Finite Group Databasebeta

← #26137615

Subgroups — #26137615

Order 12032.

1710 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689016C212yesmaximal
2#20241625C212yesmaximal
3#20241622C212yesmaximal, complemented
4#20241618C211yesmaximal, characteristic, complemented
5#20241615C211yesmaximal, characteristic, complemented
6#20241613C212yesmaximal, complemented
7#20241611C211yesmaximal, characteristic
8#20241611C211yesmaximal, characteristic
9C4×C47⋊#95C211yesmaximal, characteristic, complemented
10C4×C47⋊#95C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊C4×C47C211yesmaximal, characteristic
12C22×C4⋊Dic47C2212yes—
13C22×C22⋊Dic47C2212yes—
14C22×C22⋊Dic47C2212yes—
15C22×C4×Dic47C2212yes—
16C22×C4×Dic47C2212yes—
17#15689016C4116yescomplemented
18C2×C47⋊#95C2211yescharacteristic
19C2×C47⋊#95C2211yescharacteristic
20C2×C47⋊#95C2211yescharacteristic
21C2×C47⋊#95C2211yescharacteristic
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C24×C4×C47C2211yescharacteristic
30C22×C22⋊C4×C47C2211yescharacteristic
31C22×C22⋊C4×C47C2211yescharacteristic
32C22×C42×C47C2211yescharacteristic
33C22×C42×C47C2211yescharacteristic
34C4×C22⋊C4×C47C418yescomplemented
35C2×#95×C47C2211yescharacteristic
36C2×#95×C47C2211yescharacteristic
37C23×Dic47C2312yes—
38C23×Dic47C2312yes—
39C23×Dic47C2312yes—
40C23×Dic47C2312yes—
41C2×C4⋊Dic47—42no—
42C2×C4⋊Dic47—42no—
43C2×C4⋊Dic47C2×C418yescomplemented
44C2×C22⋊Dic47—42no—
45C2×C22⋊Dic47—42no—
46C2×C22⋊Dic47—42no—
47C2×C22⋊Dic47—42no—
48C2×C22⋊Dic47C2×C418yes—
49C2×C22⋊Dic47C2×C418yes—
50C2×C4×Dic47—24no—
51C2×C4×Dic47—42no—
52C2×C4×Dic47—42no—
53C2×C4×Dic47—42no—
54C2×C4×Dic47—42no—
55C2×C4×Dic47C2×C418yescomplemented
56C47⋊#95—24no—
57C47⋊#95—22no—
58C47⋊#95—22no—
59C47⋊#95—22no—
60C47⋊#95—22no—
61C47⋊#95—42no—
62C47⋊#95—42no—
63C47⋊#95—42no—
64C47⋊#95—42no—
65C47⋊#95C2×C418yescomplemented
66C47⋊#95C2×C418yescomplemented
67C25×C47C2311yescharacteristic
68C23×C4×C47—22no—
69C23×C4×C47—22no—
70C23×C4×C47—42no—
71C23×C4×C47—42no—
72C23×C4×C47C2×C414yes—
73C23×C4×C47C2311yescharacteristic
74C23×C4×C47C2311yescharacteristic
75C23×C4×C47C2311yescharacteristic
76C23×C4×C47C2311yescharacteristic
77C23×C4×C47C2311yescharacteristic
78C23×C4×C47C2311yescharacteristic
79C2×C22⋊C4×C47—24no—
80C2×C22⋊C4×C47C2×C414yes—
81C2×C22⋊C4×C47C2×C418yes—
82C2×C22⋊C4×C47C2×C414yes—
83C2×C42×C47—42no—
84C2×C42×C47—42no—
85C2×C42×C47C2×C414yes—
86C2×C42×C47C2×C414yes—
87#95×C47C2×C414yes—
88#95×C47C2×C414yes—
89C24×C47—22no—
90C24×C47—21no—
91C24×C47—21no—
92C24×C47—22no—
93C24×C47—22no—
94C24×C47—21no—
95C24×C47—21no—
96C24×C47—21no—
97C24×C47—21no—
98C24×C47C22×C414yes—
99C24×C47C22×C412yes—
100C24×C47C2411yescharacteristic
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—22no—
106C22×C4×C47—21no—
107C22×C4×C47—21no—
108C22×C4×C47—21no—
109C22×C4×C47—21no—
110C22×C4×C47—21no—
111C22×C4×C47—21no—
112C22×C4×C47—22no—
113C22×C4×C47—21no—
114C22×C4×C47—21no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—21no—
119C22×C4×C47—21no—
120C22×C4×C47—28no—
121C22×C4×C47—44no—
122C22×C4×C47—44no—
123C22×C4×C47—48no—
124C22×C4×C47—48no—
125C22×C4×C47C22×C412yes—
126C22×C4×C47C22×C412yes—
127C22×C4×C47C22×C414yes—
128C22×C4×C47C22×C414yes—
129C22×C4×C47C22×C412yes—
130C22×C4×C47C22×C412yes—
131C22×C4×C47C22×C412yes—
132C22×C4×C47C22×C412yes—
133C22×C4×C47#4114yes—
134C22×C4×C47#4112yes—
135C22×C4×C47#4112yes—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47C22×C414yes—
161C22×Dic47C22×C414yes—
162C22×Dic47C22×C414yes—
163C22×Dic47C22×C414yes—
164C22⋊C4×C47—28no—
165C22⋊C4×C47C42116yescomplemented
166C42×C47—44no—
167C42×C47—44no—
168C22⋊Dic47—48no—
169C22⋊Dic47—48no—
170C4⋊Dic47—48no—
171C4×Dic47—48no—
172C4×Dic47—48no—
173C23×C47—22no—
174C23×C47—22no—
175C23×C47—24no—
176C23×C47—24no—
177C23×C47—24no—
178C23×C47—24no—
179C23×C47—22no—
180C23×C47—22no—
181C23×C47—24no—
182C23×C47—24no—
183C23×C47—22no—
184C23×C47—22no—
185C23×C47—42no—
186C23×C47—42no—
187C23×C47—42no—
188C23×C47—42no—
189C23×C47—41no—
190C23×C47—42no—
191C23×C47—41no—
192C23×C47—42no—
193C23×C47—41no—
194C23×C47—41no—
195C23×C47C2×C4214yes—
196C23×C47C23×C412yes—
197C23×C47C23×C411yescharacteristic
198C23×C47C2×#4111yescharacteristic
199C23×C47C2×#4111yescharacteristic
200C23×C47C2×#4112yes—
201C23×C47H(5,𝔽2)12yes—
202C23×C47H(5,𝔽2)12yes—
203C23×C47H(5,𝔽2)11yescharacteristic
204C23×C47H(5,𝔽2)11yescharacteristic
205C23×C472−1+411yescharacteristic
206C23×C472−1+411yescharacteristic
207C2×C4×C47—24no—
208C2×C4×C47—24no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—24no—
215C2×C4×C47—24no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—24no—
220C2×C4×C47—24no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—416no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47—42no—
234C2×C4×C47C2×C4214yes—
235C2×C4×C47C2×C4214yes—
236C2×C4×C47#11714yes—
237C2×C4×C47#11714yes—
238C2×Dic47—24no—
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—24no—
244C2×Dic47—24no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—24no—
248C2×Dic47—24no—
249C2×Dic47—24no—
250C2×Dic47—42no—
251C2×Dic47—42no—
252C2×Dic47—42no—
253C2×Dic47—42no—
254C2×Dic47—42no—
255C2×Dic47—42no—
256C2×Dic47—42no—
257C2×Dic47—42no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47—42no—
263C2×Dic47—42no—
264C2×Dic47—42no—
265C2×Dic47—42no—
266#13788—471nomaximal, Sylow
267C22×C47—24no—
268C22×C47—24no—
269C22×C47—24no—
270C22×C47—42no—
271C22×C47—42no—
272C22×C47—42no—
273C22×C47—42no—
274C22×C47—44no—
275C22×C47—44no—
276C22×C47—44no—
277C22×C47—44no—
278C22×C47C22×C4211yescharacteristic
279C22×C47C2×#11711yescharacteristic
280C22×C47C2×#11711yescharacteristic
281C22×C47C4×#4112yes—
282C22×C47C4×#4112yes—
283C22×C47#51812yes—
284C22×C47#51812yes—
285C22×C47#51811yescharacteristic
286C22×C47#51811yescharacteristic
287C22×C47#52012yes—
288C22×C47#52012yes—
289C22×C47#52011yescharacteristic
290C22×C47#52011yescharacteristic
291C22×C47#53812yes—
292C22×C47#53812yes—
293C22×C47#53812yes—
294C22×C47#53912yes—
295C22×C47#54011yescharacteristic
296C22×C47#54012yes—
297C22×C47#54111yescharacteristic
298C22×C47#54212yes—
299C22×C47#54211yescharacteristic
300C22×C47#54211yescharacteristic
301C4×C47—24no—
302C4×C47—24no—
303C4×C47—24no—
304C4×C47—24no—
305C4×C47—42no—
306C4×C47—42no—
307Dic47—44no—
308Dic47—44no—
309Dic47—44no—
310Dic47—44no—
311C2×#388—472no—
312C2×C4×C22⋊C4D4711yescharacteristic
313#1437—472no—
314#1436—472no—
315C4⋊#95—471no—
316#1434—471no—
317#1434—472no—
318C22⋊#95—471no—
319C22⋊#95—471no—
320C4×#95—471no—
321C4×#95—471no—
322C2×C47—44no—
323C2×C47C4×#11711yescharacteristic
324C2×C47#227511yescharacteristic
325C2×C47#227511yescharacteristic
326C2×C47#231211yescharacteristic
327C2×C47#231411yescharacteristic
328C2×C47#231811yescharacteristic
329C2×C47#231911yescharacteristic
330C2×C47#233312yes—
331C2×C47#233412yes—
332C2×C47#233512yes—
333C2×C47#233712yes—
334C24×C4C2×D4711yescharacteristic
335C22×C4⋊C4—472no—
336C22×C22⋊C4—472no—
337C22×C22⋊C4—472no—
338C22×C22⋊C4C2×D4711yescharacteristic
339C22×C22⋊C4C2×D4711yescharacteristic
340C22×C42—472no—
341C22×C42—472no—
342C22×C42C2×D4711yescharacteristic
343C22×C42C2×D4711yescharacteristic
344#388—4716no—
345C4×C22⋊C4Dic4718yescomplemented
346C2×#95—472no—
347C2×#95—472no—
348C2×#95—471no—
349C2×#95—471no—
350C2×#95—471no—
351C2×#95—471no—
352C2×#95—472no—
353C2×#95—472no—
354C2×#95—472no—
355C2×#95—472no—
356C2×#95—472no—
357C2×#95C2×D4711yescharacteristic
358C2×#95C2×D4711yescharacteristic
359C47#1378811yescharacteristic, complemented, Sylow
360C25C22×D4711yescharacteristic
361C23×C4—22no—
362C23×C4—22no—
363C23×C4—42no—
364C23×C4—42no—
365C23×C4—472no—
366C23×C4—472no—
367C23×C4—472no—
368C23×C4—472no—
369C23×C4C2×Dic4714yes—
370C23×C4C22×D4711yescharacteristic
371C23×C4C22×D4711yescharacteristic
372C23×C4C22×D4711yescharacteristic
373C23×C4C22×D4711yescharacteristic
374C23×C4C22×D4711yescharacteristic
375C23×C4C22×D4711yescharacteristic
376C2×C4⋊C4—478no—
377C2×C4⋊C4—1882no—
378C2×C4⋊C4—1882no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4—478no—
381C2×C22⋊C4—478no—
382C2×C22⋊C4—1882no—
383C2×C22⋊C4—1882no—
384C2×C22⋊C4—1882no—
385C2×C22⋊C4—1882no—
386C2×C22⋊C4C4×D4718yes—
387C2×C22⋊C4C2×Dic4714yes—
388C2×C22⋊C4C2×Dic4714yes—
389C2×C42—42no—
390C2×C42—42no—
391C2×C42—478no—
392C2×C42—944no—
393C2×C42—1882no—
394C2×C42—1882no—
395C2×C42—1882no—
396C2×C42—1882no—
397C2×C42C2×Dic4714yes—
398C2×C42C2×Dic4714yes—
399#95—478no—
400#95—478no—
401#95—942no—
402#95—942no—
403#95—942no—
404#95—942no—
405#95—944no—
406#95—1882no—
407#95—1882no—
408#95—1882no—
409#95—1882no—
410#95C2×Dic4714yes—
411#95C2×Dic4714yes—
412C24—22no—
413C24—21no—
414C24—21no—
415C24—22no—
416C24—22no—
417C24—21no—
418C24—21no—
419C24—21no—
420C24—21no—
421C24C2×C4×D4714yes—
422C24C22×Dic4712yes—
423C24C23×D4711yescharacteristic
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—28no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—474no—
449C22×C4—474no—
450C22×C4—474no—
451C22×C4—474no—
452C22×C4—942no—
453C22×C4—942no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—942no—
457C22×C4—942no—
458C22×C4—942no—
459C22×C4—942no—
460C22×C4—942no—
461C22×C4—942no—
462C22×C4—942no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—942no—
466C22×C4—942no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—942no—
472C22×C4—942no—
473C22×C4—942no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4C2×C4×D4714yes—
477C22×C4C2×C4×D4714yes—
478C22×C4#1062535012yes—
479C22×C4#1062535012yes—
480C22×C4#1062535014yes—
481C22×C4C22×Dic4712yes—
482C22×C4C22×Dic4712yes—
483C22×C4C22×Dic4712yes—
484C22×C4C22×Dic4712yes—
485C22×C4C22×Dic4712yes—
486C22×C4C22×Dic4712yes—
487C4⋊C4—1888no—
488C22⋊C4—28no—
489C22⋊C4—1888no—
490C22⋊C4—1888no—
491C22⋊C4C4×Dic47116yescomplemented
492C42—44no—
493C42—44no—
494C42—1888no—
495C42—1888no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23C2×C4×Dic4714yes—
519C23C22×C4×D4712yes—
520C23C23×Dic4711yescharacteristic
521C23C2×#1062535011yescharacteristic
522C23C2×#1062535012yes—
523C23C2×#1062535011yescharacteristic
524C23#1309406912yes—
525C23#1309406912yes—
526C23#1309407011yescharacteristic
527C23#1309407011yescharacteristic
528C23#1309407111yescharacteristic
529C23#1309407111yescharacteristic
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—416no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—944no—
558C2×C4—944no—
559C2×C4—944no—
560C2×C4—944no—
561C2×C4—944no—
562C2×C4—944no—
563C2×C4—944no—
564C2×C4—944no—
565C2×C4—944no—
566C2×C4—944no—
567C2×C4—944no—
568C2×C4—944no—
569C2×C4—1882no—
570C2×C4—1882no—
571C2×C4—1882no—
572C2×C4—1882no—
573C2×C4—1882no—
574C2×C4—1882no—
575C2×C4—1882no—
576C2×C4—1882no—
577C2×C4—1882no—
578C2×C4—1882no—
579C2×C4—1882no—
580C2×C4—1882no—
581C2×C4—1882no—
582C2×C4—1882no—
583C2×C4—1882no—
584C2×C4—1882no—
585C2×C4C2×C4×Dic4714yes—
586C2×C4C2×C4×Dic4714yes—
587C2×C4#1309398314yes—
588C2×C4#1309398314yes—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—42no—
593C22—42no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—42no—
599C22—42no—
600C22C22×C4×Dic4711yescharacteristic
601C22C2×#1309398311yescharacteristic
602C22C2×#1309398311yescharacteristic
603C22C4×#1062535012yes—
604C22C4×#1062535012yes—
605C22#1568959512yes—
606C22#1568959512yes—
607C22#1568959811yescharacteristic
608C22#1568959811yescharacteristic
609C22#1568960212yes—
610C22#1568960212yes—
611C22#1568960611yescharacteristic
612C22#1568960611yescharacteristic
613C22#1568970612yes—
614C22#1568971112yes—
615C22#1568971112yes—
616C22#1568971512yes—
617C22#1568972412yes—
618C22#1568972811yescharacteristic
619C22#1568973511yescharacteristic
620C22#1568973812yes—
621C22#1568974411yescharacteristic
622C22#1568974411yescharacteristic
623C4—24no—
624C4—24no—
625C4—24no—
626C4—24no—
627C4—42no—
628C4—42no—
629C4—1884no—
630C4—1884no—
631C4—1884no—
632C4—1884no—
633C2—44no—
634C2C4×#1309398311yescharacteristic
635C2#2024574911yescharacteristic
636C2#2024574911yescharacteristic
637C2#2024598611yescharacteristic
638C2#2024599711yescharacteristic
639C2#2024602111yescharacteristic
640C2#2024603111yescharacteristic
641C2#2024614812yes—
642C2#2024616512yes—
643C2#2024618012yes—
644C2#2024619612yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC22×C47C22×C42
FittingC2×C4×C22⋊C4×C47C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenterC2×C4×C22⋊C4D47
hypocenterC47#13788
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #20241622C2semidirect
4. #20241618C2semidirect
5. #20241615C2semidirect
6. #20241613C2semidirect
9. C4×C47⋊#95C2semidirect
10. C4×C47⋊#95C2semidirect
17. #15689016C4semidirect
34. C4×C22⋊C4×C47C4semidirect
43. C2×C4⋊Dic47C2×C4semidirect
55. C2×C4×Dic47C2×C4semidirect
65. C47⋊#95C2×C4semidirect
66. C47⋊#95C2×C4semidirect
165. C22⋊C4×C47C42semidirect
345. C4×C22⋊C4Dic47semidirect
359. C47#13788semidirect
491. C22⋊C4C4×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156890162
#202416252
#202416222
#202416182
#202416152
#202416132
#202416112
C4×C47⋊#952
C2×C4×C22⋊C4×C472
#1378847