Finite Group Databasebeta

← #26137591

Subgroups — #26137591

Order 12032.

1862 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093989C211yesmaximal, characteristic
2C2×#15689027C212yesmaximal
3C2×C4×Dic47⋊C4C211yesmaximal, characteristic
4C2×#15688968C211yesmaximal, characteristic
5#20241623C211yesmaximal, characteristic
6#20241621C211yesmaximal, characteristic
7#20241620C211yesmaximal, characteristic
8Dic47⋊#95C212yesmaximal
9#20241614C212yesmaximal
10#20241613C212yesmaximal
11C47×#1436C211yesmaximal, characteristic
12C23×Dic94C2211yescharacteristic
13C2×#13093989—24no—
14C2×#13093989—24no—
15C2×#13093989C418yescomplemented
16C22×Dic47⋊C4C2211yescharacteristic
17C22×Dic47⋊C4C2211yescharacteristic
18C22×Dic47⋊C4C2212yes—
19C22×C4×Dic47C2211yescharacteristic
20C22×C4×Dic47C2211yescharacteristic
21C22×C4×Dic47C2212yes—
22C22×C4×Dic47C2211yescharacteristic
23C22×C4×Dic47C2211yescharacteristic
24#15689027C4116yescomplemented
25C4×Dic47⋊C4—24no—
26#15688968—24no—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2211yescharacteristic
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2212yes—
34C2×C47⋊#95C2212yes—
35C2×C47⋊#95C2212yes—
36C22×C4⋊C4×C47C2211yescharacteristic
37C22×C42×C47C2211yescharacteristic
38C2×#95×C47C2211yescharacteristic
39C2×#95×C47C2212yes—
40C2×#95×C47C2212yes—
41C22×Dic94—22no—
42C22×Dic94—22no—
43C22×Dic94—24no—
44C22×Dic94C2×C418yes—
45C22×Dic94C2×C414yes—
46C23×Dic47C2312yes—
47C23×Dic47C2312yes—
48C23×Dic47C2311yescharacteristic
49C23×Dic47C2311yescharacteristic
50C23×Dic47C2311yescharacteristic
51C23×Dic47C2311yescharacteristic
52#13093989—232no—
53C2×Dic47⋊C4—22no—
54C2×Dic47⋊C4—22no—
55C2×Dic47⋊C4—24no—
56C2×Dic47⋊C4—22no—
57C2×Dic47⋊C4—22no—
58C2×Dic47⋊C4—24no—
59C2×Dic47⋊C4C2×C418yes—
60C2×Dic47⋊C4C2×C414yes—
61C2×Dic47⋊C4C2×C414yes—
62C2×C4×Dic47—24no—
63C2×C4×Dic47—22no—
64C2×C4×Dic47—22no—
65C2×C4×Dic47—22no—
66C2×C4×Dic47—22no—
67C2×C4×Dic47—24no—
68C2×C4×Dic47—22no—
69C2×C4×Dic47—42no—
70C2×C4×Dic47C2×C418yes—
71C2×C4×Dic47C2×C414yes—
72C2×C4×Dic47C2×C414yes—
73C2×C4×Dic47C2×C414yes—
74C2×C4×Dic47D414yes—
75C2×C4×Dic47D414yes—
76C47⋊#95—22no—
77C47⋊#95—24no—
78C47⋊#95—24no—
79C47⋊#95—42no—
80C47⋊#95—42no—
81C47⋊#95—42no—
82C47⋊#95C2×C418yes—
83C47⋊#95C2×C418yes—
84C47⋊#95C2×C418yes—
85C23×C4×C47C2312yes—
86C23×C4×C47C2312yes—
87C23×C4×C47C2311yescharacteristic
88C23×C4×C47C2311yescharacteristic
89C23×C4×C47C2311yescharacteristic
90C2×C4⋊C4×C47—22no—
91C2×C4⋊C4×C47—22no—
92C2×C4⋊C4×C47C2×C414yes—
93C2×C42×C47—22no—
94C2×C42×C47—42no—
95#95×C47—22no—
96#95×C47—42no—
97#95×C47C2×C418yes—
98C24×C47C2411yescharacteristic
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—21no—
106C22×C4×C47—21no—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47C22×C412yes—
121C22×C4×C47C22×C414yes—
122C22×C4×C47C22×C414yes—
123C22×C4×C47C22×C412yes—
124C22×C4×C47C22×C414yes—
125C22×C4×C47C22×C412yes—
126C22×C4×C47C2×D412yes—
127C22×C4×C47C2×D412yes—
128C22×Dic47—21no—
129C22×Dic47—21no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—21no—
136C22×Dic47—21no—
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×Dic47C22×C414yes—
150C22×Dic47C22×C414yes—
151C22×Dic47C22×C412yes—
152C22×Dic47C22×C414yes—
153C22×Dic47C22×C412yes—
154C22×Dic47C22×C412yes—
155C22×Dic47C22×C412yes—
156C22×Dic47C22×C414yes—
157C22×Dic47C2×D412yes—
158C22×Dic47C2×D412yes—
159C22×Dic47C2×D412yes—
160C22×Dic47C2×D412yes—
161C22×Dic47#4114yes—
162C2×Dic94—28no—
163C2×Dic94—28no—
164C2×Dic94—216no—
165C2×Dic94—216no—
166C2×Dic94C42116yescomplemented
167C4⋊C4×C47—28no—
168C42×C47—44no—
169Dic47⋊C4—28no—
170Dic47⋊C4—28no—
171Dic47⋊C4—216no—
172C4×Dic47—28no—
173C4×Dic47—28no—
174C4×Dic47—216no—
175C4×Dic47—44no—
176C4×Dic47C22⋊C4116yes—
177C23×C47C23×C411yescharacteristic
178C23×C47C23×C412yes—
179C23×C47C22×D411yescharacteristic
180C23×C47C22×D411yescharacteristic
181C23×C47C2×#4112yes—
182C23×C47H(5,𝔽2)12yes—
183C23×C472−1+412yes—
184C23×C472−1+412yes—
185C23×C472−1+412yes—
186C2×C4×C47—22no—
187C2×C4×C47—22no—
188C2×C4×C47—24no—
189C2×C4×C47—24no—
190C2×C4×C47—24no—
191C2×C4×C47—22no—
192C2×C4×C47—22no—
193C2×C4×C47—24no—
194C2×C4×C47—22no—
195C2×C4×C47—24no—
196C2×C4×C47—24no—
197C2×C4×C47—24no—
198C2×C4×C47—24no—
199C2×C4×C47—24no—
200C2×C4×C47—24no—
201C2×C4×C47—24no—
202C2×C4×C47—42no—
203C2×C4×C47—41no—
204C2×C4×C47—42no—
205C2×C4×C47—41no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47C2×C4214yes—
214C2×C4×C47C2×C22⋊C414yes—
215C2×Dic47—22no—
216C2×Dic47—22no—
217C2×Dic47—24no—
218C2×Dic47—24no—
219C2×Dic47—24no—
220C2×Dic47—22no—
221C2×Dic47—24no—
222C2×Dic47—22no—
223C2×Dic47—22no—
224C2×Dic47—24no—
225C2×Dic47—24no—
226C2×Dic47—24no—
227C2×Dic47—24no—
228C2×Dic47—24no—
229C2×Dic47—24no—
230C2×Dic47—24no—
231C2×Dic47—42no—
232C2×Dic47—41no—
233C2×Dic47—41no—
234C2×Dic47—42no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47—42no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241C2×Dic47—42no—
242C2×Dic47C2×C4214yes—
243C2×Dic47C2×C4214yes—
244C2×Dic47C2×C22⋊C414yes—
245C2×Dic47C2×C22⋊C414yes—
246C2×Dic47#11714yes—
247C2×Dic47C4×D418yes—
248C2×Dic47C4×D418yes—
249Dic94—232no—
250#13786—471nomaximal, Sylow
251C22×C47C22×C4211yescharacteristic
252C22×C47C22×C22⋊C411yescharacteristic
253C22×C47C2×#11711yescharacteristic
254C22×C47C2×C4×D412yes—
255C22×C47C2×C4×D412yes—
256C22×C47#51912yes—
257C22×C47#51912yes—
258C22×C47#51912yes—
259C22×C47#52012yes—
260C22×C47#52012yes—
261C22×C47#52012yes—
262C22×C47#53511yescharacteristic
263C22×C47#53611yescharacteristic
264C22×C47#53611yescharacteristic
265C22×C47#53612yes—
266C22×C47#53611yescharacteristic
267C22×C47#53712yes—
268C22×C47#54112yes—
269C22×C47#54112yes—
270C22×C47#54212yes—
271C22×C47#54212yes—
272C4×C47—24no—
273C4×C47—24no—
274C4×C47—42no—
275C4×C47—44no—
276C4×C47—44no—
277Dic47—24no—
278Dic47—24no—
279Dic47—42no—
280Dic47—44no—
281Dic47—44no—
282Dic47C4×C22⋊C418yes—
283C22×C4×Q8—471no—
284C2×#391—472no—
285C2×C4×C4⋊C4—471no—
286C2×#376—471no—
287#1436—471no—
288#1436—471no—
289#1436—471no—
290#1436D4711yescharacteristic
291C4⋊#95—472no—
292#1434—472no—
293#1434—472no—
294C2×C47C2×C4×C22⋊C411yescharacteristic
295C2×C47#227612yes—
296C2×C47#231011yescharacteristic
297C2×C47#231111yescharacteristic
298C2×C47#232012yes—
299C2×C47#232312yes—
300C2×C47#232312yes—
301C2×C47#232612yes—
302C2×C47#232712yes—
303C23×Q8—471no—
304C2×C4×Q8—478no—
305C2×C4×Q8—944no—
306C2×C4×Q8—944no—
307C22×C4⋊C4—471no—
308C22×C4⋊C4—471no—
309C22×C4⋊C4—472no—
310C22×C4⋊C4C2×D4711yescharacteristic
311C22×C42—471no—
312C22×C42—471no—
313C22×C42—471no—
314C22×C42—472no—
315C22×C42—471no—
316C22×C42C2×D4711yescharacteristic
317#391—4716no—
318C4×C4⋊C4—944no—
319#376—944no—
320C2×#95—472no—
321C2×#95—472no—
322C2×#95—472no—
323C2×#95—472no—
324C2×#95—472no—
325C2×#95—471no—
326C2×#95—472no—
327C2×#95—472no—
328C2×#95—472no—
329C2×#95C2×D4711yescharacteristic
330C2×#95C2×D4712yes—
331C2×#95C2×D4712yes—
332C47#1378611yescharacteristic, complemented, Sylow
333C22×Q8—474no—
334C22×Q8—478no—
335C22×Q8—942no—
336C22×Q8—942no—
337C22×Q8—944no—
338C23×C4—471no—
339C23×C4—471no—
340C23×C4—471no—
341C23×C4—471no—
342C23×C4—472no—
343C23×C4—472no—
344C23×C4C22×D4711yescharacteristic
345C23×C4C22×D4712yes—
346C23×C4C22×D4712yes—
347C23×C4C22×D4711yescharacteristic
348C23×C4C22×D4711yescharacteristic
349C4×Q8—9432no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—474no—
353C2×C4⋊C4—474no—
354C2×C4⋊C4—478no—
355C2×C4⋊C4—942no—
356C2×C4⋊C4—942no—
357C2×C4⋊C4—944no—
358C2×C4⋊C4—944no—
359C2×C4⋊C4—942no—
360C2×C4⋊C4—942no—
361C2×C4⋊C4C4×D4714yes—
362C2×C42—22no—
363C2×C42—42no—
364C2×C42—474no—
365C2×C42—474no—
366C2×C42—474no—
367C2×C42—474no—
368C2×C42—474no—
369C2×C42—478no—
370C2×C42—942no—
371C2×C42—942no—
372C2×C42—942no—
373C2×C42—942no—
374C2×C42—944no—
375C2×C42—944no—
376C2×C42—942no—
377C2×C42—1882no—
378#95—22no—
379#95—42no—
380#95—478no—
381#95—478no—
382#95—478no—
383#95—944no—
384#95—944no—
385#95—942no—
386#95—1882no—
387#95—1882no—
388#95—1882no—
389#95C4×D4718yes—
390C24C23×D4711yescharacteristic
391C2×Q8—4716no—
392C2×Q8—948no—
393C2×Q8—948no—
394C2×Q8—9416no—
395C2×Q8—9416no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—472no—
418C22×C4—474no—
419C22×C4—472no—
420C22×C4—472no—
421C22×C4—474no—
422C22×C4—474no—
423C22×C4—472no—
424C22×C4—472no—
425C22×C4—472no—
426C22×C4—472no—
427C22×C4—474no—
428C22×C4—474no—
429C22×C4—472no—
430C22×C4—941no—
431C22×C4—941no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4—942no—
437C22×C4—941no—
438C22×C4—941no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4—942no—
446C22×C4—942no—
447C22×C4—942no—
448C22×C4—942no—
449C22×C4—942no—
450C22×C4—942no—
451C22×C4C2×C4×D4714yes—
452C22×C4C2×C4×D4714yes—
453C22×C4C2×C4×D4712yes—
454C22×C4C2×C4×D4714yes—
455C22×C4C2×C4×D4712yes—
456C22×C4C2×C4×D4712yes—
457C22×C4D4×D4712yes—
458C22×C4D4×D4712yes—
459C4⋊C4—28no—
460C4⋊C4—9416no—
461C4⋊C4—948no—
462C4⋊C4—948no—
463C42—44no—
464C42—4716no—
465C42—9416no—
466C42—948no—
467C42—948no—
468C42—1884no—
469C23C22×C4×D4711yescharacteristic
470C23C22×C4×D4712yes—
471C23C2×D4×D4711yescharacteristic
472C23C2×D4×D4711yescharacteristic
473C23#41×D4712yes—
474C23#1309406912yes—
475C23#1309407112yes—
476C23#1309407112yes—
477C23#1309407212yes—
478Q8—9432no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—474no—
507C2×C4—474no—
508C2×C4—474no—
509C2×C4—478no—
510C2×C4—478no—
511C2×C4—474no—
512C2×C4—474no—
513C2×C4—942no—
514C2×C4—942no—
515C2×C4—944no—
516C2×C4—944no—
517C2×C4—944no—
518C2×C4—942no—
519C2×C4—944no—
520C2×C4—942no—
521C2×C4—942no—
522C2×C4—944no—
523C2×C4—944no—
524C2×C4—944no—
525C2×C4—944no—
526C2×C4—944no—
527C2×C4—944no—
528C2×C4—944no—
529C2×C4—1882no—
530C2×C4—1882no—
531C2×C4—1882no—
532C2×C4—1882no—
533C2×C4—1882no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4—1882no—
537C2×C4—1882no—
538C2×C4—1882no—
539C2×C4—1882no—
540C2×C4C42×D4714yes—
541C2×C4C22⋊C4×D4714yes—
542C22C2×C42×D4711yescharacteristic
543C22C2×C22⋊C4×D4711yescharacteristic
544C22#117×D4711yescharacteristic
545C22C4×D4×D4712yes—
546C22C4×D4×D4712yes—
547C22#1568959912yes—
548C22#1568959912yes—
549C22#1568960012yes—
550C22#1568960212yes—
551C22#1568960212yes—
552C22#1568960412yes—
553C22#1568968811yescharacteristic
554C22#1568969311yescharacteristic
555C22#1568969511yescharacteristic
556C22#1568969612yes—
557C22#1568969811yescharacteristic
558C22#1568970012yes—
559C22#1568973012yes—
560C22#1568973212yes—
561C22#1568974112yes—
562C22#1568974312yes—
563C4—24no—
564C4—24no—
565C4—44no—
566C4—44no—
567C4—42no—
568C4—478no—
569C4—944no—
570C4—944no—
571C4—1884no—
572C4—1884no—
573C4—1882no—
574C2C4×C22⋊C4×D4711yescharacteristic
575C2#2024575212yes—
576C2#2024597111yescharacteristic
577C2#2024597511yescharacteristic
578C2#2024603712yes—
579C2#2024606312yes—
580C2#2024606512yes—
581C2#2024609412yes—
582C2#2024610312yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC22×C47C22×C42
FittingC47×#1436C2
socle, prime radical, cosocleC24×C47C24
hypercenter#1436D47
hypocenterC47#13786
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
15. C2×#13093989C4semidirect
24. #15689027C4semidirect
166. C2×Dic94C42semidirect
332. C47#13786semidirect

Maximal subgroups

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

subgroupindex
C22×#130939892
C2×#156890272
C2×C4×Dic47⋊C42
C2×#156889682
#202416232
#202416212
#202416202
Dic47⋊#952
#202416142
#202416132
C47×#14362
#1378647