Finite Group Databasebeta

← #26249324

Subgroups — #26249324

Order 12032.

1338 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
1C2×#15689451C211yesmaximal, characteristic, complemented
2C22×#13093948C211yesmaximal, characteristic, complemented
3#20247696C211yesmaximal, characteristic
4#20244631C212yesmaximal, complemented
5#20244581C212yesmaximal, complemented
6#20243812C211yesmaximal, characteristic, complemented
7#20243681C211yesmaximal, characteristic
8#20243656C212yesmaximal, complemented
9#20243323C211yesmaximal, characteristic, complemented
10#20243323C212yesmaximal, complemented
11C2×D4×Q8×C47C211yesmaximal, characteristic
12C23×C47⋊C8C2211yescharacteristic
13C2×#13094005—21no—
14C22×C4⋊Dic47C2211yescharacteristic
15#15689451—24no—
16C2×#13093955C2212yes—
17C2×#13093948—24no—
18C2×#13093948C2212yes—
19C2×#13093948C2212yes—
20C2×#13093948C2212yes—
21C2×#13093948C2212yes—
22C2×#13093948C2212yes—
23C2×#13093948C2212yes—
24C2×C22⋊(C47⋊C8)C2211yescharacteristic
25C2×C22⋊(C47⋊C8)C2212yes—
26#15689053—21no—
27#15689048—21no—
28#15689021C2212yes—
29#15688984—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—
52#13094005—44no—
53C2×C4⋊Dic47—22no—
54C2×C4⋊Dic47—22no—
55C2×C4⋊Dic47—21no—
56C2×C4⋊Dic47C2312yes—
57C2×C4⋊Dic47C2312yes—
58C2×Dic47⋊C4—22no—
59C2×C22⋊Dic47—21no—
60C2×C22⋊Dic47—22no—
61#13093955—24no—
62#13093948—24no—
63#13093948—24no—
64#13093948—24no—
65#13093948—24no—
66#13093948—42no—
67#13093948—42no—
68C22⋊(C47⋊C8)—22no—
69C22⋊(C47⋊C8)—42no—
70C47⋊#95—22no—
71C47⋊#95—21no—
72C47⋊#95—22no—
73C22×Q8×C47—22no—
74C22×Q8×C47—22no—
75C22×Q8×C47—21no—
76C22×Q8×C47—21no—
77C22×Q8×C47—22no—
78C22×Q8×C47—24no—
79C22×Q8×C47—24no—
80C22×Q8×C47—42no—
81C22×Q8×C47—42no—
82C22×Q8×C47D414yes—
83C22×Q8×C47C2311yescharacteristic
84C22×Q8×C47C2311yescharacteristic
85C22×Q8×C47C2311yescharacteristic
86C22×D4×C47D411yescharacteristic
87C23×C4×C47—21no—
88C23×C4×C47—21no—
89C23×C4×C47C2311yescharacteristic
90C23×C4×C47C2311yescharacteristic
91C4⋊Q8×C47—28no—
92C4⋊Q8×C47—22no—
93C4⋊Q8×C47—22no—
94C22⋊Q8×C47—28no—
95C22⋊Q8×C47—28no—
96C22⋊Q8×C47—22no—
97C22⋊Q8×C47—22no—
98C22⋊Q8×C47—22no—
99C22⋊Q8×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×C47—21no—
108C2×C4⋊C4×C47—21no—
109C2×C4⋊C4×C47—21no—
110C2×C4⋊C4×C47D411yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C22⋊C4×C47—21no—
113C2×C22⋊C4×C47—21no—
114C2×C22⋊C4×C47D411yescharacteristic
115C2×C22⋊C4×C47D411yescharacteristic
116C2×C42×C47—21no—
117C2×C42×C47C2311yescharacteristic
118C24×C47C2×D411yescharacteristic
119C24×C47C2×D411yescharacteristic
120C2×Q8×C47—24no—
121C2×Q8×C47—24no—
122C2×Q8×C47—21no—
123C2×Q8×C47—21no—
124C2×Q8×C47—21no—
125C2×Q8×C47—22no—
126C2×Q8×C47—28no—
127C2×Q8×C47—22no—
128C2×Q8×C47—22no—
129C2×Q8×C47—22no—
130C2×Q8×C47—22no—
131C2×Q8×C47—44no—
132C2×Q8×C47—44no—
133C2×Q8×C47—44no—
134C2×Q8×C47—48no—
135C2×Q8×C47—48no—
136C2×Q8×C47—44no—
137C2×Q8×C47—44no—
138C2×Q8×C47C2×D412yes—
139C2×Q8×C47C2×D412yes—
140C2×Q8×C47C2×D412yes—
141C2×Q8×C47#4114yes—
142C2×D4×C47—22no—
143C2×D4×C47—22no—
144C2×D4×C47SD1614yes—
145C22×C4×C47—21no—
146C22×C4×C47—22no—
147C22×C4×C47—22no—
148C22×C4×C47—24no—
149C22×C4×C47—24no—
150C22×C4×C47—21no—
151C22×C4×C47—21no—
152C22×C4×C47—21no—
153C22×C4×C47—21no—
154C22×C4×C47—21no—
155C22×C4×C47—22no—
156C22×C4×C47—22no—
157C22×C4×C47—44no—
158C22×C4×C47—44no—
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47C2×D411yescharacteristic
161C22×C4×C47C2×D411yescharacteristic
162C22×C4×C47C2×D412yes—
163C22×C4×C47C2×D412yes—
164C22×C4×C47C2×D411yescharacteristic
165C22×C4×C47C2411yescharacteristic
166C22×Dic47—22no—
167C22×Dic47—22no—
168C22×Dic47—42no—
169C22×Dic47—42no—
170C22×Dic47—42no—
171C4⋊C4×C47—24no—
172C4⋊C4×C47—24no—
173C4⋊C4×C47—24no—
174C4⋊C4×C47—24no—
175C4⋊C4×C47—24no—
176C4⋊C4×C47—22no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47C2×D412yes—
179C22⋊C4×C47—24no—
180C22⋊C4×C47—24no—
181C22⋊C4×C47—22no—
182C22⋊C4×C47—22no—
183C42×C47—24no—
184C42×C47—21no—
185C42×C47—21no—
186C22⋊Dic47—42no—
187C22⋊Dic47—44no—
188C4⋊Dic47—22no—
189C4⋊Dic47—22no—
190C4⋊Dic47—22no—
191C4⋊Dic47—22no—
192C4⋊Dic47—42no—
193Dic47⋊C4—44no—
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—21no—
206C23×C47—22no—
207C23×C47#12012yes—
208C23×C47C2×SD1612yes—
209C23×C47C2×SD1612yes—
210C23×C47#13712yes—
211C23×C47C22×D411yescharacteristic
212Q8×C47—22no—
213Q8×C47—24no—
214Q8×C47—22no—
215Q8×C47—22no—
216Q8×C47—24no—
217Q8×C47—48no—
218Q8×C47—48no—
219Q8×C47—41no—
220Q8×C47—41no—
221Q8×C47—42no—
222Q8×C47#12314yescomplemented
223D4×C47—28no—
224C2×C4×C47—21no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—24no—
228C2×C4×C47—21no—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—24no—
232C2×C4×C47—21no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—22no—
236C2×C4×C47—21no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—48no—
240C2×C4×C47—48no—
241C2×C4×C47—42no—
242C2×C4×C47—42no—
243C2×C4×C47—42no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—42no—
247C2×C4×C47#12012yes—
248C2×C4×C47C2×SD1612yes—
249C2×C4×C47C22×D411yescharacteristic
250C2×C4×C47C22×D411yescharacteristic
251C2×C4×C47C2×#4112yes—
252C2×C4×C47H(5,𝔽2)12yes—
253C2×Dic47—42no—
254C2×Dic47—42no—
255C2×Dic47—42no—
256C2×Dic47—42no—
257C2×Dic47—42no—
258C2×Dic47—42no—
259C2×Dic47—84no—
260C47⋊C8—42no—
261C47⋊C8—42no—
262#24041—471nomaximal, Sylow
263C22×C47—22no—
264C22×C47—22no—
265C22×C47—24no—
266C22×C47—24no—
267C22×C47—22no—
268C22×C47—24no—
269C22×C47—41no—
270C22×C47—41no—
271C22×C47—42no—
272C22×C47#44814yes—
273C22×C47C2×#12011yescharacteristic
274C22×C47#54612yes—
275C22×C47C22×SD1611yescharacteristic
276C22×C47C2×#13711yescharacteristic
277C22×C47#57812yes—
278C4×C47—22no—
279C4×C47—24no—
280C4×C47—22no—
281C4×C47—41no—
282C4×C47—42no—
283C4×C47—42no—
284C4×C47—41no—
285C4×C47—42no—
286C4×C47C2×#12311yescharacteristic
287C4×C47#53411yescharacteristic
288C4×C47#54612yes—
289Dic47—82no—
290Dic47—82no—
291C2×D4×Q8D4711yescharacteristic
292C2×#483—471no—
293C22×#103—471no—
294#2486—471no—
295#2043—472no—
296#2035—472no—
297#1892—471no—
298#1869—471no—
299#1865—472no—
300#1787—472no—
301#1787—471no—
302C2×C47—24no—
303C2×C47—42no—
304C2×C47#240511yescharacteristic
305C2×C47C2×#44811yescharacteristic
306C2×C47#301111yescharacteristic
307C2×C47#329812yes—
308C2×C47#331212yes—
309C23×Q8C2×D4711yescharacteristic
310C23×Q8C2×D4711yescharacteristic
311C23×C8—471no—
312D4×Q8—24no—
313D4×Q8—24no—
314C2×C4⋊Q8—21no—
315C2×C4⋊Q8C2×D4711yescharacteristic
316C2×#123—941no—
317C2×C22⋊Q8—21no—
318C2×C22⋊Q8—21no—
319C2×C22⋊Q8C2×D4711yescharacteristic
320C2×C22⋊Q8C2×D4711yescharacteristic
321C2×C4×Q8C2×D4711yescharacteristic
322C2×C4×D4—21no—
323C2×C4×D4C2×D4711yescharacteristic
324C22×C4⋊C4—471no—
325#483—944no—
326C2×#106—472no—
327C2×#103—472no—
328C2×#103—472no—
329C2×#103—472no—
330C2×#103—472no—
331C2×#103—472no—
332C2×#103—472no—
333C2×#103—944no—
334C2×C22⋊C8—471no—
335C2×C22⋊C8—472no—
336#398—941no—
337#397—941no—
338#389—472no—
339#381—941no—
340#380—471no—
341#380—472no—
342#340—472no—
343#336—472no—
344C47#2404111yescharacteristic, complemented, Sylow
345C22×Q8—22no—
346C22×Q8—22no—
347C22×Q8—21no—
348C22×Q8—21no—
349C22×Q8—22no—
350C22×Q8—24no—
351C22×Q8—24no—
352C22×Q8—42no—
353C22×Q8—42no—
354C22×Q8C22⋊D4714yes—
355C22×Q8C22×D4711yescharacteristic
356C22×Q8C22×D4711yescharacteristic
357C22×Q8C22×D4711yescharacteristic
358C22×D4C22⋊D4711yescharacteristic
359C23×C4—21no—
360C23×C4—21no—
361C23×C4—941no—
362C23×C4C22×D4711yescharacteristic
363C23×C4C22×D4711yescharacteristic
364C22×C8—472no—
365C22×C8—472no—
366C22×C8—942no—
367C22×C8—1882no—
368C4⋊Q8—28no—
369C4⋊Q8—22no—
370C4⋊Q8—22no—
371#123—1884no—
372C22⋊Q8—28no—
373C22⋊Q8—28no—
374C22⋊Q8—22no—
375C22⋊Q8—22no—
376C22⋊Q8—22no—
377C22⋊Q8C22⋊D4714yes—
378C4×Q8—22no—
379C4×Q8—22no—
380C4×D4—28no—
381C4×D4—22no—
382C4×D4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—472no—
389C2×C4⋊C4—472no—
390C2×C4⋊C4—942no—
391C2×C4⋊C4—942no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4—942no—
394C2×C4⋊C4C22⋊D4711yescharacteristic
395C2×C4⋊C4C22×D4711yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—941no—
399C2×C22⋊C4—942no—
400C2×C22⋊C4C22⋊D4711yescharacteristic
401C2×C22⋊C4C22⋊D4711yescharacteristic
402C2×C42—21no—
403C2×C42C22×D4711yescharacteristic
404#106—944no—
405#103—944no—
406#103—944no—
407#103—944no—
408#103—944no—
409#103—1882no—
410#103—1882no—
411C22⋊C8—942no—
412C22⋊C8—1882no—
413#95—942no—
414#95—941no—
415#95—942no—
416C24C2×C22⋊D4711yescharacteristic
417C24C2×C22⋊D4711yescharacteristic
418C2×Q8—24no—
419C2×Q8—24no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8—28no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8—48no—
433C2×Q8—48no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8#1062535214yes—
437C2×Q8C2×C22⋊D4712yes—
438C2×Q8C2×C22⋊D4712yes—
439C2×Q8C2×C22⋊D4712yes—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4#1062533614yes—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—942no—
458C22×C4—942no—
459C22×C4—1882no—
460C22×C4—1882no—
461C22×C4—1882no—
462C22×C4C2×C22⋊D4711yescharacteristic
463C22×C4C2×C22⋊D4711yescharacteristic
464C22×C4C2×C22⋊D4711yescharacteristic
465C22×C4C2×C22⋊D4712yes—
466C22×C4C2×C22⋊D4712yes—
467C22×C4C2×C22⋊D4711yescharacteristic
468C22×C4C23×D4711yescharacteristic
469C2×C8—942no—
470C2×C8—942no—
471C2×C8—942no—
472C2×C8—942no—
473C2×C8—942no—
474C2×C8—942no—
475C2×C8—1884no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—942no—
484C4⋊C4—942no—
485C4⋊C4—942no—
486C4⋊C4—942no—
487C4⋊C4—1882no—
488C4⋊C4—1884no—
489C4⋊C4C2×C22⋊D4712yes—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—1882no—
495C22⋊C4—1884no—
496C42—24no—
497C42—21no—
498C42—21no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23#1309399312yes—
506C23C2×#1062533612yes—
507C23C2×#1062533612yes—
508C23#1309405512yes—
509C23C22×C22⋊D4711yescharacteristic
510Q8—24no—
511Q8—22no—
512Q8—24no—
513Q8—22no—
514Q8—22no—
515Q8—48no—
516Q8—48no—
517Q8—42no—
518Q8—41no—
519Q8—41no—
520Q8#1309400514yescomplemented
521D4—28no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—3764no—
546C2×C4—1882no—
547C2×C4—1882no—
548C2×C4—1882no—
549C2×C4—1882no—
550C2×C4—1882no—
551C2×C4—1882no—
552C2×C4#1309399312yes—
553C2×C4C2×#1062533612yes—
554C2×C4C22×C22⋊D4711yescharacteristic
555C2×C4C22×C22⋊D4711yescharacteristic
556C2×C4C2×#1062535212yes—
557C2×C4#1309406912yes—
558C8—1882no—
559C8—1882no—
560C22—22no—
561C22—24no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—22no—
566C22—41no—
567C22—41no—
568C22—42no—
569C22#1568926214yes—
570C22C2×#1309399311yescharacteristic
571C22#1568976712yes—
572C22C22×#1062533611yescharacteristic
573C22C2×#1309405511yescharacteristic
574C22#1568993812yes—
575C4—22no—
576C4—22no—
577C4—24no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4—42no—
582C4—42no—
583C4—3762no—
584C4—3762no—
585C4C2×#1309400511yescharacteristic
586C4#1568968211yescharacteristic
587C4#1568976712yes—
588C2—24no—
589C2—42no—
590C2#2024671111yescharacteristic
591C2C2×#1568926211yescharacteristic
592C2#2025268311yescharacteristic
593C2#2025525312yes—
594C2#2025543912yes—
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#24041
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×#15689451C2semidirect
2. C22×#13093948C2semidirect
4. #20244631C2semidirect
5. #20244581C2semidirect
6. #20243812C2semidirect
8. #20243656C2semidirect
9. #20243323C2semidirect
10. #20243323C2semidirect
222. Q8×C47#123semidirect
344. C47#24041semidirect
520. Q8#13094005semidirect

Maximal subgroups

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

subgroupindex
C2×#156894512
C22×#130939482
#202476962
#202446312
#202445812
#202438122
#202436812
#202436562
#202433232
C2×D4×Q8×C472
#2404147