Finite Group Databasebeta

← #26243758

Subgroups — #26243758

Order 12032.

1610 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
1C2×#15689630C211yesmaximal, characteristic
2C2×#15689161C211yesmaximal, characteristic, complemented
3C2×#15689119C211yesmaximal, characteristic, complemented
4C22×(C2×D47)⋊C8C211yesmaximal, characteristic, complemented
5#20245767C211yesmaximal, characteristic, complemented
6#20244590C212yesmaximal, complemented
7#20244583C212yesmaximal, complemented
8#20244576C212yesmaximal, complemented
9#20244569C212yesmaximal, complemented
10#20243104C211yesmaximal, characteristic, complemented
11C2×C47×#416C211yesmaximal, characteristic, complemented
12C23×C4×D47C2211yescharacteristic
13C23×C47⋊C8C2211yescharacteristic
14#15689630—24no—
15#15689630—24no—
16C2×#13094005—21no—
17C2×#13094000C2212yes—
18C2×#13093995C2212yes—
19C2×#13093992—21no—
20C2×C4×C22⋊D47—22no—
21C2×#13093983C2211yescharacteristic
22C2×#13093983C2211yescharacteristic
23#15689161—24no—
24C2×#13093958C2212yescomplemented
25C2×#13093956C2212yescomplemented
26#15689119—24no—
27C2×#13093948C2212yescomplemented
28C2×#13093945C2212yescomplemented
29C2×(C2×D47)⋊C8—24no—
30C2×(C2×D47)⋊C8C2212yes—
31C2×(C2×D47)⋊C8C2212yes—
32#15688984—21no—
33#15688980C2212yes—
34#15688980C2212yes—
35#15688978—21no—
36C4×(C2×D47)⋊C4—22no—
37#15688885C2212yescomplemented
38#15688885C2212yescomplemented
39C22×#41×C47C2211yescharacteristic
40C23×C8×C47C2211yescharacteristic
41C2×#117×C47C2211yescharacteristic
42C47×#416—24no—
43C2×#103×C47C2211yescharacteristic, complemented
44C2×#103×C47C2211yescharacteristic, complemented
45C2×#102×C47C2211yescharacteristic, complemented
46C2×#102×C47C2211yescharacteristic, complemented
47C24×D47D411yescharacteristic
48C22×C22⋊D47—22no—
49C23×Dic47D411yescharacteristic
50C22×C4×D47—22no—
51C22×C4×D47—22no—
52C22×C4×D47—22no—
53C22×C4×D47—22no—
54C22×C4×D47—22no—
55C22×C4×D47—22no—
56C22×C4×D47D412yes—
57C22×C4×D47C2312yes—
58C22×C47⋊C8—22no—
59C22×C47⋊C8—22no—
60C22×C47⋊C8—22no—
61C22×C47⋊C8C2312yes—
62#13094005—28no—
63#13094000—24no—
64#13094000—24no—
65#13093995—24no—
66#13093995—24no—
67#13093992—28no—
68C4×C22⋊D47—216no—
69#13093983—22no—
70#13093983—22no—
71#13093983—22no—
72#13093983D414yes—
73C2×C4⋊Dic47C2312yes—
74C2×C4⋊Dic47C2312yes—
75C2×Dic47⋊C4—22no—
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47—22no—
78C2×C22⋊Dic47—21no—
79C2×(C2×D47)⋊C4—22no—
80C2×(C2×D47)⋊C4—22no—
81C2×(C2×D47)⋊C4—22no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84#13093958—24no—
85#13093956—24no—
86#13093948—24no—
87#13093945—24no—
88(C2×D47)⋊C8—24no—
89(C2×D47)⋊C8—24no—
90C47⋊#95—22no—
91C47⋊#95—22no—
92C2×#41×C47—22no—
93C2×#41×C47—22no—
94C2×#41×C47—22no—
95C2×#41×C47—22no—
96C2×#41×C47—22no—
97C2×#41×C47D414yes—
98C22×Q8×C47C2311yescharacteristic
99C22×D4×C47—21no—
100C22×D4×C47C2311yescharacteristic
101C23×C4×C47—21no—
102C23×C4×C47C2311yescharacteristic
103C22×C8×C47—22no—
104C22×C8×C47—22no—
105C22×C8×C47D414yescomplemented
106C22×C8×C47C2311yescharacteristic
107C22×C8×C47C2311yescharacteristic
108#117×C47—22no—
109#117×C47—22no—
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C22⋊C4×C47—21no—
113C2×C42×C47—21no—
114#103×C47—22no—
115#103×C47—22no—
116#102×C47—22no—
117#102×C47—22no—
118C24×C47—21no—
119C24×C47C2×D411yescharacteristic
120C23×D47—22no—
121C23×D47—24no—
122C23×D47—22no—
123C23×D47—22no—
124C23×D47—42no—
125C23×D47C2×D412yes—
126#41×C47—28no—
127#41×C47—28no—
128#41×C47—44no—
129#41×C47—44no—
130C2×Q8×C47—21no—
131C2×Q8×C47—21no—
132C2×Q8×C47—21no—
133C2×Q8×C47—21no—
134C2×Q8×C47—21no—
135C2×Q8×C47C2×D412yes—
136C2×D4×C47—24no—
137C2×D4×C47—21no—
138C2×D4×C47—21no—
139C2×D4×C47—21no—
140C2×D4×C47—21no—
141C2×D4×C47—21no—
142C2×D4×C47—42no—
143C2×D4×C47—42no—
144C2×D4×C47C2×D412yes—
145C22×C4×C47—21no—
146C22×C4×C47—24no—
147C22×C4×C47—21no—
148C22×C4×C47—21no—
149C22×C4×C47—42no—
150C22×C4×C47—42no—
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D412yes—
153C22×C4×C47C2×D412yes—
154C22×C4×C47C2×D412yes—
155C22×C4×C47#4112yes—
156C22×C4×C47#4112yes—
157C22×C4×C47#4112yes—
158C22×C4×C47C2411yescharacteristic
159C2×C22⋊D47—28no—
160C2×C22⋊D47—44no—
161C2×C22⋊D47—44no—
162C22×Dic47—22no—
163C22×Dic47—22no—
164C22×Dic47—22no—
165C22×Dic47—22no—
166C22×Dic47—22no—
167C22×Dic47C2×D412yes—
168C2×C4×D47—24no—
169C2×C4×D47—24no—
170C2×C4×D47—24no—
171C2×C4×D47—22no—
172C2×C4×D47—22no—
173C2×C4×D47—22no—
174C2×C4×D47—22no—
175C2×C4×D47—22no—
176C2×C4×D47—22no—
177C2×C4×D47—44no—
178C2×C4×D47—44no—
179C2×C4×D47—44no—
180C2×C4×D47—44no—
181C2×C8×C47—28no—
182C2×C8×C47—21no—
183C2×C8×C47—21no—
184C2×C8×C47—21no—
185C2×C8×C47—21no—
186C2×C8×C47C2×D412yes—
187C2×C8×C47C2×D412yes—
188C4⋊C4×C47—21no—
189C4⋊C4×C47—21no—
190C4⋊C4×C47—21no—
191C4⋊C4×C47—21no—
192C22⋊C4×C47—42no—
193C42×C47—42no—
194C22⋊Dic47—28no—
195C22⋊Dic47—24no—
196C22⋊Dic47—42no—
197(C2×D47)⋊C4—28no—
198(C2×D47)⋊C4—44no—
199(C2×D47)⋊C4—44no—
200C4⋊Dic47—22no—
201C4⋊Dic47—22no—
202C4⋊Dic47—22no—
203C4⋊Dic47C2×D414yes—
204Dic47⋊C4—28no—
205C4×Dic47—24no—
206C4×Dic47—42no—
207C2×C47⋊C8—22no—
208C2×C47⋊C8—22no—
209C2×C47⋊C8—22no—
210C2×C47⋊C8—44no—
211C23×C47—22no—
212C23×C47—22no—
213C23×C47—22no—
214C23×C47—41no—
215C23×C47—41no—
216C23×C47—42no—
217C23×C47—42no—
218C23×C47#12012yes—
219C23×C47#12112yes—
220C23×C47#12312yes—
221C23×C47C22×D411yescharacteristic
222C22×D47—28no—
223C22×D47—22no—
224C22×D47—22no—
225C22×D47—22no—
226C22×D47—22no—
227C22×D47—22no—
228C22×D47—22no—
229C22×D47—48no—
230C22×D47—48no—
231C22×D47—44no—
232C22×D47—44no—
233C22×D47#13514yes—
234Q8×C47—22no—
235Q8×C47—22no—
236Q8×C47—41no—
237Q8×C47—41no—
238D4×C47—22no—
239D4×C47—22no—
240D4×C47—48no—
241D4×C47—41no—
242D4×C47—41no—
243C2×C4×C47—22no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—48no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—42no—
250C2×C4×C47—42no—
251C2×C4×C47#12012yes—
252C2×C4×C47#12112yes—
253C2×C4×C47#12114yes—
254C2×C4×C47#12114yes—
255C2×C4×C47#12312yes—
256C2×C4×C47#12314yes—
257C2×C4×C47#12414yes—
258C2×C4×C47C22×D411yescharacteristic
259C2×C4×C47C22×D411yescharacteristic
260C2×C4×C47C22×D411yescharacteristic
261C2×C4×C47C2×#4111yescharacteristic
262C2×C4×C47C2×#4111yescharacteristic
263C2×C4×C47C2×#4111yescharacteristic
264C22⋊D47—416no—
265C2×Dic47—24no—
266C2×Dic47—22no—
267C2×Dic47—22no—
268C2×Dic47—22no—
269C2×Dic47—42no—
270C2×Dic47—42no—
271C2×Dic47—44no—
272C2×Dic47—42no—
273C2×Dic47—42no—
274C2×Dic47—42no—
275C4×D47—42no—
276C4×D47—42no—
277C4×D47—42no—
278C4×D47—42no—
279C4×D47—48no—
280C8×C47—22no—
281C8×C47—22no—
282C47⋊C8—42no—
283#23479—471nomaximal, Sylow
284C22×C47—24no—
285C22×C47—24no—
286C22×C47—22no—
287C22×C47—22no—
288C22×C47—42no—
289C22×C47—44no—
290C22×C47—82no—
291C22×C47—81no—
292C22×C47—81no—
293C22×C47#39414yes—
294C22×C47C2×#12011yescharacteristic
295C22×C47C2×#12111yescharacteristic
296C22×C47C2×#12311yescharacteristic
297C22×C47C2×#13512yes—
298C22×C47#57512yes—
299C2×D47—24no—
300C2×D47—24no—
301C2×D47—24no—
302C2×D47—416no—
303C2×D47—42no—
304C2×D47—42no—
305C2×D47—42no—
306C2×D47—42no—
307C4×C47—42no—
308C4×C47—81no—
309C4×C47—81no—
310C4×C47#39414yes—
311C4×C47C2×#12111yescharacteristic
312C4×C47C2×#12111yescharacteristic
313C4×C47C2×#12311yescharacteristic
314C4×C47C2×#12411yescharacteristic
315Dic47—42no—
316Dic47—44no—
317Dic47—82no—
318C2×#525—471no—
319C2×#427—471no—
320C2×#416—471no—
321C2×#416D4711yescharacteristic, complemented
322C22×C22⋊C8—471no—
323#2280—471no—
324#2036—472no—
325#2035—472no—
326#2034—472no—
327#2033—472no—
328#1735—471no—
329C2×C47—24no—
330C2×C47—82no—
331C2×C47C2×#39411yescharacteristic
332C2×C47#300212yes—
333C2×C47#305011yescharacteristic
334C2×C47#305411yescharacteristic
335C2×C47#309112yes—
336D47—44no—
337C22×#41C2×D4711yescharacteristic
338C24×C4—471no—
339C23×C8—471no—
340C23×C8C2×D4711yescharacteristic
341#525—944no—
342#525—944no—
343C2×#123—941no—
344C2×C22⋊Q8—472no—
345C2×#121—472no—
346C2×#120—941no—
347C2×C4×D4—942no—
348C2×#117—471no—
349C2×#117—471no—
350C2×#117C2×D4711yescharacteristic
351#427—944no—
352C2×#107—472no—
353C2×#106—472no—
354#416—24no—
355#416—944no—
356C2×#103—472no—
357C2×#103C2×D4711yescharacteristic, complemented
358C2×#103C2×D4711yescharacteristic, complemented
359C2×#102—472no—
360C2×#102C2×D4711yescharacteristic, complemented
361C2×#102C2×D4711yescharacteristic, complemented
362C2×C22⋊C8—472no—
363C2×C22⋊C8—472no—
364C2×C22⋊C8—944no—
365#381—941no—
366#380—472no—
367#380—472no—
368#379—941no—
369C4×C22⋊C4—942no—
370#340—472no—
371#340—472no—
372C47#2347911yescharacteristic, complemented, Sylow
373C25—471no—
374C2×#41—22no—
375C2×#41—22no—
376C2×#41—22no—
377C2×#41—22no—
378C2×#41—22no—
379C2×#41D18814yes—
380C22×Q8C22×D4711yescharacteristic
381C22×D4—21no—
382C22×D4—942no—
383C22×D4C22×D4711yescharacteristic
384C23×C4—21no—
385C23×C4—471no—
386C23×C4—472no—
387C23×C4—472no—
388C23×C4—942no—
389C23×C4—942no—
390C23×C4—942no—
391C23×C4—942no—
392C23×C4—942no—
393C23×C4—942no—
394C23×C4C22×D4711yescharacteristic
395C22×C8—22no—
396C22×C8—22no—
397C22×C8—472no—
398C22×C8—942no—
399C22×C8—942no—
400C22×C8—942no—
401C22×C8C22⋊D4714yescomplemented
402C22×C8C22×D4711yescharacteristic
403C22×C8C22×D4711yescharacteristic
404#123—948no—
405C22⋊Q8—944no—
406C22⋊Q8—944no—
407#121—944no—
408#121—944no—
409#120—948no—
410C4×D4—9416no—
411#117—22no—
412#117—22no—
413#117—474no—
414#117—942no—
415#117—942no—
416#117—942no—
417C2×C4⋊C4—472no—
418C2×C4⋊C4—472no—
419C2×C4⋊C4—942no—
420C2×C4⋊C4C22×D4711yescharacteristic
421C2×C4⋊C4C22×D4711yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—942no—
424C2×C22⋊C4—942no—
425C2×C22⋊C4—941no—
426C2×C22⋊C4—942no—
427C2×C22⋊C4—942no—
428C2×C22⋊C4—941no—
429C2×C42—21no—
430C2×C42—941no—
431C2×C42—941no—
432#107—944no—
433#106—944no—
434#103—22no—
435#103—22no—
436#103—944no—
437#102—22no—
438#102—22no—
439#102—944no—
440C22⋊C8—944no—
441C22⋊C8—944no—
442#95—942no—
443#95—942no—
444C24—21no—
445C24—472no—
446C24—944no—
447C24—942no—
448C24—942no—
449C24—942no—
450C24—1882no—
451C24D4×D4711yescharacteristic
452#41—28no—
453#41—28no—
454#41—44no—
455#41—44no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8C2×D18812yes—
462C2×D4—24no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—948no—
471C2×D4—1884no—
472C2×D4—1884no—
473C2×D4C2×D18812yes—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—472no—
481C22×C4—942no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—944no—
486C22×C4—944no—
487C22×C4—944no—
488C22×C4—942no—
489C22×C4—942no—
490C22×C4—942no—
491C22×C4—942no—
492C22×C4—942no—
493C22×C4—942no—
494C22×C4—942no—
495C22×C4—1884no—
496C22×C4—1884no—
497C22×C4—1884no—
498C22×C4—1884no—
499C22×C4C2×D18812yes—
500C22×C4#1062535012yes—
501C22×C4D4×D4712yes—
502C22×C4D4×D4711yescharacteristic
503C22×C4#1062535212yes—
504C22×C4#1062535212yes—
505C22×C4C2×C22⋊D4712yes—
506C22×C4C23×D4711yescharacteristic
507C2×C8—28no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—942no—
513C2×C8—942no—
514C2×C8—942no—
515C2×C8—1884no—
516C2×C8C2×C22⋊D4712yes—
517C2×C8C2×C22⋊D4712yes—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—474no—
523C4⋊C4—948no—
524C4⋊C4—942no—
525C4⋊C4—942no—
526C4⋊C4—942no—
527C22⋊C4—42no—
528C22⋊C4—948no—
529C22⋊C4—948no—
530C22⋊C4—944no—
531C22⋊C4—1884no—
532C22⋊C4—1884no—
533C22⋊C4—1882no—
534C42—42no—
535C42—944no—
536C42—1882no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—474no—
545C23—948no—
546C23—942no—
547C23—942no—
548C23—942no—
549C23—942no—
550C23—942no—
551C23—942no—
552C23—1888no—
553C23—1888no—
554C23—1884no—
555C23—1884no—
556C23#1309399412yes—
557C23#1309399512yes—
558C23#1309400612yes—
559C23C2×D4×D4711yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—41no—
563Q8—41no—
564D4—22no—
565D4—22no—
566D4—48no—
567D4—41no—
568D4—41no—
569D4—18816no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—48no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—944no—
579C2×C4—942no—
580C2×C4—942no—
581C2×C4—942no—
582C2×C4—1882no—
583C2×C4—1882no—
584C2×C4—1882no—
585C2×C4—1884no—
586C2×C4—1882no—
587C2×C4—1882no—
588C2×C4—1888no—
589C2×C4—1882no—
590C2×C4—1882no—
591C2×C4—1882no—
592C2×C4—1882no—
593C2×C4#1309399412yes—
594C2×C4#1309399512yes—
595C2×C4#1309399614yes—
596C2×C4#1309399814yes—
597C2×C4#1309400612yes—
598C2×C4#1309400814yes—
599C2×C4#1309401014yes—
600C2×C4C2×D4×D4711yescharacteristic
601C2×C4C22×C22⋊D4711yescharacteristic
602C2×C4C22×D18811yescharacteristic
603C2×C4C2×#1062535211yescharacteristic
604C2×C4C2×#1062535211yescharacteristic
605C2×C4C2×#1062535011yescharacteristic
606C8—22no—
607C8—22no—
608C8—1882no—
609C22—22no—
610C22—22no—
611C22—24no—
612C22—24no—
613C22—42no—
614C22—44no—
615C22—81no—
616C22—81no—
617C22—82no—
618C22—944no—
619C22—944no—
620C22—944no—
621C22—18816no—
622C22—1882no—
623C22—1882no—
624C22—1882no—
625C22—1882no—
626C22#1568903914yes—
627C22C2×#1309399411yescharacteristic
628C22C2×#1309399511yescharacteristic
629C22C2×#1309400611yescharacteristic
630C22#135×D4712yes—
631C22#1568992012yes—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—3762no—
636C4—1882no—
637C4—1884no—
638C4#1568903914yes—
639C4C2×#1309399611yescharacteristic
640C4C2×#1309399811yescharacteristic
641C4C2×#1309400811yescharacteristic
642C4C2×#1309401011yescharacteristic
643C2—24no—
644C2—82no—
645C2—1884no—
646C2C2×#1568903911yescharacteristic
647C2#2025260812yes—
648C2#2025296511yescharacteristic
649C2#2025299311yescharacteristic
650C2#2025329912yes—
subgroupG/H
centerC22×C4D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#416C2
socleC23×C47C22×D4
prime radicalC2×#15689630C2
hypercenterC2×#416D47
hypocenterC47#23479
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C2×#15689161C2semidirect
3. C2×#15689119C2semidirect
4. C22×(C2×D47)⋊C8C2semidirect
5. #20245767C2semidirect
6. #20244590C2semidirect
7. #20244583C2semidirect
8. #20244576C2semidirect
9. #20244569C2semidirect
10. #20243104C2semidirect
11. C2×C47×#416C2semidirect
24. C2×#13093958C22semidirect
25. C2×#13093956C22semidirect
27. C2×#13093948C22semidirect
28. C2×#13093945C22semidirect
37. #15688885C22semidirect
38. #15688885C22semidirect
43. C2×#103×C47C22semidirect
44. C2×#103×C47C22semidirect
45. C2×#102×C47C22semidirect
46. C2×#102×C47C22semidirect
105. C22×C8×C47D4semidirect
321. C2×#416D47semidirect
357. C2×#103C2×D47semidirect
358. C2×#103C2×D47semidirect
360. C2×#102C2×D47semidirect
361. C2×#102C2×D47semidirect
372. C47#23479semidirect
401. C22×C8C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896302
C2×#156891612
C2×#156891192
C22×(C2×D47)⋊C82
#202457672
#202445902
#202445832
#202445762
#202445692
#202431042
C2×C47×#4162
#2347947