Finite Group Databasebeta

← #26243134

Subgroups — #26243134

Order 12032.

2206 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094049C211yesmaximal, characteristic, complemented
2C2×#15689687C211yesmaximal, characteristic, complemented
3C2×#15689119C211yesmaximal, characteristic, complemented
4C22×#13093944C211yesmaximal, characteristic, complemented
5C2×#15689090C211yesmaximal, characteristic, complemented
6#20244389C214yesmaximal, complemented
7#20244384C214yesmaximal, complemented
8#20243107C211yesmaximal, characteristic, complemented
9C2×C47×#537C211yesmaximal, characteristic, complemented
10C23×D188C2211yescharacteristic
11C2×#13094049—24no—
12C2×#13094049—24no—
13C2×#13094049—24no—
14C22×#10625336C2212yes—
15C22×#10625334C2212yes—
16C22×#10625329C2211yescharacteristic
17C23×C47⋊C8C2211yescharacteristic
18#15689687—24no—
19#15689687—24no—
20C2×#13094020—21no—
21C2×#13094013—21no—
22C2×#13093998C2214yes—
23C2×#13093992—22no—
24C2×#13093983C2211yescharacteristic
25#15689119—24no—
26C2×#13093948C2212yescomplemented
27C2×#13093945C2212yescomplemented
28C2×#13093944—24no—
29C2×#13093944C2214yes—
30#15689090—24no—
31C2×C22⋊(C47⋊C8)C2214yescomplemented
32#15688885C2214yescomplemented
33C22×#41×C47C2211yescharacteristic
34C22×#41×C47C2211yescharacteristic
35C47×#537—24no—
36C47×#537—24no—
37C2×C22⋊Q8×C47—21no—
38C2×C22⋊Q8×C47C2212yescomplemented
39C2×#121×C47—21no—
40C2×#121×C47—22no—
41C2×#121×C47C2212yescomplemented
42C22×C4⋊C4×C47C2211yescharacteristic
43C24×D47—21no—
44C22×D188—22no—
45C22×D188—22no—
46C22×D188—22no—
47C22×D188—22no—
48C22×D188—21no—
49C22×D188—22no—
50C22×D188D414yescomplemented
51C22×D188C2312yes—
52C22×C22⋊D47—24no—
53#13094049—232no—
54C2×#10625336—24no—
55C2×#10625336—24no—
56C2×#10625336—24no—
57C2×#10625334—24no—
58C2×#10625334—24no—
59C2×#10625334—24no—
60C2×#10625329—22no—
61C2×#10625329—22no—
62C2×#10625329D414yescomplemented
63C22×C47⋊C8—22no—
64C22×C47⋊C8—22no—
65C22×C47⋊C8—22no—
66C22×C47⋊C8C2312yes—
67C22×C47⋊C8C2312yes—
68#13094020—28no—
69#13094013—28no—
70#13093998—28no—
71#13093998—28no—
72#13093992—216no—
73#13093983—22no—
74#13093983D414yescomplemented
75C2×C4⋊Dic47C2312yes—
76C2×C22⋊Dic47—21no—
77C2×(C2×D47)⋊C4—24no—
78C2×C4×Dic47—21no—
79#13093948—24no—
80#13093945—24no—
81#13093944—28no—
82C22⋊(C47⋊C8)—28no—
83C2×#41×C47—22no—
84C2×#41×C47—22no—
85C2×#41×C47—22no—
86C2×#41×C47—22no—
87C2×#41×C47—22no—
88C2×#41×C47—22no—
89C2×#41×C47—22no—
90C2×#41×C47—22no—
91C2×#41×C47—22no—
92C2×#41×C47—22no—
93C2×#41×C47—22no—
94C2×#41×C47D414yescomplemented
95C22×Q8×C47D411yescharacteristic
96C22×Q8×C47C2311yescharacteristic
97C22×D4×C47—21no—
98C22×D4×C47—22no—
99C22×D4×C47D412yes—
100C22×D4×C47D411yescharacteristic
101C22×D4×C47C2311yescharacteristic
102C23×C4×C47—21no—
103C23×C4×C47—21no—
104C23×C4×C47C2312yes—
105C23×C4×C47C2311yescharacteristic
106C22⋊Q8×C47—28no—
107C22⋊Q8×C47—24no—
108C22⋊Q8×C47—24no—
109#121×C47—216no—
110#121×C47—28no—
111#121×C47—24no—
112#121×C47—24no—
113C2×C4⋊C4×C47—21no—
114C2×C4⋊C4×C47—22no—
115C2×C4⋊C4×C47—22no—
116C2×C4⋊C4×C47D414yescomplemented
117C2×C4⋊C4×C47C2312yes—
118C2×C22⋊C4×C47—22no—
119C2×C22⋊C4×C47—22no—
120C24×C47—21no—
121C24×C47C2×D412yes—
122C24×C47C2×D411yescharacteristic
123C23×D47—22no—
124C23×D47—24no—
125C23×D47—42no—
126C23×D47—42no—
127C23×D47—82no—
128C23×D47—82no—
129#41×C47—28no—
130#41×C47—28no—
131#41×C47—44no—
132#41×C47—44no—
133#41×C47—44no—
134#41×C47—44no—
135#41×C47—44no—
136#41×C47—44no—
137C2×Q8×C47—22no—
138C2×Q8×C47—22no—
139C2×Q8×C47—22no—
140C2×Q8×C47—21no—
141C2×Q8×C47—21no—
142C2×Q8×C47—21no—
143C2×Q8×C47—21no—
144C2×Q8×C47—21no—
145C2×Q8×C47C2×D412yes—
146C2×D4×C47—24no—
147C2×D4×C47—24no—
148C2×D4×C47—24no—
149C2×D4×C47—24no—
150C2×D4×C47—28no—
151C2×D4×C47—22no—
152C2×D4×C47—22no—
153C2×D4×C47—22no—
154C2×D4×C47—21no—
155C2×D4×C47—21no—
156C2×D4×C47—21no—
157C2×D4×C47—21no—
158C2×D4×C47—21no—
159C2×D4×C47—42no—
160C2×D4×C47—42no—
161C2×D4×C47—84no—
162C2×D4×C47C2×D412yes—
163C22×C4×C47—22no—
164C22×C4×C47—21no—
165C22×C4×C47—24no—
166C22×C4×C47—24no—
167C22×C4×C47—22no—
168C22×C4×C47—22no—
169C22×C4×C47—22no—
170C22×C4×C47—22no—
171C22×C4×C47—22no—
172C22×C4×C47—22no—
173C22×C4×C47—42no—
174C22×C4×C47—42no—
175C22×C4×C47—82no—
176C22×C4×C47C2×D412yes—
177C22×C4×C47C2×D411yescharacteristic
178C22×C4×C47C2×D412yes—
179C22×C4×C47C2×D412yes—
180C22×C4×C47C2×D412yes—
181C22×C4×C47C2×D412yes—
182C22×C4×C47C2×D412yes—
183C22×C4×C47#4112yes—
184C22×C4×C47C2411yescharacteristic
185C2×C22⋊D47—216no—
186C2×C22⋊D47—84no—
187C2×C22⋊D47—84no—
188C22×Dic47—22no—
189C2×D188—28no—
190C2×D188—28no—
191C2×D188—22no—
192C2×D188—22no—
193C2×D188—22no—
194C2×D188—22no—
195C2×D188—24no—
196C2×D188—22no—
197C2×D188—42no—
198C2×D188—44no—
199C2×D188—42no—
200C2×D188—44no—
201C2×D188C2×D414yes—
202C4⋊C4×C47—28no—
203C4⋊C4×C47—24no—
204C4⋊C4×C47—22no—
205C4⋊C4×C47C2×D414yes—
206C22⋊C4×C47—28no—
207C22⋊C4×C47—28no—
208C22⋊Dic47—24no—
209#10625336—216no—
210#10625334—216no—
211(C2×D47)⋊C4—216no—
212C4⋊Dic47—22no—
213C4⋊Dic47C2×D414yes—
214C4×Dic47—24no—
215#10625329—28no—
216C2×C47⋊C8—22no—
217C2×C47⋊C8—22no—
218C2×C47⋊C8—22no—
219C2×C47⋊C8—22no—
220C2×C47⋊C8—22no—
221C2×C47⋊C8—44no—
222C2×C47⋊C8C2×D414yes—
223C23×C47—22no—
224C23×C47—22no—
225C23×C47—22no—
226C23×C47—22no—
227C23×C47—22no—
228C23×C47—22no—
229C23×C47—24no—
230C23×C47—24no—
231C23×C47—41no—
232C23×C47—41no—
233C23×C47—42no—
234C23×C47—42no—
235C23×C47#12012yes—
236C23×C47#12012yes—
237C23×C47#12112yes—
238C23×C47C22×D411yescharacteristic
239C22×D47—24no—
240C22×D47—42no—
241C22×D47—42no—
242C22×D47—44no—
243C22×D47—44no—
244C22×D47—48no—
245C22×D47—88no—
246C22×D47—84no—
247C22×D47—84no—
248C22×D47—88no—
249Q8×C47—22no—
250Q8×C47—22no—
251Q8×C47—41no—
252Q8×C47—41no—
253Q8×C47—44no—
254D4×C47—22no—
255D4×C47—22no—
256D4×C47—48no—
257D4×C47—48no—
258D4×C47—41no—
259D4×C47—41no—
260D4×C47—44no—
261D4×C47—88no—
262C2×C4×C47—22no—
263C2×C4×C47—24no—
264C2×C4×C47—22no—
265C2×C4×C47—22no—
266C2×C4×C47—22no—
267C2×C4×C47—41no—
268C2×C4×C47—41no—
269C2×C4×C47—44no—
270C2×C4×C47—42no—
271C2×C4×C47—42no—
272C2×C4×C47—42no—
273C2×C4×C47—42no—
274C2×C4×C47—48no—
275C2×C4×C47—84no—
276C2×C4×C47#12014yes—
277C2×C4×C47#12012yes—
278C2×C4×C47#12012yes—
279C2×C4×C47#12114yes—
280C2×C4×C47#12114yes—
281C2×C4×C47#12112yes—
282C2×C4×C47#12714yes—
283C2×C4×C47C22×D411yescharacteristic
284C2×C4×C47C22×D411yescharacteristic
285C2×C4×C47C22×D411yescharacteristic
286C2×C4×C47C22×D411yescharacteristic
287C2×C4×C47C22×D411yescharacteristic
288C2×C4×C47C2×#4111yescharacteristic
289C22⋊D47—816no—
290C2×Dic47—24no—
291C2×Dic47—42no—
292C2×Dic47—42no—
293D188—24no—
294D188—24no—
295D188—42no—
296D188—42no—
297D188—48no—
298C47⋊C8—24no—
299C47⋊C8—42no—
300#23393—471nomaximal, Sylow
301C22×C47—22no—
302C22×C47—22no—
303C22×C47—24no—
304C22×C47—24no—
305C22×C47—42no—
306C22×C47—42no—
307C22×C47—42no—
308C22×C47—42no—
309C22×C47—42no—
310C22×C47—42no—
311C22×C47—44no—
312C22×C47—44no—
313C22×C47—82no—
314C22×C47—82no—
315C22×C47#39214yes—
316C22×C47C2×#12011yescharacteristic
317C22×C47C2×#12011yescharacteristic
318C22×C47C2×#12111yescharacteristic
319C22×C47#57612yes—
320C22×C47#57712yes—
321C2×D47—44no—
322C2×D47—48no—
323C2×D47—816no—
324C2×D47—82no—
325C2×D47—82no—
326C2×D47—84no—
327C4×C47—42no—
328C4×C47—42no—
329C4×C47—44no—
330C4×C47#39214yes—
331C4×C47C2×#12011yescharacteristic
332C4×C47C2×#12111yescharacteristic
333C4×C47C2×#12111yescharacteristic
334C4×C47C2×#12711yescharacteristic
335Dic47—44no—
336C22×AGL(1,ℤ/8)—471no—
337C2×#537D4711yescharacteristic, complemented
338C2×#535—471no—
339C2×#416—471no—
340C22×#102—471no—
341C2×#408—471no—
342#2000—474no—
343#1999—474no—
344#1736—471no—
345C2×C47—24no—
346C2×C47—82no—
347C2×C47—82no—
348C2×C47C2×#39211yescharacteristic
349C2×C47#300811yescharacteristic
350C2×C47#301011yescharacteristic
351C2×C47#301212yes—
352C2×C47#306212yes—
353D47—84no—
354C22×#41C2×D4711yescharacteristic
355C22×#41C2×D4711yescharacteristic
356C23×D4—471no—
357C2×AGL(1,ℤ/8)—944no—
358C2×AGL(1,ℤ/8)—944no—
359C2×AGL(1,ℤ/8)—944no—
360C22×SD16—472no—
361C22×D8—472no—
362C22×M16—471no—
363C23×C8—471no—
364#537—24no—
365#537—24no—
366#535—944no—
367#535—944no—
368C2×#127—941no—
369C2×#124—941no—
370C2×C22⋊Q8—21no—
371C2×C22⋊Q8C2×D4712yescomplemented
372C2×#121—21no—
373C2×#121—22no—
374C2×#121—474no—
375C2×#121C2×D4712yescomplemented
376C2×#120—942no—
377C2×#117—471no—
378C22×C4⋊C4C2×D4711yescharacteristic
379#416—944no—
380C2×#103—472no—
381C2×#102—472no—
382C2×#102—474no—
383C2×#102—944no—
384#408—944no—
385C2×C22⋊C8—474no—
386#340—474no—
387C47#2339311yescharacteristic, complemented, Sylow
388C25—941no—
389C2×#41—22no—
390C2×#41—22no—
391C2×#41—22no—
392C2×#41—22no—
393C2×#41—22no—
394C2×#41—22no—
395C2×#41—22no—
396C2×#41—22no—
397C2×#41—22no—
398C2×#41—22no—
399C2×#41—22no—
400C2×#41C22⋊D4714yescomplemented
401C22×Q8C22⋊D4711yescharacteristic
402C22×Q8C22×D4711yescharacteristic
403C22×D4—21no—
404C22×D4—22no—
405C22×D4—474no—
406C22×D4—472no—
407C22×D4—944no—
408C22×D4—942no—
409C22×D4—942no—
410C22×D4—942no—
411C22×D4—942no—
412C22×D4—942no—
413C22×D4—941no—
414C22×D4C22⋊D4712yes—
415C22×D4C22⋊D4711yescharacteristic
416C22×D4C22×D4711yescharacteristic
417C23×C4—21no—
418C23×C4—21no—
419C23×C4C22×D4712yes—
420C23×C4C22×D4711yescharacteristic
421AGL(1,ℤ/8)—9432no—
422C2×SD16—944no—
423C2×SD16—944no—
424C2×SD16—944no—
425C2×D8—944no—
426C2×D8—944no—
427C2×D8—944no—
428C2×M16—474no—
429C2×M16—942no—
430C2×M16—942no—
431C22×C8—472no—
432C22×C8—472no—
433C22×C8—942no—
434C22×C8—942no—
435C22×C8—942no—
436#127—948no—
437#124—948no—
438C22⋊Q8—28no—
439C22⋊Q8—24no—
440C22⋊Q8—24no—
441#121—216no—
442#121—28no—
443#121—24no—
444#121—24no—
445#121—948no—
446#121—948no—
447#120—9416no—
448#117—474no—
449#117—942no—
450C2×C4⋊C4—21no—
451C2×C4⋊C4—22no—
452C2×C4⋊C4—22no—
453C2×C4⋊C4—472no—
454C2×C4⋊C4C22⋊D4714yescomplemented
455C2×C4⋊C4C22×D4712yes—
456C2×C22⋊C4—22no—
457C2×C22⋊C4—22no—
458C2×C22⋊C4—941no—
459C2×C22⋊C4—944no—
460C2×C42—941no—
461#103—944no—
462#102—944no—
463#102—948no—
464C22⋊C8—948no—
465C24—21no—
466C24—944no—
467C24—942no—
468C24—3762no—
469C24—3762no—
470C24—1882no—
471C24—1882no—
472C24C2×C22⋊D4712yes—
473C24C2×C22⋊D4711yescharacteristic
474#41—28no—
475#41—28no—
476#41—44no—
477#41—44no—
478#41—44no—
479#41—44no—
480#41—44no—
481#41—44no—
482C2×Q8—22no—
483C2×Q8—22no—
484C2×Q8—22no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8C2×C22⋊D4712yes—
491C2×D4—24no—
492C2×D4—24no—
493C2×D4—24no—
494C2×D4—24no—
495C2×D4—28no—
496C2×D4—22no—
497C2×D4—22no—
498C2×D4—22no—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—21no—
504C2×D4—42no—
505C2×D4—42no—
506C2×D4—84no—
507C2×D4—474no—
508C2×D4—9416no—
509C2×D4—948no—
510C2×D4—944no—
511C2×D4—948no—
512C2×D4—942no—
513C2×D4—942no—
514C2×D4—942no—
515C2×D4—942no—
516C2×D4—942no—
517C2×D4—3764no—
518C2×D4—3764no—
519C2×D4—1882no—
520C2×D4—1882no—
521C2×D4—1884no—
522C2×D4—1884no—
523C2×D4C2×C22⋊D4712yes—
524C22×C4—22no—
525C22×C4—21no—
526C22×C4—24no—
527C22×C4—24no—
528C22×C4—22no—
529C22×C4—22no—
530C22×C4—22no—
531C22×C4—22no—
532C22×C4—22no—
533C22×C4—22no—
534C22×C4—42no—
535C22×C4—42no—
536C22×C4—82no—
537C22×C4—942no—
538C22×C4D4×D4712yes—
539C22×C4D4×D4712yes—
540C22×C4D4×D4712yes—
541C22×C4#1062535212yes—
542C22×C4C2×C22⋊D4712yes—
543C22×C4C2×C22⋊D4711yescharacteristic
544C22×C4C2×C22⋊D4712yes—
545C22×C4C2×C22⋊D4712yes—
546C22×C4C23×D4711yescharacteristic
547SD16—9416no—
548D8—9416no—
549M16—948no—
550C2×C8—474no—
551C2×C8—942no—
552C2×C8—942no—
553C2×C8—942no—
554C2×C8—942no—
555C2×C8—942no—
556C2×C8—1884no—
557C4⋊C4—28no—
558C4⋊C4—24no—
559C4⋊C4—22no—
560C4⋊C4—474no—
561C4⋊C4—942no—
562C4⋊C4C2×C22⋊D4714yes—
563C22⋊C4—28no—
564C22⋊C4—28no—
565C22⋊C4—9416no—
566C22⋊C4—944no—
567C42—944no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—24no—
575C23—24no—
576C23—41no—
577C23—41no—
578C23—42no—
579C23—42no—
580C23—944no—
581C23—3764no—
582C23—3768no—
583C23—3764no—
584C23—3768no—
585C23—1888no—
586C23—1882no—
587C23—1882no—
588C23—1884no—
589C23—1884no—
590C23#1309399212yes—
591C23#1309399312yes—
592C23#1309399612yes—
593C23C22×C22⋊D4711yescharacteristic
594Q8—22no—
595Q8—22no—
596Q8—44no—
597Q8—41no—
598Q8—41no—
599D4—22no—
600D4—22no—
601D4—48no—
602D4—48no—
603D4—44no—
604D4—41no—
605D4—41no—
606D4—88no—
607D4—944no—
608D4—944no—
609D4—37616no—
610D4—1888no—
611D4—1882no—
612D4—1882no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—48no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—84no—
627C2×C4—944no—
628C2×C4—1882no—
629C2×C4—1882no—
630C2×C4#1309399212yes—
631C2×C4#1309399214yes—
632C2×C4#1309399312yes—
633C2×C4#1309399514yes—
634C2×C4#1309399612yes—
635C2×C4#1309399614yes—
636C2×C4#1309402014yes—
637C2×C4C2×D4×D4711yescharacteristic
638C2×C4C2×D4×D4711yescharacteristic
639C2×C4C2×D4×D4711yescharacteristic
640C2×C4C22×C22⋊D4711yescharacteristic
641C2×C4C22×C22⋊D4711yescharacteristic
642C2×C4C2×#1062535211yescharacteristic
643C8—944no—
644C8—1882no—
645C22—22no—
646C22—24no—
647C22—24no—
648C22—22no—
649C22—42no—
650C22—42no—
651C22—44no—
652C22—42no—
653C22—42no—
654C22—44no—
655C22—42no—
656C22—42no—
657C22—82no—
658C22—82no—
659C22—37616no—
660C22—3762no—
661C22—3764no—
662C22—3762no—
663C22—1888no—
664C22—1884no—
665C22#1568903114yes—
666C22C2×#1309399211yescharacteristic
667C22C2×#1309399311yescharacteristic
668C22C2×#1309399611yescharacteristic
669C22#1568992612yes—
670C22#1568993112yes—
671C4—42no—
672C4—42no—
673C4—44no—
674C4—1884no—
675C4#1568903114yes—
676C4C2×#1309399211yescharacteristic
677C4C2×#1309399511yescharacteristic
678C4C2×#1309399611yescharacteristic
679C4C2×#1309402011yescharacteristic
680C2—24no—
681C2—82no—
682C2—82no—
683C2—3764no—
684C2C2×#1568903111yescharacteristic
685C2#2025265611yescharacteristic
686C2#2025267411yescharacteristic
687C2#2025268912yes—
688C2#2025307012yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#537C2
socleC23×C47C22×D4
hypercenterC2×#537D47
hypocenterC47#23393
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. C22×#13094049C2semidirect
2. C2×#15689687C2semidirect
3. C2×#15689119C2semidirect
4. C22×#13093944C2semidirect
5. C2×#15689090C2semidirect
6. #20244389C2semidirect
7. #20244384C2semidirect
8. #20243107C2semidirect
9. C2×C47×#537C2semidirect
26. C2×#13093948C22semidirect
27. C2×#13093945C22semidirect
31. C2×C22⋊(C47⋊C8)C22semidirect
32. #15688885C22semidirect
38. C2×C22⋊Q8×C47C22semidirect
41. C2×#121×C47C22semidirect
50. C22×D188D4semidirect
62. C2×#10625329D4semidirect
74. #13093983D4semidirect
94. C2×#41×C47D4semidirect
116. C2×C4⋊C4×C47D4semidirect
337. C2×#537D47semidirect
371. C2×C22⋊Q8C2×D47semidirect
375. C2×#121C2×D47semidirect
387. C47#23393semidirect
400. C2×#41C22⋊D47semidirect
454. C2×C4⋊C4C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C22×#130940492
C2×#156896872
C2×#156891192
C22×#130939442
C2×#156890902
#202443892
#202443842
#202431072
C2×C47×#5372
#2339347