Finite Group Databasebeta

← #26243124

Subgroups — #26243124

Order 12032.

2170 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689630C211yesmaximal, characteristic
2C22×#13094000C211yesmaximal, characteristic, complemented
3C2×#15689124C211yesmaximal, characteristic, complemented
4C2×#15689132C211yesmaximal, characteristic, complemented
5C2×#15689086C211yesmaximal, characteristic, complemented
6#20244395C214yesmaximal, complemented
7#20244388C214yesmaximal, complemented
8#20243108C211yesmaximal, characteristic, complemented
9C22×#137×C47C211yesmaximal, characteristic, complemented
10C23×C4×D47C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12#15689630—24no—
13#15689630—24no—
14C2×#13094005—21no—
15C2×#13094000—22no—
16C2×#13094000—24no—
17C2×#13094000—24no—
18C2×#13094000C2214yes—
19C2×#13094000C2212yes—
20C2×#13093995C2212yes—
21C2×#13093992—21no—
22C2×C4×C22⋊D47—22no—
23C2×#13093983C2211yescharacteristic
24C22×C4⋊Dic47C2211yescharacteristic
25C22×Dic47⋊C4—21no—
26C22×(C2×D47)⋊C4—21no—
27#15689132—24no—
28#15689124—24no—
29C2×#13093948C2214yescomplemented
30C2×#13093948C2212yescomplemented
31C2×#13093945C2212yescomplemented
32#15689086—24no—
33C2×(C2×D47)⋊C8C2214yes—
34#15688886C2214yescomplemented
35C22×#41×C47C2211yescharacteristic
36C23×Q8×C47C2211yescharacteristic
37C2×#137×C47—24no—
38C2×#137×C47—24no—
39C2×#137×C47—24no—
40C22×Q16×C47C2212yescomplemented
41C22×SD16×C47C2212yescomplemented
42C22×M16×C47C2211yescharacteristic
43C24×D47D411yescharacteristic
44C22×C22⋊D47—22no—
45C23×Dic47—21no—
46C23×Dic47D411yescharacteristic
47C22×C4×D47—22no—
48C22×C4×D47—22no—
49C22×C4×D47—22no—
50C22×C4×D47—22no—
51C22×C4×D47—22no—
52C22×C4×D47—22no—
53C22×C4×D47D412yes—
54C22×C4×D47C2312yes—
55C2×#10625329—22no—
56C2×#10625329—22no—
57C2×#10625329—22no—
58C22×C47⋊C8C2312yes—
59#13094005—28no—
60#13094000—216no—
61#13094000—28no—
62#13094000—24no—
63#13094000—28no—
64#13094000—24no—
65#13093995—24no—
66#13093995—24no—
67#13093992—28no—
68C4×C22⋊D47—216no—
69#13093983—22no—
70#13093983D414yes—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47—22no—
73C2×C4⋊Dic47D414yescomplemented
74C2×C4⋊Dic47C2312yes—
75C2×C4⋊Dic47C2312yes—
76C2×Dic47⋊C4—28no—
77C2×Dic47⋊C4—24no—
78C2×Dic47⋊C4—22no—
79C2×C22⋊Dic47—21no—
80C2×C22⋊Dic47—22no—
81C2×(C2×D47)⋊C4—28no—
82C2×(C2×D47)⋊C4—24no—
83C2×(C2×D47)⋊C4—22no—
84C2×C4×Dic47—21no—
85#13093948—24no—
86#13093948—28no—
87#13093945—24no—
88(C2×D47)⋊C8—28no—
89C2×#41×C47—22no—
90C2×#41×C47—22no—
91C2×#41×C47—22no—
92C2×#41×C47—22no—
93C2×#41×C47—22no—
94C2×#41×C47D414yes—
95C22×Q8×C47—22no—
96C22×Q8×C47—21no—
97C22×Q8×C47—22no—
98C22×Q8×C47—22no—
99C22×Q8×C47—22no—
100C22×Q8×C47—22no—
101C22×Q8×C47D414yescomplemented
102C22×Q8×C47C2312yes—
103C22×Q8×C47C2311yescharacteristic
104C22×D4×C47—21no—
105C22×D4×C47C2311yescharacteristic
106C23×C4×C47—21no—
107C23×C4×C47—21no—
108C23×C4×C47C2311yescharacteristic
109#137×C47—232no—
110C2×Q16×C47—24no—
111C2×Q16×C47—24no—
112C2×Q16×C47—24no—
113C2×SD16×C47—24no—
114C2×SD16×C47—24no—
115C2×SD16×C47—24no—
116C2×M16×C47—22no—
117C2×M16×C47—22no—
118C2×M16×C47D414yescomplemented
119C22×C8×C47C2312yes—
120C24×C47—21no—
121C24×C47C2×D411yescharacteristic
122C23×D47—22no—
123C23×D47—24no—
124C23×D47—22no—
125C23×D47—22no—
126C23×D47—42no—
127C23×D47C2×D412yes—
128#41×C47—28no—
129#41×C47—28no—
130#41×C47—44no—
131#41×C47—44no—
132C2×Q8×C47—24no—
133C2×Q8×C47—22no—
134C2×Q8×C47—21no—
135C2×Q8×C47—28no—
136C2×Q8×C47—28no—
137C2×Q8×C47—22no—
138C2×Q8×C47—22no—
139C2×Q8×C47—21no—
140C2×Q8×C47—21no—
141C2×Q8×C47—22no—
142C2×Q8×C47—21no—
143C2×Q8×C47—22no—
144C2×Q8×C47—21no—
145C2×Q8×C47—42no—
146C2×Q8×C47—42no—
147C2×Q8×C47—44no—
148C2×Q8×C47—44no—
149C2×Q8×C47C2×D414yes—
150C2×Q8×C47C2×D412yes—
151C2×D4×C47—24no—
152C2×D4×C47—21no—
153C2×D4×C47—21no—
154C2×D4×C47—21no—
155C2×D4×C47—21no—
156C2×D4×C47—21no—
157C2×D4×C47—42no—
158C2×D4×C47—42no—
159C2×D4×C47C2×D412yes—
160C22×C4×C47—24no—
161C22×C4×C47—21no—
162C22×C4×C47—22no—
163C22×C4×C47—24no—
164C22×C4×C47—42no—
165C22×C4×C47—42no—
166C22×C4×C47—42no—
167C22×C4×C47—42no—
168C22×C4×C47C2×D411yescharacteristic
169C22×C4×C47C2×D412yes—
170C22×C4×C47C2×D412yes—
171C22×C4×C47C2×D412yes—
172C22×C4×C47C2×D412yes—
173C22×C4×C47C2×D412yes—
174C22×C4×C47#4112yes—
175C22×C4×C47C2411yescharacteristic
176C2×C22⋊D47—28no—
177C2×C22⋊D47—44no—
178C2×C22⋊D47—44no—
179C22×Dic47—24no—
180C22×Dic47—22no—
181C22×Dic47—22no—
182C22×Dic47—22no—
183C22×Dic47—22no—
184C22×Dic47—22no—
185C22×Dic47—44no—
186C22×Dic47C2×D412yes—
187C2×C4×D47—24no—
188C2×C4×D47—24no—
189C2×C4×D47—24no—
190C2×C4×D47—22no—
191C2×C4×D47—22no—
192C2×C4×D47—22no—
193C2×C4×D47—22no—
194C2×C4×D47—22no—
195C2×C4×D47—22no—
196C2×C4×D47—44no—
197C2×C4×D47—44no—
198C2×C4×D47—44no—
199C2×C4×D47—44no—
200Q16×C47—216no—
201SD16×C47—216no—
202M16×C47—28no—
203C2×C8×C47—22no—
204C2×C8×C47—22no—
205C2×C8×C47C2×D414yes—
206C22⋊Dic47—28no—
207C22⋊Dic47—24no—
208(C2×D47)⋊C4—216no—
209(C2×D47)⋊C4—28no—
210C4⋊Dic47—24no—
211C4⋊Dic47—22no—
212C4⋊Dic47—22no—
213C4⋊Dic47C2×D414yes—
214C4⋊Dic47C2×D414yes—
215Dic47⋊C4—216no—
216Dic47⋊C4—28no—
217C4×Dic47—24no—
218#10625329—44no—
219C2×C47⋊C8—22no—
220C2×C47⋊C8—22no—
221C2×C47⋊C8—22no—
222C23×C47—22no—
223C23×C47—22no—
224C23×C47—22no—
225C23×C47—41no—
226C23×C47—41no—
227C23×C47—42no—
228C23×C47—42no—
229C23×C47#12012yes—
230C23×C47#12012yes—
231C23×C47#12112yes—
232C23×C47C22×D411yescharacteristic
233C22×D47—28no—
234C22×D47—22no—
235C22×D47—22no—
236C22×D47—22no—
237C22×D47—22no—
238C22×D47—22no—
239C22×D47—22no—
240C22×D47—48no—
241C22×D47—48no—
242C22×D47—44no—
243C22×D47—44no—
244C22×D47#13714yescomplemented
245Q8×C47—24no—
246Q8×C47—22no—
247Q8×C47—24no—
248Q8×C47—22no—
249Q8×C47—42no—
250Q8×C47—41no—
251Q8×C47—42no—
252Q8×C47—41no—
253Q8×C47—48no—
254D4×C47—22no—
255D4×C47—22no—
256D4×C47—48no—
257D4×C47—41no—
258D4×C47—41no—
259C2×C4×C47—22no—
260C2×C4×C47—24no—
261C2×C4×C47—41no—
262C2×C4×C47—42no—
263C2×C4×C47—41no—
264C2×C4×C47—42no—
265C2×C4×C47—48no—
266C2×C4×C47—48no—
267C2×C4×C47#12012yes—
268C2×C4×C47#12012yes—
269C2×C4×C47#12014yes—
270C2×C4×C47#12112yes—
271C2×C4×C47#12114yes—
272C2×C4×C47#12114yes—
273C2×C4×C47#12714yes—
274C2×C4×C47C22×D411yescharacteristic
275C2×C4×C47C22×D411yescharacteristic
276C2×C4×C47C22×D411yescharacteristic
277C2×C4×C47C22×D411yescharacteristic
278C2×C4×C47C22×D411yescharacteristic
279C2×C4×C47C2×#4111yescharacteristic
280C22⋊D47—416no—
281C2×Dic47—24no—
282C2×Dic47—22no—
283C2×Dic47—24no—
284C2×Dic47—22no—
285C2×Dic47—22no—
286C2×Dic47—48no—
287C2×Dic47—42no—
288C2×Dic47—44no—
289C2×Dic47—42no—
290C2×Dic47—44no—
291C4×D47—42no—
292C4×D47—42no—
293C4×D47—42no—
294C4×D47—42no—
295C4×D47—48no—
296C8×C47—24no—
297C47⋊C8—42no—
298#23392—471nomaximal, Sylow
299C22×C47—22no—
300C22×C47—22no—
301C22×C47—24no—
302C22×C47—24no—
303C22×C47—42no—
304C22×C47—44no—
305C22×C47—81no—
306C22×C47—81no—
307C22×C47—82no—
308C22×C47#39214yes—
309C22×C47C2×#12011yescharacteristic
310C22×C47C2×#12011yescharacteristic
311C22×C47C2×#12111yescharacteristic
312C22×C47C2×#13712yes—
313C22×C47#57512yes—
314C2×D47—24no—
315C2×D47—24no—
316C2×D47—24no—
317C2×D47—416no—
318C2×D47—42no—
319C2×D47—42no—
320C2×D47—42no—
321C2×D47—42no—
322C4×C47—42no—
323C4×C47—44no—
324C4×C47#39214yes—
325C4×C47C2×#12011yescharacteristic
326C4×C47C2×#12111yescharacteristic
327C4×C47C2×#12111yescharacteristic
328C4×C47C2×#12711yescharacteristic
329Dic47—42no—
330Dic47—44no—
331Dic47—44no—
332C22×#137D4711yescharacteristic, complemented
333C2×#525—471no—
334C22×C22⋊Q8—471no—
335C2×#417—471no—
336C2×#419—471no—
337C2×#407—471no—
338#2001—474no—
339#2000—474no—
340#1736—471no—
341C2×C47—24no—
342C2×C47—82no—
343C2×C47C2×#39211yescharacteristic
344C2×C47#300411yescharacteristic
345C2×C47#300511yescharacteristic
346C2×C47#300712yes—
347C2×C47#305612yes—
348D47—44no—
349C22×#41C2×D4711yescharacteristic
350C23×Q8C2×D4711yescharacteristic
351C24×C4—471no—
352C2×#137—24no—
353C2×#137—24no—
354C2×#137—24no—
355C22×Q16C2×D4712yescomplemented
356C22×SD16C2×D4712yescomplemented
357C22×M16—471no—
358C22×M16C2×D4711yescharacteristic
359#525—944no—
360#525—944no—
361C2×#123—941no—
362C2×C22⋊Q8—474no—
363C2×C22⋊Q8—472no—
364C2×C22⋊Q8—944no—
365C2×C22⋊Q8—942no—
366C2×C22⋊Q8—944no—
367C2×#121—472no—
368C2×#120—941no—
369C2×C4×D4—942no—
370C2×#117—471no—
371C22×C4⋊C4—471no—
372C22×C4⋊C4—941no—
373C22×C22⋊C4—941no—
374#419—944no—
375#417—944no—
376C2×#103—474no—
377C2×#103—472no—
378C2×#102—472no—
379#407—944no—
380C2×C22⋊C8—474no—
381#340—474no—
382C47#2339211yescharacteristic, complemented, Sylow
383C25—471no—
384C2×#41—22no—
385C2×#41—22no—
386C2×#41—22no—
387C2×#41—22no—
388C2×#41—22no—
389C2×#41C22⋊D4714yes—
390C22×Q8—22no—
391C22×Q8—21no—
392C22×Q8—22no—
393C22×Q8—22no—
394C22×Q8—22no—
395C22×Q8—22no—
396C22×Q8C22⋊D4714yescomplemented
397C22×Q8C22×D4712yes—
398C22×Q8C22×D4711yescharacteristic
399C22×D4—21no—
400C22×D4—942no—
401C22×D4C22×D4711yescharacteristic
402C23×C4—21no—
403C23×C4—21no—
404C23×C4—471no—
405C23×C4—472no—
406C23×C4—472no—
407C23×C4—941no—
408C23×C4—942no—
409C23×C4—942no—
410C23×C4—942no—
411C23×C4—942no—
412C23×C4—942no—
413C23×C4—942no—
414C23×C4C22×D4711yescharacteristic
415#137—232no—
416C2×Q16—24no—
417C2×Q16—24no—
418C2×Q16—24no—
419C2×SD16—24no—
420C2×SD16—24no—
421C2×SD16—24no—
422C2×M16—22no—
423C2×M16—22no—
424C2×M16—942no—
425C2×M16—942no—
426C2×M16—942no—
427C2×M16C22⋊D4714yescomplemented
428C22×C8—472no—
429C22×C8C22×D4712yes—
430#123—948no—
431C22⋊Q8—948no—
432C22⋊Q8—944no—
433C22⋊Q8—9416no—
434C22⋊Q8—948no—
435C22⋊Q8—944no—
436#121—944no—
437#121—944no—
438#120—948no—
439C4×D4—9416no—
440#117—474no—
441#117—942no—
442C2×C4⋊C4—474no—
443C2×C4⋊C4—472no—
444C2×C4⋊C4—472no—
445C2×C4⋊C4—944no—
446C2×C4⋊C4—942no—
447C2×C4⋊C4—948no—
448C2×C4⋊C4—941no—
449C2×C4⋊C4—942no—
450C2×C22⋊C4—944no—
451C2×C22⋊C4—942no—
452C2×C22⋊C4—948no—
453C2×C22⋊C4—942no—
454C2×C22⋊C4—941no—
455C2×C42—941no—
456#103—944no—
457#103—948no—
458#102—944no—
459C22⋊C8—948no—
460C24—21no—
461C24—472no—
462C24—944no—
463C24—942no—
464C24—942no—
465C24—942no—
466C24—1882no—
467C24D4×D4711yescharacteristic
468#41—28no—
469#41—28no—
470#41—44no—
471#41—44no—
472C2×Q8—24no—
473C2×Q8—22no—
474C2×Q8—21no—
475C2×Q8—28no—
476C2×Q8—28no—
477C2×Q8—22no—
478C2×Q8—21no—
479C2×Q8—22no—
480C2×Q8—22no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—22no—
484C2×Q8—21no—
485C2×Q8—42no—
486C2×Q8—42no—
487C2×Q8—44no—
488C2×Q8—44no—
489C2×Q8C2×C22⋊D4714yes—
490C2×Q8C2×C22⋊D4712yes—
491C2×D4—24no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—42no—
498C2×D4—42no—
499C2×D4—948no—
500C2×D4—1884no—
501C2×D4—1884no—
502C2×D4C2×C22⋊D4712yes—
503C22×C4—24no—
504C22×C4—21no—
505C22×C4—22no—
506C22×C4—24no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—472no—
512C22×C4—944no—
513C22×C4—942no—
514C22×C4—942no—
515C22×C4—942no—
516C22×C4—942no—
517C22×C4—942no—
518C22×C4—944no—
519C22×C4—944no—
520C22×C4—944no—
521C22×C4—942no—
522C22×C4—942no—
523C22×C4—942no—
524C22×C4—942no—
525C22×C4—942no—
526C22×C4—942no—
527C22×C4—1884no—
528C22×C4—1884no—
529C22×C4—1884no—
530C22×C4—1884no—
531C22×C4—1884no—
532C22×C4D4×D4711yescharacteristic
533C22×C4D4×D4712yes—
534C22×C4D4×D4712yes—
535C22×C4#1062535212yes—
536C22×C4C2×C22⋊D4712yes—
537C22×C4C2×C22⋊D4712yes—
538C22×C4C2×C22⋊D4712yes—
539C22×C4C23×D4711yescharacteristic
540Q16—216no—
541SD16—216no—
542M16—28no—
543M16—1884no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—942no—
547C2×C8—942no—
548C2×C8—942no—
549C2×C8C2×C22⋊D4714yes—
550C4⋊C4—474no—
551C4⋊C4—474no—
552C4⋊C4—9416no—
553C4⋊C4—948no—
554C4⋊C4—944no—
555C4⋊C4—942no—
556C4⋊C4—942no—
557C22⋊C4—9416no—
558C22⋊C4—948no—
559C22⋊C4—948no—
560C22⋊C4—944no—
561C42—944no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—41no—
566C23—41no—
567C23—42no—
568C23—42no—
569C23—474no—
570C23—948no—
571C23—942no—
572C23—942no—
573C23—942no—
574C23—942no—
575C23—942no—
576C23—942no—
577C23—1888no—
578C23—1888no—
579C23—1884no—
580C23—1884no—
581C23#1309399212yes—
582C23#1309399212yes—
583C23#1309399512yes—
584C23C2×D4×D4711yescharacteristic
585Q8—24no—
586Q8—22no—
587Q8—24no—
588Q8—22no—
589Q8—48no—
590Q8—42no—
591Q8—41no—
592Q8—42no—
593Q8—41no—
594D4—22no—
595D4—22no—
596D4—48no—
597D4—41no—
598D4—41no—
599D4—18816no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—48no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—48no—
608C2×C4—944no—
609C2×C4—944no—
610C2×C4—942no—
611C2×C4—942no—
612C2×C4—942no—
613C2×C4—1888no—
614C2×C4—1884no—
615C2×C4—1882no—
616C2×C4—1884no—
617C2×C4—1882no—
618C2×C4—1888no—
619C2×C4—1882no—
620C2×C4—1882no—
621C2×C4—1882no—
622C2×C4—1882no—
623C2×C4#1309399212yes—
624C2×C4#1309399212yes—
625C2×C4#1309399314yes—
626C2×C4#1309399512yes—
627C2×C4#1309399614yes—
628C2×C4#1309399614yes—
629C2×C4#1309402014yes—
630C2×C4C2×D4×D4711yescharacteristic
631C2×C4C2×D4×D4711yescharacteristic
632C2×C4C22×C22⋊D4711yescharacteristic
633C2×C4C22×C22⋊D4711yescharacteristic
634C2×C4C22×C22⋊D4711yescharacteristic
635C2×C4C2×#1062535211yescharacteristic
636C8—24no—
637C8—1882no—
638C22—24no—
639C22—22no—
640C22—22no—
641C22—24no—
642C22—42no—
643C22—44no—
644C22—82no—
645C22—81no—
646C22—81no—
647C22—944no—
648C22—944no—
649C22—944no—
650C22—18816no—
651C22—1882no—
652C22—1882no—
653C22—1882no—
654C22—1882no—
655C22#1568903114yes—
656C22C2×#1309399211yescharacteristic
657C22C2×#1309399211yescharacteristic
658C22C2×#1309399511yescharacteristic
659C22#137×D4712yes—
660C22#1568992212yes—
661C4—42no—
662C4—44no—
663C4—1882no—
664C4—1884no—
665C4—1884no—
666C4#1568903114yes—
667C4C2×#1309399311yescharacteristic
668C4C2×#1309399611yescharacteristic
669C4C2×#1309399611yescharacteristic
670C4C2×#1309402011yescharacteristic
671C2—24no—
672C2—82no—
673C2—1884no—
674C2C2×#1568903111yescharacteristic
675C2#2025262011yescharacteristic
676C2#2025262911yescharacteristic
677C2#2025264412yes—
678C2#2025301212yes—
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC22×#137×C47C2
socleC23×C47C22×D4
prime radicalC2×#15689630C2
hypercenterC22×#137D47
hypocenterC47#23392
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
2. C22×#13094000C2semidirect
3. C2×#15689124C2semidirect
4. C2×#15689132C2semidirect
5. C2×#15689086C2semidirect
6. #20244395C2semidirect
7. #20244388C2semidirect
8. #20243108C2semidirect
9. C22×#137×C47C2semidirect
29. C2×#13093948C22semidirect
30. C2×#13093948C22semidirect
31. C2×#13093945C22semidirect
34. #15688886C22semidirect
40. C22×Q16×C47C22semidirect
41. C22×SD16×C47C22semidirect
73. C2×C4⋊Dic47D4semidirect
101. C22×Q8×C47D4semidirect
118. C2×M16×C47D4semidirect
244. C22×D47#137semidirect
332. C22×#137D47semidirect
355. C22×Q16C2×D47semidirect
356. C22×SD16C2×D47semidirect
382. C47#23392semidirect
396. C22×Q8C22⋊D47semidirect
427. C2×M16C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896302
C22×#130940002
C2×#156891242
C2×#156891322
C2×#156890862
#202443952
#202443882
#202431082
C22×#137×C472
#2339247