Finite Group Databasebeta

← #26284380

Subgroups — #26284380

Order 12032.

748 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246391C211yesmaximal, characteristic
2#20244710C211yesmaximal, characteristic, complemented
3#20244584C211yesmaximal, characteristic
4#20244536C211yesmaximal, characteristic, complemented
5#20244396C211yesmaximal, characteristic
6#20244346C211yesmaximal, characteristic, complemented
7#20244067C211yesmaximal, characteristic, complemented
8#20244019C211yesmaximal, characteristic, complemented
9#20243996C211yesmaximal, characteristic
10#20243973C211yesmaximal, characteristic, complemented
11#20243732C211yesmaximal, characteristic, complemented
12#20243696C211yesmaximal, characteristic
13#20243472C211yesmaximal, characteristic
14C4×#13093947C211yesmaximal, characteristic, complemented
15C47×#2489C211yesmaximal, characteristic
16C22×#10625337C2211yescharacteristic
17C2×#13094003C2211yescharacteristic
18C2×C4×Dic94C2211yescharacteristic
19C22×C4⋊Dic47C2211yescharacteristic
20C2×#13093955C2211yescharacteristic
21C2×C4⋊(C47⋊C8)C2211yescharacteristic
22C2×#13093948C2211yescharacteristic
23C2×#13093948C2211yescharacteristic
24C2×#13093948C2211yescharacteristic
25C2×#13093947C2211yescharacteristic
26C2×#13093947C2211yescharacteristic
27C2×#13093947C2211yescharacteristic
28C2×C22⋊(C47⋊C8)C2211yescharacteristic
29C2×C22⋊(C47⋊C8)C2211yescharacteristic
30C2×C4×C47⋊C8C2211yescharacteristic
31#15689029C2211yescharacteristic
32#15689021C2211yescharacteristic
33#15689020—21no—
34#15689010—21no—
35#15688995—21no—
36C4×C4⋊Dic47C2211yescharacteristic
37C22⋊C4×Dic47—21no—
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C2×#125×C47C2211yescharacteristic
49C2×C22⋊Q8×C47C2211yescharacteristic
50C47×#400—21no—
51C47×#397—21no—
52C47×#391C2211yescharacteristic
53C47×#391C2211yescharacteristic
54C47×#388—21no—
55C47×#382—21no—
56C47×#380C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C4×C22⋊C4×C47C2211yescharacteristic
59C22×Dic94C2311yescharacteristic
60C23×Dic47—21no—
61C2×#10625337—22no—
62C2×#10625337—41no—
63C2×#10625337—41no—
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68#13094003—22no—
69#13094003—22no—
70C4×Dic94—22no—
71C4×Dic94—41no—
72C2×C4⋊Dic47—22no—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47—41no—
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×Dic47⋊C4—21no—
81C2×C22⋊Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84#13093955—22no—
85C4⋊(C47⋊C8)—41no—
86#13093948—22no—
87#13093948—22no—
88#13093948—41no—
89#13093947—22no—
90#13093947—22no—
91#13093947—41no—
92C22⋊(C47⋊C8)—41no—
93C22⋊(C47⋊C8)—41no—
94C4×C47⋊C8—41no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C22×Q8×C47C2311yescharacteristic
100C23×C4×C47C2311yescharacteristic
101#125×C47—22no—
102#125×C47—41no—
103C22⋊Q8×C47—22no—
104C22⋊Q8×C47—41no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C22⋊C4×C47—21no—
112C2×C22⋊C4×C47—21no—
113C2×C22⋊C4×C47D411yescharacteristic
114C2×C22⋊C4×C47D411yescharacteristic
115C2×C42×C47—21no—
116C2×C42×C47C2311yescharacteristic
117C2×C42×C47C2311yescharacteristic
118#95×C47—21no—
119#95×C47—21no—
120#95×C47—21no—
121#95×C47—21no—
122#95×C47—21no—
123#95×C47—21no—
124#95×C47D411yescharacteristic
125#95×C47D411yescharacteristic
126C24×C47C2×D411yescharacteristic
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47—21no—
131C2×Q8×C47—21no—
132C2×Q8×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—21no—
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2411yescharacteristic
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—42no—
154C22×Dic47—42no—
155C22×Dic47—81no—
156C2×Dic94—21no—
157C2×Dic94—21no—
158C2×Dic94—21no—
159C2×Dic94—21no—
160C2×Dic94—21no—
161C2×Dic94#4112yes—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—42no—
168C4⋊C4×C47—42no—
169C4⋊C4×C47—42no—
170C4⋊C4×C47#4112yes—
171C22⋊C4×C47—22no—
172C22⋊C4×C47—22no—
173C22⋊C4×C47—42no—
174C22⋊C4×C47—42no—
175C42×C47—21no—
176C42×C47—21no—
177C42×C47—21no—
178C42×C47—21no—
179C42×C47—42no—
180C22⋊Dic47—24no—
181#10625337—44no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47—42no—
187C4⋊Dic47#4112yes—
188C4⋊Dic47#4112yes—
189Dic47⋊C4—24no—
190Dic47⋊C4—42no—
191Dic47⋊C4—81no—
192C4×Dic47—42no—
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—21no—
207C23×C47—21no—
208C23×C47—21no—
209C23×C47—21no—
210C23×C47—21no—
211C23×C47—21no—
212C23×C47#13512yes—
213C23×C47C22×D411yescharacteristic
214Q8×C47—41no—
215Q8×C47—41no—
216Q8×C47—41no—
217Q8×C47—41no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—22no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
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—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47#13712yes—
249C2×C4×C47C2×#4111yescharacteristic
250C2×C4×C47C2×#4111yescharacteristic
251C2×C4×C47C2×#4111yescharacteristic
252C2×C4×C47C2×#4111yescharacteristic
253C2×C4×C47H(5,𝔽2)11yescharacteristic
254C2×C4×C47H(5,𝔽2)11yescharacteristic
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—84no—
268Dic94—22no—
269Dic94—22no—
270Dic94—41no—
271Dic94—41no—
272C47⋊C8—41no—
273C47⋊C8—41no—
274C47⋊C8—41no—
275C47⋊C8—41no—
276#26725—471nomaximal, Sylow
277C22×C47—22no—
278C22×C47—22no—
279C22×C47—22no—
280C22×C47—41no—
281C22×C47—41no—
282C22×C47—41no—
283C22×C47—41no—
284C22×C47#52511yescharacteristic
285C22×C47#54611yescharacteristic
286C22×C47#54611yescharacteristic
287C22×C47C2×#13511yescharacteristic
288C22×C47C2×#13711yescharacteristic
289C22×C47#57711yescharacteristic
290C22×C47#57811yescharacteristic
291C4×C47—22no—
292C4×C47—41no—
293C4×C47—42no—
294C4×C47—41no—
295C4×C47—41no—
296C4×C47—41no—
297C4×C47—81no—
298C4×C47—81no—
299C4×C47—81no—
300C4×C47#52911yescharacteristic
301C4×C47#53811yescharacteristic
302C4×C47#55511yescharacteristic
303C4×C47#55911yescharacteristic
304Dic47—42no—
305Dic47—42no—
306Dic47—42no—
307Dic47—81no—
308#2489D4711yescharacteristic
309#2360—471no—
310#2056—471no—
311#2035—471no—
312#2026—471no—
313#2001—471no—
314#1989—471no—
315#1933—471no—
316#1927—471no—
317#1923—471no—
318#1920—471no—
319#1878—471no—
320#1872—471no—
321#1823—471no—
322C4×#103—471no—
323C2×C47—42no—
324C2×C47#245911yescharacteristic
325C2×C47#310811yescharacteristic
326C2×C47#312011yescharacteristic
327C2×C47#329611yescharacteristic
328C2×C47#330011yescharacteristic
329C2×C47#331211yescharacteristic
330C2×C47#333111yescharacteristic
331C22×Q16—471no—
332C2×#125C2×D4711yescharacteristic
333C2×C22⋊Q8—471no—
334C2×C22⋊Q8C2×D4711yescharacteristic
335C2×C4×Q8—471no—
336C22×C4⋊C4—471no—
337C2×#106—471no—
338C2×C4⋊C8—471no—
339C2×#103—471no—
340C2×#103—471no—
341C2×#103—471no—
342C2×#103—471no—
343C2×#103—471no—
344C2×#103—471no—
345C2×C22⋊C8—471no—
346C2×C22⋊C8—471no—
347C2×C4×C8—471no—
348#400—21no—
349#397—21no—
350#391—471no—
351#391C2×D4711yescharacteristic
352#391C2×D4711yescharacteristic
353#389—471no—
354#389—941no—
355#388—21no—
356#387—941no—
357#385—941no—
358#382—21no—
359#380C2×D4711yescharacteristic
360C4×C4⋊C4—471no—
361C4×C4⋊C4C2×D4711yescharacteristic
362C4×C22⋊C4—941no—
363C4×C22⋊C4C2×D4711yescharacteristic
364#340—471no—
365#340—471no—
366#340—471no—
367#340—471no—
368#340—471no—
369#340—471no—
370#340—471no—
371#340—471no—
372#336—471no—
373#336—471no—
374C47#2672511yescharacteristic, complemented, Sylow
375C22×Q8—471no—
376C22×Q8C22×D4711yescharacteristic
377C23×C4—941no—
378C23×C4C22×D4711yescharacteristic
379C2×Q16—942no—
380C2×Q16—1881no—
381C2×Q16—1881no—
382C22×C8—471no—
383C22×C8—471no—
384C22×C8—471no—
385C22×C8—471no—
386#125—22no—
387#125—41no—
388C22⋊Q8—22no—
389C22⋊Q8—41no—
390C22⋊Q8—942no—
391C22⋊Q8—942no—
392C4×Q8—942no—
393C4×Q8—1881no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—471no—
398C2×C4⋊C4—471no—
399C2×C4⋊C4—471no—
400C2×C4⋊C4—942no—
401C2×C4⋊C4—941no—
402C2×C4⋊C4—941no—
403C2×C4⋊C4—941no—
404C2×C4⋊C4—941no—
405C2×C4⋊C4—1881no—
406C2×C4⋊C4C22×D4711yescharacteristic
407C2×C4⋊C4C22×D4711yescharacteristic
408C2×C4⋊C4C22×D4711yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—941no—
412C2×C22⋊C4C22⋊D4711yescharacteristic
413C2×C22⋊C4C22⋊D4711yescharacteristic
414C2×C42—21no—
415C2×C42—941no—
416C2×C42—941no—
417C2×C42C22×D4711yescharacteristic
418C2×C42C22×D4711yescharacteristic
419#106—942no—
420C4⋊C8—1881no—
421#103—942no—
422#103—942no—
423#103—942no—
424#103—942no—
425#103—1881no—
426#103—1881no—
427C22⋊C8—1881no—
428C22⋊C8—1881no—
429C4×C8—1881no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—941no—
437#95—941no—
438#95—941no—
439#95—941no—
440#95C22⋊D4711yescharacteristic
441#95C22⋊D4711yescharacteristic
442C24C2×C22⋊D4711yescharacteristic
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—472no—
450C2×Q8—941no—
451C2×Q8—941no—
452C2×Q8—941no—
453C2×Q8—941no—
454C2×Q8—941no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—941no—
466C22×C4—941no—
467C22×C4—941no—
468C22×C4—941no—
469C22×C4—3761no—
470C22×C4—1882no—
471C22×C4—1882no—
472C22×C4C2×C22⋊D4711yescharacteristic
473C22×C4C2×C22⋊D4711yescharacteristic
474C22×C4C2×C22⋊D4711yescharacteristic
475C22×C4C2×C22⋊D4711yescharacteristic
476C22×C4C2×C22⋊D4711yescharacteristic
477C22×C4C23×D4711yescharacteristic
478Q16—1884no—
479C2×C8—941no—
480C2×C8—941no—
481C2×C8—941no—
482C2×C8—941no—
483C2×C8—941no—
484C2×C8—941no—
485C2×C8—941no—
486C2×C8—941no—
487C2×C8—941no—
488C2×C8—941no—
489C2×C8—941no—
490C2×C8—941no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—472no—
500C4⋊C4—472no—
501C4⋊C4—944no—
502C4⋊C4—941no—
503C4⋊C4—941no—
504C4⋊C4—941no—
505C4⋊C4—941no—
506C4⋊C4—3761no—
507C4⋊C4—1882no—
508C4⋊C4—1882no—
509C4⋊C4#1062535212yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—944no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—3761no—
521C42—1882no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#1309404412yes—
529C23C22×C22⋊D4711yescharacteristic
530Q8—41no—
531Q8—41no—
532Q8—41no—
533Q8—41no—
534Q8—942no—
535Q8—942no—
536Q8—1881no—
537Q8—1881no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—942no—
569C2×C4—942no—
570C2×C4—942no—
571C2×C4—3764no—
572C2×C4—1881no—
573C2×C4—1881no—
574C2×C4—1881no—
575C2×C4—1881no—
576C2×C4—1881no—
577C2×C4—1881no—
578C2×C4—1881no—
579C2×C4—1881no—
580C2×C4—1881no—
581C2×C4#1309405412yes—
582C2×C4#41×D4711yescharacteristic
583C2×C4#41×D4711yescharacteristic
584C2×C4#41×D4711yescharacteristic
585C2×C4C2×#1062535211yescharacteristic
586C2×C4#1309406911yescharacteristic
587C2×C4#1309407011yescharacteristic
588C8—1881no—
589C8—1881no—
590C8—1881no—
591C8—1881no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22#1568963011yescharacteristic
600C22#1568976111yescharacteristic
601C22#1568976711yescharacteristic
602C22C2×#1309404411yescharacteristic
603C22C2×#1309405411yescharacteristic
604C22#1568993111yescharacteristic
605C22#1568993811yescharacteristic
606C4—22no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—42no—
611C4—41no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4—3761no—
616C4—1882no—
617C4—1882no—
618C4—1882no—
619C4#1568965211yescharacteristic
620C4#1568970711yescharacteristic
621C4#1568983511yescharacteristic
622C4#1568985511yescharacteristic
623C2—42no—
624C2#2024732111yescharacteristic
625C2#2025346411yescharacteristic
626C2#2025359611yescharacteristic
627C2#2025521711yescharacteristic
628C2#2025527511yescharacteristic
629C2#2025543311yescharacteristic
630C2#2025570411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2489C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenter#2489D47
hypocenterC47#26725
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2489

Complements

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

NH ≅ G/Ntype
2. #20244710C2semidirect
4. #20244536C2semidirect
6. #20244346C2semidirect
7. #20244067C2semidirect
8. #20244019C2semidirect
10. #20243973C2semidirect
11. #20243732C2semidirect
14. C4×#13093947C2semidirect
374. C47#26725semidirect

Maximal subgroups

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

subgroupindex
#202463912
#202447102
#202445842
#202445362
#202443962
#202443462
#202440672
#202440192
#202439962
#202439732
#202437322
#202436962
#202434722
C4×#130939472
C47×#24892
#2672547