Finite Group Databasebeta

← #26221747

Subgroups — #26221747

Order 12032.

688 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20245883C211yesmaximal, characteristic
2#20244001C211yesmaximal, characteristic
3#20244001C211yesmaximal, characteristic
4#20243994C211yesmaximal, characteristic
5#20243994C211yesmaximal, characteristic
6#20243952C211yesmaximal, characteristic, complemented
7#20243952C211yesmaximal, characteristic, complemented
8#20243945C211yesmaximal, characteristic, complemented
9#20243945C211yesmaximal, characteristic, complemented
10#20243242C211yesmaximal, characteristic, complemented
11C4×Dic47⋊C8C211yesmaximal, characteristic, complemented
12#20243157C211yesmaximal, characteristic
13#20243155C211yesmaximal, characteristic
14#20243140C211yesmaximal, characteristic, complemented
15C47×#1726C211yesmaximal, characteristic, complemented
16C2×#13093981C2211yescharacteristic
17C2×Dic47⋊C8C2211yescharacteristic
18C2×Dic47⋊C8C2211yescharacteristic
19C2×Dic47⋊C8C2211yescharacteristic
20C2×Dic47⋊C8C2211yescharacteristic
21C2×(C2×D47)⋊C8C2211yescharacteristic
22C2×(C2×D47)⋊C8C2211yescharacteristic
23C2×(C2×D47)⋊C8C2211yescharacteristic
24C2×(C2×D47)⋊C8C2211yescharacteristic
25C2×#13093935C2211yescharacteristic
26C2×C4×C47⋊C8C2211yescharacteristic
27#15689012C411yescharacteristic
28#15689012C411yescharacteristic
29#15689012C411yescharacteristic
30#15689012C411yescharacteristic
31#15689005C411yescharacteristic
32#15689005C411yescharacteristic
33#15689005C411yescharacteristic
34#15689005C411yescharacteristic
35C4×Dic47⋊C4C2211yescharacteristic
36C4×Dic47⋊C4C2211yescharacteristic
37C4×(C2×D47)⋊C4C2211yescharacteristic
38C4×(C2×D47)⋊C4C2211yescharacteristic
39C42×Dic47C2211yescharacteristic
40#15688878C2211yescharacteristic
41#15688878C2211yescharacteristic
42#15688878C2211yescharacteristic
43#15688878C2211yescharacteristic
44#15688877C2211yescharacteristic
45#15688877C2211yescharacteristic
46#15688877C2211yescharacteristic
47#15688877C2211yescharacteristic
48#15688877C2211yescharacteristic
49#15688877C2211yescharacteristic
50#15688877C2211yescharacteristic
51#15688877C2211yescharacteristic
52C2×#97×C47C2211yescharacteristic
53C2×#97×C47C2211yescharacteristic
54C2×#97×C47C2211yescharacteristic
55C2×C4×C8×C47C2211yescharacteristic
56C2×C4×C8×C47C2211yescharacteristic
57C2×C4×C8×C47C2211yescharacteristic
58C47×#376C2211yescharacteristic
59C22×C4×D47C2311yescharacteristic
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64#13093981—22no—
65#13093981—22no—
66C2×Dic47⋊C4C2×C411yescharacteristic
67C2×Dic47⋊C4C2×C411yescharacteristic
68C2×Dic47⋊C4C2×C411yescharacteristic
69C2×Dic47⋊C4C2×C411yescharacteristic
70C2×Dic47⋊C4C2×C411yescharacteristic
71C2×Dic47⋊C4C2×C411yescharacteristic
72C2×(C2×D47)⋊C4C2×C411yescharacteristic
73C2×(C2×D47)⋊C4C2×C411yescharacteristic
74C2×(C2×D47)⋊C4C2×C411yescharacteristic
75C2×(C2×D47)⋊C4C2×C411yescharacteristic
76C2×(C2×D47)⋊C4C2×C411yescharacteristic
77C2×(C2×D47)⋊C4C2×C411yescharacteristic
78C2×C4×Dic47C2×C411yescharacteristic
79C2×C4×Dic47C2×C411yescharacteristic
80C2×C4×Dic47C2×C411yescharacteristic
81C2×C4×Dic47C2×C411yescharacteristic
82C2×C4×Dic47C2311yescharacteristic
83C2×C4×Dic47C2311yescharacteristic
84C2×C4×Dic47C2311yescharacteristic
85Dic47⋊C8—22no—
86Dic47⋊C8—22no—
87Dic47⋊C8—22no—
88Dic47⋊C8—22no—
89(C2×D47)⋊C8—22no—
90(C2×D47)⋊C8—22no—
91(C2×D47)⋊C8—22no—
92(C2×D47)⋊C8—22no—
93#13093935—22no—
94C4×C47⋊C8—22no—
95C47⋊#95C2×C411yescharacteristic
96C47⋊#95C2×C411yescharacteristic
97C47⋊#95C2×C411yescharacteristic
98C47⋊#95C2×C411yescharacteristic
99C47⋊#95C2×C411yescharacteristic
100C47⋊#95C2×C411yescharacteristic
101C47⋊#95C2×C411yescharacteristic
102C47⋊#95C2×C411yescharacteristic
103C22×C8×C47C2311yescharacteristic
104C22×C8×C47C2311yescharacteristic
105C22×C8×C47C2311yescharacteristic
106C22×C8×C47C2311yescharacteristic
107C2×C42×C47C2311yescharacteristic
108C2×C42×C47C2311yescharacteristic
109C2×C42×C47C2311yescharacteristic
110#97×C47—22no—
111#97×C47—22no—
112#97×C47—22no—
113C4×C8×C47—22no—
114C4×C8×C47—22no—
115C4×C8×C47—22no—
116#95×C47C2×C411yescharacteristic
117#95×C47C2×C411yescharacteristic
118#95×C47C2×C411yescharacteristic
119#95×C47C2×C411yescharacteristic
120C23×D47C22×C411yescharacteristic
121C22×C4×C47C22×C411yescharacteristic
122C22×C4×C47C22×C411yescharacteristic
123C22×C4×C47C22×C411yescharacteristic
124C22×C4×C47C22×C411yescharacteristic
125C22×C4×C47C22×C411yescharacteristic
126C22×C4×C47C22×C411yescharacteristic
127C22×C4×C47C2411yescharacteristic
128C22×Dic47C22×C411yescharacteristic
129C22×Dic47C22×C411yescharacteristic
130C22×Dic47C22×C411yescharacteristic
131C22×Dic47C22×C411yescharacteristic
132C22×Dic47C22×C411yescharacteristic
133C22×Dic47C22×C411yescharacteristic
134C22×Dic47C22×C411yescharacteristic
135C2×C4×D47—21no—
136C2×C4×D47—21no—
137C2×C4×D47—21no—
138C2×C4×D47—21no—
139C2×C4×D47—21no—
140C2×C4×D47—21no—
141C2×C8×C47—21no—
142C2×C8×C47—21no—
143C2×C8×C47—21no—
144C2×C8×C47—21no—
145C2×C8×C47—21no—
146C2×C8×C47—21no—
147C2×C8×C47—21no—
148C2×C8×C47—21no—
149C2×C8×C47#4112yes—
150C2×C8×C47#4112yes—
151C2×C8×C47#4112yes—
152C2×C8×C47#4112yes—
153C42×C47—21no—
154C42×C47—21no—
155C42×C47—21no—
156C42×C47—21no—
157C42×C47#4112yes—
158C42×C47#4112yes—
159(C2×D47)⋊C4—22no—
160(C2×D47)⋊C4—22no—
161(C2×D47)⋊C4—22no—
162(C2×D47)⋊C4—22no—
163(C2×D47)⋊C4—41no—
164(C2×D47)⋊C4—41no—
165Dic47⋊C4—22no—
166Dic47⋊C4—22no—
167Dic47⋊C4—22no—
168Dic47⋊C4—22no—
169Dic47⋊C4—41no—
170Dic47⋊C4—41no—
171C4×Dic47—21no—
172C4×Dic47—21no—
173C4×Dic47—21no—
174C4×Dic47—21no—
175C4×Dic47—21no—
176C4×Dic47—21no—
177C4×Dic47—22no—
178C4×Dic47—22no—
179C4×Dic47—41no—
180C4×Dic47—41no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—21no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—21no—
192C2×C47⋊C8—21no—
193C23×C47C23×C411yescharacteristic
194C22×D47—21no—
195C22×D47—21no—
196C22×D47—21no—
197C22×D47—21no—
198C22×D47—21no—
199C22×D47—21no—
200C22×D47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47#11712yes—
218C2×C4×C47#11712yes—
219C2×C4×C47C2×#4111yescharacteristic
220C2×C4×C47C2×#4111yescharacteristic
221C2×C4×C47C2×#4111yescharacteristic
222C2×C4×C47C2×#4111yescharacteristic
223C2×C4×C47C2×#4111yescharacteristic
224C2×C4×C47C2×#4111yescharacteristic
225C2×Dic47—21no—
226C2×Dic47—21no—
227C2×Dic47—21no—
228C2×Dic47—21no—
229C2×Dic47—21no—
230C2×Dic47—21no—
231C2×Dic47—21no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—21no—
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47#13112yes—
245C2×Dic47#13112yes—
246C4×D47—41no—
247C4×D47—41no—
248C4×D47—41no—
249C4×D47—41no—
250C8×C47—22no—
251C8×C47—22no—
252C8×C47—22no—
253C8×C47—22no—
254C47⋊C8—41no—
255C47⋊C8—41no—
256C47⋊C8—41no—
257C47⋊C8—41no—
258#21436—471nomaximal, Sylow
259C22×C47C2×#11711yescharacteristic
260C22×C47C4×#4111yescharacteristic
261C22×C47C4×#4111yescharacteristic
262C22×C47C2×#13111yescharacteristic
263C22×C47C2×#13111yescharacteristic
264C22×C47#56811yescharacteristic
265C22×C47#56811yescharacteristic
266C2×D47—41no—
267C2×D47—41no—
268C2×D47—41no—
269C2×D47—41no—
270C2×D47—41no—
271C2×D47—41no—
272C2×D47—41no—
273C4×C47—22no—
274C4×C47—22no—
275C4×C47—41no—
276C4×C47—41no—
277C4×C47—41no—
278C4×C47—41no—
279C4×C47#52911yescharacteristic
280C4×C47#52911yescharacteristic
281C4×C47#52911yescharacteristic
282C4×C47#52911yescharacteristic
283Dic47—22no—
284Dic47—22no—
285Dic47—41no—
286Dic47—41no—
287Dic47—41no—
288Dic47—41no—
289Dic47—41no—
290#2298—471no—
291#1924—471no—
292#1924—471no—
293#1923—471no—
294#1923—471no—
295#1917—471no—
296#1917—471no—
297#1916—471no—
298#1916—471no—
299#1767—471no—
300C4×C4⋊C8—471no—
301#1750—471no—
302#1750—471no—
303#1746—471no—
304#1726D4711yescharacteristic, complemented
305C2×C47#229811yescharacteristic
306C2×C47#292511yescharacteristic
307C2×C47#292511yescharacteristic
308C2×C47#296611yescharacteristic
309C2×C47#296711yescharacteristic
310C2×C47#296811yescharacteristic
311C2×C47#296911yescharacteristic
312D47—81no—
313C2×#117—471no—
314C2×C4⋊C8—471no—
315C2×C4⋊C8—471no—
316C2×C4⋊C8—471no—
317C2×C4⋊C8—471no—
318C2×C22⋊C8—471no—
319C2×C22⋊C8—471no—
320C2×C22⋊C8—471no—
321C2×C22⋊C8—471no—
322C2×#97—471no—
323C2×#97C2×D4711yescharacteristic
324C2×#97C2×D4711yescharacteristic
325C2×#97C2×D4711yescharacteristic
326C2×C4×C8—471no—
327C2×C4×C8C2×D4711yescharacteristic
328C2×C4×C8C2×D4711yescharacteristic
329C2×C4×C8C2×D4711yescharacteristic
330#388—471no—
331#388—471no—
332#388—471no—
333#388—471no—
334#387—471no—
335#387—471no—
336#387—471no—
337#387—471no—
338C4×C4⋊C4—471no—
339C4×C4⋊C4—471no—
340C4×C22⋊C4—471no—
341C4×C22⋊C4—471no—
342#376C2×D4711yescharacteristic
343C43—471no—
344#336—471no—
345#336—471no—
346#336—471no—
347#336—471no—
348#336—471no—
349#336—471no—
350#336—471no—
351#336—471no—
352#336—471no—
353#336—471no—
354#336—471no—
355#336—471no—
356C47#2143611yescharacteristic, complemented, Sylow
357C23×C4—471no—
358C22×C8—471no—
359C22×C8—471no—
360C22×C8—471no—
361C22×C8—471no—
362C22×C8C22×D4711yescharacteristic
363C22×C8C22×D4711yescharacteristic
364C22×C8C22×D4711yescharacteristic
365C22×C8C22×D4711yescharacteristic
366#117—942no—
367#117—942no—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—471no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—471no—
374C2×C22⋊C4—471no—
375C2×C22⋊C4—471no—
376C2×C22⋊C4—471no—
377C2×C22⋊C4—471no—
378C2×C22⋊C4—471no—
379C2×C22⋊C4—471no—
380C2×C42—471no—
381C2×C42—471no—
382C2×C42—471no—
383C2×C42—471no—
384C2×C42—471no—
385C2×C42—471no—
386C2×C42—471no—
387C2×C42C22×D4711yescharacteristic
388C2×C42C22×D4711yescharacteristic
389C2×C42C22×D4711yescharacteristic
390C4⋊C8—942no—
391C4⋊C8—942no—
392C4⋊C8—942no—
393C4⋊C8—942no—
394C22⋊C8—942no—
395C22⋊C8—942no—
396C22⋊C8—942no—
397C22⋊C8—942no—
398#97—22no—
399#97—22no—
400#97—22no—
401#97—942no—
402C4×C8—22no—
403C4×C8—22no—
404C4×C8—22no—
405C4×C8—942no—
406#95—471no—
407#95—471no—
408#95—471no—
409#95—471no—
410#95—471no—
411#95—471no—
412#95—471no—
413#95—471no—
414#95C4×D4711yescharacteristic
415#95C4×D4711yescharacteristic
416#95C4×D4711yescharacteristic
417#95C4×D4711yescharacteristic
418C24—471no—
419C22×C4—471no—
420C22×C4—471no—
421C22×C4—471no—
422C22×C4—471no—
423C22×C4—471no—
424C22×C4—471no—
425C22×C4—471no—
426C22×C4—941no—
427C22×C4—941no—
428C22×C4—941no—
429C22×C4—941no—
430C22×C4—941no—
431C22×C4—941no—
432C22×C4C2×C4×D4711yescharacteristic
433C22×C4C2×C4×D4711yescharacteristic
434C22×C4C2×C4×D4711yescharacteristic
435C22×C4C2×C4×D4711yescharacteristic
436C22×C4C2×C4×D4711yescharacteristic
437C22×C4C2×C4×D4711yescharacteristic
438C22×C4C23×D4711yescharacteristic
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—941no—
448C2×C8—941no—
449C2×C8—941no—
450C2×C8—941no—
451C2×C8—941no—
452C2×C8—941no—
453C2×C8—941no—
454C2×C8—941no—
455C2×C8—941no—
456C2×C8—941no—
457C2×C8—941no—
458C2×C8—941no—
459C2×C8#1062535012yes—
460C2×C8#1062535012yes—
461C2×C8#1062535012yes—
462C2×C8#1062535012yes—
463C4⋊C4—942no—
464C4⋊C4—942no—
465C4⋊C4—942no—
466C4⋊C4—942no—
467C4⋊C4—1881no—
468C4⋊C4—1881no—
469C22⋊C4—942no—
470C22⋊C4—942no—
471C22⋊C4—942no—
472C22⋊C4—942no—
473C22⋊C4—1881no—
474C22⋊C4—1881no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—942no—
480C42—942no—
481C42—941no—
482C42—941no—
483C42—941no—
484C42—941no—
485C42—941no—
486C42—941no—
487C42—1881no—
488C42—1881no—
489C42#1062535012yes—
490C42#1062535012yes—
491C23—941no—
492C23—941no—
493C23—941no—
494C23—941no—
495C23—941no—
496C23—941no—
497C23—941no—
498C23C22×C4×D4711yescharacteristic
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—472no—
516C2×C4—472no—
517C2×C4—941no—
518C2×C4—941no—
519C2×C4—941no—
520C2×C4—941no—
521C2×C4—941no—
522C2×C4—941no—
523C2×C4—941no—
524C2×C4—941no—
525C2×C4—941no—
526C2×C4—941no—
527C2×C4—941no—
528C2×C4—941no—
529C2×C4—941no—
530C2×C4—941no—
531C2×C4—941no—
532C2×C4—941no—
533C2×C4—941no—
534C2×C4—941no—
535C2×C4—941no—
536C2×C4—1881no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4—1881no—
540C2×C4#1309398112yes—
541C2×C4#1309398112yes—
542C2×C4C2×#1062535011yescharacteristic
543C2×C4C2×#1062535011yescharacteristic
544C2×C4C2×#1062535011yescharacteristic
545C2×C4C2×#1062535011yescharacteristic
546C2×C4C2×#1062535011yescharacteristic
547C2×C4C2×#1062535011yescharacteristic
548C8—22no—
549C8—22no—
550C8—22no—
551C8—22no—
552C8—1881no—
553C8—1881no—
554C8—1881no—
555C8—1881no—
556C22—1881no—
557C22—1881no—
558C22—1881no—
559C22—1881no—
560C22—1881no—
561C22—1881no—
562C22—1881no—
563C22C2×#1309398111yescharacteristic
564C22C4×#1062535011yescharacteristic
565C22C4×#1062535011yescharacteristic
566C22#131×D4711yescharacteristic
567C22#131×D4711yescharacteristic
568C22#1568988811yescharacteristic
569C22#1568988811yescharacteristic
570C4—22no—
571C4—22no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—942no—
577C4—942no—
578C4—1881no—
579C4—1881no—
580C4—1881no—
581C4—1881no—
582C4—1881no—
583C4#1568965711yescharacteristic
584C4#1568965711yescharacteristic
585C4#1568965711yescharacteristic
586C4#1568965711yescharacteristic
587C2—3761no—
588C2#2024588511yescharacteristic
589C2#2025205211yescharacteristic
590C2#2025205211yescharacteristic
591C2#2025235911yescharacteristic
592C2#2025236911yescharacteristic
593C2#2025237811yescharacteristic
594C2#2025238811yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#1726C2
prime radicalC23×D47C22×C4
cosocleC22×C4×C47C24
hypercenter#1726D47
hypocenterC47#21436
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1726

Complements

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

NH ≅ G/Ntype
6. #20243952C2semidirect
7. #20243952C2semidirect
8. #20243945C2semidirect
9. #20243945C2semidirect
10. #20243242C2semidirect
11. C4×Dic47⋊C8C2semidirect
14. #20243140C2semidirect
15. C47×#1726C2semidirect
304. #1726D47semidirect
356. C47#21436semidirect

Maximal subgroups

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

subgroupindex
#202458832
#202440012
#202439942
#202439522
#202439452
#202432422
C4×Dic47⋊C82
#202431572
#202431552
#202431402
C47×#17262
#2143647