Finite Group Databasebeta

← #26249335

Subgroups — #26249335

Order 12032.

1318 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689456C211yesmaximal, characteristic, complemented
2C22×#13093948C211yesmaximal, characteristic, complemented
3#20247705C211yesmaximal, characteristic
4#20244631C212yesmaximal, complemented
5#20244588C212yesmaximal, complemented
6#20243812C211yesmaximal, characteristic, complemented
7#20243681C211yesmaximal, characteristic
8#20243642C212yesmaximal, complemented
9#20243323C211yesmaximal, characteristic, complemented
10#20243323C212yesmaximal, complemented
11C2×D4×Q8×C47C211yesmaximal, characteristic
12C23×C47⋊C8C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14#15689456—24no—
15C2×#13093957C2212yes—
16C2×#13093948—24no—
17C2×#13093948C2212yes—
18C2×#13093948C2212yes—
19C2×#13093948C2212yes—
20C2×#13093948C2212yes—
21C2×#13093948C2212yes—
22C2×#13093948C2212yes—
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#15688983C2211yescharacteristic
31#15688983C2212yes—
32#15688885C2212yes—
33#15688878C2212yes—
34C23×Q8×C47C2211yescharacteristic
35C23×Q8×C47C2211yescharacteristic
36D4×Q8×C47—24no—
37D4×Q8×C47—24no—
38C2×C4⋊Q8×C47—21no—
39C2×C4⋊Q8×C47C2211yescharacteristic
40C2×C22⋊Q8×C47—21no—
41C2×C22⋊Q8×C47—21no—
42C2×C22⋊Q8×C47C2211yescharacteristic
43C2×C22⋊Q8×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#13093957—24no—
61#13093948—24no—
62#13093948—24no—
63#13093948—24no—
64#13093948—24no—
65#13093948—42no—
66#13093948—42no—
67C22⋊(C47⋊C8)—22no—
68C22⋊(C47⋊C8)—42no—
69C47⋊#95—22no—
70C47⋊#95—21no—
71C47⋊#95—22no—
72C22×Q8×C47—22no—
73C22×Q8×C47—22no—
74C22×Q8×C47—21no—
75C22×Q8×C47—21no—
76C22×Q8×C47—22no—
77C22×Q8×C47—24no—
78C22×Q8×C47—24no—
79C22×Q8×C47—42no—
80C22×Q8×C47—42no—
81C22×Q8×C47D414yes—
82C22×Q8×C47C2311yescharacteristic
83C22×Q8×C47C2311yescharacteristic
84C22×Q8×C47C2311yescharacteristic
85C22×D4×C47D411yescharacteristic
86C23×C4×C47—21no—
87C23×C4×C47—21no—
88C23×C4×C47C2311yescharacteristic
89C23×C4×C47C2311yescharacteristic
90C4⋊Q8×C47—28no—
91C4⋊Q8×C47—22no—
92C4⋊Q8×C47—22no—
93C22⋊Q8×C47—28no—
94C22⋊Q8×C47—28no—
95C22⋊Q8×C47—22no—
96C22⋊Q8×C47—22no—
97C22⋊Q8×C47—22no—
98C22⋊Q8×C47D414yes—
99C4×Q8×C47—22no—
100C4×Q8×C47—22no—
101C4×D4×C47—28no—
102C4×D4×C47—22no—
103C4×D4×C47—22no—
104C2×C4⋊C4×C47—21no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47—21no—
109C2×C4⋊C4×C47D411yescharacteristic
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
117C24×C47C2×D411yescharacteristic
118C24×C47C2×D411yescharacteristic
119C2×Q8×C47—24no—
120C2×Q8×C47—24no—
121C2×Q8×C47—21no—
122C2×Q8×C47—21no—
123C2×Q8×C47—21no—
124C2×Q8×C47—22no—
125C2×Q8×C47—28no—
126C2×Q8×C47—22no—
127C2×Q8×C47—22no—
128C2×Q8×C47—22no—
129C2×Q8×C47—22no—
130C2×Q8×C47—44no—
131C2×Q8×C47—44no—
132C2×Q8×C47—44no—
133C2×Q8×C47—48no—
134C2×Q8×C47—48no—
135C2×Q8×C47—44no—
136C2×Q8×C47—44no—
137C2×Q8×C47C2×D412yes—
138C2×Q8×C47C2×D412yes—
139C2×Q8×C47C2×D412yes—
140C2×Q8×C47#4114yes—
141C2×D4×C47—22no—
142C2×D4×C47—22no—
143C2×D4×C47Q1614yes—
144C22×C4×C47—21no—
145C22×C4×C47—22no—
146C22×C4×C47—22no—
147C22×C4×C47—24no—
148C22×C4×C47—24no—
149C22×C4×C47—21no—
150C22×C4×C47—21no—
151C22×C4×C47—21no—
152C22×C4×C47—21no—
153C22×C4×C47—21no—
154C22×C4×C47—22no—
155C22×C4×C47—22no—
156C22×C4×C47—44no—
157C22×C4×C47—44no—
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47C2×D411yescharacteristic
161C22×C4×C47C2×D412yes—
162C22×C4×C47C2×D412yes—
163C22×C4×C47C2×D411yescharacteristic
164C22×C4×C47C2411yescharacteristic
165C22×Dic47—22no—
166C22×Dic47—22no—
167C22×Dic47—42no—
168C22×Dic47—82no—
169C4⋊C4×C47—24no—
170C4⋊C4×C47—24no—
171C4⋊C4×C47—24no—
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—22no—
180C22⋊C4×C47—22no—
181C42×C47—24no—
182C42×C47—21no—
183C42×C47—21no—
184C22⋊Dic47—42no—
185C22⋊Dic47—82no—
186C4⋊Dic47—22no—
187C4⋊Dic47—22no—
188C4⋊Dic47—22no—
189C4⋊Dic47—22no—
190C4⋊Dic47—42no—
191Dic47⋊C4—82no—
192C2×C47⋊C8—22no—
193C2×C47⋊C8—22no—
194C2×C47⋊C8—22no—
195C2×C47⋊C8—22no—
196C2×C47⋊C8—22no—
197C2×C47⋊C8—22no—
198C2×C47⋊C8—44no—
199C23×C47—22no—
200C23×C47—22no—
201C23×C47—21no—
202C23×C47—22no—
203C23×C47—21no—
204C23×C47—22no—
205C23×C47#12012yes—
206C23×C47C2×Q1612yes—
207C23×C47C2×Q1612yes—
208C23×C47#13712yes—
209C23×C47C22×D411yescharacteristic
210Q8×C47—22no—
211Q8×C47—24no—
212Q8×C47—22no—
213Q8×C47—22no—
214Q8×C47—24no—
215Q8×C47—48no—
216Q8×C47—48no—
217Q8×C47—41no—
218Q8×C47—41no—
219Q8×C47—42no—
220Q8×C47#12314yes—
221D4×C47—28no—
222C2×C4×C47—21no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—24no—
226C2×C4×C47—21no—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—24no—
230C2×C4×C47—21no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—21no—
235C2×C4×C47—22no—
236C2×C4×C47—22no—
237C2×C4×C47—48no—
238C2×C4×C47—48no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—42no—
245C2×C4×C47#12012yes—
246C2×C4×C47C2×Q1612yes—
247C2×C4×C47C22×D411yescharacteristic
248C2×C4×C47C22×D411yescharacteristic
249C2×C4×C47C2×#4112yes—
250C2×C4×C47H(5,𝔽2)12yes—
251C2×Dic47—42no—
252C2×Dic47—42no—
253C2×Dic47—42no—
254C2×Dic47—42no—
255C2×Dic47—42no—
256C2×Dic47—42no—
257C2×Dic47—84no—
258C47⋊C8—42no—
259C47⋊C8—42no—
260#24042—471nomaximal, Sylow
261C22×C47—22no—
262C22×C47—22no—
263C22×C47—24no—
264C22×C47—24no—
265C22×C47—22no—
266C22×C47—24no—
267C22×C47—41no—
268C22×C47—41no—
269C22×C47—42no—
270C22×C47#45114yes—
271C22×C47C2×#12011yescharacteristic
272C22×C47#54612yes—
273C22×C47C22×Q1611yescharacteristic
274C22×C47C2×#13711yescharacteristic
275C22×C47#57712yes—
276C4×C47—22no—
277C4×C47—24no—
278C4×C47—22no—
279C4×C47—41no—
280C4×C47—42no—
281C4×C47—42no—
282C4×C47—41no—
283C4×C47—42no—
284C4×C47C2×#12311yescharacteristic
285C4×C47#53411yescharacteristic
286C4×C47#54612yes—
287Dic47—82no—
288Dic47—82no—
289C2×D4×Q8D4711yescharacteristic
290C2×#484—471no—
291C22×#103—471no—
292#2487—471no—
293#2043—472no—
294#2036—472no—
295#1892—471no—
296#1869—471no—
297#1863—472no—
298#1787—472no—
299#1787—471no—
300C2×C47—24no—
301C2×C47—42no—
302C2×C47#240511yescharacteristic
303C2×C47C2×#45111yescharacteristic
304C2×C47#301011yescharacteristic
305C2×C47#329912yes—
306C2×C47#332012yes—
307C23×Q8C2×D4711yescharacteristic
308C23×Q8C2×D4711yescharacteristic
309C23×C8—471no—
310D4×Q8—24no—
311D4×Q8—24no—
312C2×C4⋊Q8—21no—
313C2×C4⋊Q8C2×D4711yescharacteristic
314C2×C22⋊Q8—21no—
315C2×C22⋊Q8—21no—
316C2×C22⋊Q8C2×D4711yescharacteristic
317C2×C22⋊Q8C2×D4711yescharacteristic
318C2×C4×Q8C2×D4711yescharacteristic
319C2×C4×D4—21no—
320C2×C4×D4C2×D4711yescharacteristic
321C22×C4⋊C4—471no—
322#484—944no—
323C2×#107—472no—
324C2×#103—472no—
325C2×#103—472no—
326C2×#103—472no—
327C2×#103—472no—
328C2×#103—472no—
329C2×#103—472no—
330C2×#103—944no—
331C2×C22⋊C8—471no—
332C2×C22⋊C8—472no—
333#400—941no—
334#399—941no—
335#389—472no—
336#386—941no—
337#385—941no—
338#380—471no—
339#380—472no—
340#340—472no—
341#336—472no—
342C47#2404211yescharacteristic, complemented, Sylow
343C22×Q8—22no—
344C22×Q8—22no—
345C22×Q8—21no—
346C22×Q8—21no—
347C22×Q8—22no—
348C22×Q8—24no—
349C22×Q8—24no—
350C22×Q8—42no—
351C22×Q8—42no—
352C22×Q8C22⋊D4714yes—
353C22×Q8C22×D4711yescharacteristic
354C22×Q8C22×D4711yescharacteristic
355C22×Q8C22×D4711yescharacteristic
356C22×D4C22⋊D4711yescharacteristic
357C23×C4—21no—
358C23×C4—21no—
359C23×C4—941no—
360C23×C4C22×D4711yescharacteristic
361C23×C4C22×D4711yescharacteristic
362C22×C8—472no—
363C22×C8—472no—
364C22×C8—942no—
365C22×C8—1882no—
366C4⋊Q8—28no—
367C4⋊Q8—22no—
368C4⋊Q8—22no—
369C22⋊Q8—28no—
370C22⋊Q8—28no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374C22⋊Q8C22⋊D4714yes—
375C4×Q8—22no—
376C4×Q8—22no—
377C4×D4—28no—
378C4×D4—22no—
379C4×D4—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—472no—
386C2×C4⋊C4—472no—
387C2×C4⋊C4—942no—
388C2×C4⋊C4—942no—
389C2×C4⋊C4—942no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4C22⋊D4711yescharacteristic
392C2×C4⋊C4C22×D4711yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—942no—
396C2×C22⋊C4—941no—
397C2×C22⋊C4C22⋊D4711yescharacteristic
398C2×C22⋊C4C22⋊D4711yescharacteristic
399C2×C42—21no—
400C2×C42C22×D4711yescharacteristic
401#107—944no—
402#103—944no—
403#103—944no—
404#103—944no—
405#103—944no—
406#103—1882no—
407#103—1882no—
408C22⋊C8—942no—
409C22⋊C8—1882no—
410#95—942no—
411#95—941no—
412#95—942no—
413C24C2×C22⋊D4711yescharacteristic
414C24C2×C22⋊D4711yescharacteristic
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—28no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—48no—
430C2×Q8—48no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8#1062535214yes—
434C2×Q8C2×C22⋊D4712yes—
435C2×Q8C2×C22⋊D4712yes—
436C2×Q8C2×C22⋊D4712yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4#1062533714yes—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—3762no—
457C22×C4—1882no—
458C22×C4C2×C22⋊D4711yescharacteristic
459C22×C4C2×C22⋊D4711yescharacteristic
460C22×C4C2×C22⋊D4711yescharacteristic
461C22×C4C2×C22⋊D4712yes—
462C22×C4C2×C22⋊D4712yes—
463C22×C4C2×C22⋊D4711yescharacteristic
464C22×C4C23×D4711yescharacteristic
465C2×C8—942no—
466C2×C8—942no—
467C2×C8—942no—
468C2×C8—942no—
469C2×C8—942no—
470C2×C8—942no—
471C2×C8—1884no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—942no—
480C4⋊C4—942no—
481C4⋊C4—942no—
482C4⋊C4—942no—
483C4⋊C4—3762no—
484C4⋊C4—1882no—
485C4⋊C4C2×C22⋊D4712yes—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—3762no—
491C22⋊C4—1882no—
492C42—24no—
493C42—21no—
494C42—21no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23#1309399312yes—
502C23C2×#1062533712yes—
503C23C2×#1062533712yes—
504C23#1309405512yes—
505C23C22×C22⋊D4711yescharacteristic
506Q8—24no—
507Q8—22no—
508Q8—24no—
509Q8—22no—
510Q8—22no—
511Q8—48no—
512Q8—48no—
513Q8—42no—
514Q8—41no—
515Q8—41no—
516Q8#1309400514yes—
517D4—28no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—3764no—
542C2×C4—1882no—
543C2×C4—1882no—
544C2×C4—1882no—
545C2×C4—1882no—
546C2×C4—1882no—
547C2×C4—1882no—
548C2×C4#1309399312yes—
549C2×C4C2×#1062533712yes—
550C2×C4C22×C22⋊D4711yescharacteristic
551C2×C4C22×C22⋊D4711yescharacteristic
552C2×C4C2×#1062535212yes—
553C2×C4#1309406912yes—
554C8—1882no—
555C8—1882no—
556C22—22no—
557C22—24no—
558C22—24no—
559C22—22no—
560C22—24no—
561C22—22no—
562C22—41no—
563C22—42no—
564C22—41no—
565C22#1568927914yes—
566C22C2×#1309399311yescharacteristic
567C22#1568976712yes—
568C22C22×#1062533711yescharacteristic
569C22C2×#1309405511yescharacteristic
570C22#1568993112yes—
571C4—22no—
572C4—22no—
573C4—24no—
574C4—41no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—42no—
579C4—3762no—
580C4—3762no—
581C4C2×#1309400511yescharacteristic
582C4#1568968211yescharacteristic
583C4#1568976712yes—
584C2—24no—
585C2—42no—
586C2#2024671111yescharacteristic
587C2C2×#1568927911yescharacteristic
588C2#2025267411yescharacteristic
589C2#2025526412yes—
590C2#2025555912yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×D4×Q8×C47C2
socleC23×C47C22×D4
prime radicalC22×D4×C47D4
hypercenterC2×D4×Q8D47
hypocenterC47#24042
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#15689456C2semidirect
2. C22×#13093948C2semidirect
4. #20244631C2semidirect
5. #20244588C2semidirect
6. #20243812C2semidirect
8. #20243642C2semidirect
9. #20243323C2semidirect
10. #20243323C2semidirect
342. C47#24042semidirect

Maximal subgroups

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

subgroupindex
C2×#156894562
C22×#130939482
#202477052
#202446312
#202445882
#202438122
#202436812
#202436422
#202433232
C2×D4×Q8×C472
#2404247