Finite Group Databasebeta

← #26187165

Subgroups — #26187165

Order 12032.

748 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253386C211yesmaximal, characteristic
2#20252133C211yesmaximal, characteristic, complemented
3D4×#10625324C211yesmaximal, characteristic, complemented
4#20242272C211yesmaximal, characteristic, complemented
5#20242271C211yesmaximal, characteristic, complemented
6#20242219C211yesmaximal, characteristic, complemented
7#20242214C211yesmaximal, characteristic, complemented
8#20242214C211yesmaximal, characteristic, complemented
9D8×C47⋊C8C211yesmaximal, characteristic, complemented
10C8×#10625334C211yesmaximal, characteristic, complemented
11#20242161C211yesmaximal, characteristic, complemented
12#20241837C211yesmaximal, characteristic, complemented
13D4⋊C8×D47C211yesmaximal, characteristic, complemented
14#20241667C211yesmaximal, characteristic, complemented
15C47×#1581C211yesmaximal, characteristic, complemented
16C2×#13094032—21no—
17C22×#10625324—21no—
18#15689670C2211yescharacteristic
19C4×D4×D47C2211yescharacteristic
20#15689438C411yescharacteristic
21#15689435C411yescharacteristic
22#15689402C411yescharacteristic
23#15689401C411yescharacteristic
24#15689317C411yescharacteristic
25#15689316C411yescharacteristic
26#15689258C411yescharacteristic
27#15689255C411yescharacteristic
28C4×D376C2211yescharacteristic
29C4×#10625334C2211yescharacteristic
30C4×#10625334C2211yescharacteristic
31#15689202C2211yescharacteristic
32#15689200C2211yescharacteristic
33#15689200C2211yescharacteristic, complemented
34#15689194—21no—
35#15689192—21no—
36#15689191—21no—
37#15689191—21no—
38C8×D188C2211yescharacteristic, complemented
39D4×C47⋊C8C2211yescharacteristic
40D4×C47⋊C8C2211yescharacteristic, complemented
41#15689174C2211yescharacteristic
42#15689145C2211yescharacteristic
43C4⋊C8×D47C2211yescharacteristic
44#15689088—21no—
45#15689085—21no—
46C4×#10625324C2211yescharacteristic
47C4×C8×D47C2211yescharacteristic
48#15688876C2211yescharacteristic, complemented
49#15688875C2211yescharacteristic, complemented
50#15688875C2211yescharacteristic, complemented
51#15688852C2211yescharacteristic
52#15688852C2211yescharacteristic, complemented
53D4⋊(C47⋊C8)C2211yescharacteristic
54D4⋊(C47⋊C8)C2211yescharacteristic, complemented
55#15688850C2211yescharacteristic
56#15688850C2211yescharacteristic, complemented
57#15688845C2211yescharacteristic, complemented
58C8×C47⋊C8C2211yescharacteristic, complemented
59C8×C47⋊C8C2211yescharacteristic, complemented
60C4×D8×C47C2211yescharacteristic
61C47×#436C2211yescharacteristic, complemented
62C8×D4×C47C2211yescharacteristic
63C47×#335C2211yescharacteristic, complemented
64C47×D4⋊C8C2211yescharacteristic
65C47×D4⋊C8C2211yescharacteristic, complemented
66C47×#322C2211yescharacteristic, complemented
67C2×#10625350—21no—
68C2×D4×D47C2×C411yescharacteristic
69C22×C4×D47—21no—
70C2×D376C2×C411yescharacteristic
71C2×#10625334C2×C411yescharacteristic
72C2×#10625334C2×C411yescharacteristic
73#13094032—44no—
74C2×#10625324—21no—
75C2×#10625324—21no—
76C2×#10625324—21no—
77C2×#10625324—21no—
78C2×#10625324—44no—
79C2×#10625324D411yescharacteristic
80C2×#10625324D411yescharacteristic
81C22×C47⋊C8—21no—
82C22×C47⋊C8—21no—
83C2×C8×D47—21no—
84C2×C8×D47—21no—
85C2×C8×D47—21no—
86#13094023C2×C411yescharacteristic
87#13094020C2×C411yescharacteristic
88#13094010—21no—
89#13093999C2×C411yescharacteristic
90#13093997—21no—
91#13093995C2×C411yescharacteristic
92C4×D188C2311yescharacteristic
93C4×D188C2311yescharacteristic
94D4×Dic47C2×C411yescharacteristic
95#13093986C2×C411yescharacteristic
96C4×C22⋊D47—21no—
97C4×C22⋊D47—21no—
98(C22⋊D47)⋊C4—21no—
99C4⋊C4×D47C2×C411yescharacteristic
100C22⋊C4×D47—21no—
101C42×D47C2311yescharacteristic
102#13093958C2×C411yescharacteristic
103#13093957C2×C411yescharacteristic
104#13093957C2×C411yescharacteristic
105C4⋊(C47⋊C8)C2311yescharacteristic
106C4⋊(C47⋊C8)C2311yescharacteristic
107Dic47⋊C8—21no—
108Dic47⋊C8—21no—
109Dic47⋊C8C2×C411yescharacteristic
110Dic47⋊C8C2×C411yescharacteristic
111#13093946C2×C411yescharacteristic
112#13093946C2×C411yescharacteristic
113#13093945C2×C411yescharacteristic
114#13093945C2×C411yescharacteristic
115#13093944C2×C411yescharacteristic
116#13093944C2×C411yescharacteristic
117C22⋊(C47⋊C8)—21no—
118C22⋊(C47⋊C8)—21no—
119(C2×D47)⋊C8—21no—
120(C2×D47)⋊C8—21no—
121(C2×D47)⋊C8—21no—
122(C2×D47)⋊C8—21no—
123(C2×D47)⋊C8C2×C411yescharacteristic
124(C2×D47)⋊C8C2×C411yescharacteristic
125#13093934D411yescharacteristic
126#13093934D411yescharacteristic
127C4×C47⋊C8C2311yescharacteristic
128C4×C47⋊C8C2311yescharacteristic
129C4×C47⋊C8C2311yescharacteristic
130C8×Dic47—21no—
131C2×D8×C47C2×C411yescharacteristic
132C2×M16×C47—21no—
133C22×C8×C47—21no—
134C4×D4×C47C2311yescharacteristic
135C4×D4×C47C2311yescharacteristic
136#107×C47C2×C411yescharacteristic
137C4⋊C8×C47C2311yescharacteristic
138C4⋊C8×C47C2311yescharacteristic
139#102×C47C2×C411yescharacteristic
140#102×C47C2×C411yescharacteristic
141C22⋊C8×C47—21no—
142C22⋊C8×C47—21no—
143C4×C8×C47C2311yescharacteristic
144C4×C8×C47C2311yescharacteristic
145C4×C8×C47C2311yescharacteristic
146C23×D47—21no—
147C2×D4×C47C22×C411yescharacteristic
148C2×D4×C47C22×C411yescharacteristic
149C22×C4×C47—21no—
150C22×C4×C47—21no—
151C2×C22⋊D47—21no—
152C2×C22⋊D47—21no—
153C2×C22⋊D47—21no—
154C2×C22⋊D47—21no—
155C22×Dic47—21no—
156D4×D47—22no—
157D4×D47—41no—
158#10625350—42no—
159#10625350—42no—
160C2×D188—21no—
161C2×D188C22×C411yescharacteristic
162C2×D188C22×C411yescharacteristic
163C2×C4×D47—21no—
164C2×C4×D47—21no—
165C2×C4×D47—21no—
166C2×C4×D47—21no—
167C2×C4×D47—42no—
168C2×C4×D47—42no—
169C2×C4×D47C22×C411yescharacteristic
170C2×C4×D47C2×D411yescharacteristic
171C2×C4×D47#4111yescharacteristic
172C2×Dic94—21no—
173D8×C47—22no—
174M16×C47—42no—
175C2×C8×C47—21no—
176C2×C8×C47—21no—
177C2×C8×C47—21no—
178C2×C8×C47—42no—
179C2×C8×C47C22×C411yescharacteristic
180C2×C8×C47C22×C411yescharacteristic
181C2×C8×C47C2×D411yescharacteristic
182C2×C8×C47C2×D411yescharacteristic
183C4⋊C4×C47C22×C411yescharacteristic
184C4⋊C4×C47C22×C411yescharacteristic
185C22⋊C4×C47—21no—
186C22⋊C4×C47—21no—
187C42×C47C2411yescharacteristic
188C22⋊Dic47—21no—
189C22⋊Dic47—21no—
190#10625334—22no—
191#10625334—41no—
192(C2×D47)⋊C4—21no—
193(C2×D47)⋊C4—21no—
194(C2×D47)⋊C4—21no—
195(C2×D47)⋊C4—21no—
196C4⋊Dic47C22×C411yescharacteristic
197C4⋊Dic47C22×C411yescharacteristic
198Dic47⋊C4—21no—
199Dic47⋊C4—21no—
200C4×Dic47C22×C411yescharacteristic
201C4×Dic47C2×D411yescharacteristic
202C4×Dic47#4111yescharacteristic
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—42no—
207C2×C47⋊C8—42no—
208C2×C47⋊C8C22×C411yescharacteristic
209C2×C47⋊C8C22×C411yescharacteristic
210C2×C47⋊C8C2×D411yescharacteristic
211C2×C47⋊C8C2×D411yescharacteristic
212D376—41no—
213#10625324—22no—
214#10625324—42no—
215#10625324—42no—
216#10625324—42no—
217#10625324—42no—
218#10625324D814yescomplemented
219C8×D47—42no—
220C8×D47—42no—
221C8×D47—42no—
222C23×C47—21no—
223C23×C47—21no—
224C22×D47—21no—
225C22×D47—21no—
226C22×D47—42no—
227C22×D47—42no—
228C22×D47—81no—
229C22×D47C4×D411yescharacteristic
230D4×C47—21no—
231D4×C47—21no—
232D4×C47—21no—
233D4×C47#13112yescomplemented
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47C23×C411yescharacteristic
241C2×C4×C47C22×D411yescharacteristic
242C2×C4×C47C2×#4111yescharacteristic
243C22⋊D47—42no—
244C22⋊D47—42no—
245C22⋊D47—42no—
246C22⋊D47—81no—
247C2×Dic47—21no—
248C2×Dic47—21no—
249C2×Dic47—42no—
250C2×Dic47C4×D411yescharacteristic
251C2×Dic47C4×D411yescharacteristic
252C2×Dic47C4×D411yescharacteristic
253D188—21no—
254D188—21no—
255D188—21no—
256D188—21no—
257D188—42no—
258C4×D47—21no—
259C4×D47—21no—
260C4×D47—41no—
261C4×D47—42no—
262C4×D47—41no—
263C4×D47—41no—
264C4×D47—41no—
265C4×D47—41no—
266C4×D47—42no—
267C4×D47C2×D812yes—
268C4×D47#13512yes—
269Dic94—42no—
270C8×C47—21no—
271C8×C47—21no—
272C8×C47—21no—
273C8×C47—41no—
274C8×C47—41no—
275C8×C47—41no—
276C8×C47C2×D812yes—
277C47⋊C8—22no—
278C47⋊C8—21no—
279C47⋊C8—21no—
280C47⋊C8—21no—
281C47⋊C8—41no—
282C47⋊C8—41no—
283C47⋊C8C2×D812yes—
284#17994—471nomaximal, Sylow
285C22×C47—22no—
286C22×C47—41no—
287C22×C47—41no—
288C22×C47—41no—
289C22×C47—41no—
290C22×C47—41no—
291C22×C47C2×C4×D411yescharacteristic
292C2×D47—21no—
293C2×D47—21no—
294C2×D47—41no—
295C2×D47—41no—
296C2×D47—41no—
297C2×D47—41no—
298C2×D47—41no—
299C2×D47—41no—
300C2×D47—84no—
301C2×D47C4×D812yes—
302C4×C47—21no—
303C4×C47—41no—
304C4×C47—41no—
305C4×C47C2×#13111yescharacteristic
306C4×C47#56811yescharacteristic
307C4×C47C22×D811yescharacteristic
308C4×C47C2×#13511yescharacteristic
309Dic47—21no—
310Dic47—21no—
311Dic47—41no—
312Dic47—41no—
313Dic47C4×D812yes—
314#3100—471no—
315#2935—471no—
316D4×M16—471no—
317#1589—471no—
318#1589—471no—
319#1581—471no—
320#1581—471no—
321#1581—471no—
322#1581D4711yescharacteristic, complemented
323C8×D8—471no—
324C8×D8—471no—
325#1571—471no—
326#1492—471no—
327C2×D4⋊C8—471no—
328C8×M16—471no—
329C2×C47—42no—
330C2×C47—81no—
331C2×C47#293311yescharacteristic
332C2×C47C2×C4×D811yescharacteristic
333C2×C47#295611yescharacteristic
334D47—22no—
335D47—81no—
336D47—81no—
337C2×#131—941no—
338C22×M16—941no—
339#532—471no—
340C2×C4×D4—471no—
341#480—471no—
342#480—471no—
343#474—471no—
344#474—471no—
345#459—471no—
346#459—471no—
347#447—471no—
348#447—471no—
349C4×D8—471no—
350C4×D8—471no—
351C4×D8—471no—
352C4×D8C2×D4711yescharacteristic
353#436—471no—
354#436—471no—
355#436—471no—
356#436C2×D4711yescharacteristic, complemented
357#435—941no—
358#435—941no—
359#435—941no—
360#435—941no—
361C8×D4—471no—
362C8×D4—471no—
363C8×D4—471no—
364C8×D4C2×D4711yescharacteristic
365#431—471no—
366#423—471no—
367C2×C4⋊C8—471no—
368#408—941no—
369#407—941no—
370C4×M16—471no—
371C2×C4×C8—471no—
372#335—471no—
373#335—471no—
374#335—471no—
375#335C2×D4711yescharacteristic, complemented
376D4⋊C8—471no—
377D4⋊C8—471no—
378D4⋊C8—471no—
379D4⋊C8—471no—
380D4⋊C8—471no—
381D4⋊C8—471no—
382D4⋊C8C2×D4711yescharacteristic
383D4⋊C8C2×D4711yescharacteristic, complemented
384#322—471no—
385#322C2×D4711yescharacteristic, complemented
386C82—471no—
387C82—471no—
388C47#1799411yescharacteristic, complemented, Sylow
389C2×#41—941no—
390C22×D4—471no—
391C23×C4—941no—
392C2×D8—471no—
393C2×D8—471no—
394C2×D8—471no—
395C2×D8C4×D4711yescharacteristic
396#131—1884no—
397C2×M16—21no—
398C2×M16—471no—
399C2×M16—471no—
400C2×M16—941no—
401C2×M16—941no—
402C2×M16—941no—
403C2×M16—941no—
404C2×M16—1884no—
405C22×C8—21no—
406C22×C8—941no—
407C22×C8—941no—
408C22×C8—941no—
409C22×C8—941no—
410C22×C8—941no—
411C4⋊Q8—471no—
412#127—471no—
413#124—941no—
414#121—471no—
415#121—471no—
416#121—941no—
417C4×D4—471no—
418C4×D4—471no—
419C4×D4—471no—
420C4×D4—471no—
421C4×D4—941no—
422C4×D4—941no—
423C4×D4—941no—
424C4×D4C22×D4711yescharacteristic
425C4×D4C22×D4711yescharacteristic
426C2×C4⋊C4—471no—
427C2×C22⋊C4—941no—
428C2×C42—471no—
429#107—471no—
430#107—471no—
431#107—471no—
432#107C4×D4711yescharacteristic
433C4⋊C8—471no—
434C4⋊C8—471no—
435C4⋊C8—471no—
436C4⋊C8—471no—
437C4⋊C8—941no—
438C4⋊C8—941no—
439C4⋊C8C22×D4711yescharacteristic
440C4⋊C8C22×D4711yescharacteristic
441#102—471no—
442#102—471no—
443#102—471no—
444#102—471no—
445#102—471no—
446#102—471no—
447#102C4×D4711yescharacteristic
448#102C4×D4711yescharacteristic
449C22⋊C8—21no—
450C22⋊C8—21no—
451C22⋊C8—471no—
452C22⋊C8—471no—
453C22⋊C8—941no—
454C22⋊C8—941no—
455C22⋊C8—941no—
456C22⋊C8—941no—
457C22⋊C8—941no—
458C22⋊C8—941no—
459#97—471no—
460#97—471no—
461C4×C8—471no—
462C4×C8—471no—
463C4×C8—471no—
464C4×C8—941no—
465C4×C8C22×D4711yescharacteristic
466C4×C8C22×D4711yescharacteristic
467C4×C8C22×D4711yescharacteristic
468C24—941no—
469#41—1882no—
470#41—1882no—
471C2×Q8—941no—
472C2×D4—471no—
473C2×D4—471no—
474C2×D4—941no—
475C2×D4—941no—
476C2×D4—941no—
477C2×D4—941no—
478C2×D4—942no—
479C2×D4—941no—
480C2×D4—1881no—
481C2×D4C2×C4×D4711yescharacteristic
482C2×D4C2×C4×D4711yescharacteristic
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—471no—
486C22×C4—471no—
487C22×C4—471no—
488C22×C4—941no—
489C22×C4—941no—
490C22×C4—941no—
491C22×C4—941no—
492C22×C4—941no—
493C22×C4—1882no—
494C22×C4—1882no—
495D8—22no—
496D8—942no—
497D8—1881no—
498D8—1881no—
499M16—42no—
500M16—474no—
501M16—942no—
502M16—1882no—
503M16—1882no—
504M16—1882no—
505M16—1882no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—42no—
510C2×C8—471no—
511C2×C8—471no—
512C2×C8—471no—
513C2×C8—471no—
514C2×C8—941no—
515C2×C8—941no—
516C2×C8—941no—
517C2×C8—1882no—
518C2×C8—1882no—
519C2×C8—1882no—
520C2×C8—1882no—
521C2×C8—1882no—
522C2×C8C2×C4×D4711yescharacteristic
523C2×C8C2×C4×D4711yescharacteristic
524C2×C8D4×D4711yescharacteristic
525C2×C8D4×D4711yescharacteristic
526C4⋊C4—471no—
527C4⋊C4—471no—
528C4⋊C4—941no—
529C4⋊C4—941no—
530C4⋊C4C2×C4×D4711yescharacteristic
531C4⋊C4C2×C4×D4711yescharacteristic
532C22⋊C4—21no—
533C22⋊C4—21no—
534C22⋊C4—941no—
535C22⋊C4—941no—
536C22⋊C4—941no—
537C22⋊C4—941no—
538C22⋊C4—941no—
539C22⋊C4—941no—
540C42—471no—
541C42—471no—
542C42—471no—
543C42C23×D4711yescharacteristic
544C23—21no—
545C23—21no—
546C23—471no—
547C23—941no—
548C23—941no—
549C23—3761no—
550C23—1882no—
551C23—1882no—
552Q8—1882no—
553D4—21no—
554D4—21no—
555D4—21no—
556D4—941no—
557D4—941no—
558D4—941no—
559D4—941no—
560D4—3761no—
561D4—1882no—
562D4—1882no—
563D4—1882no—
564D4—1882no—
565D4#1309403212yescomplemented
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—471no—
573C2×C4—472no—
574C2×C4—472no—
575C2×C4—471no—
576C2×C4—471no—
577C2×C4—941no—
578C2×C4—941no—
579C2×C4—941no—
580C2×C4—941no—
581C2×C4—1881no—
582C2×C4—1882no—
583C2×C4—1882no—
584C2×C4—1882no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4—1881no—
589C2×C4C22×C4×D4711yescharacteristic
590C2×C4C2×D4×D4711yescharacteristic
591C2×C4#41×D4711yescharacteristic
592C8—21no—
593C8—21no—
594C8—21no—
595C8—41no—
596C8—41no—
597C8—41no—
598C8—472no—
599C8—942no—
600C8—941no—
601C8—941no—
602C8—941no—
603C8—1881no—
604C8—1881no—
605C8D8×D4712yes—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—472no—
613C22—941no—
614C22—941no—
615C22—3764no—
616C22—1881no—
617C22—1881no—
618C22—1881no—
619C22—1881no—
620C22—1881no—
621C22—1881no—
622C22C4×D4×D4711yescharacteristic
623C4—21no—
624C4—41no—
625C4—41no—
626C4—472no—
627C4—941no—
628C4—941no—
629C4—1881no—
630C4—1881no—
631C4C2×#1309403211yescharacteristic
632C4#1568988811yescharacteristic
633C4C2×D8×D4711yescharacteristic
634C4#135×D4711yescharacteristic
635C2—42no—
636C2—81no—
637C2—942no—
638C2—3761no—
639C2—3761no—
640C2#2025211411yescharacteristic
641C2C4×D8×D4711yescharacteristic
642C2#2025229611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1581C2
socleC22×C47C2×C4×D4
prime radical#20253386C2
cosocleC42×C47C24
hypercenter#1581D47
hypocenterC47#17994
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1581

Complements

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

NH ≅ G/Ntype
2. #20252133C2semidirect
3. D4×#10625324C2semidirect
4. #20242272C2semidirect
5. #20242271C2semidirect
6. #20242219C2semidirect
7. #20242214C2semidirect
8. #20242214C2semidirect
9. D8×C47⋊C8C2semidirect
10. C8×#10625334C2semidirect
11. #20242161C2semidirect
12. #20241837C2semidirect
13. D4⋊C8×D47C2semidirect
14. #20241667C2semidirect
15. C47×#1581C2semidirect
33. #15689200C22semidirect
38. C8×D188C22semidirect
40. D4×C47⋊C8C22semidirect
48. #15688876C22semidirect
49. #15688875C22semidirect
50. #15688875C22semidirect
52. #15688852C22semidirect
54. D4⋊(C47⋊C8)C22semidirect
56. #15688850C22semidirect
57. #15688845C22semidirect
58. C8×C47⋊C8C22semidirect
59. C8×C47⋊C8C22semidirect
61. C47×#436C22semidirect
63. C47×#335C22semidirect
65. C47×D4⋊C8C22semidirect
66. C47×#322C22semidirect
218. #10625324D8semidirect
233. D4×C47#131semidirect
322. #1581D47semidirect
356. #436C2×D47semidirect
375. #335C2×D47semidirect
383. D4⋊C8C2×D47semidirect
385. #322C2×D47semidirect
388. C47#17994semidirect
565. D4#13094032semidirect

Maximal subgroups

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

subgroupindex
#202533862
#202521332
D4×#106253242
#202422722
#202422712
#202422192
#202422142
D8×C47⋊C82
C8×#106253342
#202421612
#202418372
D4⋊C8×D472
#202416672
C47×#15812
#1799447