Finite Group Databasebeta

← #26250165

Subgroups — #26250165

Order 12032.

1434 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689400C211yesmaximal, characteristic, complemented
2C2×C4×#10625334C211yesmaximal, characteristic, complemented
3#20248301C211yesmaximal, characteristic, complemented
4#20244567C212yesmaximal, complemented
5#20244384C212yesmaximal, complemented
6#20244350C211yesmaximal, characteristic, complemented
7#20244183C211yesmaximal, characteristic, complemented
8#20243705C212yesmaximal, complemented
9#20243576C212yesmaximal, complemented
10#20243388C211yesmaximal, characteristic, complemented
11C2×C47×#547C211yesmaximal, characteristic, complemented
12C22×#10625334C2211yescharacteristic
13C2×#13093998C2212yes—
14C2×#13093995—21no—
15C2×C4×D188C2211yescharacteristic
16#15689400—24no—
17C4×#10625334—24no—
18C2×#13093957C2212yes—
19C2×#13093957C2211yescharacteristic
20C2×C4⋊(C47⋊C8)C2211yescharacteristic
21C2×#13093945C2211yescharacteristic
22C2×#13093945C2212yes—
23C2×#13093944C2212yes—
24C2×#13093944C2211yescharacteristic
25C2×C22⋊(C47⋊C8)C2212yes—
26C2×C22⋊(C47⋊C8)C2212yes—
27C2×C4×C47⋊C8C2211yescharacteristic
28#15689039—22no—
29#15688993C2211yescharacteristic
30#15688983C2212yes—
31#15688980—21no—
32#15688885C2212yes—
33#15688885C2212yes—
34#15688885C2212yes—
35C22×#41×C47—21no—
36C47×#547—24no—
37C47×#547—24no—
38C2×C4⋊Q8×C47C2211yescharacteristic
39C2×#123×C47—22no—
40C2×C22⋊Q8×C47—21no—
41C2×#121×C47C2212yes—
42C2×C4×D4×C47C2211yescharacteristic
43C2×C4×D4×C47C2211yescharacteristic
44C22×C4⋊C4×C47C2212yes—
45C22×D188C2311yescharacteristic
46C22×C22⋊D47—22no—
47C22×C4×D47—21no—
48C2×#10625334—22no—
49C2×#10625334—22no—
50C2×#10625334—22no—
51C22×C47⋊C8C2311yescharacteristic
52C22×C47⋊C8C2312yes—
53C22×C47⋊C8C2311yescharacteristic
54#13093998—24no—
55#13093998—42no—
56#13093995—44no—
57C4×D188—22no—
58C4×D188—22no—
59C2×C4⋊Dic47—21no—
60C2×C4⋊Dic47—21no—
61C2×C4⋊Dic47C2311yescharacteristic
62C2×C4⋊Dic47C2312yes—
63C2×C22⋊Dic47—22no—
64C2×(C2×D47)⋊C4—22no—
65C2×(C2×D47)⋊C4—21no—
66#13093957—24no—
67#13093957—22no—
68C4⋊(C47⋊C8)—22no—
69#13093945—22no—
70#13093945—42no—
71#13093944—24no—
72#13093944—22no—
73C22⋊(C47⋊C8)—24no—
74C22⋊(C47⋊C8)—42no—
75C4×C47⋊C8—22no—
76C47⋊#95—22no—
77C2×#41×C47—28no—
78C2×#41×C47—44no—
79C2×#41×C47—44no—
80C22×Q8×C47—21no—
81C22×D4×C47—22no—
82C22×D4×C47D411yescharacteristic
83C22×D4×C47C2311yescharacteristic
84C23×C4×C47—22no—
85C23×C4×C47—21no—
86C23×C4×C47C2312yes—
87C4⋊Q8×C47—22no—
88C4⋊Q8×C47D414yescomplemented
89#123×C47—216no—
90C22⋊Q8×C47—28no—
91#121×C47—24no—
92#121×C47—24no—
93C4×D4×C47—22no—
94C4×D4×C47—22no—
95C4×D4×C47—22no—
96C4×D4×C47D414yescomplemented
97C2×C4⋊C4×C47—22no—
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47—24no—
100C2×C4⋊C4×C47—24no—
101C2×C4⋊C4×C47D411yescharacteristic
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C4⋊C4×C47C2312yes—
104C2×C22⋊C4×C47—21no—
105C2×C22⋊C4×C47—22no—
106C2×C22⋊C4×C47D412yes—
107C2×C42×C47C2311yescharacteristic
108C24×C47—21no—
109C24×C47C2×D412yes—
110C23×D47—21no—
111#41×C47—432no—
112C2×Q8×C47—24no—
113C2×Q8×C47—42no—
114C2×Q8×C47—42no—
115C2×D4×C47—28no—
116C2×D4×C47—22no—
117C2×D4×C47—22no—
118C2×D4×C47—21no—
119C2×D4×C47—21no—
120C2×D4×C47—21no—
121C2×D4×C47—21no—
122C2×D4×C47—21no—
123C2×D4×C47—44no—
124C2×D4×C47—44no—
125C2×D4×C47D814yescomplemented
126C2×D4×C47C2×D412yes—
127C22×C4×C47—28no—
128C22×C4×C47—21no—
129C22×C4×C47—22no—
130C22×C4×C47—24no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—22no—
136C22×C4×C47—22no—
137C22×C4×C47—44no—
138C22×C4×C47—44no—
139C22×C4×C47—42no—
140C22×C4×C47—42no—
141C22×C4×C47C2×D412yes—
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2411yescharacteristic
145C2×C22⋊D47—44no—
146C2×C22⋊D47—44no—
147C2×C22⋊D47—82no—
148C22×Dic47—21no—
149C22×Dic47—22no—
150C2×D188—21no—
151C2×D188—21no—
152C2×D188—21no—
153C2×D188—21no—
154C2×D188—21no—
155C2×D188#4112yescomplemented
156C2×C4×D47—42no—
157C2×C4×D47—42no—
158C2×C4×D47—42no—
159C4⋊C4×C47—28no—
160C4⋊C4×C47—24no—
161C4⋊C4×C47—22no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47C2×D412yes—
165C4⋊C4×C47C2×D414yes—
166C22⋊C4×C47—28no—
167C22⋊C4×C47—24no—
168C22⋊C4×C47—24no—
169C42×C47C2×D412yes—
170C42×C47C2×D412yes—
171C22⋊Dic47—44no—
172#10625334—28no—
173(C2×D47)⋊C4—44no—
174(C2×D47)⋊C4—42no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—22no—
178C4⋊Dic47—22no—
179C4⋊Dic47—42no—
180C4⋊Dic47—81no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—21no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—22no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8#4112yescomplemented
190C23×C47—22no—
191C23×C47—22no—
192C23×C47—22no—
193C23×C47—22no—
194C23×C47—22no—
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—41no—
198C23×C47—41no—
199C23×C47—41no—
200C23×C47—41no—
201C23×C47—41no—
202C23×C47—41no—
203C23×C47C2×D814yes—
204C23×C47C22×D411yescharacteristic
205C22×D47—22no—
206C22×D47—41no—
207C22×D47—41no—
208C22×D47—41no—
209C22×D47—41no—
210C22×D47—41no—
211C22×D47—41no—
212Q8×C47—48no—
213D4×C47—22no—
214D4×C47—22no—
215D4×C47—28no—
216D4×C47—416no—
217D4×C47—41no—
218D4×C47—41no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—24no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—416no—
227C2×C4×C47—41no—
228C2×C4×C47—42no—
229C2×C4×C47—41no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47—42no—
234C2×C4×C47—42no—
235C2×C4×C47—48no—
236C2×C4×C47#12012yes—
237C2×C4×C47#12012yes—
238C2×C4×C47C2×D812yes—
239C2×C4×C47AGL(1,ℤ/8)12yes—
240C2×C4×C47C22×D411yescharacteristic
241C2×C4×C47C22×D411yescharacteristic
242C2×C4×C47C2×#4111yescharacteristic
243C2×C4×C47C2×#4111yescharacteristic
244C2×C4×C47H(5,𝔽2)11yescharacteristic
245C2×C4×C472−1+411yescharacteristic
246C22⋊D47—88no—
247C2×Dic47—41no—
248C2×Dic47—42no—
249C2×Dic47—42no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—42no—
253D188—22no—
254D188—22no—
255D188—41no—
256D188—41no—
257C4×D47—84no—
258C47⋊C8—22no—
259C47⋊C8—41no—
260C47⋊C8—42no—
261#24116—471nomaximal, Sylow
262C22×C47—24no—
263C22×C47—24no—
264C22×C47—24no—
265C22×C47—42no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—42no—
269C22×C47—42no—
270C22×C47—42no—
271C22×C47—42no—
272C22×C47—81no—
273C22×C47—81no—
274C22×C47—81no—
275C22×C47—81no—
276C22×C47C2×#12011yescharacteristic
277C22×C47#54611yescharacteristic
278C22×C47#54711yescharacteristic
279C22×C47C22×D811yescharacteristic
280C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C47#57711yescharacteristic
282C22×C47#57811yescharacteristic
283C2×D47—42no—
284C2×D47—42no—
285C2×D47—42no—
286C2×D47—81no—
287C2×D47—81no—
288C2×D47—81no—
289C2×D47—81no—
290C4×C47—22no—
291C4×C47—42no—
292C4×C47—42no—
293C4×C47—44no—
294C4×C47#44714yes—
295C4×C47#52511yescharacteristic
296C4×C47#53711yescharacteristic
297C4×C47#54811yescharacteristic
298C4×C47#55011yescharacteristic
299Dic47—81no—
300Dic47—82no—
301C2×#547D4711yescharacteristic, complemented
302C2×#474—471no—
303C2×C4×D8—471no—
304#2537—471no—
305#2033—472no—
306#1999—472no—
307#1990—471no—
308#1954—471no—
309#1874—472no—
310#1852—472no—
311#1802—471no—
312C2×C47—44no—
313C2×C47—82no—
314C2×C47#240711yescharacteristic
315C2×C47C2×#44711yescharacteristic
316C2×C47#301311yescharacteristic
317C2×C47#329411yescharacteristic
318C2×C47#333011yescharacteristic
319C2×C47#333411yescharacteristic
320C2×C47#335411yescharacteristic
321D47—82no—
322C22×#41—21no—
323C22×D8—471no—
324#547—24no—
325#547—24no—
326C2×C4⋊Q8C2×D4711yescharacteristic
327C2×#123—22no—
328C2×C22⋊Q8—21no—
329C2×#121—472no—
330C2×#121—941no—
331C2×#121C2×D4712yes—
332C2×C4×D4—471no—
333C2×C4×D4C2×D4711yescharacteristic
334C2×C4×D4C2×D4711yescharacteristic
335C22×C4⋊C4C2×D4712yes—
336#474—944no—
337C4×D8—944no—
338C2×#107—471no—
339C2×#107—472no—
340C2×C4⋊C8—471no—
341C2×#102—471no—
342C2×#102—471no—
343C2×#102—472no—
344C2×#102—472no—
345C2×C22⋊C8—472no—
346C2×C22⋊C8—472no—
347C2×C4×C8—471no—
348#394—942no—
349#384—471no—
350#380—472no—
351#380—941no—
352#340—472no—
353#340—472no—
354#340—472no—
355C47#2411611yescharacteristic, complemented, Sylow
356C2×#41—28no—
357C2×#41—44no—
358C2×#41—44no—
359C22×Q8—21no—
360C22×D4—22no—
361C22×D4—471no—
362C22×D4—942no—
363C22×D4C22⋊D4711yescharacteristic
364C22×D4C22×D4711yescharacteristic
365C23×C4—22no—
366C23×C4—21no—
367C23×C4—941no—
368C23×C4C22×D4712yes—
369C2×D8—942no—
370C2×D8—942no—
371C2×D8—942no—
372C22×C8—471no—
373C22×C8—471no—
374C22×C8—472no—
375C4⋊Q8—22no—
376C4⋊Q8C22⋊D4714yescomplemented
377#123—216no—
378C22⋊Q8—28no—
379#121—24no—
380#121—24no—
381#121—944no—
382#121—1882no—
383#121—1884no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—942no—
388C4×D4—942no—
389C4×D4C22⋊D4714yescomplemented
390C2×C4⋊C4—22no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—24no—
394C2×C4⋊C4—471no—
395C2×C4⋊C4—472no—
396C2×C4⋊C4—941no—
397C2×C4⋊C4—941no—
398C2×C4⋊C4C22⋊D4711yescharacteristic
399C2×C4⋊C4C22×D4711yescharacteristic
400C2×C4⋊C4C22×D4712yes—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—942no—
404C2×C22⋊C4—941no—
405C2×C22⋊C4—942no—
406C2×C22⋊C4C22⋊D4712yes—
407C2×C42C22×D4711yescharacteristic
408#107—942no—
409#107—944no—
410C4⋊C8—942no—
411#102—942no—
412#102—942no—
413#102—944no—
414#102—1882no—
415C22⋊C8—944no—
416C22⋊C8—1882no—
417C4×C8—942no—
418#95—942no—
419C24—21no—
420C24—941no—
421C24C2×C22⋊D4712yes—
422#41—432no—
423C2×Q8—24no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×D4—28no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—472no—
437C2×D4—941no—
438C2×D4—941no—
439C2×D4—941no—
440C2×D4—941no—
441C2×D4—941no—
442C2×D4—3762no—
443C2×D4—1884no—
444C2×D4—1884no—
445C2×D4#1062533414yescomplemented
446C2×D4C2×C22⋊D4712yes—
447C22×C4—28no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—941no—
462C22×C4—942no—
463C22×C4—1882no—
464C22×C4—1882no—
465C22×C4—1882no—
466C22×C4C2×C22⋊D4712yes—
467C22×C4C2×C22⋊D4711yescharacteristic
468C22×C4C2×C22⋊D4711yescharacteristic
469C22×C4C23×D4711yescharacteristic
470D8—948no—
471C2×C8—472no—
472C2×C8—941no—
473C2×C8—941no—
474C2×C8—941no—
475C2×C8—942no—
476C2×C8—942no—
477C2×C8—941no—
478C2×C8—942no—
479C2×C8—941no—
480C4⋊C4—28no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—941no—
486C4⋊C4—941no—
487C4⋊C4—942no—
488C4⋊C4—942no—
489C4⋊C4—3761no—
490C4⋊C4—1882no—
491C4⋊C4C2×C22⋊D4712yes—
492C4⋊C4C2×C22⋊D4714yes—
493C22⋊C4—28no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—1884no—
497C22⋊C4—1884no—
498C22⋊C4—1882no—
499C42C2×C22⋊D4712yes—
500C42C2×C22⋊D4712yes—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—942no—
515C23—1881no—
516C23—1881no—
517C23—1881no—
518C23—1881no—
519C23—1881no—
520C23—1881no—
521C23C2×#1062533414yes—
522C23C22×C22⋊D4711yescharacteristic
523Q8—48no—
524D4—28no—
525D4—22no—
526D4—22no—
527D4—416no—
528D4—41no—
529D4—41no—
530D4—942no—
531D4—942no—
532D4—3768no—
533D4—1881no—
534D4—1881no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—416no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—3764no—
553C2×C4—1882no—
554C2×C4—1881no—
555C2×C4—1881no—
556C2×C4—1882no—
557C2×C4—1881no—
558C2×C4—1882no—
559C2×C4#1309399312yes—
560C2×C4#1309399312yes—
561C2×C4C2×#1062533412yes—
562C2×C4#1309405012yes—
563C2×C4C22×C22⋊D4711yescharacteristic
564C2×C4C22×C22⋊D4711yescharacteristic
565C2×C4#41×D4711yescharacteristic
566C2×C4#41×D4711yescharacteristic
567C2×C4#1309407011yescharacteristic
568C2×C4#1309407111yescharacteristic
569C8—942no—
570C8—1882no—
571C8—1881no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—3761no—
587C22—3761no—
588C22—3761no—
589C22—3761no—
590C22—1882no—
591C22—1882no—
592C22—1882no—
593C22C2×#1309399311yescharacteristic
594C22#1568976111yescharacteristic
595C22#1568977211yescharacteristic
596C22C22×#1062533411yescharacteristic
597C22C2×#1309405011yescharacteristic
598C22#1568993111yescharacteristic
599C22#1568993811yescharacteristic
600C4—22no—
601C4—42no—
602C4—44no—
603C4—42no—
604C4—3761no—
605C4—3762no—
606C4#1568925714yes—
607C4#1568963011yescharacteristic
608C4#1568970111yescharacteristic
609C4#1568978011yescharacteristic
610C4#1568979211yescharacteristic
611C2—44no—
612C2—82no—
613C2—3762no—
614C2#2024672711yescharacteristic
615C2C2×#1568925711yescharacteristic
616C2#2025270311yescharacteristic
617C2#2025519511yescharacteristic
618C2#2025568711yescharacteristic
619C2#2025573111yescharacteristic
620C2#2025598311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#547C2
socleC23×C47C22×D4
hypercenterC2×#547D47
hypocenterC47#24116
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#15689400C2semidirect
2. C2×C4×#10625334C2semidirect
3. #20248301C2semidirect
4. #20244567C2semidirect
5. #20244384C2semidirect
6. #20244350C2semidirect
7. #20244183C2semidirect
8. #20243705C2semidirect
9. #20243576C2semidirect
10. #20243388C2semidirect
11. C2×C47×#547C2semidirect
88. C4⋊Q8×C47D4semidirect
96. C4×D4×C47D4semidirect
125. C2×D4×C47D8semidirect
155. C2×D188#41semidirect
189. C2×C47⋊C8#41semidirect
301. C2×#547D47semidirect
355. C47#24116semidirect
376. C4⋊Q8C22⋊D47semidirect
389. C4×D4C22⋊D47semidirect
445. C2×D4#10625334semidirect

Maximal subgroups

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

subgroupindex
C2×#156894002
C2×C4×#106253342
#202483012
#202445672
#202443842
#202443502
#202441832
#202437052
#202435762
#202433882
C2×C47×#5472
#2411647