Finite Group Databasebeta

← #26224287

Subgroups — #26224287

Order 12032.

890 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689468C211yesmaximal, characteristic
2#20246132C211yesmaximal, characteristic, complemented
3#20246110C211yesmaximal, characteristic
4#20244359C211yesmaximal, characteristic, complemented
5#20244210C211yesmaximal, characteristic
6#20244086C211yesmaximal, characteristic, complemented
7#20244082C211yesmaximal, characteristic, complemented
8#20244073C211yesmaximal, characteristic, complemented
9#20244069C211yesmaximal, characteristic, complemented
10#20243517C211yesmaximal, characteristic, complemented
11#20243216C211yesmaximal, characteristic, complemented
12#20243213C211yesmaximal, characteristic, complemented
13C4×#13093946C211yesmaximal, characteristic, complemented
14#20243074C211yesmaximal, characteristic, complemented
15C4×#102×C47C211yesmaximal, characteristic, complemented
16C2×#13094023C2211yescharacteristic
17C2×#13094020C2211yescharacteristic
18C2×C4×D188C2211yescharacteristic
19#15689468C418yescomplemented
20C2×#13093946C2211yescharacteristic
21C2×#13093946C2211yescharacteristic
22C2×#13093945C2211yescharacteristic
23C2×#13093945C2211yescharacteristic
24C2×#13093944C2211yescharacteristic
25C2×#13093944C2211yescharacteristic
26C2×#13093935C2211yescharacteristic
27C2×#13093934C2211yescharacteristic
28C2×#13093934C2211yescharacteristic
29C2×C8×Dic47C2211yescharacteristic
30C2×C8×Dic47C2211yescharacteristic
31#15689029—21no—
32#15689027—21no—
33#15689026C2211yescharacteristic
34#15689024C2211yescharacteristic
35#15689022C2211yescharacteristic
36#15689019C2211yescharacteristic
37#15689010—21no—
38#15689007—21no—
39#15689002—21no—
40#15689002—21no—
41#15688999—21no—
42#15688999—21no—
43#15688988C2211yescharacteristic
44#15688988C2211yescharacteristic
45C4×C4⋊Dic47C2211yescharacteristic
46#15688969C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C2×#102×C47C2211yescharacteristic
55C2×#102×C47C2211yescharacteristic
56C2×C4×C8×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×Dic94—21no—
61C22×D188C2311yescharacteristic
62C22×C22⋊D47—21no—
63C22×C22⋊D47—21no—
64C22×C4×D47—21no—
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67#13094023—22no—
68#13094023C2×C414yescomplemented
69#13094020—22no—
70#13094020C2×C414yes—
71C4×D188—22no—
72C4×D188—22no—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×C22⋊Dic47—21no—
80C2×C22⋊Dic47—21no—
81C2×(C2×D47)⋊C4—21no—
82C2×(C2×D47)⋊C4—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47C2311yescharacteristic
86C2×C4×Dic47C2311yescharacteristic
87#13093946—22no—
88#13093946C2×C414yescomplemented
89#13093945—22no—
90#13093945C2×C414yescomplemented
91#13093944—22no—
92#13093944C2×C414yescomplemented
93#13093935—22no—
94#13093934—22no—
95#13093934—22no—
96C8×Dic47—22no—
97C8×Dic47C2×C414yescomplemented
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95D411yescharacteristic
106C47⋊#95D411yescharacteristic
107C47⋊#95D411yescharacteristic
108C47⋊#95D411yescharacteristic
109C22×D4×C47C2311yescharacteristic
110C23×C4×C47—21no—
111C22×C8×C47C2311yescharacteristic
112C22×C8×C47C2311yescharacteristic
113C4×D4×C47—22no—
114C4×D4×C47—22no—
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C22⋊C4×C47—21no—
119C2×C42×C47—21no—
120C2×C42×C47C2311yescharacteristic
121#102×C47—22no—
122#102×C47C2×C414yescomplemented
123C4×C8×C47—22no—
124#95×C47—21no—
125C24×C47—21no—
126C23×D47—21no—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47C22×C412yes—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—42no—
137C22×C4×C47—81no—
138C22×C4×C47—81no—
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2411yescharacteristic
142C2×C22⋊D47—24no—
143C2×C22⋊D47—24no—
144C2×C22⋊D47—81no—
145C2×C22⋊D47—81no—
146C2×C22⋊D47—81no—
147C2×C22⋊D47—81no—
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47C2×D411yescharacteristic
152C22×Dic47C2×D411yescharacteristic
153C22×Dic47C2×D411yescharacteristic
154C22×Dic47C2×D411yescharacteristic
155C2×D188—21no—
156C2×D188—21no—
157C2×D188—21no—
158C2×D188—21no—
159C2×D188—21no—
160C2×D188C22×C412yes—
161C2×C4×D47—42no—
162C2×C4×D47—81no—
163C2×C4×D47—81no—
164C2×Dic94—24no—
165C2×Dic94—81no—
166C2×Dic94—81no—
167C2×C8×C47—21no—
168C2×C8×C47—21no—
169C2×C8×C47—21no—
170C2×C8×C47—21no—
171C2×C8×C47C22×C412yes—
172C2×C8×C47#4112yes—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47C22×C412yes—
179C22⋊C4×C47—42no—
180C42×C47—81no—
181C42×C47#4112yes—
182C42×C47#4112yes—
183C22⋊Dic47—81no—
184C22⋊Dic47—81no—
185(C2×D47)⋊C4—42no—
186(C2×D47)⋊C4—81no—
187(C2×D47)⋊C4—81no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—21no—
193C4⋊Dic47—81no—
194C4⋊Dic47C22×C412yes—
195Dic47⋊C4—24no—
196Dic47⋊C4—81no—
197C4×Dic47—21no—
198C4×Dic47—21no—
199C4×Dic47—81no—
200C4×Dic47C22×C412yes—
201C4×Dic47#4112yes—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8C22×C412yes—
207C2×C47⋊C8#4112yes—
208C23×C47—22no—
209C23×C47—41no—
210C23×C47—41no—
211C23×C47—41no—
212C23×C47—41no—
213C23×C47—41no—
214C23×C47—41no—
215C23×C47C22×D411yescharacteristic
216C22×D47—22no—
217C22×D47—41no—
218C22×D47—41no—
219C22×D47—41no—
220C22×D47—41no—
221C22×D47—41no—
222C22×D47—41no—
223D4×C47—22no—
224D4×C47—22no—
225D4×C47—41no—
226D4×C47—41no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—22no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47—84no—
241C2×C4×C47#12412yes—
242C2×C4×C47#12412yes—
243C2×C4×C47C23×C411yescharacteristic
244C2×C4×C47C2×#4111yescharacteristic
245C2×C4×C47C2×#4111yescharacteristic
246C2×C4×C47C2×#4111yescharacteristic
247C2×C4×C47C2×#4111yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C22⋊D47—84no—
250C22⋊D47—84no—
251C2×Dic47—21no—
252C2×Dic47—21no—
253C2×Dic47—21no—
254C2×Dic47—21no—
255C2×Dic47—22no—
256C2×Dic47—21no—
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47C4×D412yes—
271C2×Dic47C4×D412yes—
272D188—22no—
273D188—22no—
274D188—41no—
275D188—41no—
276C4×D47—84no—
277Dic94—84no—
278C8×C47—22no—
279C8×C47—22no—
280C47⋊C8—22no—
281C47⋊C8—22no—
282#21644—471nomaximal, Sylow
283C22×C47—42no—
284C22×C47—42no—
285C22×C47—42no—
286C22×C47—81no—
287C22×C47—81no—
288C22×C47—81no—
289C22×C47—81no—
290C22×C47C2×C4×D411yescharacteristic
291C22×C47C2×#12411yescharacteristic
292C22×C47#53211yescharacteristic
293C22×C47#57611yescharacteristic
294C22×C47#57611yescharacteristic
295C22×C47#57711yescharacteristic
296C22×C47#57711yescharacteristic
297C2×D47—42no—
298C2×D47—42no—
299C2×D47—42no—
300C2×D47—81no—
301C2×D47—81no—
302C2×D47—81no—
303C2×D47—81no—
304C4×C47—22no—
305C4×C47—22no—
306C4×C47—42no—
307C4×C47—81no—
308C4×C47—81no—
309C4×C47C4×#4111yescharacteristic
310C4×C47C4×#4111yescharacteristic
311C4×C47#52911yescharacteristic
312C4×C47#52911yescharacteristic
313Dic47—22no—
314Dic47—22no—
315Dic47—41no—
316Dic47—41no—
317Dic47—42no—
318Dic47—81no—
319Dic47—81no—
320C2×#486—471no—
321#2331—471no—
322#2328—471no—
323#1993—471no—
324#1961—471no—
325#1936—471no—
326#1936—471no—
327#1934—471no—
328#1934—471no—
329#1835—471no—
330#1762—471no—
331#1762—471no—
332C4×#102—471no—
333C4×#102D4711yescharacteristic, complemented
334#1726—471no—
335C2×C47—82no—
336C2×C47C4×#12411yescharacteristic
337C2×C47#294811yescharacteristic
338C2×C47#294911yescharacteristic
339C2×C47#313611yescharacteristic
340C2×C47#313611yescharacteristic
341C2×C47#317911yescharacteristic
342C2×C47#317911yescharacteristic
343D47—82no—
344C2×C4⋊Q8—471no—
345C2×#127—471no—
346C2×C4×D4—471no—
347C2×C4×D4C2×D4711yescharacteristic
348#486—478no—
349C2×#102—471no—
350C2×#102—471no—
351C2×#102—471no—
352C2×#102—471no—
353C2×#102—471no—
354C2×#102—471no—
355C2×#102C2×D4711yescharacteristic
356C2×#102C2×D4711yescharacteristic
357C2×#97—471no—
358C2×#97—471no—
359C2×#97—471no—
360C2×C4×C8—471no—
361C2×C4×C8—471no—
362C2×C4×C8C2×D4711yescharacteristic
363#391—941no—
364#391—941no—
365#390—471no—
366#390—471no—
367#389—471no—
368#389—471no—
369#387—941no—
370#387—941no—
371#386—941no—
372#386—941no—
373#386—941no—
374#386—941no—
375#382—471no—
376#382—471no—
377C4×C4⋊C4—471no—
378C4×C4⋊C4C2×D4711yescharacteristic
379#376—471no—
380#340—471no—
381#340—471no—
382#340—471no—
383#340—471no—
384#340—471no—
385#340—471no—
386#340C2×D4711yescharacteristic
387#340C2×D4711yescharacteristic
388C47#2164411yescharacteristic, complemented, Sylow
389C22×Q8—941no—
390C22×D4—471no—
391C22×D4—941no—
392C22×D4—941no—
393C22×D4C22×D4711yescharacteristic
394C23×C4—21no—
395C23×C4—941no—
396C22×C8—471no—
397C22×C8—471no—
398C22×C8C22×D4711yescharacteristic
399C22×C8C22×D4711yescharacteristic
400C4⋊Q8—474no—
401C4⋊Q8—942no—
402#127—474no—
403#127—942no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—942no—
407C4×D4—942no—
408C2×C4⋊C4—471no—
409C2×C4⋊C4—471no—
410C2×C4⋊C4—471no—
411C2×C4⋊C4—941no—
412C2×C4⋊C4—941no—
413C2×C4⋊C4—941no—
414C2×C4⋊C4C22×D4711yescharacteristic
415C2×C4⋊C4C22×D4711yescharacteristic
416C2×C4⋊C4C22×D4711yescharacteristic
417C2×C22⋊C4—21no—
418C2×C22⋊C4—941no—
419C2×C22⋊C4—941no—
420C2×C22⋊C4—941no—
421C2×C22⋊C4—941no—
422C2×C22⋊C4—941no—
423C2×C42—21no—
424C2×C42—471no—
425C2×C42—471no—
426C2×C42—941no—
427C2×C42C22×D4711yescharacteristic
428#102—22no—
429#102—474no—
430#102—474no—
431#102—474no—
432#102—942no—
433#102—942no—
434#102—942no—
435#102C4×D4714yescomplemented
436#97—942no—
437#97—942no—
438#97—942no—
439C4×C8—22no—
440C4×C8—474no—
441C4×C8—942no—
442#95—21no—
443#95—471no—
444#95—471no—
445#95—471no—
446#95—471no—
447#95—941no—
448#95—941no—
449#95—941no—
450#95—941no—
451#95—941no—
452#95—941no—
453#95—941no—
454C24—21no—
455C24—941no—
456C2×Q8—944no—
457C2×Q8—3761no—
458C2×Q8—3761no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—472no—
465C2×D4—944no—
466C2×D4—944no—
467C2×D4—941no—
468C2×D4—941no—
469C2×D4—941no—
470C2×D4—941no—
471C2×D4—941no—
472C2×D4—3761no—
473C2×D4—3761no—
474C2×D4—3761no—
475C2×D4—3761no—
476C2×D4C2×C4×D4712yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—81no—
482C22×C4—81no—
483C22×C4—471no—
484C22×C4—471no—
485C22×C4—471no—
486C22×C4—471no—
487C22×C4—941no—
488C22×C4—941no—
489C22×C4—941no—
490C22×C4—3761no—
491C22×C4—3761no—
492C22×C4—1882no—
493C22×C4D4×D4711yescharacteristic
494C22×C4D4×D4711yescharacteristic
495C22×C4C23×D4711yescharacteristic
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—472no—
501C2×C8—472no—
502C2×C8—941no—
503C2×C8—941no—
504C2×C8—941no—
505C2×C8—941no—
506C2×C8C2×C4×D4712yes—
507C2×C8#1062535012yes—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—472no—
514C4⋊C4—944no—
515C4⋊C4—941no—
516C4⋊C4—941no—
517C4⋊C4—941no—
518C4⋊C4—941no—
519C4⋊C4—941no—
520C4⋊C4—3761no—
521C4⋊C4—3761no—
522C4⋊C4C2×C4×D4712yes—
523C22⋊C4—42no—
524C22⋊C4—3761no—
525C22⋊C4—3761no—
526C22⋊C4—3761no—
527C22⋊C4—3761no—
528C22⋊C4—1882no—
529C42—81no—
530C42—472no—
531C42—472no—
532C42—941no—
533C42—941no—
534C42—3761no—
535C42#1062535012yes—
536C42#1062535212yes—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—942no—
545C23—1881no—
546C23—1881no—
547C23—1881no—
548C23—1881no—
549C23—1881no—
550C23—1881no—
551C23C2×D4×D4711yescharacteristic
552Q8—3764no—
553D4—22no—
554D4—22no—
555D4—41no—
556D4—41no—
557D4—942no—
558D4—942no—
559D4—3764no—
560D4—3764no—
561D4—1881no—
562D4—1881no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—84no—
577C2×C4—472no—
578C2×C4—472no—
579C2×C4—941no—
580C2×C4—941no—
581C2×C4—941no—
582C2×C4—941no—
583C2×C4—942no—
584C2×C4—941no—
585C2×C4—941no—
586C2×C4—941no—
587C2×C4—941no—
588C2×C4—941no—
589C2×C4—941no—
590C2×C4—3764no—
591C2×C4—1881no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4—1881no—
595C2×C4—1881no—
596C2×C4—1881no—
597C2×C4—1881no—
598C2×C4—1881no—
599C2×C4#1309401012yes—
600C2×C4#1309401012yes—
601C2×C4C22×C4×D4711yescharacteristic
602C2×C4#41×D4711yescharacteristic
603C2×C4#41×D4711yescharacteristic
604C2×C4C2×#1062535211yescharacteristic
605C2×C4C2×#1062535011yescharacteristic
606C2×C4C2×#1062535011yescharacteristic
607C8—22no—
608C8—22no—
609C8—942no—
610C8—942no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—3761no—
619C22—3761no—
620C22—3761no—
621C22—3761no—
622C22—1882no—
623C22—1882no—
624C22—1882no—
625C22C4×D4×D4711yescharacteristic
626C22C2×#1309401011yescharacteristic
627C22#1568967011yescharacteristic
628C22#1568992411yescharacteristic
629C22#1568992511yescharacteristic
630C22#1568992911yescharacteristic
631C22#1568993011yescharacteristic
632C4—22no—
633C4—22no—
634C4—42no—
635C4—81no—
636C4—81no—
637C4—942no—
638C4—942no—
639C4—3761no—
640C4—3761no—
641C4—1882no—
642C4—1881no—
643C4—1881no—
644C4C4×#1062535211yescharacteristic
645C4C4×#1062535011yescharacteristic
646C4#1568965111yescharacteristic
647C4#1568965311yescharacteristic
648C2—82no—
649C2—3762no—
650C2C4×#1309401011yescharacteristic
651C2#2025223511yescharacteristic
652C2#2025224211yescharacteristic
653C2#2025374011yescharacteristic
654C2#2025374611yescharacteristic
655C2#2025406711yescharacteristic
656C2#2025407511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×#102×C47C2
socleC23×C47C22×D4
prime radicalC2×#13094020C22
cosocleC22×C4×C47C24
hypercenterC4×#102D47
hypocenterC47#21644
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. #20246132C2semidirect
4. #20244359C2semidirect
6. #20244086C2semidirect
7. #20244082C2semidirect
8. #20244073C2semidirect
9. #20244069C2semidirect
10. #20243517C2semidirect
11. #20243216C2semidirect
12. #20243213C2semidirect
13. C4×#13093946C2semidirect
14. #20243074C2semidirect
15. C4×#102×C47C2semidirect
19. #15689468C4semidirect
68. #13094023C2×C4semidirect
88. #13093946C2×C4semidirect
90. #13093945C2×C4semidirect
92. #13093944C2×C4semidirect
97. C8×Dic47C2×C4semidirect
122. #102×C47C2×C4semidirect
333. C4×#102D47semidirect
388. C47#21644semidirect
435. #102C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156894682
#202461322
#202461102
#202443592
#202442102
#202440862
#202440822
#202440732
#202440692
#202435172
#202432162
#202432132
C4×#130939462
#202430742
C4×#102×C472
#2164447