Finite Group Databasebeta

← #26284813

Subgroups — #26284813

Order 12032.

812 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20247915C211yesmaximal, characteristic, complemented
2#20244791C211yesmaximal, characteristic, complemented
3#20244762C211yesmaximal, characteristic, complemented
4#20244619C211yesmaximal, characteristic
5#20244619C211yesmaximal, characteristic
6#20244074C211yesmaximal, characteristic, complemented
7#20244012C211yesmaximal, characteristic, complemented
8#20243973C211yesmaximal, characteristic, complemented
9#20243973C211yesmaximal, characteristic, complemented
10#20243947C211yesmaximal, characteristic, complemented
11#20243649C211yesmaximal, characteristic, complemented
12#20243649C211yesmaximal, characteristic, complemented
13#20243388C211yesmaximal, characteristic, complemented
14C4×#13093945C211yesmaximal, characteristic, complemented
15C47×#2366C211yesmaximal, characteristic
16C2×#13093955C2211yescharacteristic
17C2×#13093955C2211yescharacteristic
18C2×C4⋊(C47⋊C8)C2211yescharacteristic
19C2×#13093945C2211yescharacteristic
20C2×#13093945C2211yescharacteristic
21C2×#13093945C2211yescharacteristic
22C2×#13093945C2211yescharacteristic
23C2×#13093945C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×C4×C47⋊C8C2211yescharacteristic
28#15689057—21no—
29#15689056—21no—
30#15689021C2211yescharacteristic
31#15689021C2211yescharacteristic
32#15689021C2211yescharacteristic
33#15689021C2211yescharacteristic
34#15689020—21no—
35#15689009—21no—
36#15688993C2211yescharacteristic
37C4×C4⋊Dic47C2211yescharacteristic
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688878C2211yescharacteristic
45#15688878C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C2×#127×C47C2211yescharacteristic
49C2×C4×D4×C47—21no—
50C2×C4×D4×C47C2211yescharacteristic
51C2×C4×D4×C47C2211yescharacteristic
52C47×#390—21no—
53C47×#390C2211yescharacteristic
54C47×#390C2211yescharacteristic
55C47×#386—21no—
56C47×#386—21no—
57C4×C4⋊C4×C47C2211yescharacteristic
58C4×C4⋊C4×C47C2211yescharacteristic
59C22×C47⋊C8C2311yescharacteristic
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47C2311yescharacteristic
66C2×C4⋊Dic47C2311yescharacteristic
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×Dic47⋊C4—21no—
70C2×Dic47⋊C4—21no—
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×C4×Dic47—21no—
74#13093955—22no—
75#13093955—22no—
76C4⋊(C47⋊C8)—41no—
77#13093945—22no—
78#13093945—22no—
79#13093945—22no—
80#13093945—22no—
81#13093945—22no—
82#13093945—22no—
83#13093945—41no—
84#13093945—41no—
85C4×C47⋊C8—41no—
86C47⋊#95—21no—
87C47⋊#95—21no—
88C47⋊#95—21no—
89C47⋊#95—21no—
90C47⋊#95—21no—
91C22×D4×C47—21no—
92C22×D4×C47—21no—
93C22×D4×C47C2311yescharacteristic
94C22×D4×C47C2311yescharacteristic
95C23×C4×C47—21no—
96C23×C4×C47—21no—
97#127×C47—22no—
98#127×C47—22no—
99C4×D4×C47—22no—
100C4×D4×C47—22no—
101C4×D4×C47—41no—
102C4×D4×C47—41no—
103C4×D4×C47—44no—
104C2×C4⋊C4×C47D411yescharacteristic
105C2×C4⋊C4×C47D411yescharacteristic
106C2×C4⋊C4×C47C2311yescharacteristic
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C22⋊C4×C47—21no—
109C2×C22⋊C4×C47—21no—
110C2×C22⋊C4×C47—21no—
111C2×C22⋊C4×C47—21no—
112C2×C22⋊C4×C47—21no—
113C2×C22⋊C4×C47—21no—
114C2×C42×C47—21no—
115C2×C42×C47C2311yescharacteristic
116C2×C42×C47C2311yescharacteristic
117C2×C42×C47C2311yescharacteristic
118#95×C47—21no—
119#95×C47D411yescharacteristic
120#95×C47D411yescharacteristic
121C24×C47—21no—
122C24×C47—21no—
123C2×D4×C47—24no—
124C2×D4×C47—24no—
125C2×D4×C47—21no—
126C2×D4×C47—21no—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47—21no—
133C2×D4×C47—21no—
134C2×D4×C47—21no—
135C2×D4×C47—42no—
136C2×D4×C47—42no—
137C2×D4×C47—81no—
138C2×D4×C47—81no—
139C2×D4×C47#4112yes—
140C2×D4×C47#4112yes—
141C22×C4×C47—21no—
142C22×C4×C47—21no—
143C22×C4×C47—42no—
144C22×C4×C47—42no—
145C22×C4×C47—42no—
146C22×C4×C47—42no—
147C22×C4×C47—81no—
148C22×C4×C47—81no—
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2411yescharacteristic
156C22×Dic47—21no—
157C22×Dic47—21no—
158C22×Dic47—21no—
159C22×Dic47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—41no—
164C4⋊C4×C47SD1614yes—
165C4⋊C4×C47#4112yes—
166C22⋊C4×C47—42no—
167C22⋊C4×C47—42no—
168C22⋊C4×C47—42no—
169C22⋊C4×C47—42no—
170C22⋊C4×C47—81no—
171C22⋊C4×C47—81no—
172C42×C47—21no—
173C42×C47—21no—
174C42×C47—21no—
175C42×C47—21no—
176C42×C47—21no—
177C42×C47—21no—
178C42×C47—42no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47—81no—
187C4⋊Dic47—81no—
188C4⋊Dic47#4112yes—
189Dic47⋊C4—42no—
190Dic47⋊C4—81no—
191Dic47⋊C4—81no—
192Dic47⋊C4—81no—
193C4×Dic47—81no—
194C2×C47⋊C8—21no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C23×C47—22no—
207C23×C47—22no—
208C23×C47—41no—
209C23×C47—41no—
210C23×C47—41no—
211C23×C47—41no—
212C23×C47—41no—
213C23×C47—41no—
214C23×C47—41no—
215C23×C47—41no—
216C23×C47—41no—
217C23×C47—41no—
218C23×C47—41no—
219C23×C47—41no—
220C23×C47C22×D411yescharacteristic
221D4×C47—22no—
222D4×C47—22no—
223D4×C47—22no—
224D4×C47—22no—
225D4×C47—48no—
226D4×C47—41no—
227D4×C47—41no—
228D4×C47—41no—
229D4×C47—41no—
230D4×C47—84no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—22no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—21no—
246C2×C4×C47—21no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—84no—
253C2×C4×C47—84no—
254C2×C4×C47C2×SD1612yes—
255C2×C4×C47C2×SD1612yes—
256C2×C4×C47C2×SD1612yes—
257C2×C4×C47C2×#4111yescharacteristic
258C2×C4×C47C2×#4111yescharacteristic
259C2×C4×C47C2×#4111yescharacteristic
260C2×C4×C47C2×#4111yescharacteristic
261C2×C4×C47H(5,𝔽2)11yescharacteristic
262C2×C4×C472−1+411yescharacteristic
263C2×Dic47—22no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C47⋊C8—41no—
276C47⋊C8—41no—
277C47⋊C8—41no—
278C47⋊C8—41no—
279#26754—471nomaximal, Sylow
280C22×C47—42no—
281C22×C47—42no—
282C22×C47—42no—
283C22×C47—42no—
284C22×C47—42no—
285C22×C47—42no—
286C22×C47—81no—
287C22×C47—81no—
288C22×C47—81no—
289C22×C47—81no—
290C22×C47—81no—
291C22×C47—81no—
292C22×C47—81no—
293C22×C47—81no—
294C22×C47#52511yescharacteristic
295C22×C47#54611yescharacteristic
296C22×C47#54811yescharacteristic
297C22×C47C22×SD1611yescharacteristic
298C22×C47#57511yescharacteristic
299C22×C47#57611yescharacteristic
300C22×C47#57611yescharacteristic
301C4×C47—22no—
302C4×C47—22no—
303C4×C47—22no—
304C4×C47—41no—
305C4×C47—42no—
306C4×C47—41no—
307C4×C47—41no—
308C4×C47—81no—
309C4×C47#52911yescharacteristic
310C4×C47#53911yescharacteristic
311C4×C47#55211yescharacteristic
312C4×C47#55311yescharacteristic
313Dic47—42no—
314Dic47—81no—
315Dic47—81no—
316Dic47—81no—
317#2507—471no—
318#2366D4711yescharacteristic
319#2071—471no—
320#2065—471no—
321#2041—471no—
322#2041—471no—
323#1934—471no—
324#1926—471no—
325#1920—471no—
326#1920—471no—
327#1916—471no—
328#1864—471no—
329#1864—471no—
330#1802—471no—
331C4×#102—471no—
332C2×C47—82no—
333C2×C47—82no—
334C2×C47#246011yescharacteristic
335C2×C47#310111yescharacteristic
336C2×C47#312211yescharacteristic
337C2×C47#329511yescharacteristic
338C2×C47#330911yescharacteristic
339C2×C47#336211yescharacteristic
340C2×C47#337011yescharacteristic
341C2×#127C2×D4711yescharacteristic
342C2×C4×D4—21no—
343C2×C4×D4C2×D4711yescharacteristic
344C2×C4×D4C2×D4711yescharacteristic
345C2×#106—471no—
346C2×#106—471no—
347C2×C4⋊C8—471no—
348C2×#102—471no—
349C2×#102—471no—
350C2×#102—471no—
351C2×#102—471no—
352C2×#102—471no—
353C2×#102—471no—
354C2×#102—471no—
355C2×#102—471no—
356C2×C4×C8—471no—
357#398—941no—
358#398—941no—
359#390—21no—
360#390C2×D4711yescharacteristic
361#390C2×D4711yescharacteristic
362#389—471no—
363#389—471no—
364#389—471no—
365#389—471no—
366#389—941no—
367#387—941no—
368#386—21no—
369#386—21no—
370#384—471no—
371C4×C4⋊C4—471no—
372C4×C4⋊C4C2×D4711yescharacteristic
373C4×C4⋊C4C2×D4711yescharacteristic
374#340—471no—
375#340—471no—
376#340—471no—
377#340—471no—
378#340—471no—
379#340—471no—
380#336—471no—
381#336—471no—
382#336—471no—
383#336—471no—
384C47#2675411yescharacteristic, complemented, Sylow
385C22×D4—21no—
386C22×D4—21no—
387C22×D4C22×D4711yescharacteristic
388C22×D4C22×D4711yescharacteristic
389C23×C4—21no—
390C23×C4—21no—
391C22×C8—471no—
392C22×C8—471no—
393C22×C8—471no—
394C22×C8—471no—
395#127—22no—
396#127—22no—
397C4×D4—22no—
398C4×D4—22no—
399C4×D4—41no—
400C4×D4—41no—
401C4×D4—44no—
402C2×C4⋊C4—471no—
403C2×C4⋊C4—471no—
404C2×C4⋊C4—471no—
405C2×C4⋊C4—471no—
406C2×C4⋊C4—941no—
407C2×C4⋊C4—941no—
408C2×C4⋊C4—941no—
409C2×C4⋊C4—941no—
410C2×C4⋊C4—941no—
411C2×C4⋊C4—941no—
412C2×C4⋊C4C22⋊D4711yescharacteristic
413C2×C4⋊C4C22⋊D4711yescharacteristic
414C2×C4⋊C4C22×D4711yescharacteristic
415C2×C4⋊C4C22×D4711yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C42—21no—
423C2×C42—941no—
424C2×C42C22×D4711yescharacteristic
425C2×C42C22×D4711yescharacteristic
426C2×C42C22×D4711yescharacteristic
427#106—942no—
428#106—942no—
429C4⋊C8—1881no—
430#102—942no—
431#102—942no—
432#102—942no—
433#102—942no—
434#102—942no—
435#102—942no—
436#102—1881no—
437#102—1881no—
438C4×C8—1881no—
439#95—21no—
440#95—941no—
441#95—941no—
442#95—941no—
443#95—941no—
444#95—941no—
445#95C22⋊D4711yescharacteristic
446#95C22⋊D4711yescharacteristic
447C24—21no—
448C24—21no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—81no—
464C2×D4—81no—
465C2×D4#1062535012yes—
466C2×D4#1062535012yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—941no—
476C22×C4—941no—
477C22×C4—941no—
478C22×C4—941no—
479C22×C4C2×C22⋊D4711yescharacteristic
480C22×C4C2×C22⋊D4711yescharacteristic
481C22×C4C2×C22⋊D4711yescharacteristic
482C22×C4C2×C22⋊D4711yescharacteristic
483C22×C4C2×C22⋊D4711yescharacteristic
484C22×C4C2×C22⋊D4711yescharacteristic
485C22×C4C23×D4711yescharacteristic
486C2×C8—941no—
487C2×C8—941no—
488C2×C8—941no—
489C2×C8—941no—
490C2×C8—941no—
491C2×C8—941no—
492C2×C8—941no—
493C2×C8—941no—
494C2×C8—941no—
495C2×C8—941no—
496C2×C8—941no—
497C2×C8—941no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—41no—
502C4⋊C4—472no—
503C4⋊C4—941no—
504C4⋊C4—941no—
505C4⋊C4—941no—
506C4⋊C4—941no—
507C4⋊C4—941no—
508C4⋊C4—941no—
509C4⋊C4—941no—
510C4⋊C4—3761no—
511C4⋊C4—3761no—
512C4⋊C4—3761no—
513C4⋊C4—3761no—
514C4⋊C4—3761no—
515C4⋊C4—1882no—
516C4⋊C4#1062533514yes—
517C4⋊C4#1062535212yes—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—81no—
523C22⋊C4—81no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—3761no—
532C23—22no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C22×C22⋊D4711yescharacteristic
547D4—22no—
548D4—22no—
549D4—22no—
550D4—22no—
551D4—48no—
552D4—41no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—84no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—942no—
581C2×C4—1881no—
582C2×C4—1881no—
583C2×C4—1881no—
584C2×C4—1881no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4—1881no—
589C2×C4—1881no—
590C2×C4—1881no—
591C2×C4—1881no—
592C2×C4C2×#1062533512yes—
593C2×C4C2×#1062533512yes—
594C2×C4C2×#1062533512yes—
595C2×C4#41×D4711yescharacteristic
596C2×C4C2×#1062535211yescharacteristic
597C2×C4C2×#1062535011yescharacteristic
598C2×C4C2×#1062535011yescharacteristic
599C2×C4#1309406911yescharacteristic
600C2×C4#1309407111yescharacteristic
601C8—1881no—
602C8—1881no—
603C8—1881no—
604C8—1881no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22#1568963411yescharacteristic
620C22#1568976711yescharacteristic
621C22#1568978011yescharacteristic
622C22C22×#1062533511yescharacteristic
623C22#1568992111yescharacteristic
624C22#1568992611yescharacteristic
625C22#1568992611yescharacteristic
626C4—22no—
627C4—22no—
628C4—22no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—42no—
633C4—81no—
634C4—3761no—
635C4—3761no—
636C4—3761no—
637C4—1882no—
638C4#1568965311yescharacteristic
639C4#1568971511yescharacteristic
640C4#1568981311yescharacteristic
641C4#1568982211yescharacteristic
642C2—82no—
643C2—82no—
644C2#2024733411yescharacteristic
645C2#2025339611yescharacteristic
646C2#2025363111yescharacteristic
647C2#2025521211yescharacteristic
648C2#2025539411yescharacteristic
649C2#2025605511yescharacteristic
650C2#2025613511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2366C2
socleC23×C47C22×D4
prime radicalC2×#127×C47C22
hypercenter#2366D47
hypocenterC47#26754
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2366

Complements

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

NH ≅ G/Ntype
1. #20247915C2semidirect
2. #20244791C2semidirect
3. #20244762C2semidirect
6. #20244074C2semidirect
7. #20244012C2semidirect
8. #20243973C2semidirect
9. #20243973C2semidirect
10. #20243947C2semidirect
11. #20243649C2semidirect
12. #20243649C2semidirect
13. #20243388C2semidirect
14. C4×#13093945C2semidirect
384. C47#26754semidirect

Maximal subgroups

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

subgroupindex
#202479152
#202447912
#202447622
#202446192
#202440742
#202440122
#202439732
#202439472
#202436492
#202433882
C4×#130939452
C47×#23662
#2675447