Finite Group Databasebeta

← #26226883

Subgroups — #26226883

Order 12032.

762 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×#15689184C211yesmaximal, characteristic
2#20246011C211yesmaximal, characteristic
3#20244156C211yesmaximal, characteristic, complemented
4#20244156C211yesmaximal, characteristic, complemented
5#20244151C211yesmaximal, characteristic, complemented
6#20244146C211yesmaximal, characteristic, complemented
7#20243585C211yesmaximal, characteristic
8#20243585C211yesmaximal, characteristic
9#20243580C211yesmaximal, characteristic
10#20243575C211yesmaximal, characteristic
11#20243557C211yesmaximal, characteristic, complemented
12#20243550C211yesmaximal, characteristic, complemented
13#20243523C211yesmaximal, characteristic, complemented
14#20243515C211yesmaximal, characteristic, complemented
15C47×#2314C211yesmaximal, characteristic
16C2×#13093983C2211yescharacteristic
17C22×C4⋊Dic47C2211yescharacteristic
18#15689184C418yescomplemented
19C2×#13093957C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
22C2×#13093957C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×#13093955C2211yescharacteristic
27C2×C4⋊(C47⋊C8)C2211yescharacteristic
28C2×C4⋊(C47⋊C8)C2211yescharacteristic
29C2×C22⋊(C47⋊C8)C2211yescharacteristic
30C2×C22⋊(C47⋊C8)C2211yescharacteristic
31C2×#13093935C2211yescharacteristic
32C2×C4×C47⋊C8C2211yescharacteristic
33#15689022—21no—
34#15689022—21no—
35#15688998—21no—
36#15688998—21no—
37#15688993C2211yescharacteristic
38#15688993C2211yescharacteristic
39#15688989C2211yescharacteristic
40#15688989C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49C2×#117×C47C2211yescharacteristic
50C47×#388—21no—
51C47×#388—21no—
52C47×#387—21no—
53C47×#387—21no—
54C47×#384C2211yescharacteristic
55C47×#382C2211yescharacteristic
56C47×#380C2211yescharacteristic
57C47×#380C2211yescharacteristic
58C4×C4⋊C4×C47C2211yescharacteristic
59C4×C4⋊C4×C47C2211yescharacteristic
60C23×Dic47—21no—
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65#13093983—41no—
66#13093983—41no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47—41no—
73C2×C4⋊Dic47—41no—
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×C22⋊Dic47—21no—
79C2×C4×Dic47—21no—
80#13093957—41no—
81#13093957—41no—
82#13093957—41no—
83#13093957—41no—
84#13093955—41no—
85#13093955—41no—
86#13093955—41no—
87#13093955—41no—
88C4⋊(C47⋊C8)C2×C414yescomplemented
89C4⋊(C47⋊C8)C2×C414yescomplemented
90C22⋊(C47⋊C8)C2×C414yes—
91C22⋊(C47⋊C8)C2×C414yes—
92#13093935C2×C414yescomplemented
93C4×C47⋊C8C2×C414yescomplemented
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C23×C4×C47C2311yescharacteristic
99#117×C47C2×C414yes—
100#117×C47Q814yes—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47D411yescharacteristic
104C2×C4⋊C4×C47D411yescharacteristic
105C2×C4⋊C4×C47C2311yescharacteristic
106C2×C4⋊C4×C47C2311yescharacteristic
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C22⋊C4×C47—21no—
110C2×C22⋊C4×C47—21no—
111C2×C22⋊C4×C47D411yescharacteristic
112C2×C22⋊C4×C47D411yescharacteristic
113C2×C42×C47—21no—
114C2×C42×C47—21no—
115C2×C42×C47C2311yescharacteristic
116C2×C42×C47C2311yescharacteristic
117#95×C47—21no—
118#95×C47—21no—
119#95×C47—21no—
120#95×C47—21no—
121#95×C47—21no—
122#95×C47—21no—
123C24×C47C2×D411yescharacteristic
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47C22×C412yes—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×Q812yes—
139C22×C4×C47C2411yescharacteristic
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—42no—
145C22×Dic47—81no—
146C22×Dic47—81no—
147C4⋊C4×C47—21no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—81no—
156C4⋊C4×C47—81no—
157C4⋊C4×C47C4⋊C414yes—
158C4⋊C4×C47C4⋊C414yes—
159C22⋊C4×C47—81no—
160C22⋊C4×C47—81no—
161C22⋊C4×C47C4⋊C414yes—
162C22⋊C4×C47C4⋊C414yes—
163C42×C47—81no—
164C42×C47—81no—
165C42×C47C22×C412yes—
166C42×C47C22×C412yes—
167C42×C47C2×Q812yes—
168C42×C47C2×Q812yes—
169C22⋊Dic47—42no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—42no—
179C4⋊Dic47—81no—
180C4⋊Dic47—81no—
181C4⋊Dic47—81no—
182C4⋊Dic47—81no—
183C4×Dic47—42no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—21no—
192C2×C47⋊C8C22×C412yes—
193C2×C47⋊C8C22×C412yes—
194C2×C47⋊C8C22×C412yes—
195C2×C47⋊C8C22×C412yes—
196C23×C47—21no—
197C23×C47—21no—
198C23×C47—21no—
199C23×C47—21no—
200C23×C47—21no—
201C23×C47—21no—
202C23×C47C2×C4⋊C412yes—
203C23×C47C22×D411yescharacteristic
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47C2×C4⋊C412yes—
231C2×C4×C47C2×C4⋊C412yes—
232C2×C4×C47C2×C4⋊C412yes—
233C2×C4×C47C2×C4⋊C412yes—
234C2×C4×C47C2×C4⋊C412yes—
235C2×C4×C47C23×C411yescharacteristic
236C2×C4×C47C22×Q811yescharacteristic
237C2×C4×C47H(5,𝔽2)11yescharacteristic
238C2×C4×C47H(5,𝔽2)11yescharacteristic
239C2×C4×C472−1+411yescharacteristic
240C2×C4×C472−1+411yescharacteristic
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—84no—
254C47⋊C8—22no—
255C47⋊C8—22no—
256C47⋊C8—22no—
257C47⋊C8—22no—
258#21992—471nomaximal, Sylow
259C22×C47—22no—
260C22×C47—22no—
261C22×C47—22no—
262C22×C47—41no—
263C22×C47—41no—
264C22×C47—41no—
265C22×C47—41no—
266C22×C47C22×C4⋊C411yescharacteristic
267C22×C47#53511yescharacteristic
268C22×C47#53611yescharacteristic
269C22×C47#57611yescharacteristic
270C22×C47#57711yescharacteristic
271C22×C47#57711yescharacteristic
272C22×C47#57811yescharacteristic
273C4×C47—22no—
274C4×C47—22no—
275C4×C47—22no—
276C4×C47—22no—
277C4×C47—22no—
278C4×C47—81no—
279C4×C47—81no—
280C4×C47—81no—
281C4×C47—81no—
282C4×C47#52011yescharacteristic
283C4×C47#52011yescharacteristic
284C4×C47#54411yescharacteristic
285C4×C47#54411yescharacteristic
286Dic47—81no—
287Dic47—81no—
288Dic47—81no—
289Dic47—81no—
290C2×#433—471no—
291#2317—471no—
292#2314D4711yescharacteristic
293#1949—471no—
294#1949—471no—
295#1948—471no—
296#1947—471no—
297#1854—471no—
298#1854—471no—
299#1853—471no—
300#1852—471no—
301#1847—471no—
302#1846—471no—
303#1836—471no—
304#1834—471no—
305C2×C47—42no—
306C2×C47#231711yescharacteristic
307C2×C47#291211yescharacteristic
308C2×C47#291311yescharacteristic
309C2×C47#322411yescharacteristic
310C2×C47#322511yescharacteristic
311C2×C47#324111yescharacteristic
312C2×C47#324211yescharacteristic
313C2×#117—471no—
314C2×#117C2×D4711yescharacteristic
315C22×C4⋊C4—471no—
316#433—478no—
317C2×#107—471no—
318C2×#107—471no—
319C2×#107—471no—
320C2×#107—471no—
321C2×#106—471no—
322C2×#106—471no—
323C2×#106—471no—
324C2×#106—471no—
325C2×C4⋊C8—471no—
326C2×C4⋊C8—471no—
327C2×C22⋊C8—471no—
328C2×C22⋊C8—471no—
329C2×#97—471no—
330C2×C4×C8—471no—
331#389—941no—
332#389—941no—
333#388—21no—
334#388—21no—
335#387—21no—
336#387—21no—
337#385—941no—
338#385—941no—
339#384—471no—
340#384—471no—
341#384C2×D4711yescharacteristic
342#382—471no—
343#382—471no—
344#382C2×D4711yescharacteristic
345#380C2×D4711yescharacteristic
346#380C2×D4711yescharacteristic
347C4×C4⋊C4C2×D4711yescharacteristic
348C4×C4⋊C4C2×D4711yescharacteristic
349#340—471no—
350#340—471no—
351#340—471no—
352#340—471no—
353#340—471no—
354#340—471no—
355#340—471no—
356#340—471no—
357C47#2199211yescharacteristic, complemented, Sylow
358C23×C4—941no—
359C23×C4C22×D4711yescharacteristic
360C22×C8—471no—
361C22×C8—471no—
362C22×C8—471no—
363C22×C8—471no—
364#117—1881no—
365#117—1881no—
366#117Dic9414yes—
367#117C4×D4714yes—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—471no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4—941no—
377C2×C4⋊C4—941no—
378C2×C4⋊C4—941no—
379C2×C4⋊C4—1881no—
380C2×C4⋊C4—1881no—
381C2×C4⋊C4C22⋊D4711yescharacteristic
382C2×C4⋊C4C22⋊D4711yescharacteristic
383C2×C4⋊C4C22×D4711yescharacteristic
384C2×C4⋊C4C22×D4711yescharacteristic
385C2×C4⋊C4C22×D4711yescharacteristic
386C2×C4⋊C4C22×D4711yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—941no—
390C2×C22⋊C4C22⋊D4711yescharacteristic
391C2×C22⋊C4C22⋊D4711yescharacteristic
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—941no—
395C2×C42C22×D4711yescharacteristic
396C2×C42C22×D4711yescharacteristic
397#107—1881no—
398#107—1881no—
399#107—1881no—
400#107—1881no—
401#106—1881no—
402#106—1881no—
403#106—1881no—
404#106—1881no—
405C4⋊C8—474no—
406C4⋊C8—474no—
407C22⋊C8—474no—
408C22⋊C8—474no—
409#97—474no—
410C4×C8—474no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—941no—
418#95—941no—
419#95—941no—
420#95—941no—
421C24C2×C22⋊D4711yescharacteristic
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—941no—
431C22×C4—941no—
432C22×C4—941no—
433C22×C4—941no—
434C22×C4—3761no—
435C22×C4—3761no—
436C22×C4—1882no—
437C22×C4C2×Dic9412yes—
438C22×C4C2×C4×D4712yes—
439C22×C4C2×C22⋊D4711yescharacteristic
440C22×C4C2×C22⋊D4711yescharacteristic
441C22×C4C2×C22⋊D4711yescharacteristic
442C22×C4C2×C22⋊D4711yescharacteristic
443C22×C4C2×C22⋊D4711yescharacteristic
444C22×C4C23×D4711yescharacteristic
445C2×C8—472no—
446C2×C8—472no—
447C2×C8—472no—
448C2×C8—472no—
449C2×C8—941no—
450C2×C8—941no—
451C2×C8—941no—
452C2×C8—941no—
453C2×C8—941no—
454C2×C8—941no—
455C2×C8—941no—
456C2×C8—941no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—941no—
468C4⋊C4—941no—
469C4⋊C4—941no—
470C4⋊C4—941no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—941no—
475C4⋊C4—3761no—
476C4⋊C4—3761no—
477C4⋊C4—3761no—
478C4⋊C4—3761no—
479C4⋊C4—1882no—
480C4⋊C4Dic47⋊C414yes—
481C4⋊C4Dic47⋊C414yes—
482C22⋊C4—81no—
483C22⋊C4—81no—
484C22⋊C4—1882no—
485C22⋊C4Dic47⋊C414yes—
486C22⋊C4Dic47⋊C414yes—
487C42—81no—
488C42—81no—
489C42—1882no—
490C42C2×Dic9412yes—
491C42C2×Dic9412yes—
492C42C2×C4×D4712yes—
493C42C2×C4×D4712yes—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C2×Dic47⋊C412yes—
501C23C22×C22⋊D4711yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—3764no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4—1881no—
540C2×C4—1881no—
541C2×C4C2×Dic47⋊C412yes—
542C2×C4C2×Dic47⋊C412yes—
543C2×C4C2×Dic47⋊C412yes—
544C2×C4C2×Dic47⋊C412yes—
545C2×C4C2×Dic47⋊C412yes—
546C2×C4C22×C4×D4711yescharacteristic
547C2×C4C22×Dic9411yescharacteristic
548C2×C4#1309407011yescharacteristic
549C2×C4#1309407011yescharacteristic
550C2×C4#1309407111yescharacteristic
551C2×C4#1309407111yescharacteristic
552C8—942no—
553C8—942no—
554C8—942no—
555C8—942no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C22×Dic47⋊C411yescharacteristic
564C22#1568968711yescharacteristic
565C22#1568969411yescharacteristic
566C22#1568992611yescharacteristic
567C22#1568993111yescharacteristic
568C22#1568993411yescharacteristic
569C22#1568993811yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—3761no—
580C4—3761no—
581C4—3761no—
582C4—3761no—
583C4#1568960311yescharacteristic
584C4#1568960311yescharacteristic
585C4#1568975111yescharacteristic
586C4#1568975111yescharacteristic
587C2—42no—
588C2#2024600811yescharacteristic
589C2#2025198111yescharacteristic
590C2#2025198611yescharacteristic
591C2#2025447611yescharacteristic
592C2#2025449111yescharacteristic
593C2#2025463511yescharacteristic
594C2#2025465011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2314C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#2314D47
hypocenterC47#21992
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2314

Complements

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

NH ≅ G/Ntype
3. #20244156C2semidirect
4. #20244156C2semidirect
5. #20244151C2semidirect
6. #20244146C2semidirect
11. #20243557C2semidirect
12. #20243550C2semidirect
13. #20243523C2semidirect
14. #20243515C2semidirect
18. #15689184C4semidirect
88. C4⋊(C47⋊C8)C2×C4semidirect
89. C4⋊(C47⋊C8)C2×C4semidirect
92. #13093935C2×C4semidirect
93. C4×C47⋊C8C2×C4semidirect
357. C47#21992semidirect

Maximal subgroups

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

subgroupindex
C2×#156891842
#202460112
#202441562
#202441512
#202441462
#202435852
#202435802
#202435752
#202435572
#202435502
#202435232
#202435152
C47×#23142
#2199247