Finite Group Databasebeta

← #26137875

Subgroups — #26137875

Order 12032.

2022 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689016C212yesmaximal
2C2×C4⋊C4×Dic47C212yesmaximal, complemented
3C2×C4×C22⋊Dic47C212yesmaximal
4#20241625C211yesmaximal, characteristic
5#20241625C211yesmaximal, characteristic
6#20241622C212yesmaximal, complemented
7#20241620C212yesmaximal, complemented
8#95×Dic47C212yesmaximal, complemented
9C2×C47×#385C211yesmaximal, characteristic
10C22×C4⋊Dic47C2212yes—
11C22×C22⋊Dic47C2212yes—
12C22×C22⋊Dic47C2212yes—
13C22×C4×Dic47C2211yescharacteristic
14C22×C4×Dic47C2212yes—
15C22×C4×Dic47C2212yes—
16C22×C4×Dic47C2211yescharacteristic
17C22×C4×Dic47C2212yes—
18#15689016C4116yescomplemented
19C4⋊C4×Dic47—28no—
20C4×C22⋊Dic47—28no—
21C2×C47⋊#95C2212yes—
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2211yescharacteristic
25C2×C47⋊#95C2211yescharacteristic
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C24×C4×C47C2211yescharacteristic
30C22×C4⋊C4×C47C2212yes—
31C22×C22⋊C4×C47C2211yescharacteristic
32C22×C22⋊C4×C47C2211yescharacteristic
33C47×#385C418yescomplemented
34C2×#95×C47C2212yes—
35C23×Dic47C2312yes—
36C23×Dic47C2312yes—
37C23×Dic47C2312yes—
38C23×Dic47C2312yes—
39C2×C4⋊Dic47—24no—
40C2×C4⋊Dic47—24no—
41C2×C4⋊Dic47C2×C418yescomplemented
42C2×C22⋊Dic47—24no—
43C2×C22⋊Dic47—24no—
44C2×C22⋊Dic47—24no—
45C2×C22⋊Dic47—24no—
46C2×C22⋊Dic47C2×C418yes—
47C2×C22⋊Dic47C2×C418yes—
48C2×C4×Dic47—24no—
49C2×C4×Dic47—24no—
50C2×C4×Dic47—24no—
51C2×C4×Dic47—24no—
52C2×C4×Dic47—24no—
53C2×C4×Dic47—24no—
54C2×C4×Dic47—24no—
55C2×C4×Dic47—24no—
56C2×C4×Dic47—41no—
57C2×C4×Dic47—41no—
58C2×C4×Dic47—42no—
59C2×C4×Dic47—42no—
60C2×C4×Dic47C2×C418yescomplemented
61C47⋊#95—24no—
62C47⋊#95—24no—
63C47⋊#95—24no—
64C47⋊#95—41no—
65C47⋊#95—41no—
66C47⋊#95C2×C418yescomplemented
67C47⋊#95C2×C418yescomplemented
68C47⋊#95D418yes—
69C25×C47C2311yescharacteristic
70C23×C4×C47—24no—
71C23×C4×C47—24no—
72C23×C4×C47—41no—
73C23×C4×C47—41no—
74C23×C4×C47C2×C414yes—
75C23×C4×C47C2311yescharacteristic
76C23×C4×C47C2311yescharacteristic
77C23×C4×C47C2312yes—
78C23×C4×C47C2312yes—
79C2×C4⋊C4×C47—24no—
80C2×C4⋊C4×C47—24no—
81C2×C4⋊C4×C47C2×C418yes—
82C2×C22⋊C4×C47—41no—
83C2×C22⋊C4×C47—41no—
84C2×C22⋊C4×C47C2×C414yes—
85C2×C22⋊C4×C47C2×C418yes—
86C2×C22⋊C4×C47C2×C414yes—
87#95×C47C2×C418yes—
88C24×C47—22no—
89C24×C47—22no—
90C24×C47—22no—
91C24×C47—22no—
92C24×C47—21no—
93C24×C47—21no—
94C24×C47—22no—
95C24×C47C22×C414yes—
96C24×C47C22×C412yes—
97C24×C47C2411yescharacteristic
98C22×C4×C47—22no—
99C22×C4×C47—22no—
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—21no—
111C22×C4×C47—21no—
112C22×C4×C47—21no—
113C22×C4×C47—21no—
114C22×C4×C47—216no—
115C22×C4×C47—44no—
116C22×C4×C47—44no—
117C22×C4×C47—48no—
118C22×C4×C47—44no—
119C22×C4×C47C22×C412yes—
120C22×C4×C47C22×C412yes—
121C22×C4×C47C22×C414yes—
122C22×C4×C47C22×C414yes—
123C22×C4×C47C22×C414yes—
124C22×C4×C47C22×C414yes—
125C22×C4×C47C2×D414yes—
126C22×C4×C47#4114yes—
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×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47C22×C414yes—
148C22×Dic47C22×C414yes—
149C22×Dic47C22×C414yes—
150C22×Dic47C22×C414yes—
151C22×Dic47C2×D414yes—
152C22×Dic47C2×D414yes—
153C22×Dic47#4114yes—
154C22×Dic47#4114yes—
155C4⋊C4×C47—216no—
156C22⋊C4×C47—44no—
157C22⋊C4×C47C42116yescomplemented
158C22⋊Dic47—216no—
159C22⋊Dic47—216no—
160C4⋊Dic47—216no—
161C4×Dic47—216no—
162C4×Dic47—44no—
163C4×Dic47—44no—
164C4×Dic47—48no—
165C4×Dic47—48no—
166C23×C47—24no—
167C23×C47—24no—
168C23×C47—24no—
169C23×C47—24no—
170C23×C47—24no—
171C23×C47—24no—
172C23×C47—22no—
173C23×C47—22no—
174C23×C47—24no—
175C23×C47—24no—
176C23×C47—41no—
177C23×C47—42no—
178C23×C47—42no—
179C23×C47—42no—
180C23×C47—41no—
181C23×C47—41no—
182C23×C47—42no—
183C23×C47—42no—
184C23×C47—42no—
185C23×C47—41no—
186C23×C47C2×C4214yes—
187C23×C47C23×C412yes—
188C23×C47C23×C411yescharacteristic
189C23×C47C22×D412yes—
190C23×C47C2×#4112yes—
191C23×C47C2×#4112yes—
192C23×C47C2×#4112yes—
193C23×C47H(5,𝔽2)11yescharacteristic
194C23×C47H(5,𝔽2)11yescharacteristic
195C23×C472−1+412yes—
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—24no—
203C2×C4×C47—24no—
204C2×C4×C47—24no—
205C2×C4×C47—24no—
206C2×C4×C47—24no—
207C2×C4×C47—24no—
208C2×C4×C47—24no—
209C2×C4×C47—24no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—416no—
213C2×C4×C47—41no—
214C2×C4×C47—42no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—42no—
218C2×C4×C47—41no—
219C2×C4×C47C2×C4218yes—
220C2×C4×C47C4×D418yes—
221C2×Dic47—24no—
222C2×Dic47—24no—
223C2×Dic47—24no—
224C2×Dic47—24no—
225C2×Dic47—24no—
226C2×Dic47—24no—
227C2×Dic47—24no—
228C2×Dic47—24no—
229C2×Dic47—24no—
230C2×Dic47—24no—
231C2×Dic47—24no—
232C2×Dic47—24no—
233C2×Dic47—24no—
234C2×Dic47—24no—
235C2×Dic47—24no—
236C2×Dic47—24no—
237C2×Dic47—24no—
238C2×Dic47—24no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241C2×Dic47—42no—
242C2×Dic47—42no—
243C2×Dic47—42no—
244C2×Dic47—42no—
245C2×Dic47—42no—
246C2×Dic47—42no—
247C2×Dic47C4×D418yes—
248C2×Dic47C4×D418yes—
249#13823—471nomaximal, Sylow
250C22×C47—28no—
251C22×C47—24no—
252C22×C47—42no—
253C22×C47—44no—
254C22×C47—44no—
255C22×C47—44no—
256C22×C47—44no—
257C22×C47—44no—
258C22×C47—42no—
259C22×C47C22×C4211yescharacteristic
260C22×C47C2×C4×D412yes—
261C22×C47C2×C4×D412yes—
262C22×C47C2×C4×D412yes—
263C22×C47C4×#4112yes—
264C22×C47C4×#4112yes—
265C22×C47C4×#4112yes—
266C22×C47#51811yescharacteristic
267C22×C47#51911yescharacteristic
268C22×C47#52012yes—
269C22×C47#52012yes—
270C22×C47D4211yescharacteristic
271C22×C47#54612yes—
272C22×C47#54712yes—
273C22×C47#54812yes—
274C22×C47#55111yescharacteristic
275C22×C47#55111yescharacteristic
276C22×C47#55212yes—
277C22×C47#55312yes—
278C22×C47#55612yes—
279C22×C47#55911yescharacteristic
280C4×C47—28no—
281C4×C47—28no—
282C4×C47—42no—
283C4×C47—42no—
284Dic47—28no—
285Dic47—28no—
286Dic47—44no—
287Dic47—44no—
288C2×#388—472no—
289C2×#385D4711yescharacteristic
290C2×C4×C4⋊C4—472no—
291C2×C4×C22⋊C4—472no—
292#1437—471no—
293#1437—471no—
294#1436—472no—
295#1436—472no—
296C4×#95—472no—
297C2×C47—44no—
298C2×C47C42×D412yes—
299C2×C47#227411yescharacteristic
300C2×C47#235811yescharacteristic
301C2×C47#235912yes—
302C2×C47#236112yes—
303C2×C47#236212yes—
304C2×C47#236712yes—
305C2×C47#237312yes—
306C2×C47#237411yescharacteristic
307C24×C4C2×D4711yescharacteristic
308C22×C4⋊C4—472no—
309C22×C4⋊C4C2×D4712yes—
310C22×C22⋊C4—472no—
311C22×C22⋊C4—472no—
312C22×C22⋊C4C2×D4711yescharacteristic
313C22×C22⋊C4C2×D4711yescharacteristic
314C22×C42—472no—
315C22×C42—471no—
316C22×C42—472no—
317C22×C42—472no—
318C22×C42—471no—
319#388—4716no—
320#385Dic4718yescomplemented
321C4×C4⋊C4—948no—
322C4×C22⋊C4—948no—
323C2×#95—472no—
324C2×#95—472no—
325C2×#95—472no—
326C2×#95—472no—
327C2×#95—472no—
328C2×#95—472no—
329C2×#95—471no—
330C2×#95—471no—
331C2×#95C2×D4712yes—
332C47#1382311yescharacteristic, complemented, Sylow
333C25C22×D4711yescharacteristic
334C23×C4—24no—
335C23×C4—24no—
336C23×C4—41no—
337C23×C4—41no—
338C23×C4—472no—
339C23×C4—472no—
340C23×C4—472no—
341C23×C4—472no—
342C23×C4C2×Dic4714yes—
343C23×C4C22×D4711yescharacteristic
344C23×C4C22×D4711yescharacteristic
345C23×C4C22×D4712yes—
346C23×C4C22×D4712yes—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—478no—
350C2×C4⋊C4—944no—
351C2×C4⋊C4—944no—
352C2×C4⋊C4C2×Dic4718yes—
353C2×C22⋊C4—41no—
354C2×C22⋊C4—41no—
355C2×C22⋊C4—478no—
356C2×C22⋊C4—478no—
357C2×C22⋊C4—944no—
358C2×C22⋊C4—944no—
359C2×C22⋊C4—944no—
360C2×C22⋊C4—944no—
361C2×C22⋊C4C4×D4718yes—
362C2×C22⋊C4C2×Dic4714yes—
363C2×C22⋊C4C2×Dic4714yes—
364C2×C42—478no—
365C2×C42—944no—
366C2×C42—944no—
367C2×C42—944no—
368C2×C42—944no—
369C2×C42—944no—
370C2×C42—944no—
371C2×C42—944no—
372C2×C42—944no—
373C2×C42—1882no—
374C2×C42—1882no—
375C2×C42—1881no—
376C2×C42—1881no—
377#95—478no—
378#95—478no—
379#95—478no—
380#95—944no—
381#95—944no—
382#95—944no—
383#95—1881no—
384#95—1881no—
385#95C2×Dic4718yes—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—21no—
391C24—21no—
392C24—22no—
393C24C2×C4×D4714yes—
394C24C22×Dic4712yes—
395C24C23×D4711yescharacteristic
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—216no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—474no—
418C22×C4—474no—
419C22×C4—474no—
420C22×C4—474no—
421C22×C4—474no—
422C22×C4—474no—
423C22×C4—474no—
424C22×C4—474no—
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×C4—942no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4C2×C4×D4714yes—
446C22×C4C2×C4×D4714yes—
447C22×C4D4×D4714yes—
448C22×C4#1062535214yes—
449C22×C4C22×Dic4712yes—
450C22×C4C22×Dic4712yes—
451C22×C4C22×Dic4714yes—
452C22×C4C22×Dic4714yes—
453C4⋊C4—216no—
454C4⋊C4—9416no—
455C22⋊C4—44no—
456C22⋊C4—9416no—
457C22⋊C4—9416no—
458C22⋊C4C4×Dic47116yescomplemented
459C42—9416no—
460C42—1888no—
461C42—1888no—
462C42—1884no—
463C42—1884no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23C2×C4×Dic4714yes—
485C23C22×C4×D4712yes—
486C23C23×Dic4711yescharacteristic
487C23C2×D4×D4712yes—
488C23#41×D4712yes—
489C23#41×D4712yes—
490C23C2×#1062535212yes—
491C23#1309406911yescharacteristic
492C23#1309406911yescharacteristic
493C23#1309407212yes—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—416no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—478no—
518C2×C4—478no—
519C2×C4—944no—
520C2×C4—944no—
521C2×C4—944no—
522C2×C4—944no—
523C2×C4—944no—
524C2×C4—944no—
525C2×C4—944no—
526C2×C4—944no—
527C2×C4—944no—
528C2×C4—944no—
529C2×C4—944no—
530C2×C4—944no—
531C2×C4—944no—
532C2×C4—944no—
533C2×C4—944no—
534C2×C4—944no—
535C2×C4—944no—
536C2×C4—944no—
537C2×C4—1882no—
538C2×C4—1882no—
539C2×C4—1882no—
540C2×C4—1882no—
541C2×C4—1882no—
542C2×C4—1882no—
543C2×C4—1882no—
544C2×C4—1882no—
545C2×C4C2×C4×Dic4718yes—
546C2×C4D4×Dic4718yes—
547C22—28no—
548C22—24no—
549C22—42no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—42no—
556C22C22×C4×Dic4711yescharacteristic
557C22C4×D4×D4712yes—
558C22C4×D4×D4712yes—
559C22C2×D4×Dic4712yes—
560C22C4×#1062535212yes—
561C22C4×#1062535212yes—
562C22#41×Dic4712yes—
563C22#1568959811yescharacteristic
564C22#1568960111yescharacteristic
565C22#1568960412yes—
566C22#1568960412yes—
567C22#1568975511yescharacteristic
568C22#1568976012yes—
569C22#1568977512yes—
570C22#1568978312yes—
571C22#1568980111yescharacteristic
572C22#1568980411yescharacteristic
573C22#1568980712yes—
574C22#1568981812yes—
575C22#1568984412yes—
576C22#1568985711yescharacteristic
577C4—28no—
578C4—28no—
579C4—42no—
580C4—42no—
581C4—948no—
582C4—948no—
583C4—1884no—
584C4—1884no—
585C2—44no—
586C2C4×D4×Dic4712yes—
587C2#2024574511yescharacteristic
588C2#2024637211yescharacteristic
589C2#2024638312yes—
590C2#2024640712yes—
591C2#2024641912yes—
592C2#2024645612yes—
593C2#2024650312yes—
594C2#2024652411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC22×C47C22×C42
FittingC2×C47×#385C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenterC2×#385D47
hypocenterC47#13823
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×Dic47C2semidirect
6. #20241622C2semidirect
7. #20241620C2semidirect
8. #95×Dic47C2semidirect
18. #15689016C4semidirect
33. C47×#385C4semidirect
41. C2×C4⋊Dic47C2×C4semidirect
60. C2×C4×Dic47C2×C4semidirect
66. C47⋊#95C2×C4semidirect
67. C47⋊#95C2×C4semidirect
157. C22⋊C4×C47C42semidirect
320. #385Dic47semidirect
332. C47#13823semidirect
458. 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
C2×C4⋊C4×Dic472
C2×C4×C22⋊Dic472
#202416252
#202416222
#202416202
#95×Dic472
C2×C47×#3852
#1382347