Finite Group Databasebeta

← #26224115

Subgroups — #26224115

Order 12032.

738 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#20253172C211yesmaximal, characteristic
2#20252320C211yesmaximal, characteristic, complemented
3#131×Dic47C211yesmaximal, characteristic
4#20244330C211yesmaximal, characteristic, complemented
5#20244162C211yesmaximal, characteristic, complemented
6#20243990C211yesmaximal, characteristic, complemented
7#20243986C211yesmaximal, characteristic, complemented
8#20243665C211yesmaximal, characteristic
9#20243661C211yesmaximal, characteristic
10#20243406C211yesmaximal, characteristic
11#20243273C211yesmaximal, characteristic, complemented
12#20243272C211yesmaximal, characteristic, complemented
13#20243256C211yesmaximal, characteristic, complemented
14#20243230C211yesmaximal, characteristic
15C47×#1765C211yesmaximal, characteristic
16C2×#13094031C2211yescharacteristic
17#15689676C2211yescharacteristic
18#41×Dic47C2211yescharacteristic
19#15689432C411yescharacteristic, complemented
20#15689428C411yescharacteristic, complemented
21#15689425C411yescharacteristic, complemented
22#15689421C411yescharacteristic, complemented
23#15689416C411yescharacteristic, complemented
24#15689411C411yescharacteristic, complemented
25#15689406C411yescharacteristic, complemented
26#15689401C411yescharacteristic, complemented
27#15689251—21no—
28#15689251—21no—
29C8×Dic94—21no—
30C8×Dic94—21no—
31C2×#13093960C2211yescharacteristic
32C2×#13093959C2211yescharacteristic
33#15689161C2211yescharacteristic
34#15689159C2211yescharacteristic
35#15689151C2211yescharacteristic
36#15689150C2211yescharacteristic
37#15689146C2211yescharacteristic
38C2×Dic47⋊C8C2211yescharacteristic
39C2×#13093951C2211yescharacteristic
40#15689119C2211yescharacteristic
41#15689081—21no—
42#15689081C2211yescharacteristic
43#15689078—21no—
44#15689078C2211yescharacteristic
45#15689078C2211yescharacteristic
46M16×Dic47—21no—
47M16×Dic47C2211yescharacteristic
48C2×#13093934C2211yescharacteristic
49C2×C8×Dic47—21no—
50C2×C8×Dic47C2211yescharacteristic
51C4×Dic47⋊C4C2211yescharacteristic
52#15688887C2211yescharacteristic
53#15688884C2211yescharacteristic
54#15688884C2211yescharacteristic
55#15688883C2211yescharacteristic
56#15688883C2211yescharacteristic
57#15688882C2211yescharacteristic
58#15688878C2211yescharacteristic
59#15688877C2211yescharacteristic
60C2×#131×C47C2211yescharacteristic
61C2×#104×C47C2211yescharacteristic
62C47×#416C2211yescharacteristic
63C47×#405C2211yescharacteristic
64C2×#97×C47C2211yescharacteristic
65C47×#341C2211yescharacteristic
66C47×#339C2211yescharacteristic
67#13094031—22no—
68#13094031—22no—
69C2×#10625329—21no—
70C2×#10625329C2311yescharacteristic
71C2×#10625329C2311yescharacteristic
72C22×C47⋊C8—21no—
73C22×C47⋊C8C2311yescharacteristic
74C22×C47⋊C8C2311yescharacteristic
75#13094023C2×C411yescharacteristic
76#13094023C2×C411yescharacteristic
77#13094015C2×C411yescharacteristic
78#13094015C2×C411yescharacteristic
79#13094001—21no—
80#13094001—21no—
81Q8×Dic47C2×C411yescharacteristic
82Q8×Dic47C2×C411yescharacteristic
83C4×Dic94—21no—
84C4×Dic94—21no—
85D4×Dic47—21no—
86D4×Dic47—21no—
87D4×Dic47C2×C411yescharacteristic
88D4×Dic47C2×C411yescharacteristic
89#13093983—21no—
90#13093983C2311yescharacteristic
91#13093983C2311yescharacteristic
92C2×Dic47⋊C4—21no—
93C2×C4×Dic47—21no—
94C2×C4×Dic47C2311yescharacteristic
95C2×C4×Dic47C2311yescharacteristic
96#13093960—22no—
97#13093959—22no—
98#13093958C2×C411yescharacteristic
99#13093958C2×C411yescharacteristic
100#13093957C2×C411yescharacteristic
101#13093957C2×C411yescharacteristic
102#13093956C2×C411yescharacteristic
103#13093956C2×C411yescharacteristic
104#13093955C2×C411yescharacteristic
105#13093955C2×C411yescharacteristic
106C4⋊(C47⋊C8)—21no—
107C4⋊(C47⋊C8)—21no—
108Dic47⋊C8—21no—
109Dic47⋊C8—21no—
110Dic47⋊C8—21no—
111Dic47⋊C8—21no—
112Dic47⋊C8C2×C411yescharacteristic
113Dic47⋊C8C2×C411yescharacteristic
114Dic47⋊C8C2×C411yescharacteristic
115Dic47⋊C8C2×C411yescharacteristic
116#13093951—22no—
117#13093948C2×C411yescharacteristic
118#13093948C2×C411yescharacteristic
119#13093945C2×C411yescharacteristic
120#13093945C2×C411yescharacteristic
121#13093934—21no—
122#13093934—21no—
123#13093934—22no—
124#13093934—21no—
125#13093934—21no—
126C8×Dic47—21no—
127C8×Dic47—21no—
128C8×Dic47—21no—
129C8×Dic47—21no—
130C8×Dic47D411yescharacteristic
131C8×Dic47D411yescharacteristic
132C8×Dic47D411yescharacteristic
133C8×Dic47D411yescharacteristic
134C47⋊#95—21no—
135C2×#41×C47C2311yescharacteristic
136#131×C47—22no—
137#131×C47Q814yescomplemented
138C2×M16×C47—21no—
139C2×M16×C47C2311yescharacteristic
140C2×M16×C47C2311yescharacteristic
141C22×C8×C47—21no—
142C22×C8×C47C2311yescharacteristic
143C22×C8×C47C2311yescharacteristic
144#117×C47C2311yescharacteristic
145C2×C42×C47C2311yescharacteristic
146#104×C47—22no—
147#103×C47C2×C411yescharacteristic
148#103×C47C2×C411yescharacteristic
149#102×C47C2×C411yescharacteristic
150#102×C47C2×C411yescharacteristic
151#97×C47—21no—
152#97×C47—22no—
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×C47C2411yescharacteristic
164C22×Dic47—21no—
165C22×Dic47—21no—
166C22×Dic47—21no—
167C22×Dic47#4111yescharacteristic
168C22×Dic47#4111yescharacteristic
169C2×Dic94—21no—
170C2×Dic94—21no—
171M16×C47—21no—
172M16×C47—24no—
173M16×C47—21no—
174M16×C47—21no—
175M16×C47C2×Q812yes—
176C2×C8×C47—21no—
177C2×C8×C47—21no—
178C2×C8×C47—21no—
179C2×C8×C47—24no—
180C2×C8×C47—21no—
181C2×C8×C47C22×C411yescharacteristic
182C2×C8×C47C22×C411yescharacteristic
183C2×C8×C47C2×D411yescharacteristic
184C2×C8×C47C2×D411yescharacteristic
185C2×C8×C47C2×Q812yes—
186C2×C8×C47#4112yes—
187C4⋊C4×C47C22×C411yescharacteristic
188C4⋊C4×C47C22×C411yescharacteristic
189C22⋊C4×C47—21no—
190C42×C47—21no—
191C42×C47—21no—
192C42×C47—21no—
193C22⋊Dic47—21no—
194C22⋊Dic47—21no—
195C22⋊Dic47—21no—
196C22⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47—21no—
200C4⋊Dic47—21no—
201C4⋊Dic47C22×C411yescharacteristic
202C4⋊Dic47C22×C411yescharacteristic
203C4⋊Dic47C22×C411yescharacteristic
204C4⋊Dic47C22×C411yescharacteristic
205Dic47⋊C4—21no—
206Dic47⋊C4—21no—
207Dic47⋊C4—21no—
208Dic47⋊C4—21no—
209Dic47⋊C4—42no—
210C4×Dic47—21no—
211C4×Dic47—21no—
212C4×Dic47—21no—
213C4×Dic47—21no—
214C4×Dic47—21no—
215C4×Dic47—21no—
216C4×Dic47C22×C411yescharacteristic
217C4×Dic47C22×C411yescharacteristic
218C4×Dic47C2×D411yescharacteristic
219C4×Dic47C2×D411yescharacteristic
220#10625329—24no—
221#10625329—21no—
222#10625329—21no—
223#10625329—21no—
224#10625329#4112yes—
225C2×C47⋊C8—24no—
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⋊C8C2×D411yescharacteristic
235C2×C47⋊C8C2×D411yescharacteristic
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—41no—
252C2×C4×C47C23×C411yescharacteristic
253C2×C4×C47C22×D411yescharacteristic
254C2×C4×C47C22×Q811yescharacteristic
255C2×C4×C47C2×#4111yescharacteristic
256C2×C4×C47C2×#4111yescharacteristic
257C2×C4×C47C2×#4111yescharacteristic
258C2×Dic47—22no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—21no—
263C2×Dic47—41no—
264C2×Dic47—42no—
265C2×Dic47—42no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47C4×D411yescharacteristic
270C2×Dic47C4×D411yescharacteristic
271C2×Dic47C4×D411yescharacteristic
272C2×Dic47C4×D411yescharacteristic
273Dic94—42no—
274Dic94—42no—
275C8×C47—21no—
276C8×C47—21no—
277C8×C47—22no—
278C8×C47—41no—
279C8×C47C22⋊Q812yes—
280C8×C47C22⋊Q812yes—
281C47⋊C8—21no—
282C47⋊C8—21no—
283C47⋊C8—21no—
284C47⋊C8—22no—
285C47⋊C8—21no—
286C47⋊C8—41no—
287#21628—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—22no—
297C4×C47—41no—
298C4×C47—41no—
299C4×C47—81no—
300C4×C47C2×C4×Q811yescharacteristic
301C4×C47C4×#4111yescharacteristic
302C4×C47C2×C22⋊Q811yescharacteristic
303C4×C47#52511yescharacteristic
304Dic47—21no—
305Dic47—21no—
306Dic47—21no—
307Dic47—41no—
308Dic47—41no—
309Dic47—41no—
310Dic47—41no—
311Dic47—81no—
312Dic47#44412yes—
313#3076—471no—
314#2960—471no—
315C4×#131—471no—
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#1773—471no—
325#1770—471no—
326#1765—471no—
327#1765D4711yescharacteristic
328C2×C47—21no—
329C2×C47—21no—
330C2×C47—42no—
331C2×C47C4×C22⋊Q811yescharacteristic
332C2×C47C2×#44411yescharacteristic
333C2×C47#295511yescharacteristic
334C2×#131—471no—
335C2×#131C2×D4711yescharacteristic
336#533—471no—
337C4×#41—471no—
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—
348C8×Q8—941no—
349C8×Q8—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×#104C2×D4711yescharacteristic
360#416—471no—
361#416C2×D4711yescharacteristic
362#405—471no—
363#405—471no—
364#405—471no—
365#405—941no—
366#405—941no—
367#405C2×D4711yescharacteristic
368C4×M16—471no—
369C4×M16—941no—
370C2×#97—471no—
371C2×#97C2×D4711yescharacteristic
372C2×C4×C8—471no—
373C2×C4×C8—941no—
374C4×C4⋊C4—471no—
375#341—471no—
376#341C2×D4711yescharacteristic
377#339—471no—
378#339—471no—
379#339—471no—
380#339—471no—
381#339—471no—
382#339C2×D4711yescharacteristic
383#336—471no—
384#336—471no—
385C47#2162811yescharacteristic, complemented, Sylow
386C2×#41C22×D4711yescharacteristic
387#131—22no—
388#131—942no—
389#131—942no—
390#131Dic9414yescomplemented
391C2×M16—21no—
392C2×M16—471no—
393C2×M16—471no—
394C2×M16—941no—
395C2×M16C22×D4711yescharacteristic
396C2×M16C22×D4711yescharacteristic
397C22×C8—21no—
398C22×C8—471no—
399C22×C8—471no—
400C22×C8—941no—
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—471no—
410C4×Q8—471no—
411C4×Q8—941no—
412C4×Q8—941no—
413C4×D4—471no—
414C4×D4—471no—
415C4×D4—941no—
416C4×D4—941no—
417#117—471no—
418#117—471no—
419#117—941no—
420#117C22×D4711yescharacteristic
421C2×C4⋊C4—941no—
422C2×C42—471no—
423C2×C42—471no—
424C2×C42—941no—
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—22no—
447#104—942no—
448#103—471no—
449#103—471no—
450#103C4×D4711yescharacteristic
451#103C4×D4711yescharacteristic
452#102—471no—
453#102—471no—
454#102C4×D4711yescharacteristic
455#102C4×D4711yescharacteristic
456#97—21no—
457#97—22no—
458#97—941no—
459#97—941no—
460#97—941no—
461#97—941no—
462#97—942no—
463C4×C8—21no—
464C4×C8—471no—
465C4×C8—471no—
466C4×C8—471no—
467C4×C8—471no—
468C4×C8—941no—
469C4×C8—941no—
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—471no—
485C22×C4—471no—
486C22×C4—941no—
487C22×C4—941no—
488C22×C4—941no—
489C22×C4C23×D4711yescharacteristic
490M16—21no—
491M16—24no—
492M16—21no—
493M16—21no—
494M16—472no—
495M16—944no—
496M16—941no—
497M16—941no—
498M16—941no—
499M16C2×Dic9412yes—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—24no—
504C2×C8—21no—
505C2×C8—471no—
506C2×C8—471no—
507C2×C8—471no—
508C2×C8—471no—
509C2×C8—944no—
510C2×C8—941no—
511C2×C8—941no—
512C2×C8—941no—
513C2×C8—941no—
514C2×C8—941no—
515C2×C8—941no—
516C2×C8C2×Dic9412yes—
517C2×C8C2×C4×D4711yescharacteristic
518C2×C8C2×C4×D4711yescharacteristic
519C2×C8D4×D4711yescharacteristic
520C2×C8D4×D4711yescharacteristic
521C2×C8#1062535212yes—
522C4⋊C4—471no—
523C4⋊C4—471no—
524C4⋊C4—471no—
525C4⋊C4—471no—
526C4⋊C4—941no—
527C4⋊C4—941no—
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—941no—
539C22⋊C4—941no—
540C22⋊C4—941no—
541C22⋊C4—941no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—471no—
546C42—471no—
547C42—471no—
548C42—471no—
549C42—941no—
550C42—941no—
551C42—941no—
552C42—941no—
553C42—941no—
554C42—941no—
555C23—21no—
556C23#41×D4711yescharacteristic
557Q8—21no—
558Q8—1882no—
559Q8—1882no—
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—41no—
573C2×C4—471no—
574C2×C4—471no—
575C2×C4—471no—
576C2×C4—471no—
577C2×C4—942no—
578C2×C4—941no—
579C2×C4—941no—
580C2×C4—941no—
581C2×C4—941no—
582C2×C4—1882no—
583C2×C4—1881no—
584C2×C4—1882no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4C22×C4×D4711yescharacteristic
589C2×C4C2×D4×D4711yescharacteristic
590C2×C4C22×Dic9411yescharacteristic
591C2×C4#41×D4711yescharacteristic
592C2×C4C2×#1062535211yescharacteristic
593C2×C4C2×#1062535011yescharacteristic
594C8—21no—
595C8—21no—
596C8—22no—
597C8—41no—
598C8—941no—
599C8—941no—
600C8—941no—
601C8—941no—
602C8—942no—
603C8—1881no—
604C8#1309400112yes—
605C8#1309400112yes—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22C4×D4×D4711yescharacteristic
612C22#1568965311yescharacteristic
613C22#1568967611yescharacteristic
614C4—22no—
615C4—41no—
616C4—41no—
617C4—81no—
618C4—472no—
619C4—941no—
620C4—941no—
621C4—941no—
622C4—3761no—
623C4—1881no—
624C4—1881no—
625C4—1881no—
626C4—1881no—
627C4C2×C4×Dic9411yescharacteristic
628C4C4×#1062535211yescharacteristic
629C4C2×#1309400111yescharacteristic
630C4#1568963211yescharacteristic
631C2—21no—
632C2—21no—
633C2—42no—
634C2C4×#1309400111yescharacteristic
635C2#444×D4711yescharacteristic
636C2#2025228811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1765C2
socleC22×C47C2×C4×D4
prime radicalC2×#41×C47C23
cosocleC22×C4×C47C24
hypercenter#1765D47
hypocenterC47#21628
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1765

Complements

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

NH ≅ G/Ntype
2. #20252320C2semidirect
4. #20244330C2semidirect
5. #20244162C2semidirect
6. #20243990C2semidirect
7. #20243986C2semidirect
11. #20243273C2semidirect
12. #20243272C2semidirect
13. #20243256C2semidirect
19. #15689432C4semidirect
20. #15689428C4semidirect
21. #15689425C4semidirect
22. #15689421C4semidirect
23. #15689416C4semidirect
24. #15689411C4semidirect
25. #15689406C4semidirect
26. #15689401C4semidirect
137. #131×C47Q8semidirect
385. C47#21628semidirect
390. #131Dic94semidirect

Maximal subgroups

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

subgroupindex
#202531722
#202523202
#131×Dic472
#202443302
#202441622
#202439902
#202439862
#202436652
#202436612
#202434062
#202432732
#202432722
#202432562
#202432302
C47×#17652
#2162847