Finite Group Databasebeta

← #26217810

Subgroups — #26217810

Order 12032.

1862 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094049C211yesmaximal, characteristic
2C2×#15689140C211yesmaximal, characteristic, complemented
3C2×#15689127C211yesmaximal, characteristic, complemented
4#20245968C211yesmaximal, characteristic, complemented
5#20243764C218yesmaximal, complemented
6#20243127C211yesmaximal, characteristic, complemented
7#20243097C211yesmaximal, characteristic, complemented
8C47×#3428C211yesmaximal, characteristic, complemented
9C23×D188C2211yescharacteristic
10C2×#13094049—24no—
11C2×#13094049—22no—
12C2×#13094049C418yescomplemented
13C2×#13094049C2214yes—
14C22×#10625336—21no—
15C22×#10625334—21no—
16C22×#10625329C2211yescharacteristic
17C22×#10625329C2211yescharacteristic
18C2×#13093983C2211yescharacteristic
19#15689140—24no—
20C2×#13093951C2214yescomplemented
21#15689127—22no—
22#15689127C2214yescomplemented
23C2×#13093944—22no—
24#15689025—22no—
25#15689024C2214yescomplemented
26#15688978—22no—
27#15688892C2214yescomplemented
28#15688882C2214yescomplemented
29C22×#41×C47C2211yescharacteristic
30C47×#520—22no—
31C47×#520C2214yescomplemented
32C47×#519—21no—
33C47×#518—21no—
34C47×#518—22no—
35C4×#41×C47—22no—
36C4×#41×C47—22no—
37C2×#117×C47—22no—
38C2×#117×C47C2211yescharacteristic
39C2×#117×C47C2211yescharacteristic
40C24×D47—21no—
41C22×D188—21no—
42C22×D188—21no—
43C22×D188—21no—
44C22×D188—22no—
45C22×D188—21no—
46C22×D188—22no—
47C22×D188—21no—
48C22×D188—22no—
49C22×D188C2×C414yescomplemented
50C22×D188C2312yes—
51#13094049—216no—
52#13094049—28no—
53#13094049C2×C4116yescomplemented
54C2×#10625336—28no—
55C2×#10625336—24no—
56C2×#10625334—28no—
57C2×#10625334—24no—
58C2×#10625329—22no—
59C2×#10625329—21no—
60C2×#10625329—21no—
61C2×#10625329—22no—
62C2×#10625329—22no—
63C2×#10625329C2×C414yescomplemented
64C2×#10625329C2312yes—
65C2×#10625329C2312yes—
66C22×C47⋊C8—21no—
67C22×C47⋊C8—21no—
68#13093983—22no—
69#13093983—22no—
70C2×C4⋊Dic47—21no—
71C2×C22⋊Dic47—21no—
72C2×(C2×D47)⋊C4—28no—
73C2×C4×Dic47C2312yes—
74#13093951—28no—
75#13093944—28no—
76C2×#41×C47—21no—
77C2×#41×C47—21no—
78C2×#41×C47—22no—
79C2×#41×C47—22no—
80C2×#41×C47C2×C414yes—
81C2×#41×C47D414yes—
82C2×#41×C47D412yes—
83C2×#41×C47C2312yes—
84C22×Q8×C47D411yescharacteristic
85C22×D4×C47—21no—
86C22×D4×C47D411yescharacteristic
87C23×C4×C47—21no—
88C23×C4×C47C2311yescharacteristic
89C4×Q8×C47—24no—
90C4×Q8×C47—24no—
91C4×D4×C47—28no—
92C4×D4×C47—24no—
93C4×D4×C47—24no—
94C4×D4×C47—28no—
95#117×C47—24no—
96#117×C47—24no—
97#117×C47—21no—
98#117×C47—21no—
99#117×C47—21no—
100#117×C47—21no—
101#117×C47—21no—
102#117×C47—21no—
103#117×C47—24no—
104#117×C47D412yes—
105#117×C47C2312yes—
106C2×C4⋊C4×C47—24no—
107C2×C4⋊C4×C47D412yes—
108C2×C4⋊C4×C47C2312yes—
109C2×C22⋊C4×C47—24no—
110C2×C22⋊C4×C47—21no—
111C2×C22⋊C4×C47D412yes—
112C2×C42×C47—24no—
113C2×C42×C47—21no—
114C2×C42×C47D412yes—
115C24×C47—21no—
116C24×C47C2×D411yescharacteristic
117C23×D47—22no—
118C23×D47—24no—
119C23×D47—41no—
120C23×D47—41no—
121C23×D47—42no—
122C23×D47—84no—
123#41×C47—24no—
124#41×C47—24no—
125#41×C47—24no—
126#41×C47—24no—
127#41×C47—24no—
128#41×C47—22no—
129#41×C47—22no—
130#41×C47C22⋊C418yescomplemented
131#41×C47C22×C414yes—
132#41×C47C2×D414yes—
133C2×Q8×C47—21no—
134C2×Q8×C47—22no—
135C2×Q8×C47C22⋊C414yescomplemented
136C2×Q8×C47C2×D412yes—
137C2×D4×C47—24no—
138C2×D4×C47—22no—
139C2×D4×C47—24no—
140C2×D4×C47—22no—
141C2×D4×C47—21no—
142C2×D4×C47—22no—
143C2×D4×C47C22⋊C414yescomplemented
144C2×D4×C47C2×D412yes—
145C22×C4×C47—21no—
146C22×C4×C47—24no—
147C22×C4×C47—22no—
148C22×C4×C47—24no—
149C22×C4×C47—22no—
150C22×C4×C47—28no—
151C22×C4×C47—22no—
152C22×C4×C47C22×C412yes—
153C22×C4×C47C2×D414yes—
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D412yes—
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47#4112yes—
161C22×C4×C47#4112yes—
162C22×C4×C47C2411yescharacteristic
163C22×Dic47—22no—
164C2×D188—24no—
165C2×D188—24no—
166C2×D188—24no—
167C2×D188—24no—
168C2×D188—22no—
169C2×D188—22no—
170C2×D188—22no—
171C2×D188—22no—
172C2×D188—24no—
173C2×D188—22no—
174C2×D188—41no—
175C2×D188—41no—
176C2×D188—42no—
177C2×D188—42no—
178C2×D188—42no—
179C2×D188—42no—
180C2×D188—42no—
181C2×D188C22×C414yes—
182C4⋊C4×C47—28no—
183C4⋊C4×C47—22no—
184C4⋊C4×C47—22no—
185C4⋊C4×C47—22no—
186C4⋊C4×C47C2×D414yes—
187C22⋊C4×C47—28no—
188C22⋊C4×C47—22no—
189C22⋊C4×C47—22no—
190C22⋊C4×C47—22no—
191C22⋊C4×C47—22no—
192C42×C47—28no—
193C42×C47—22no—
194C42×C47—22no—
195C42×C47—22no—
196C42×C47—22no—
197C22⋊Dic47—42no—
198#10625336—216no—
199#10625334—216no—
200(C2×D47)⋊C4—88no—
201C4⋊Dic47—42no—
202C4×Dic47—22no—
203C4×Dic47—22no—
204#10625329—24no—
205#10625329—22no—
206#10625329—22no—
207#10625329—22no—
208#10625329—42no—
209#10625329C22×C414yes—
210C2×C47⋊C8—24no—
211C2×C47⋊C8—22no—
212C2×C47⋊C8—22no—
213C2×C47⋊C8—42no—
214C23×C47—22no—
215C23×C47—24no—
216C23×C47—42no—
217C23×C47—82no—
218C23×C47C4×D412yes—
219C23×C47#12011yescharacteristic
220C23×C47#12011yescharacteristic
221C23×C47#12112yes—
222C23×C47C22×D411yescharacteristic
223C22×D47—24no—
224C22×D47—48no—
225C22×D47—42no—
226C22×D47—42no—
227C22×D47—44no—
228C22×D47—44no—
229C22×D47—88no—
230C22×D47—84no—
231C22×D47—84no—
232C22×D47—88no—
233Q8×C47—22no—
234Q8×C47—24no—
235Q8×C47C2×C22⋊C414yes—
236D4×C47—24no—
237D4×C47—24no—
238D4×C47—28no—
239D4×C47—22no—
240D4×C47—24no—
241D4×C47C2×C22⋊C414yes—
242C2×C4×C47—22no—
243C2×C4×C47—24no—
244C2×C4×C47—24no—
245C2×C4×C47—24no—
246C2×C4×C47—28no—
247C2×C4×C47—24no—
248C2×C4×C47—24no—
249C2×C4×C47—44no—
250C2×C4×C47—82no—
251C2×C4×C47—88no—
252C2×C4×C47C2×C22⋊C412yes—
253C2×C4×C47C2×C22⋊C412yes—
254C2×C4×C47C4×D412yes—
255C2×C4×C47C4×D412yes—
256C2×C4×C47C4×D412yes—
257C2×C4×C47#12011yescharacteristic
258C2×C4×C47#12011yescharacteristic
259C2×C4×C47#12112yes—
260C2×C4×C47#12112yes—
261C2×C4×C47#12112yes—
262C2×C4×C47#12312yes—
263C2×C4×C47#12312yes—
264C2×C4×C47C23×C411yescharacteristic
265C2×C4×C47C22×D411yescharacteristic
266C2×C4×C47C22×D411yescharacteristic
267C2×C4×C47C22×D411yescharacteristic
268C2×C4×C47C2×#4111yescharacteristic
269C2×C4×C47C2×#4111yescharacteristic
270C2×Dic47—44no—
271C2×Dic47—42no—
272D188—24no—
273D188—24no—
274D188—42no—
275D188—42no—
276D188—44no—
277D188—44no—
278C47⋊C8—24no—
279C47⋊C8—42no—
280#20945—471nomaximal, Sylow
281C22×C47—22no—
282C22×C47—22no—
283C22×C47—22no—
284C22×C47—24no—
285C22×C47—42no—
286C22×C47—42no—
287C22×C47—44no—
288C22×C47—84no—
289C22×C47#38612yes—
290C22×C47#38612yes—
291C22×C47C2×C4×D411yescharacteristic
292C22×C47C2×#12011yescharacteristic
293C22×C47C2×#12111yescharacteristic
294C2×D47—48no—
295C2×D47—44no—
296C2×D47—84no—
297C2×D47—88no—
298C2×D47—88no—
299C2×D47—84no—
300C4×C47—24no—
301C4×C47—44no—
302C4×C47—44no—
303C4×C47#38612yes—
304C4×C47#38612yes—
305C4×C47C22×C22⋊C411yescharacteristic
306C4×C47C2×C4×D411yescharacteristic
307C4×C47C2×#12111yescharacteristic
308C4×C47C2×#12311yescharacteristic
309Dic47—82no—
310C22×AGL(1,ℤ/8)—471no—
311#3428D4711yescharacteristic, complemented
312C2×#421—471no—
313C2×#418—471no—
314#2309—471no—
315#1883—478no—
316#1742—471no—
317#1733—471no—
318C2×C47—21no—
319C2×C47—21no—
320C2×C47—22no—
321C2×C47—22no—
322C2×C47—44no—
323C2×C47C2×#38611yescharacteristic
324C2×C47#302112yes—
325D47—84no—
326C22×#41C2×D4711yescharacteristic
327C23×D4—471no—
328C2×AGL(1,ℤ/8)—478no—
329C2×AGL(1,ℤ/8)—474no—
330C2×AGL(1,ℤ/8)—944no—
331C2×AGL(1,ℤ/8)—942no—
332C22×SD16—941no—
333C22×D8—941no—
334C22×M16—471no—
335C22×M16—471no—
336#520—22no—
337#520C2×D4714yescomplemented
338#519—21no—
339#518—21no—
340#518—22no—
341C4×#41—22no—
342C4×#41—22no—
343C2×#117—22no—
344C2×#117—471no—
345C2×#117C2×D4711yescharacteristic
346C2×#117C2×D4711yescharacteristic
347#421—944no—
348C2×#104—474no—
349#418—474no—
350#418—942no—
351C2×#102—942no—
352#390—474no—
353#390—942no—
354#379—942no—
355#343—474no—
356#339—474no—
357C47#2094511yescharacteristic, complemented, Sylow
358C25—941no—
359C2×#41—22no—
360C2×#41—22no—
361C2×#41—21no—
362C2×#41—21no—
363C2×#41C4×D4714yes—
364C2×#41D18814yes—
365C2×#41C22⋊D4712yes—
366C2×#41C22×D4712yes—
367C22×Q8C22⋊D4711yescharacteristic
368C22×D4—21no—
369C22×D4—474no—
370C22×D4—472no—
371C22×D4—942no—
372C22×D4—942no—
373C22×D4—942no—
374C22×D4—941no—
375C22×D4—941no—
376C22×D4—941no—
377C22×D4—941no—
378C22×D4—941no—
379C22×D4C22⋊D4711yescharacteristic
380C23×C4—21no—
381C23×C4C22×D4711yescharacteristic
382AGL(1,ℤ/8)—4716no—
383AGL(1,ℤ/8)—9416no—
384AGL(1,ℤ/8)—948no—
385C2×SD16—948no—
386C2×SD16—944no—
387C2×D8—948no—
388C2×D8—944no—
389C2×M16—474no—
390C2×M16—472no—
391C2×M16—472no—
392C2×M16—942no—
393C2×M16—941no—
394C2×M16—941no—
395C2×M16—942no—
396C2×M16—942no—
397C22×C8—941no—
398C22×C8—941no—
399C4×Q8—24no—
400C4×Q8—24no—
401C4×D4—28no—
402C4×D4—24no—
403C4×D4—24no—
404C4×D4—28no—
405#117—24no—
406#117—24no—
407#117—21no—
408#117—21no—
409#117—21no—
410#117—21no—
411#117—21no—
412#117—21no—
413#117—24no—
414#117—942no—
415#117—942no—
416#117C22⋊D4712yes—
417#117C22×D4712yes—
418C2×C4⋊C4—24no—
419C2×C4⋊C4—941no—
420C2×C4⋊C4C22⋊D4712yes—
421C2×C4⋊C4C22×D4712yes—
422C2×C22⋊C4—24no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—948no—
425C2×C22⋊C4—941no—
426C2×C22⋊C4C22⋊D4712yes—
427C2×C42—24no—
428C2×C42—21no—
429C2×C42—472no—
430C2×C42C22⋊D4712yes—
431#104—948no—
432#102—948no—
433C24—21no—
434C24—944no—
435C24—942no—
436C24—3764no—
437C24—1881no—
438C24—1881no—
439C24—1882no—
440C24C2×C22⋊D4711yescharacteristic
441#41—24no—
442#41—24no—
443#41—24no—
444#41—24no—
445#41—24no—
446#41—22no—
447#41—22no—
448#41(C2×D47)⋊C418yescomplemented
449#41C2×C4×D4714yes—
450#41C2×D18814yes—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8(C2×D47)⋊C414yescomplemented
454C2×Q8C2×C22⋊D4712yes—
455C2×D4—24no—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4—474no—
462C2×D4—944no—
463C2×D4—944no—
464C2×D4—944no—
465C2×D4—944no—
466C2×D4—944no—
467C2×D4—942no—
468C2×D4—942no—
469C2×D4—942no—
470C2×D4—942no—
471C2×D4—942no—
472C2×D4—1881no—
473C2×D4—1881no—
474C2×D4—1882no—
475C2×D4—1882no—
476C2×D4—1882no—
477C2×D4—1882no—
478C2×D4—1882no—
479C2×D4(C2×D47)⋊C414yescomplemented
480C2×D4C2×C22⋊D4712yes—
481C22×C4—21no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—28no—
487C22×C4—22no—
488C22×C4—942no—
489C22×C4C2×C4×D4712yes—
490C22×C4C2×D18812yes—
491C22×C4#1062535012yes—
492C22×C4#1062535012yes—
493C22×C4C2×C22⋊D4714yes—
494C22×C4C2×C22⋊D4711yescharacteristic
495C22×C4C2×C22⋊D4711yescharacteristic
496C22×C4C2×C22⋊D4711yescharacteristic
497C22×C4C2×C22⋊D4711yescharacteristic
498C22×C4C2×C22⋊D4711yescharacteristic
499C22×C4C23×D4711yescharacteristic
500SD16—9416no—
501D8—9416no—
502M16—474no—
503M16—944no—
504M16—942no—
505M16—942no—
506M16—942no—
507M16—1882no—
508C2×C8—944no—
509C2×C8—942no—
510C2×C8—942no—
511C2×C8—1882no—
512C4⋊C4—28no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—1882no—
517C4⋊C4C2×C22⋊D4714yes—
518C22⋊C4—28no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—3768no—
524C22⋊C4—1882no—
525C42—28no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—942no—
531C42—942no—
532C23—22no—
533C23—24no—
534C23—42no—
535C23—82no—
536C23—944no—
537C23—3764no—
538C23—3764no—
539C23—3768no—
540C23—3768no—
541C23—1888no—
542C23—1884no—
543C23—1884no—
544C23—1882no—
545C23—1882no—
546C23C4×C22⋊D4712yes—
547C23#1309399311yescharacteristic
548C23#1309399311yescharacteristic
549C23#1309399812yes—
550C23C22×C22⋊D4711yescharacteristic
551Q8—24no—
552Q8—22no—
553Q8C2×(C2×D47)⋊C414yes—
554D4—28no—
555D4—24no—
556D4—24no—
557D4—24no—
558D4—22no—
559D4—944no—
560D4—944no—
561D4—1884no—
562D4—1884no—
563D4—1882no—
564D4—1882no—
565D4C2×(C2×D47)⋊C414yes—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—44no—
574C2×C4—82no—
575C2×C4—88no—
576C2×C4—1884no—
577C2×C4—1882no—
578C2×C4C2×(C2×D47)⋊C412yes—
579C2×C4C2×(C2×D47)⋊C412yes—
580C2×C4C4×C22⋊D4712yes—
581C2×C4C4×C22⋊D4712yes—
582C2×C4C4×C22⋊D4712yes—
583C2×C4#1309399311yescharacteristic
584C2×C4#1309399311yescharacteristic
585C2×C4#1309399812yes—
586C2×C4#1309399812yes—
587C2×C4#1309399812yes—
588C2×C4#1309400712yes—
589C2×C4#1309400712yes—
590C2×C4C22×C4×D4711yescharacteristic
591C2×C4C22×C22⋊D4711yescharacteristic
592C2×C4C22×C22⋊D4711yescharacteristic
593C2×C4C22×D18811yescharacteristic
594C2×C4C2×#1062535011yescharacteristic
595C2×C4C2×#1062535011yescharacteristic
596C8—944no—
597C8—1882no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—24no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22—84no—
606C22—3768no—
607C22—3768no—
608C22—3764no—
609C22—3764no—
610C22—1888no—
611C22—1884no—
612C22#1568900212yes—
613C22#1568900212yes—
614C22C2×C4×C22⋊D4711yescharacteristic
615C22C2×#1309399311yescharacteristic
616C22C2×#1309399811yescharacteristic
617C4—24no—
618C4—44no—
619C4—44no—
620C4—3762no—
621C4#1568900212yes—
622C4#1568900212yes—
623C4C22×(C2×D47)⋊C411yescharacteristic
624C4C2×C4×C22⋊D4711yescharacteristic
625C4C2×#1309399811yescharacteristic
626C4C2×#1309400711yescharacteristic
627C2—21no—
628C2—22no—
629C2—22no—
630C2—21no—
631C2—44no—
632C2—3764no—
633C2C2×#1568900211yescharacteristic
634C2#2025276012yes—
subgroupG/H
centerC22C2×#13093993
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#3428C2
socleC22×C47C2×#120
prime radicalC22×#13094049C2
cosocleC22×C4×C47C24
hypercenter#3428D47
hypocenterC47#20945
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3428

Complements

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

NH ≅ G/Ntype
2. C2×#15689140C2semidirect
3. C2×#15689127C2semidirect
4. #20245968C2semidirect
5. #20243764C2semidirect
6. #20243127C2semidirect
7. #20243097C2semidirect
8. C47×#3428C2semidirect
12. C2×#13094049C4semidirect
20. C2×#13093951C22semidirect
22. #15689127C22semidirect
25. #15689024C22semidirect
27. #15688892C22semidirect
28. #15688882C22semidirect
31. C47×#520C22semidirect
49. C22×D188C2×C4semidirect
53. #13094049C2×C4semidirect
63. C2×#10625329C2×C4semidirect
130. #41×C47C22⋊C4semidirect
135. C2×Q8×C47C22⋊C4semidirect
143. C2×D4×C47C22⋊C4semidirect
311. #3428D47semidirect
337. #520C2×D47semidirect
357. C47#20945semidirect
448. #41(C2×D47)⋊C4semidirect
453. C2×Q8(C2×D47)⋊C4semidirect
479. C2×D4(C2×D47)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#130940492
C2×#156891402
C2×#156891272
#202459682
#202437642
#202431272
#202430972
C47×#34282
#2094547