Finite Group Databasebeta

← #26249261

Subgroups — #26249261

Order 12032.

1344 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689439C211yesmaximal, characteristic, complemented
2C22×#13093948C211yesmaximal, characteristic, complemented
3#20247705C211yesmaximal, characteristic, complemented
4#20244624C212yesmaximal, complemented
5#20244581C212yesmaximal, complemented
6#20243817C211yesmaximal, characteristic, complemented
7#20243681C211yesmaximal, characteristic, complemented
8#20243656C212yesmaximal, complemented
9#20243328C212yesmaximal, complemented
10#20243323C211yesmaximal, characteristic, complemented
11C2×C47×#548C211yesmaximal, characteristic
12C23×C47⋊C8C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14#15689439—24no—
15C2×#13093955C2212yes—
16C2×#13093948—24no—
17C2×#13093948C2212yes—
18C2×#13093948C2212yes—
19C2×#13093948C2212yes—
20C2×#13093948C2212yes—
21C2×#13093945C2212yes—
22C2×#13093945C2212yes—
23C2×C22⋊(C47⋊C8)C2211yescharacteristic
24C2×C22⋊(C47⋊C8)C2212yes—
25#15689062—21no—
26#15689059—21no—
27#15689021C2212yes—
28#15689003—21no—
29#15688995—21no—
30#15688983C2212yes—
31#15688983C2211yescharacteristic
32#15688885C2212yes—
33#15688878C2212yes—
34C22×#41×C47C2211yescharacteristic
35C23×Q8×C47C2211yescharacteristic
36C47×#548—24no—
37C47×#548—24no—
38C2×#124×C47—21no—
39C2×#124×C47C2211yescharacteristic
40C2×C22⋊Q8×C47—21no—
41C2×C22⋊Q8×C47C2211yescharacteristic
42C2×#121×C47—21no—
43C2×#121×C47C2211yescharacteristic
44C2×C4×Q8×C47C2211yescharacteristic
45C2×C4×D4×C47—21no—
46C2×C4×D4×C47C2211yescharacteristic
47C23×Dic47—21no—
48C22×C47⋊C8—22no—
49C22×C47⋊C8—42no—
50C22×C47⋊C8C2312yes—
51C22×C47⋊C8C2312yes—
52C2×C4⋊Dic47—22no—
53C2×C4⋊Dic47—22no—
54C2×C4⋊Dic47—21no—
55C2×C4⋊Dic47C2312yes—
56C2×C4⋊Dic47C2312yes—
57C2×Dic47⋊C4—22no—
58C2×C22⋊Dic47—21no—
59C2×C22⋊Dic47—22no—
60#13093955—24no—
61#13093948—24no—
62#13093948—24no—
63#13093948—24no—
64#13093948—42no—
65#13093945—24no—
66#13093945—42no—
67C22⋊(C47⋊C8)—22no—
68C22⋊(C47⋊C8)—42no—
69C47⋊#95—22no—
70C47⋊#95—21no—
71C47⋊#95—22no—
72C2×#41×C47—22no—
73C2×#41×C47—22no—
74C2×#41×C47—24no—
75C2×#41×C47—42no—
76C22×Q8×C47—21no—
77C22×Q8×C47—22no—
78C22×Q8×C47—24no—
79C22×Q8×C47—42no—
80C22×Q8×C47D414yes—
81C22×Q8×C47C2311yescharacteristic
82C22×Q8×C47C2311yescharacteristic
83C22×D4×C47—21no—
84C22×D4×C47—21no—
85C22×D4×C47D411yescharacteristic
86C22×D4×C47C2311yescharacteristic
87C23×C4×C47—21no—
88C23×C4×C47—21no—
89C23×C4×C47C2311yescharacteristic
90C23×C4×C47C2311yescharacteristic
91#124×C47—28no—
92#124×C47—22no—
93#124×C47—22no—
94C22⋊Q8×C47—28no—
95C22⋊Q8×C47—22no—
96C22⋊Q8×C47—22no—
97#121×C47—28no—
98#121×C47—22no—
99#121×C47D414yes—
100C4×Q8×C47—22no—
101C4×Q8×C47—22no—
102C4×D4×C47—28no—
103C4×D4×C47—22no—
104C4×D4×C47—22no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47D411yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C22⋊C4×C47—21no—
110C2×C22⋊C4×C47—21no—
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
117C24×C47—21no—
118C24×C47C2×D411yescharacteristic
119C24×C47C2×D411yescharacteristic
120#41×C47—48no—
121#41×C47—44no—
122#41×C47—44no—
123C2×Q8×C47—24no—
124C2×Q8×C47—21no—
125C2×Q8×C47—21no—
126C2×Q8×C47—22no—
127C2×Q8×C47—28no—
128C2×Q8×C47—22no—
129C2×Q8×C47—22no—
130C2×Q8×C47—44no—
131C2×Q8×C47—44no—
132C2×Q8×C47—48no—
133C2×Q8×C47C2×D412yes—
134C2×Q8×C47C2×D412yes—
135C2×Q8×C47#4114yes—
136C2×D4×C47—24no—
137C2×D4×C47—22no—
138C2×D4×C47—24no—
139C2×D4×C47—22no—
140C2×D4×C47—22no—
141C2×D4×C47—21no—
142C2×D4×C47—22no—
143C2×D4×C47—22no—
144C2×D4×C47—44no—
145C2×D4×C47—42no—
146C2×D4×C47—42no—
147C2×D4×C47C2×D412yes—
148C22×C4×C47—21no—
149C22×C4×C47—22no—
150C22×C4×C47—22no—
151C22×C4×C47—24no—
152C22×C4×C47—24no—
153C22×C4×C47—21no—
154C22×C4×C47—21no—
155C22×C4×C47—21no—
156C22×C4×C47—21no—
157C22×C4×C47—22no—
158C22×C4×C47—22no—
159C22×C4×C47—44no—
160C22×C4×C47—44no—
161C22×C4×C47C2×D411yescharacteristic
162C22×C4×C47C2×D411yescharacteristic
163C22×C4×C47C2×D411yescharacteristic
164C22×C4×C47C2×D411yescharacteristic
165C22×C4×C47C2×D412yes—
166C22×C4×C47C2×D412yes—
167C22×C4×C47C2411yescharacteristic
168C22×Dic47—22no—
169C22×Dic47—22no—
170C22×Dic47—42no—
171C22×Dic47—82no—
172C4⋊C4×C47—24no—
173C4⋊C4×C47—24no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47C2×D412yes—
177C22⋊C4×C47—24no—
178C22⋊C4×C47—24no—
179C22⋊C4×C47—24no—
180C22⋊C4×C47—22no—
181C22⋊C4×C47—24no—
182C22⋊C4×C47—22no—
183C42×C47—24no—
184C42×C47—21no—
185C42×C47—21no—
186C22⋊Dic47—42no—
187C22⋊Dic47—82no—
188C4⋊Dic47—22no—
189C4⋊Dic47—22no—
190C4⋊Dic47—22no—
191C4⋊Dic47—22no—
192C4⋊Dic47—42no—
193Dic47⋊C4—82no—
194C2×C47⋊C8—22no—
195C2×C47⋊C8—22no—
196C2×C47⋊C8—22no—
197C2×C47⋊C8—22no—
198C2×C47⋊C8—22no—
199C2×C47⋊C8—22no—
200C2×C47⋊C8—44no—
201C23×C47—22no—
202C23×C47—22no—
203C23×C47—21no—
204C23×C47—22no—
205C23×C47—22no—
206C23×C47—21no—
207C23×C47—21no—
208C23×C47—22no—
209C23×C47—42no—
210C23×C47—41no—
211C23×C47—41no—
212C23×C47—42no—
213C23×C47#12012yes—
214C23×C47#13512yes—
215C23×C47#13712yes—
216C23×C47C22×D411yescharacteristic
217Q8×C47—22no—
218Q8×C47—24no—
219Q8×C47—22no—
220Q8×C47—22no—
221Q8×C47—48no—
222Q8×C47—42no—
223Q8×C47#12314yes—
224D4×C47—24no—
225D4×C47—48no—
226D4×C47—44no—
227D4×C47—48no—
228D4×C47—41no—
229D4×C47—41no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—24no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—24no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—21no—
241C2×C4×C47—22no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—22no—
245C2×C4×C47—48no—
246C2×C4×C47—48no—
247C2×C4×C47—42no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—42no—
253C2×C4×C47#12012yes—
254C2×C4×C47C22×D411yescharacteristic
255C2×C4×C47C22×D411yescharacteristic
256C2×C4×C47C2×#4112yes—
257C2×C4×C47H(5,𝔽2)12yes—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47—42no—
263C2×Dic47—42no—
264C2×Dic47—84no—
265C47⋊C8—42no—
266C47⋊C8—42no—
267#24036—471nomaximal, Sylow
268C22×C47—22no—
269C22×C47—22no—
270C22×C47—24no—
271C22×C47—24no—
272C22×C47—42no—
273C22×C47—41no—
274C22×C47—41no—
275C22×C47—42no—
276C22×C47—41no—
277C22×C47—42no—
278C22×C47—42no—
279C22×C47—42no—
280C22×C47—82no—
281C22×C47—82no—
282C22×C47#45214yes—
283C22×C47C2×#12011yescharacteristic
284C22×C47#54612yes—
285C22×C47C2×#13511yescharacteristic
286C22×C47C2×#13711yescharacteristic
287C22×C47#57712yes—
288C4×C47—22no—
289C4×C47—24no—
290C4×C47—41no—
291C4×C47—42no—
292C4×C47—41no—
293C4×C47—42no—
294C4×C47—41no—
295C4×C47C2×#12311yescharacteristic
296C4×C47#53411yescharacteristic
297C4×C47#54612yes—
298Dic47—82no—
299Dic47—82no—
300C2×#548D4711yescharacteristic
301C2×#481—471no—
302C22×#103—471no—
303#2487—471no—
304#2042—472no—
305#2035—472no—
306#1893—471no—
307#1869—471no—
308#1865—472no—
309#1788—472no—
310#1787—471no—
311C2×C47—24no—
312C2×C47—81no—
313C2×C47—82no—
314C2×C47#240511yescharacteristic
315C2×C47C2×#45211yescharacteristic
316C2×C47#301011yescharacteristic
317C2×C47#330112yes—
318C2×C47#331612yes—
319C22×#41C2×D4711yescharacteristic
320C23×Q8C2×D4711yescharacteristic
321C23×C8—471no—
322#548—24no—
323#548—24no—
324C2×#124—21no—
325C2×#124C2×D4711yescharacteristic
326C2×C22⋊Q8—21no—
327C2×C22⋊Q8C2×D4711yescharacteristic
328C2×#121—21no—
329C2×#121C2×D4711yescharacteristic
330C2×C4×Q8C2×D4711yescharacteristic
331C2×C4×D4—21no—
332C2×C4×D4C2×D4711yescharacteristic
333C22×C4⋊C4—471no—
334#481—944no—
335C2×#106—472no—
336C2×#103—472no—
337C2×#103—472no—
338C2×#103—472no—
339C2×#103—472no—
340C2×#103—944no—
341C2×#102—472no—
342C2×#102—472no—
343C2×C22⋊C8—471no—
344C2×C22⋊C8—472no—
345#400—941no—
346#399—941no—
347#389—472no—
348#386—941no—
349#385—941no—
350#380—471no—
351#380—472no—
352#340—472no—
353#336—472no—
354C47#2403611yescharacteristic, complemented, Sylow
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—24no—
358C2×#41—42no—
359C22×Q8—21no—
360C22×Q8—22no—
361C22×Q8—24no—
362C22×Q8—42no—
363C22×Q8C22⋊D4714yes—
364C22×Q8C22×D4711yescharacteristic
365C22×Q8C22×D4711yescharacteristic
366C22×D4—21no—
367C22×D4—21no—
368C22×D4C22⋊D4711yescharacteristic
369C22×D4C22×D4711yescharacteristic
370C23×C4—21no—
371C23×C4—21no—
372C23×C4—941no—
373C23×C4C22×D4711yescharacteristic
374C23×C4C22×D4711yescharacteristic
375C22×C8—472no—
376C22×C8—472no—
377C22×C8—942no—
378C22×C8—1882no—
379#124—28no—
380#124—22no—
381#124—22no—
382C22⋊Q8—28no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385#121—28no—
386#121—22no—
387#121C22⋊D4714yes—
388C4×Q8—22no—
389C4×Q8—22no—
390C4×D4—28no—
391C4×D4—22no—
392C4×D4—22no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—472no—
396C2×C4⋊C4—472no—
397C2×C4⋊C4—942no—
398C2×C4⋊C4—942no—
399C2×C4⋊C4—942no—
400C2×C4⋊C4—941no—
401C2×C4⋊C4C22⋊D4711yescharacteristic
402C2×C4⋊C4C22×D4711yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—942no—
408C2×C22⋊C4—941no—
409C2×C22⋊C4C22⋊D4711yescharacteristic
410C2×C22⋊C4C22⋊D4711yescharacteristic
411C2×C42—21no—
412C2×C42C22×D4711yescharacteristic
413#106—944no—
414#103—944no—
415#103—944no—
416#103—944no—
417#103—1882no—
418#102—944no—
419#102—1882no—
420C22⋊C8—942no—
421C22⋊C8—1882no—
422#95—942no—
423#95—941no—
424#95—942no—
425C24—21no—
426C24C2×C22⋊D4711yescharacteristic
427C24C2×C22⋊D4711yescharacteristic
428#41—48no—
429#41—44no—
430#41—44no—
431C2×Q8—24no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—28no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—48no—
441C2×Q8#1062535214yes—
442C2×Q8C2×C22⋊D4712yes—
443C2×Q8C2×C22⋊D4712yes—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—44no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4C2×C22⋊D4712yes—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—3762no—
472C22×C4—1882no—
473C22×C4C2×C22⋊D4711yescharacteristic
474C22×C4C2×C22⋊D4711yescharacteristic
475C22×C4C2×C22⋊D4711yescharacteristic
476C22×C4C2×C22⋊D4711yescharacteristic
477C22×C4C2×C22⋊D4712yes—
478C22×C4C2×C22⋊D4712yes—
479C22×C4C23×D4711yescharacteristic
480C2×C8—942no—
481C2×C8—942no—
482C2×C8—942no—
483C2×C8—942no—
484C2×C8—942no—
485C2×C8—942no—
486C2×C8—1884no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—942no—
492C4⋊C4—942no—
493C4⋊C4—942no—
494C4⋊C4—942no—
495C4⋊C4—3762no—
496C4⋊C4—1882no—
497C4⋊C4C2×C22⋊D4712yes—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—22no—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—3762no—
505C22⋊C4—1882no—
506C42—24no—
507C42—21no—
508C42—21no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—21no—
513C23—22no—
514C23—21no—
515C23—21no—
516C23—22no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23#1309399312yes—
522C23#1309404412yes—
523C23#1309405412yes—
524C23C22×C22⋊D4711yescharacteristic
525Q8—24no—
526Q8—22no—
527Q8—22no—
528Q8—22no—
529Q8—48no—
530Q8—42no—
531Q8#1309400514yes—
532D4—24no—
533D4—48no—
534D4—44no—
535D4—48no—
536D4—41no—
537D4—41no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—48no—
554C2×C4—48no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—3764no—
562C2×C4—1882no—
563C2×C4—1882no—
564C2×C4—1882no—
565C2×C4—1882no—
566C2×C4—1882no—
567C2×C4—1882no—
568C2×C4#1309399312yes—
569C2×C4C22×C22⋊D4711yescharacteristic
570C2×C4C22×C22⋊D4711yescharacteristic
571C2×C4C2×#1062535212yes—
572C2×C4#1309406912yes—
573C8—1882no—
574C8—1882no—
575C22—22no—
576C22—24no—
577C22—24no—
578C22—22no—
579C22—41no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—41no—
584C22—42no—
585C22—42no—
586C22—41no—
587C22—82no—
588C22—82no—
589C22#1568928514yes—
590C22C2×#1309399311yescharacteristic
591C22#1568976712yes—
592C22C2×#1309404411yescharacteristic
593C22C2×#1309405411yescharacteristic
594C22#1568993112yes—
595C4—22no—
596C4—24no—
597C4—41no—
598C4—41no—
599C4—42no—
600C4—41no—
601C4—42no—
602C4—3762no—
603C4—3762no—
604C4C2×#1309400511yescharacteristic
605C4#1568968211yescharacteristic
606C4#1568976712yes—
607C2—24no—
608C2—81no—
609C2—82no—
610C2#2024671111yescharacteristic
611C2C2×#1568928511yescharacteristic
612C2#2025267411yescharacteristic
613C2#2025529012yes—
614C2#2025549912yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#548C2
socleC23×C47C22×D4
hypercenterC2×#548D47
hypocenterC47#24036
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#548

Complements

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

NH ≅ G/Ntype
1. C2×#15689439C2semidirect
2. C22×#13093948C2semidirect
3. #20247705C2semidirect
4. #20244624C2semidirect
5. #20244581C2semidirect
6. #20243817C2semidirect
7. #20243681C2semidirect
8. #20243656C2semidirect
9. #20243328C2semidirect
10. #20243323C2semidirect
354. C47#24036semidirect

Maximal subgroups

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

subgroupindex
C2×#156894392
C22×#130939482
#202477052
#202446242
#202445812
#202438172
#202436812
#202436562
#202433282
#202433232
C2×C47×#5482
#2403647