Finite Group Databasebeta

← #26227409

Subgroups — #26227409

Order 12032.

1308 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689257C211yesmaximal, characteristic
2C4×#13093998C211yesmaximal, characteristic, complemented
3#20244253C212yesmaximal, complemented
4#20244074C212yesmaximal, complemented
5#20244005C212yesmaximal, complemented
6#20243739C212yesmaximal, complemented
7#20243705C211yesmaximal, characteristic, complemented
8#20243576C211yesmaximal, characteristic, complemented
9#20243319C211yesmaximal, characteristic, complemented
10C4×C22⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
11C2×C47×#390C211yesmaximal, characteristic, complemented
12C22×#10625334C2212yes—
13C2×#13093998C2211yescharacteristic
14C2×C4×D188C2211yescharacteristic, complemented
15C2×C4×C22⋊D47—21no—
16#15689257C418yescomplemented
17C2×#13093957C2212yescomplemented
18C2×#13093945C2212yes—
19C2×#13093945C2212yes—
20C2×#13093945C2212yes—
21C2×#13093944C2212yescomplemented
22C2×C22⋊(C47⋊C8)C2211yescharacteristic
23C2×C22⋊(C47⋊C8)C2211yescharacteristic
24C2×C4×C47⋊C8C2212yescomplemented
25#15689025C2211yescharacteristic, complemented
26#15689021C2211yescharacteristic, complemented
27#15689014—21no—
28#15689002—21no—
29#15688983C2211yescharacteristic
30C4×C4⋊Dic47C2211yescharacteristic, complemented
31C4×(C2×D47)⋊C4—21no—
32#15688885C2212yescomplemented
33#15688885C2212yescomplemented
34#15688878C2212yescomplemented
35C23×D4×C47C2211yescharacteristic
36C22×C4⋊C4×C47C2211yescharacteristic
37C22×C22⋊C4×C47—21no—
38C22×C22⋊C4×C47—21no—
39C22×C42×C47C2211yescharacteristic
40C47×#390—21no—
41C47×#390—21no—
42C47×#390C2212yes—
43C47×#390C2212yes—
44C22×D188C2311yescharacteristic
45C22×C22⋊D47—21no—
46C22×C4×D47—21no—
47C2×#10625334—42no—
48C2×#10625334—42no—
49C2×#10625334C2×C418yescomplemented
50C22×C47⋊C8C2312yes—
51C22×C47⋊C8C2312yes—
52#13093998—22no—
53#13093998C2×C414yes—
54C4×D188—42no—
55C4×C22⋊D47—44no—
56C2×C4⋊Dic47C2311yescharacteristic
57C2×C4⋊Dic47C2311yescharacteristic
58C2×C4⋊Dic47C2311yescharacteristic
59C2×Dic47⋊C4—21no—
60C2×C22⋊Dic47—21no—
61C2×(C2×D47)⋊C4—21no—
62C2×(C2×D47)⋊C4—21no—
63C2×(C2×D47)⋊C4—21no—
64C2×(C2×D47)⋊C4—21no—
65C2×C4×Dic47—21no—
66#13093957—42no—
67#13093945—42no—
68#13093945—42no—
69#13093945C2×C418yescomplemented
70#13093944—42no—
71C22⋊(C47⋊C8)—22no—
72C22⋊(C47⋊C8)C2×C414yescomplemented
73C4×C47⋊C8—42no—
74C47⋊#95—21no—
75C47⋊#95—21no—
76C47⋊#95—21no—
77C25×C47—21no—
78C22×D4×C47—21no—
79C22×D4×C47—22no—
80C22×D4×C47—42no—
81C22×D4×C47—42no—
82C22×D4×C47C2×C414yescomplemented
83C22×D4×C47C2312yes—
84C23×C4×C47—21no—
85C23×C4×C47D411yescharacteristic
86C23×C4×C47D411yescharacteristic
87C23×C4×C47C2311yescharacteristic
88C2×C4⋊C4×C47—21no—
89C2×C4⋊C4×C47—42no—
90C2×C4⋊C4×C47C2312yes—
91C2×C22⋊C4×C47—22no—
92C2×C22⋊C4×C47—22no—
93C2×C22⋊C4×C47—22no—
94C2×C22⋊C4×C47—22no—
95C2×C22⋊C4×C47—44no—
96C2×C22⋊C4×C47—44no—
97C2×C42×C47—42no—
98C2×C42×C47D411yescharacteristic
99C2×C42×C47D411yescharacteristic
100C2×C42×C47C2311yescharacteristic
101C2×C42×C47C2311yescharacteristic
102C24×C47—24no—
103C24×C47—22no—
104C24×C47—42no—
105C24×C47—42no—
106C24×C47—42no—
107C24×C47—42no—
108C24×C47—42no—
109C24×C47—42no—
110C24×C47C2×D411yescharacteristic
111C23×D47—21no—
112C2×D4×C47—24no—
113C2×D4×C47—22no—
114C2×D4×C47—22no—
115C2×D4×C47—22no—
116C2×D4×C47—22no—
117C2×D4×C47—22no—
118C2×D4×C47—48no—
119C2×D4×C47—42no—
120C2×D4×C47—42no—
121C2×D4×C47—48no—
122C2×D4×C47C22×C414yes—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—41no—
129C22×C4×C47—41no—
130C22×C4×C47—41no—
131C22×C4×C47—42no—
132C22×C4×C47—42no—
133C22×C4×C47—42no—
134C22×C4×C47D814yescomplemented
135C22×C4×C47C22×C412yes—
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47#4112yes—
142C22×C4×C47C2411yescharacteristic
143C2×C22⋊D47—24no—
144C2×C22⋊D47—42no—
145C2×C22⋊D47—42no—
146C22×Dic47—21no—
147C22×Dic47—21no—
148C22×Dic47—21no—
149C2×D188—22no—
150C2×D188—22no—
151C2×D188—21no—
152C2×D188C22×C412yes—
153C2×C4×D47—42no—
154C2×C4×D47—42no—
155C2×C4×D47—42no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47—22no—
158C4⋊C4×C47—42no—
159C22⋊C4×C47—44no—
160C22⋊C4×C47—44no—
161C22⋊C4×C47—44no—
162C22⋊C4×C47—44no—
163C42×C47—22no—
164C42×C47—22no—
165C42×C47—41no—
166C42×C47—41no—
167C22⋊Dic47—42no—
168#10625334—48no—
169(C2×D47)⋊C4—24no—
170(C2×D47)⋊C4—42no—
171(C2×D47)⋊C4—42no—
172(C2×D47)⋊C4—42no—
173C4⋊Dic47—21no—
174C4⋊Dic47—22no—
175C4⋊Dic47—22no—
176C4⋊Dic47C22×C412yes—
177Dic47⋊C4—42no—
178C4×Dic47—42no—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8—22no—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8C22×C414yes—
185C23×C47—21no—
186C23×C47—21no—
187C23×C47—21no—
188C23×C47—21no—
189C23×C47—24no—
190C23×C47—48no—
191C23×C47—48no—
192C23×C47—42no—
193C23×C47—42no—
194C23×C47—42no—
195C23×C47—42no—
196C23×C47—42no—
197C23×C47—42no—
198C23×C47—48no—
199C23×C47—84no—
200C23×C47—84no—
201C23×C47—84no—
202C23×C47—84no—
203C23×C47C4×D412yes—
204C23×C47C2×D812yes—
205C23×C47#13512yes—
206C23×C47C22×D411yescharacteristic
207C22×D47—22no—
208C22×D47—41no—
209C22×D47—41no—
210C22×D47—41no—
211C22×D47—41no—
212C22×D47—41no—
213C22×D47—41no—
214D4×C47—24no—
215D4×C47—24no—
216D4×C47—48no—
217D4×C47—42no—
218D4×C47—42no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—24no—
228C2×C4×C47—24no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—44no—
233C2×C4×C47—44no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47—84no—
238C2×C4×C47C4×D412yes—
239C2×C4×C47C2×D812yes—
240C2×C4×C47C2×D812yes—
241C2×C4×C47AGL(1,ℤ/8)12yes—
242C2×C4×C47C23×C411yescharacteristic
243C2×C4×C47C2×#4111yescharacteristic
244C2×C4×C47H(5,𝔽2)12yes—
245C2×C4×C47H(5,𝔽2)12yes—
246C22⋊D47—48no—
247C2×Dic47—22no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256D188—24no—
257D188—42no—
258C4×D47—84no—
259C47⋊C8—24no—
260C47⋊C8—42no—
261#22040—471nomaximal, Sylow
262C22×C47—22no—
263C22×C47—22no—
264C22×C47—22no—
265C22×C47—22no—
266C22×C47—22no—
267C22×C47—22no—
268C22×C47—44no—
269C22×C47—44no—
270C22×C47—44no—
271C22×C47—816no—
272C22×C47—82no—
273C22×C47—82no—
274C22×C47—82no—
275C22×C47—82no—
276C22×C47C4×D814yes—
277C22×C47C2×C4×D411yescharacteristic
278C22×C47#53411yescharacteristic
279C22×C47#53511yescharacteristic
280C22×C47C22×D811yescharacteristic
281C22×C47C2×#13511yescharacteristic
282C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C47#57511yescharacteristic
284C2×D47—42no—
285C2×D47—42no—
286C2×D47—42no—
287C2×D47—81no—
288C2×D47—81no—
289C2×D47—81no—
290C2×D47—81no—
291C4×C47—22no—
292C4×C47—22no—
293C4×C47—22no—
294C4×C47—22no—
295C4×C47—41no—
296C4×C47—82no—
297C4×C47#51812yes—
298C4×C47#53812yes—
299Dic47—42no—
300Dic47—81no—
301Dic47—81no—
302C2×#447—471no—
303C4×#121—471no—
304C2×#390D4711yescharacteristic, complemented
305#1970—472no—
306#1934—472no—
307#1925—472no—
308#1879—472no—
309#1874—471no—
310#1852—471no—
311#1786—471no—
312C4×C22⋊C8—471no—
313C2×C47—24no—
314C2×C47—84no—
315C2×C47#232111yescharacteristic
316C2×C47C2×C4×D811yescharacteristic
317C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
318C2×C47#318611yescharacteristic
319C2×C47#318811yescharacteristic
320C2×C47#320511yescharacteristic
321C2×C47#321111yescharacteristic
322D47—82no—
323C23×D4C2×D4711yescharacteristic
324C22×D8—472no—
325C2×#121—471no—
326C2×C4×D4—471no—
327C2×C4×D4—941no—
328C22×C4⋊C4C2×D4711yescharacteristic
329C22×C22⋊C4—21no—
330C22×C22⋊C4—21no—
331C22×C42C2×D4711yescharacteristic
332#447—478no—
333C2×#107—472no—
334C2×#102—472no—
335C2×#102—472no—
336C2×#102—472no—
337C2×#102—472no—
338C2×C22⋊C8—471no—
339C2×C22⋊C8—471no—
340C2×C4×C8—472no—
341#390—21no—
342#390—21no—
343#390—471no—
344#390C2×D4712yes—
345#390C2×D4712yes—
346#389—471no—
347#388—941no—
348#386—941no—
349#380—471no—
350C4×C4⋊C4—471no—
351C4×C22⋊C4—941no—
352#340—472no—
353#340—472no—
354#336—472no—
355C47#2204011yescharacteristic, complemented, Sylow
356C25—21no—
357C22×D4—22no—
358C22×D4—21no—
359C22×D4—42no—
360C22×D4—42no—
361C22×D4—471no—
362C22×D4—941no—
363C22×D4C4×D4714yescomplemented
364C22×D4C22×D4712yes—
365C23×C4—21no—
366C23×C4—941no—
367C23×C4C22⋊D4711yescharacteristic
368C23×C4C22⋊D4711yescharacteristic
369C23×C4C22×D4711yescharacteristic
370C2×D8—478no—
371C2×D8—1882no—
372C2×D8—1882no—
373C22×C8—472no—
374C22×C8—472no—
375#121—474no—
376#121—942no—
377C4×D4—1882no—
378C4×D4—1884no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—42no—
381C2×C4⋊C4—471no—
382C2×C4⋊C4—471no—
383C2×C4⋊C4—471no—
384C2×C4⋊C4—941no—
385C2×C4⋊C4C22×D4712yes—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—44no—
391C2×C22⋊C4—44no—
392C2×C22⋊C4—941no—
393C2×C22⋊C4—941no—
394C2×C22⋊C4—941no—
395C2×C22⋊C4—941no—
396C2×C22⋊C4—941no—
397C2×C42—42no—
398C2×C42—941no—
399C2×C42C22⋊D4711yescharacteristic
400C2×C42C22⋊D4711yescharacteristic
401C2×C42C22×D4711yescharacteristic
402C2×C42C22×D4711yescharacteristic
403#107—1882no—
404#102—478no—
405#102—1882no—
406#102—1882no—
407#102—1882no—
408C22⋊C8—474no—
409C22⋊C8—942no—
410C4×C8—1882no—
411#95—941no—
412#95—941no—
413#95—941no—
414C24—24no—
415C24—22no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—941no—
423C24C2×C22⋊D4711yescharacteristic
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—48no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—48no—
434C2×D4—472no—
435C2×D4—944no—
436C2×D4—941no—
437C2×D4—942no—
438C2×D4—942no—
439C2×D4—1882no—
440C2×D4—1882no—
441C2×D4C2×C4×D4714yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—941no—
454C22×C4—941no—
455C22×C4—941no—
456C22×C4—1882no—
457C22×C4—1882no—
458C22×C4—1882no—
459C22×C4#1062533414yescomplemented
460C22×C4C2×C4×D4712yes—
461C22×C4#1062535012yes—
462C22×C4C2×C22⋊D4711yescharacteristic
463C22×C4C2×C22⋊D4711yescharacteristic
464C22×C4C2×C22⋊D4711yescharacteristic
465C22×C4C2×C22⋊D4711yescharacteristic
466C22×C4C2×C22⋊D4711yescharacteristic
467C22×C4C23×D4711yescharacteristic
468D8—1888no—
469C2×C8—474no—
470C2×C8—942no—
471C2×C8—942no—
472C2×C8—942no—
473C2×C8—942no—
474C2×C8—942no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—42no—
478C4⋊C4—472no—
479C4⋊C4—941no—
480C4⋊C4—942no—
481C4⋊C4—942no—
482C4⋊C4—1882no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—944no—
488C22⋊C4—1882no—
489C22⋊C4—1882no—
490C22⋊C4—1882no—
491C22⋊C4—1882no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—41no—
496C42—1882no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—24no—
502C23—48no—
503C23—48no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—48no—
511C23—84no—
512C23—84no—
513C23—84no—
514C23—84no—
515C23—942no—
516C23—1881no—
517C23—1881no—
518C23—1881no—
519C23—1881no—
520C23—1881no—
521C23—1881no—
522C23C4×C22⋊D4712yes—
523C23C2×#1062533412yes—
524C23#1309404412yes—
525C23C22×C22⋊D4711yescharacteristic
526D4—24no—
527D4—24no—
528D4—48no—
529D4—42no—
530D4—42no—
531D4—944no—
532D4—1888no—
533D4—1882no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—84no—
553C2×C4—942no—
554C2×C4—3764no—
555C2×C4—1881no—
556C2×C4—1881no—
557C2×C4—1881no—
558C2×C4—1881no—
559C2×C4—1881no—
560C2×C4—1881no—
561C2×C4—1881no—
562C2×C4—1881no—
563C2×C4C4×C22⋊D4712yes—
564C2×C4C2×#1062533412yes—
565C2×C4C2×#1062533412yes—
566C2×C4#1309405012yes—
567C2×C4C22×C4×D4711yescharacteristic
568C2×C4C2×#1062535011yescharacteristic
569C2×C4#1309406912yes—
570C2×C4#1309406912yes—
571C8—944no—
572C8—1882no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—816no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—3761no—
588C22—3761no—
589C22—3761no—
590C22—3761no—
591C22—1882no—
592C22—1882no—
593C22—1882no—
594C22C4×#1062533414yes—
595C22C2×C4×C22⋊D4711yescharacteristic
596C22#1568968211yescharacteristic
597C22#1568969011yescharacteristic
598C22C22×#1062533411yescharacteristic
599C22C2×#1309404411yescharacteristic
600C22C2×#1309405011yescharacteristic
601C22#1568992111yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—41no—
607C4—82no—
608C4—3761no—
609C4—3761no—
610C4—1882no—
611C4#1568959612yes—
612C4#1568971112yes—
613C2—24no—
614C2—84no—
615C2—3762no—
616C2#2024604611yescharacteristic
617C2C2×C4×#1062533411yescharacteristic
618C2C4×#1309405011yescharacteristic
619C2#2025413611yescharacteristic
620C2#2025415311yescharacteristic
621C2#2025430611yescharacteristic
622C2#2025435311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#390C2
socleC23×C47C22×D4
prime radicalC2×#15689257C2
cosocleC22×C4×C47C24
hypercenterC2×#390D47
hypocenterC47#22040
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C4×#13093998C2semidirect
3. #20244253C2semidirect
4. #20244074C2semidirect
5. #20244005C2semidirect
6. #20243739C2semidirect
7. #20243705C2semidirect
8. #20243576C2semidirect
9. #20243319C2semidirect
10. C4×C22⋊(C47⋊C8)C2semidirect
11. C2×C47×#390C2semidirect
14. C2×C4×D188C22semidirect
16. #15689257C4semidirect
17. C2×#13093957C22semidirect
21. C2×#13093944C22semidirect
24. C2×C4×C47⋊C8C22semidirect
25. #15689025C22semidirect
26. #15689021C22semidirect
30. C4×C4⋊Dic47C22semidirect
32. #15688885C22semidirect
33. #15688885C22semidirect
34. #15688878C22semidirect
49. C2×#10625334C2×C4semidirect
69. #13093945C2×C4semidirect
72. C22⋊(C47⋊C8)C2×C4semidirect
82. C22×D4×C47C2×C4semidirect
134. C22×C4×C47D8semidirect
304. C2×#390D47semidirect
355. C47#22040semidirect
363. C22×D4C4×D47semidirect
459. C22×C4#10625334semidirect

Maximal subgroups

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

subgroupindex
C2×#156892572
C4×#130939982
#202442532
#202440742
#202440052
#202437392
#202437052
#202435762
#202433192
C4×C22⋊(C47⋊C8)2
C2×C47×#3902
#2204047