—21no—
137C4×D4×C47—21no—
138C4×D4×C47C2×C411yescharacteristic
139C4×D4×C47C2×C411yescharacteristic
140#117×C47—21no—
141#117×C47C2311yescharacteristic
142C2×C42×C47—21no—
143C2×C42×C47C2311yescharacteristic
144#97×C47—21no—
145#97×C47—21no—
146#97×C47—21no—
147#97×C47D411yescharacteristic
148#97×C47D411yescharacteristic
149#97×C47D411yescharacteristic
150#97×C47D411yescharacteristic
151C4×C8×C47—21no—
152C4×C8×C47—21no—
153C4×C8×C47—21no—
154#41×C47—21no—
155#41×C47—21no—
156#41×C47C2×Q812yes—
157#41×C47#4112yes—
158C2×Q8×C47C22×C411yescharacteristic
159C2×D4×C47—21no—
160C2×D4×C47C22×C411yescharacteristic
161C22×C4×C47—21no—
162C22×C4×C47—21no—
163C22×C4×C47—21no—
164C22×C4×C47#4111yescharacteristic
165C22×C4×C47#4111yescharacteristic
166C22×C4×C47C2411yescharacteristic
167C22×Dic47—21no—
168C22×Dic47—21no—
169C2×Dic94—21no—
170C2×Dic94—21no—
171M16×C47—24no—
172M16×C47—21no—
173M16×C47—21no—
174M16×C47—24no—
175M16×C47C2×Q812yes—
176M16×C47#4112yes—
177C2×C8×C47—24no—
178C2×C8×C47—21no—
179C2×C8×C47—21no—
180C2×C8×C47—21no—
181C2×C8×C47—21no—
182C2×C8×C47—24no—
183C2×C8×C47C2×D411yescharacteristic
184C2×C8×C47C2×D411yescharacteristic
185C2×C8×C47C2×D411yescharacteristic
186C2×C8×C47C2×D411yescharacteristic
187C2×C8×C47C2×Q812yes—
188C2×C8×C47#4112yes—
189C4⋊C4×C47—21no—
190C4⋊C4×C47—21no—
191C4⋊C4×C47C22×C411yescharacteristic
192C4⋊C4×C47C22×C411yescharacteristic
193C22⋊C4×C47—21no—
194C22⋊C4×C47—21no—
195C22⋊C4×C47—21no—
196C42×C47—21no—
197C42×C47—21no—
198C42×C47—21no—
199C42×C47C22×C411yescharacteristic
200C42×C47C22×C411yescharacteristic
201C42×C47C2×D411yescharacteristic
202C42×C47C2×D411yescharacteristic
203C22⋊Dic47—21no—
204C22⋊Dic47—21no—
205C4⋊Dic47—21no—
206C4⋊Dic47—21no—
207C4⋊Dic47—42no—
208C4⋊Dic47C22×C411yescharacteristic
209C4⋊Dic47C22×C411yescharacteristic
210C4⋊Dic47C22×C411yescharacteristic
211C4⋊Dic47C22×C411yescharacteristic
212Dic47⋊C4—21no—
213Dic47⋊C4—21no—
214Dic47⋊C4—21no—
215Dic47⋊C4—21no—
216C4×Dic47—21no—
217C4×Dic47—21no—
218C4×Dic47—21no—
219C4×Dic47—21no—
220C4×Dic47—21no—
221C4×Dic47—21no—
222#10625329—21no—
223#10625329—21no—
224#10625329—21no—
225#10625329—21no—
226C2×C47⋊C8—21no—
227C2×C47⋊C8—21no—
228C2×C47⋊C8—21no—
229C2×C47⋊C8—21no—
230C2×C47⋊C8—21no—
231C2×C47⋊C8—21no—
232C2×C47⋊C8C22×C411yescharacteristic
233C2×C47⋊C8C22×C411yescharacteristic
234C2×C47⋊C8C22×C411yescharacteristic
235C2×C47⋊C8C22×C411yescharacteristic
236C23×C47—21no—
237C23×C47C2×#4111yescharacteristic
238Q8×C47—21no—
239Q8×C47C4×Q812yes—
240D4×C47—22no—
241D4×C47—22no—
242D4×C47—21no—
243D4×C47C4×Q812yes—
244C2×C4×C47—21no—
245C2×C4×C47—22no—
246C2×C4×C47—22no—
247C2×C4×C47—21no—
248C2×C4×C47—21no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—81no—
252C2×C4×C47—81no—
253C2×C4×C47C4×D411yescharacteristic
254C2×C4×C47C4×D411yescharacteristic
255C2×C4×C47C4×D411yescharacteristic
256C2×C4×C47C4×D411yescharacteristic
257C2×C4×C47C23×C411yescharacteristic
258C2×C4×C47C22×D411yescharacteristic
259C2×C4×C47C22×Q811yescharacteristic
260C2×C4×C47C2×#4111yescharacteristic
261C2×C4×C47C2×#4111yescharacteristic
262C2×C4×C47C2×#4111yescharacteristic
263C2×Dic47—22no—
264C2×Dic47—21no—
265C2×Dic47—21no—
266C2×Dic47—21no—
267C2×Dic47—21no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273Dic94—81no—
274Dic94—81no—
275C8×C47—22no—
276C8×C47—21no—
277C8×C47—22no—
278C8×C47—21no—
279C8×C47C22⋊Q812yes—
280C8×C47C22⋊Q812yes—
281C47⋊C8—21no—
282C47⋊C8—21no—
283C47⋊C8—21no—
284C47⋊C8—21no—
285C47⋊C8—41no—
286C47⋊C8—41no—
287#21629—471nomaximal, Sylow
288C22×C47—21no—
289C22×C47—21no—
290C22×C47—22no—
291C22×C47—41no—
292C22×C47—41no—
293C22×C47C2×C4×D411yescharacteristic
294C22×C47#52911yescharacteristic
295C22×C47#53311yescharacteristic
296C4×C47—21no—
297C4×C47—21no—
298C4×C47—21no—
299C4×C47—21no—
300C4×C47—22no—
301C4×C47—41no—
302C4×C47—41no—
303C4×C47C2×C4×Q811yescharacteristic
304C4×C47C4×#4111yescharacteristic
305C4×C47C2×C22⋊Q811yescharacteristic
306C4×C47#52511yescharacteristic
307Dic47—42no—
308Dic47—41no—
309Dic47—41no—
310Dic47—41no—
311Dic47—41no—
312Dic47—81no—
313#3076—471no—
314#2961—471no—
315#2875D4711yescharacteristic
316#1986—471no—
317#1950—471no—
318#1922—471no—
319#1922—471no—
320#1866—471no—
321#1866—471no—
322#1806—471no—
323#1773—471no—
324#1772—471no—
325#1771—471no—
326#1765—471no—
327#1764—471no—
328C2×C47—21no—
329C2×C47—21no—
330C2×C47—42no—
331C2×C47C4×C22⋊Q811yescharacteristic
332C2×C47#295611yescharacteristic
333C2×C47#295711yescharacteristic
334C2×#131C2×D4711yescharacteristic
335C2×#131C2×D4711yescharacteristic
336#533—471no—
337C4×#41C2×D4711yescharacteristic
338#479—471no—
339#479—471no—
340#478—471no—
341#478—471no—
342#477—471no—
343#476—471no—
344#475—471no—
345#474—471no—
346#446—941no—
347#446—941no—
348#446—941no—
349#446—941no—
350C2×#108—471no—
351C2×#108—471no—
352#427—471no—
353#427—471no—
354#424—471no—
355#424—471no—
356#423—471no—
357C2×C4⋊C8—471no—
358C2×#104—471no—
359C2×#104—471no—
360#416—471no—
361#416—471no—
362#405—21no—
363#405—21no—
364#405—471no—
365#405—471no—
366#405C2×D4711yescharacteristic
367#405C2×D4711yescharacteristic
368C4×M16—21no—
369C4×M16C2×D4711yescharacteristic
370C2×#97—21no—
371C2×#97—471no—
372C2×#97C2×D4711yescharacteristic
373C2×C4×C8—471no—
374C4×C4⋊C4—471no—
375#341—471no—
376#341—471no—
377#339—471no—
378#339—471no—
379#339—471no—
380#339—471no—
381#339—471no—
382#339—471no—
383#336—471no—
384#336—471no—
385C47#2162911yescharacteristic, complemented, Sylow
386C2×#41C22×D4711yescharacteristic
387#131—22no—
388#131—22no—
389#131—22no—
390#131Dic9414yescomplemented
391C2×M16—21no—
392C2×M16—21no—
393C2×M16—471no—
394C2×M16—471no—
395C2×M16C22×D4711yescharacteristic
396C2×M16C22×D4711yescharacteristic
397C22×C8—21no—
398C22×C8—21no—
399C22×C8—471no—
400C22×C8—471no—
401C22×C8C22×D4711yescharacteristic
402C22×C8C22×D4711yescharacteristic
403C4⋊Q8—471no—
404C4⋊Q8—471no—
405#125—471no—
406#125—471no—
407C22⋊Q8—941no—
408C22⋊Q8—941no—
409C4×Q8—941no—
410C4×Q8—941no—
411C4×Q8C4×D4711yescharacteristic
412C4×Q8C4×D4711yescharacteristic
413C4×D4—21no—
414C4×D4—21no—
415C4×D4C4×D4711yescharacteristic
416C4×D4C4×D4711yescharacteristic
417#117—21no—
418#117—471no—
419#117—471no—
420#117C22×D4711yescharacteristic
421C2×C4⋊C4—941no—
422C2×C42—21no—
423C2×C42—471no—
424C2×C42—471no—
425C2×C42C22×D4711yescharacteristic
426#108—942no—
427#108—942no—
428#107—471no—
429#107—471no—
430#107—471no—
431#107—471no—
432#106—471no—
433#106—471no—
434#106—471no—
435#106—471no—
436C4⋊C8—471no—
437C4⋊C8—471no—
438C4⋊C8—471no—
439C4⋊C8—471no—
440C4⋊C8—941no—
441C4⋊C8—941no—
442C4⋊C8—941no—
443C4⋊C8—941no—
444C4⋊C8—941no—
445C4⋊C8—941no—
446#104—942no—
447#104—942no—
448#103—471no—
449#103—471no—
450#103—471no—
451#103—471no—
452#102—471no—
453#102—471no—
454#102—471no—
455#102—471no—
456#97—21no—
457#97—21no—
458#97—21no—
459#97—942no—
460#97—941no—
461#97—941no—
462#97C22⋊D4711yescharacteristic
463#97C22⋊D4711yescharacteristic
464#97C22⋊D4711yescharacteristic
465#97C22⋊D4711yescharacteristic
466C4×C8—21no—
467C4×C8—21no—
468C4×C8—21no—
469C4×C8—942no—
470C4×C8—941no—
471C4×C8—941no—
472#95—941no—
473#41—21no—
474#41—21no—
475#41C2×Dic9412yes—
476#41#1062535012yes—
477C2×Q8—941no—
478C2×Q8—941no—
479C2×Q8C2×C4×D4711yescharacteristic
480C2×D4—21no—
481C2×D4C2×C4×D4711yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—941no—
486C22×C4—941no—
487C22×C4#1062535011yescharacteristic
488C22×C4#1062535011yescharacteristic
489C22×C4C23×D4711yescharacteristic
490M16—24no—
491M16—21no—
492M16—21no—
493M16—24no—
494M16—941no—
495M16—941no—
496M16—941no—
497M16—941no—
498M16C2×Dic9412yes—
499M16#1062535012yes—
500C2×C8—24no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—24no—
506C2×C8—471no—
507C2×C8—471no—
508C2×C8—471no—
509C2×C8—471no—
510C2×C8—941no—
511C2×C8—941no—
512C2×C8—941no—
513C2×C8—941no—
514C2×C8—941no—
515C2×C8—941no—
516C2×C8C2×Dic9412yes—
517C2×C8#1062535012yes—
518C2×C8C2×C22⋊D4711yescharacteristic
519C2×C8C2×C22⋊D4711yescharacteristic
520C2×C8C2×C22⋊D4711yescharacteristic
521C2×C8C2×C22⋊D4711yescharacteristic
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—471no—
525C4⋊C4—471no—
526C4⋊C4—471no—
527C4⋊C4—471no—
528C4⋊C4—941no—
529C4⋊C4—941no—
530C4⋊C4—941no—
531C4⋊C4—941no—
532C4⋊C4—941no—
533C4⋊C4—941no—
534C4⋊C4—1882no—
535C4⋊C4C2×C4×D4711yescharacteristic
536C4⋊C4C2×C4×D4711yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—941no—
541C22⋊C4—941no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—941no—
546C42—941no—
547C42—941no—
548C42—941no—
549C42—941no—
550C42—941no—
551C42C2×C4×D4711yescharacteristic
552C42C2×C4×D4711yescharacteristic
553C42C2×C22⋊D4711yescharacteristic
554C42C2×C22⋊D4711yescharacteristic
555C23—21no—
556C23C2×#1062535011yescharacteristic
557Q8—21no—
558Q8—3761no—
559Q8—3761no—
560Q8C4×Dic9412yes—
561D4—22no—
562D4—22no—
563D4—21no—
564D4C4×Dic9412yes—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—942no—
575C2×C4—941no—
576C2×C4—941no—
577C2×C4—941no—
578C2×C4—941no—
579C2×C4—1881no—
580C2×C4—1881no—
581C2×C4—1881no—
582C2×C4—1881no—
583C2×C4—1881no—
584C2×C4C4×C22⋊D4711yescharacteristic
585C2×C4C4×C22⋊D4711yescharacteristic
586C2×C4C4×C22⋊D4711yescharacteristic
587C2×C4C4×C22⋊D4711yescharacteristic
588C2×C4C22×C4×D4711yescharacteristic
589C2×C4C22×C22⋊D4711yescharacteristic
590C2×C4C22×Dic9411yescharacteristic
591C2×C4C2×#1062535011yescharacteristic
592C2×C4C2×#1062535011yescharacteristic
593C2×C4C2×#1062535011yescharacteristic
594C8—22no—
595C8—21no—
596C8—22no—
597C8—21no—
598C8—941no—
599C8—941no—
600C8—941no—
601C8—941no—
602C8—1881no—
603C8—1881no—
604C8#1309400312yes—
605C8#1309400312yes—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22C2×C4×C22⋊D4711yescharacteristic
612C22#1568965711yescharacteristic
613C22#1568968011yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—22no—
619C4—41no—
620C4—41no—
621C4—3761no—
622C4—1882no—
623C4—1881no—
624C4—1881no—
625C4—1881no—
626C4—1881no—
627C4C2×C4×Dic9411yescharacteristic
628C4C4×#1062535011yescharacteristic
629C4C2×#1309400311yescharacteristic
630C4#1568963411yescharacteristic
631C2—21no—
632C2—21no—
633C2—42no—
634C2C4×#1309400311yescharacteristic
635C2#2025229511yescharacteristic
636C2#2025230311yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2875C2
socleC22×C47C2×C4×D4
prime radicalC2×#41×C47C23
cosocleC22×C4×C47C24
hypercenter#2875D47
hypocenterC47#21629
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×#97 ◁ #2875

Complements

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

NH ≅ G/Ntype
2. #20252328C2semidirect
3. #20244332C2semidirect
4. #20244163C2semidirect
5. #20243987C2semidirect
6. #20243987C2semidirect
10. #20243273C2semidirect
11. #20243268C2semidirect
12. #20243260C2semidirect
17. #15689430C4semidirect
18. #15689430C4semidirect
19. #15689423C4semidirect
20. #15689423C4semidirect
21. #15689418C4semidirect
22. #15689413C4semidirect
23. #15689408C4semidirect
24. #15689403C4semidirect
125. #131×C47Q8semidirect
385. C47#21629semidirect
390. #131Dic94semidirect

Maximal subgroups

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

subgroupindex
#202531762
#202523282
#202443322
#202441632
#202439872
#202436632
#202434082
#202432732
#202432682
#202432602
#202432302
#202432232
C47×#28752
#2162947