Finite Group Databasebeta

← #26137909

Subgroups — #26137909

Order 12032.

1630 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689016C212yesmaximal
2#20241625C211yesmaximal, characteristic
3#20241625C211yesmaximal, characteristic
4#20241622C212yesmaximal, complemented
5#20241620C212yesmaximal, complemented
6#20241617C212yesmaximal, complemented
7#20241613C212yesmaximal, complemented
8#20241611C212yesmaximal
9C2×C47×#385C211yesmaximal, characteristic
10C22×C4⋊Dic47C2212yes—
11C22×C22⋊Dic47C2212yes—
12C22×C22⋊Dic47C2212yes—
13C22×C4×Dic47C2211yescharacteristic
14C22×C4×Dic47C2211yescharacteristic
15C22×C4×Dic47C2212yes—
16#15689016C4116yescomplemented
17C2×C47⋊#95C2212yes—
18C2×C47⋊#95C2212yes—
19C2×C47⋊#95C2212yes—
20C2×C47⋊#95C2212yes—
21C2×C47⋊#95C2211yescharacteristic
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2211yescharacteristic
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C24×C4×C47C2211yescharacteristic
28C22×C4⋊C4×C47C2212yes—
29C22×C22⋊C4×C47C2211yescharacteristic
30C22×C22⋊C4×C47C2211yescharacteristic
31C47×#385C418yescomplemented
32C2×#95×C47C2212yes—
33C23×Dic47C2312yes—
34C23×Dic47C2312yes—
35C23×Dic47C2312yes—
36C23×Dic47C2312yes—
37C2×C4⋊Dic47—42no—
38C2×C4⋊Dic47—42no—
39C2×C4⋊Dic47C2×C418yescomplemented
40C2×C22⋊Dic47—42no—
41C2×C22⋊Dic47—42no—
42C2×C22⋊Dic47—42no—
43C2×C22⋊Dic47—42no—
44C2×C22⋊Dic47C2×C418yes—
45C2×C22⋊Dic47C2×C418yes—
46C2×C4×Dic47—42no—
47C2×C4×Dic47—42no—
48C2×C4×Dic47—41no—
49C2×C4×Dic47—41no—
50C2×C4×Dic47—42no—
51C2×C4×Dic47—42no—
52C2×C4×Dic47C2×C418yescomplemented
53C47⋊#95—41no—
54C47⋊#95—41no—
55C47⋊#95—42no—
56C47⋊#95—42no—
57C47⋊#95—42no—
58C47⋊#95—42no—
59C47⋊#95—42no—
60C47⋊#95—42no—
61C47⋊#95C2×C418yescomplemented
62C47⋊#95C2×C418yescomplemented
63C25×C47C2311yescharacteristic
64C23×C4×C47—42no—
65C23×C4×C47—42no—
66C23×C4×C47—41no—
67C23×C4×C47—41no—
68C23×C4×C47C2×C414yes—
69C23×C4×C47C2312yes—
70C23×C4×C47C2311yescharacteristic
71C23×C4×C47C2311yescharacteristic
72C23×C4×C47C2312yes—
73C2×C4⋊C4×C47—42no—
74C2×C4⋊C4×C47—42no—
75C2×C4⋊C4×C47C2×C418yes—
76C2×C22⋊C4×C47—41no—
77C2×C22⋊C4×C47—41no—
78C2×C22⋊C4×C47C2×C414yes—
79C2×C22⋊C4×C47C2×C418yes—
80C2×C22⋊C4×C47C2×C414yes—
81#95×C47C2×C418yes—
82C24×C47—21no—
83C24×C47—21no—
84C24×C47—22no—
85C24×C47—22no—
86C24×C47—22no—
87C24×C47—22no—
88C24×C47—22no—
89C24×C47C22×C414yes—
90C24×C47C22×C412yes—
91C24×C47C2411yescharacteristic
92C22×C4×C47—22no—
93C22×C4×C47—22no—
94C22×C4×C47—22no—
95C22×C4×C47—22no—
96C22×C4×C47—21no—
97C22×C4×C47—21no—
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—22no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—48no—
111C22×C4×C47—44no—
112C22×C4×C47—44no—
113C22×C4×C47—48no—
114C22×C4×C47—44no—
115C22×C4×C47C22×C412yes—
116C22×C4×C47C22×C412yes—
117C22×C4×C47C22×C414yes—
118C22×C4×C47C22×C414yes—
119C22×C4×C47C22×C414yes—
120C22×C4×C47C22×C414yes—
121C22×Dic47—22no—
122C22×Dic47—22no—
123C22×Dic47—22no—
124C22×Dic47—22no—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
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×Dic47C22×C414yes—
146C22×Dic47C22×C414yes—
147C22×Dic47C22×C414yes—
148C22×Dic47C22×C414yes—
149C4⋊C4×C47—48no—
150C22⋊C4×C47—44no—
151C22⋊C4×C47C42116yescomplemented
152C22⋊Dic47—48no—
153C22⋊Dic47—48no—
154C4⋊Dic47—48no—
155C4×Dic47—48no—
156C4×Dic47—44no—
157C4×Dic47—44no—
158C23×C47—24no—
159C23×C47—24no—
160C23×C47—22no—
161C23×C47—24no—
162C23×C47—22no—
163C23×C47—24no—
164C23×C47—24no—
165C23×C47—24no—
166C23×C47—24no—
167C23×C47—24no—
168C23×C47—41no—
169C23×C47—42no—
170C23×C47—41no—
171C23×C47—41no—
172C23×C47—42no—
173C23×C47—42no—
174C23×C47—42no—
175C23×C47—42no—
176C23×C47—42no—
177C23×C47—41no—
178C23×C47C2×C4214yes—
179C23×C47C23×C412yes—
180C23×C47C23×C411yescharacteristic
181C23×C47H(5,𝔽2)12yes—
182C23×C47H(5,𝔽2)12yes—
183C23×C47H(5,𝔽2)11yescharacteristic
184C23×C47H(5,𝔽2)11yescharacteristic
185C23×C47H(5,𝔽2)12yes—
186C23×C472−1+412yes—
187C23×C472−1+412yes—
188C2×C4×C47—24no—
189C2×C4×C47—24no—
190C2×C4×C47—24no—
191C2×C4×C47—24no—
192C2×C4×C47—24no—
193C2×C4×C47—24no—
194C2×C4×C47—22no—
195C2×C4×C47—22no—
196C2×C4×C47—24no—
197C2×C4×C47—24no—
198C2×C4×C47—24no—
199C2×C4×C47—24no—
200C2×C4×C47—24no—
201C2×C4×C47—416no—
202C2×C4×C47—41no—
203C2×C4×C47—41no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—41no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—41no—
212C2×C4×C47C2×C4218yes—
213C2×Dic47—24no—
214C2×Dic47—24no—
215C2×Dic47—24no—
216C2×Dic47—24no—
217C2×Dic47—24no—
218C2×Dic47—24no—
219C2×Dic47—24no—
220C2×Dic47—24no—
221C2×Dic47—24no—
222C2×Dic47—24no—
223C2×Dic47—24no—
224C2×Dic47—24no—
225C2×Dic47—42no—
226C2×Dic47—42no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47—42no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241#13831—471nomaximal, Sylow
242C22×C47—24no—
243C22×C47—28no—
244C22×C47—42no—
245C22×C47—42no—
246C22×C47—44no—
247C22×C47—44no—
248C22×C47—44no—
249C22×C47—44no—
250C22×C47—44no—
251C22×C47C22×C4211yescharacteristic
252C22×C47#51812yes—
253C22×C47#51812yes—
254C22×C47#51811yescharacteristic
255C22×C47#51812yes—
256C22×C47#51911yescharacteristic
257C22×C47#52012yes—
258C22×C47#52012yes—
259C22×C47#52012yes—
260C22×C47#52012yes—
261C22×C47#52012yes—
262C22×C47#56011yescharacteristic
263C22×C47#56012yes—
264C22×C47#56011yescharacteristic
265C22×C47H(3,𝔽4)11yescharacteristic
266C22×C47#56212yes—
267C22×C47#56212yes—
268C22×C47#56212yes—
269C22×C47#56312yes—
270C22×C47#56312yes—
271C22×C47#56311yescharacteristic
272C4×C47—28no—
273C4×C47—42no—
274C4×C47—44no—
275C4×C47—42no—
276Dic47—44no—
277Dic47—44no—
278Dic47—44no—
279Dic47—44no—
280C2×#388—472no—
281C2×#385D4711yescharacteristic
282#1437—471no—
283#1437—471no—
284#1436—472no—
285#1436—472no—
286C4⋊#95—472no—
287#1434—472no—
288C22⋊#95—472no—
289C2×C47—44no—
290C2×C47#227411yescharacteristic
291C2×C47#227512yes—
292C2×C47#237511yescharacteristic
293C2×C47#237611yescharacteristic
294C2×C47#237612yes—
295C2×C47#237712yes—
296C2×C47#237812yes—
297C2×C47#237912yes—
298C2×C47#238312yes—
299C24×C4C2×D4711yescharacteristic
300C22×C4⋊C4—472no—
301C22×C4⋊C4C2×D4712yes—
302C22×C22⋊C4—472no—
303C22×C22⋊C4—472no—
304C22×C22⋊C4C2×D4711yescharacteristic
305C22×C22⋊C4C2×D4711yescharacteristic
306C22×C42—472no—
307C22×C42—471no—
308C22×C42—471no—
309#388—4716no—
310#385Dic4718yescomplemented
311C2×#95—472no—
312C2×#95—472no—
313C2×#95—472no—
314C2×#95—472no—
315C2×#95—472no—
316C2×#95—472no—
317C2×#95—471no—
318C2×#95—472no—
319C2×#95—471no—
320C2×#95—472no—
321C2×#95C2×D4712yes—
322C47#1383111yescharacteristic, complemented, Sylow
323C25C22×D4711yescharacteristic
324C23×C4—41no—
325C23×C4—41no—
326C23×C4—42no—
327C23×C4—42no—
328C23×C4—472no—
329C23×C4—472no—
330C23×C4—472no—
331C23×C4—472no—
332C23×C4C2×Dic4714yes—
333C23×C4C22×D4711yescharacteristic
334C23×C4C22×D4711yescharacteristic
335C23×C4C22×D4712yes—
336C23×C4C22×D4712yes—
337C2×C4⋊C4—42no—
338C2×C4⋊C4—42no—
339C2×C4⋊C4—478no—
340C2×C4⋊C4—1882no—
341C2×C4⋊C4—1882no—
342C2×C4⋊C4C2×Dic4718yes—
343C2×C22⋊C4—41no—
344C2×C22⋊C4—41no—
345C2×C22⋊C4—478no—
346C2×C22⋊C4—478no—
347C2×C22⋊C4—1882no—
348C2×C22⋊C4—1882no—
349C2×C22⋊C4—1882no—
350C2×C22⋊C4—1882no—
351C2×C22⋊C4C4×D4718yes—
352C2×C22⋊C4C2×Dic4714yes—
353C2×C22⋊C4C2×Dic4714yes—
354C2×C42—478no—
355C2×C42—1882no—
356C2×C42—1882no—
357C2×C42—1881no—
358C2×C42—1881no—
359C2×C42—1882no—
360C2×C42—1882no—
361#95—478no—
362#95—478no—
363#95—1881no—
364#95—1881no—
365#95—1882no—
366#95—1882no—
367#95—1882no—
368#95—1882no—
369#95—1882no—
370#95—1882no—
371#95C2×Dic4718yes—
372C24—21no—
373C24—21no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24C2×C4×D4714yes—
380C24C22×Dic4712yes—
381C24C23×D4711yescharacteristic
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—48no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—48no—
404C22×C4—44no—
405C22×C4—474no—
406C22×C4—474no—
407C22×C4—474no—
408C22×C4—474no—
409C22×C4—942no—
410C22×C4—942no—
411C22×C4—942no—
412C22×C4—942no—
413C22×C4—942no—
414C22×C4—942no—
415C22×C4—942no—
416C22×C4—942no—
417C22×C4—942no—
418C22×C4—942no—
419C22×C4—942no—
420C22×C4—942no—
421C22×C4—942no—
422C22×C4—942no—
423C22×C4—942no—
424C22×C4—942no—
425C22×C4—942no—
426C22×C4—942no—
427C22×C4—942no—
428C22×C4—942no—
429C22×C4—942no—
430C22×C4—942no—
431C22×C4—942no—
432C22×C4—942no—
433C22×C4C2×C4×D4714yes—
434C22×C4C2×C4×D4714yes—
435C22×C4C22×Dic4712yes—
436C22×C4C22×Dic4714yes—
437C22×C4C22×Dic4712yes—
438C22×C4C22×Dic4714yes—
439C4⋊C4—48no—
440C4⋊C4—1888no—
441C22⋊C4—44no—
442C22⋊C4—1888no—
443C22⋊C4—1888no—
444C22⋊C4C4×Dic47116yescomplemented
445C42—1888no—
446C42—1884no—
447C42—1884no—
448C23—24no—
449C23—24no—
450C23—22no—
451C23—24no—
452C23—22no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—41no—
459C23—42no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—41no—
468C23C2×C4×Dic4714yes—
469C23C22×C4×D4712yes—
470C23C23×Dic4711yescharacteristic
471C23#1309406912yes—
472C23#1309406912yes—
473C23#1309406911yescharacteristic
474C23#1309406911yescharacteristic
475C23#1309407012yes—
476C23#1309407112yes—
477C23#1309407212yes—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—416no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—944no—
503C2×C4—944no—
504C2×C4—944no—
505C2×C4—944no—
506C2×C4—944no—
507C2×C4—944no—
508C2×C4—944no—
509C2×C4—944no—
510C2×C4—944no—
511C2×C4—944no—
512C2×C4—944no—
513C2×C4—944no—
514C2×C4—1882no—
515C2×C4—1882no—
516C2×C4—1882no—
517C2×C4—1882no—
518C2×C4—1882no—
519C2×C4—1882no—
520C2×C4—1882no—
521C2×C4—1882no—
522C2×C4—1882no—
523C2×C4—1882no—
524C2×C4—1882no—
525C2×C4—1882no—
526C2×C4—1882no—
527C2×C4—1882no—
528C2×C4—1882no—
529C2×C4—1882no—
530C2×C4C2×C4×Dic4718yes—
531C22—24no—
532C22—28no—
533C22—42no—
534C22—42no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—44no—
539C22—44no—
540C22C22×C4×Dic4711yescharacteristic
541C22#1568959512yes—
542C22#1568959512yes—
543C22#1568959811yescharacteristic
544C22#1568959812yes—
545C22#1568960111yescharacteristic
546C22#1568960212yes—
547C22#1568960212yes—
548C22#1568960412yes—
549C22#1568960412yes—
550C22#1568960612yes—
551C22#1568986211yescharacteristic
552C22#1568986212yes—
553C22#1568986311yescharacteristic
554C22#1568986611yescharacteristic
555C22#1568986912yes—
556C22#1568986912yes—
557C22#1568987012yes—
558C22#1568987412yes—
559C22#1568987412yes—
560C22#1568987611yescharacteristic
561C4—28no—
562C4—44no—
563C4—42no—
564C4—42no—
565C4—1884no—
566C4—1884no—
567C4—1884no—
568C4—1884no—
569C2—44no—
570C2#2024574511yescharacteristic
571C2#2024574912yes—
572C2#2024652911yescharacteristic
573C2#2024653611yescharacteristic
574C2#2024653712yes—
575C2#2024654412yes—
576C2#2024655412yes—
577C2#2024656112yes—
578C2#2024657612yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC22×C47C22×C42
FittingC2×C47×#385C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenterC2×#385D47
hypocenterC47#13831
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
4. #20241622C2semidirect
5. #20241620C2semidirect
6. #20241617C2semidirect
7. #20241613C2semidirect
16. #15689016C4semidirect
31. C47×#385C4semidirect
39. C2×C4⋊Dic47C2×C4semidirect
52. C2×C4×Dic47C2×C4semidirect
61. C47⋊#95C2×C4semidirect
62. C47⋊#95C2×C4semidirect
151. C22⋊C4×C47C42semidirect
310. #385Dic47semidirect
322. C47#13831semidirect
444. 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
#202416202
#202416172
#202416132
#202416112
C2×C47×#3852
#1383147