Finite Group Databasebeta

← #26316198

Subgroups — #26316198

Order 12032.

734 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20249507C211yesmaximal, characteristic, complemented
2#20244539C211yesmaximal, characteristic, complemented
3#20244534C211yesmaximal, characteristic
4#20244529C211yesmaximal, characteristic, complemented
5#20244524C211yesmaximal, characteristic
6#20244205C211yesmaximal, characteristic
7#20244200C211yesmaximal, characteristic
8#20243523C211yesmaximal, characteristic, complemented
9#20243515C211yesmaximal, characteristic, complemented
10#20243403C211yesmaximal, characteristic, complemented
11#20243398C211yesmaximal, characteristic
12#20243393C211yesmaximal, characteristic, complemented
13#20243388C211yesmaximal, characteristic
14#20243245C211yesmaximal, characteristic, complemented
15C47×#2326C211yesmaximal, characteristic
16C2×#13094024C2211yescharacteristic
17C2×#13094016C2211yescharacteristic
18C2×C4⋊(C47⋊C8)C2211yescharacteristic
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×#13093948C2211yescharacteristic
21C2×#13093948C2211yescharacteristic
22C2×#13093948C2211yescharacteristic
23C2×#13093948C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×#13093945C2211yescharacteristic
28C2×#13093935C2211yescharacteristic
29C2×C4×C47⋊C8C2211yescharacteristic
30#15689055—21no—
31#15689055—21no—
32#15689044—21no—
33#15689044—21no—
34#15688993C2211yescharacteristic
35#15688993C2211yescharacteristic
36#15688989C2211yescharacteristic
37#15688989C2211yescharacteristic
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C2×#124×C47C2211yescharacteristic
49C2×C4×Q8×C47C2211yescharacteristic
50C2×C4×D4×C47C2211yescharacteristic
51C47×#391C2211yescharacteristic
52C47×#390C2211yescharacteristic
53C47×#388—21no—
54C47×#388—21no—
55C47×#385—21no—
56C47×#385—21no—
57C47×#384C2211yescharacteristic
58C47×#376C2211yescharacteristic
59C22×Dic94—21no—
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64#13094024—41no—
65#13094024—41no—
66#13094016—41no—
67#13094016—41no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C4⋊(C47⋊C8)—22no—
80C4⋊(C47⋊C8)—22no—
81#13093948—41no—
82#13093948—41no—
83#13093948—41no—
84#13093948—41no—
85#13093945—41no—
86#13093945—41no—
87#13093945—41no—
88#13093945—41no—
89#13093935—22no—
90C4×C47⋊C8—22no—
91C47⋊#95—21no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C22×Q8×C47C2311yescharacteristic
96C22×D4×C47C2311yescharacteristic
97C23×C4×C47—21no—
98#124×C47—22no—
99#124×C47Q814yescomplemented
100C4×Q8×C47—22no—
101C4×Q8×C47—22no—
102C4×D4×C47—22no—
103C4×D4×C47—22no—
104C2×C4⋊C4×C47—21no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C22⋊C4×C47—21no—
110C2×C22⋊C4×C47—21no—
111C2×C22⋊C4×C47—21no—
112C2×C22⋊C4×C47—21no—
113C2×C22⋊C4×C47—21no—
114C2×C42×C47—21no—
115C2×C42×C47C2311yescharacteristic
116C2×C42×C47C2311yescharacteristic
117C2×C42×C47C2311yescharacteristic
118#95×C47—21no—
119#95×C47—21no—
120#95×C47D411yescharacteristic
121#95×C47D411yescharacteristic
122#95×C47D411yescharacteristic
123#95×C47D411yescharacteristic
124C24×C47—21no—
125C2×Q8×C47—21no—
126C2×Q8×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47C2×Q812yes—
131C2×D4×C47—21no—
132C2×D4×C47—21no—
133C2×D4×C47—21no—
134C2×D4×C47—21no—
135C2×D4×C47—21no—
136C2×D4×C47C2×Q812yes—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—42no—
141C22×C4×C47—81no—
142C22×C4×C47—81no—
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2411yescharacteristic
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—21no—
154C2×Dic94—42no—
155C2×Dic94—81no—
156C2×Dic94—81no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—42no—
162C4⋊C4×C47—81no—
163C4⋊C4×C47—81no—
164C22⋊C4×C47—24no—
165C22⋊C4×C47—24no—
166C22⋊C4×C47—42no—
167C22⋊C4×C47—81no—
168C22⋊C4×C47—81no—
169C42×C47—21no—
170C42×C47—21no—
171C42×C47—42no—
172C42×C47C2×Q812yes—
173C42×C47#4112yes—
174C42×C47#4112yes—
175C42×C47#4112yes—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47—42no—
185C4⋊Dic47—81no—
186C4⋊Dic47—81no—
187C4⋊Dic47—81no—
188C4⋊Dic47—81no—
189Dic47⋊C4—42no—
190Dic47⋊C4—42no—
191C2×C47⋊C8—21no—
192C2×C47⋊C8—21no—
193C2×C47⋊C8—21no—
194C2×C47⋊C8—21no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C23×C47—22no—
204C23×C47—41no—
205C23×C47—41no—
206C23×C47—41no—
207C23×C47—41no—
208C23×C47—41no—
209C23×C47—41no—
210C23×C47C22×D411yescharacteristic
211Q8×C47—22no—
212Q8×C47—22no—
213Q8×C47—41no—
214Q8×C47—41no—
215D4×C47—22no—
216D4×C47—22no—
217D4×C47—41no—
218D4×C47—41no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—22no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—84no—
243C2×C4×C47C22⋊Q812yes—
244C2×C4×C47C22⋊Q812yes—
245C2×C4×C47#12412yes—
246C2×C4×C47#12412yes—
247C2×C4×C47C22×Q811yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C2×C4×C47C2×#4111yescharacteristic
250C2×C4×C47C2×#4111yescharacteristic
251C2×C4×C47H(5,𝔽2)11yescharacteristic
252C2×C4×C47H(5,𝔽2)11yescharacteristic
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265Dic94—84no—
266C47⋊C8—41no—
267C47⋊C8—41no—
268C47⋊C8—41no—
269C47⋊C8—41no—
270#29151—471nomaximal, Sylow
271C22×C47—42no—
272C22×C47—42no—
273C22×C47—42no—
274C22×C47—81no—
275C22×C47—81no—
276C22×C47—81no—
277C22×C47—81no—
278C22×C47C2×C22⋊Q811yescharacteristic
279C22×C47C2×#12411yescharacteristic
280C22×C47#53511yescharacteristic
281C22×C47#57611yescharacteristic
282C22×C47#57711yescharacteristic
283C22×C47#57711yescharacteristic
284C22×C47#57811yescharacteristic
285C4×C47—22no—
286C4×C47—22no—
287C4×C47—22no—
288C4×C47—22no—
289C4×C47—42no—
290C4×C47—41no—
291C4×C47—41no—
292C4×C47—81no—
293C4×C47—81no—
294C4×C47#55411yescharacteristic
295C4×C47#55411yescharacteristic
296C4×C47#55511yescharacteristic
297C4×C47#55511yescharacteristic
298Dic47—81no—
299Dic47—81no—
300Dic47—81no—
301Dic47—81no—
302#2660—471no—
303#2326D4711yescharacteristic
304#2026—471no—
305#2025—471no—
306#2024—471no—
307#2023—471no—
308#1959—471no—
309#1958—471no—
310#1836—471no—
311#1834—471no—
312#1805—471no—
313#1804—471no—
314#1803—471no—
315#1802—471no—
316#1768—471no—
317C2×C47—82no—
318C2×C47#256311yescharacteristic
319C2×C47#308311yescharacteristic
320C2×C47#308411yescharacteristic
321C2×C47#313611yescharacteristic
322C2×C47#313711yescharacteristic
323C2×C47#322411yescharacteristic
324C2×C47#322511yescharacteristic
325C2×C4⋊Q8—471no—
326C2×#125—471no—
327C2×#124C2×D4711yescharacteristic
328C2×C4×Q8C2×D4711yescharacteristic
329C2×C4×D4C2×D4711yescharacteristic
330C2×C4⋊C8—471no—
331C2×C4⋊C8—471no—
332C2×#103—471no—
333C2×#103—471no—
334C2×#103—471no—
335C2×#103—471no—
336C2×#102—471no—
337C2×#102—471no—
338C2×#102—471no—
339C2×#102—471no—
340C2×#97—471no—
341C2×C4×C8—471no—
342#398—941no—
343#398—941no—
344#395—941no—
345#395—941no—
346#391C2×D4711yescharacteristic
347#390C2×D4711yescharacteristic
348#388—21no—
349#388—21no—
350#385—21no—
351#385—21no—
352#384—471no—
353#384—471no—
354#384C2×D4711yescharacteristic
355#382—471no—
356#382—471no—
357#376C2×D4711yescharacteristic
358#340—471no—
359#340—471no—
360#340—471no—
361#340—471no—
362#340—471no—
363#340—471no—
364#340—471no—
365#340—471no—
366#336—471no—
367#336—471no—
368C47#2915111yescharacteristic, complemented, Sylow
369C22×Q8—941no—
370C22×Q8C22×D4711yescharacteristic
371C22×D4C22×D4711yescharacteristic
372C23×C4—21no—
373C22×C8—471no—
374C22×C8—471no—
375C22×C8—471no—
376C22×C8—471no—
377C4⋊Q8—1881no—
378C4⋊Q8—1881no—
379#125—1881no—
380#125—1881no—
381#124—22no—
382#124Dic9414yescomplemented
383C4×Q8—22no—
384C4×Q8—22no—
385C4×D4—22no—
386C4×D4—22no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—471no—
391C2×C4⋊C4—471no—
392C2×C4⋊C4—471no—
393C2×C4⋊C4—471no—
394C2×C4⋊C4—941no—
395C2×C4⋊C4—941no—
396C2×C4⋊C4—941no—
397C2×C4⋊C4—941no—
398C2×C4⋊C4—941no—
399C2×C4⋊C4—941no—
400C2×C4⋊C4—941no—
401C2×C4⋊C4C22×D4711yescharacteristic
402C2×C4⋊C4C22×D4711yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C42—21no—
409C2×C42C22×D4711yescharacteristic
410C2×C42C22×D4711yescharacteristic
411C2×C42C22×D4711yescharacteristic
412C4⋊C8—942no—
413C4⋊C8—942no—
414#103—1881no—
415#103—1881no—
416#103—1881no—
417#103—1881no—
418#102—1881no—
419#102—1881no—
420#102—1881no—
421#102—1881no—
422#97—942no—
423C4×C8—942no—
424#95—21no—
425#95—21no—
426#95—941no—
427#95—941no—
428#95—941no—
429#95—941no—
430#95C22⋊D4711yescharacteristic
431#95C22⋊D4711yescharacteristic
432#95C22⋊D4711yescharacteristic
433#95C22⋊D4711yescharacteristic
434C24—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—3761no—
441C2×Q8—3761no—
442C2×Q8—1882no—
443C2×Q8C2×Dic9412yes—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4C2×Dic9412yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—42no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4—941no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—941no—
460C22×C4C2×C22⋊D4711yescharacteristic
461C22×C4C2×C22⋊D4711yescharacteristic
462C22×C4C2×C22⋊D4711yescharacteristic
463C22×C4C2×C22⋊D4711yescharacteristic
464C22×C4C2×C22⋊D4711yescharacteristic
465C22×C4C2×C22⋊D4711yescharacteristic
466C22×C4C23×D4711yescharacteristic
467C2×C8—941no—
468C2×C8—941no—
469C2×C8—941no—
470C2×C8—941no—
471C2×C8—941no—
472C2×C8—941no—
473C2×C8—941no—
474C2×C8—941no—
475C2×C8—941no—
476C2×C8—941no—
477C2×C8—941no—
478C2×C8—941no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—941no—
487C4⋊C4—941no—
488C4⋊C4—941no—
489C4⋊C4—941no—
490C4⋊C4—941no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4—941no—
494C4⋊C4—3761no—
495C4⋊C4—3761no—
496C4⋊C4—3761no—
497C4⋊C4—3761no—
498C4⋊C4—1882no—
499C4⋊C4—1882no—
500C4⋊C4—1882no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—42no—
504C22⋊C4—81no—
505C22⋊C4—81no—
506C42—21no—
507C42—21no—
508C42—42no—
509C42C2×Dic9412yes—
510C42#1062535012yes—
511C42#1062535212yes—
512C42#1062535212yes—
513C23—22no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23C22×C22⋊D4711yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—41no—
524Q8—41no—
525Q8—3764no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—84no—
554C2×C4—1881no—
555C2×C4—1881no—
556C2×C4—1881no—
557C2×C4—1881no—
558C2×C4—1881no—
559C2×C4—1881no—
560C2×C4—1881no—
561C2×C4—1881no—
562C2×C4—1881no—
563C2×C4—1881no—
564C2×C4—1881no—
565C2×C4—1881no—
566C2×C4#1309400312yes—
567C2×C4#1309400312yes—
568C2×C4#1309401112yes—
569C2×C4#1309401112yes—
570C2×C4C22×Dic9411yescharacteristic
571C2×C4C2×#1062535211yescharacteristic
572C2×C4C2×#1062535211yescharacteristic
573C2×C4C2×#1062535011yescharacteristic
574C2×C4#1309407011yescharacteristic
575C2×C4#1309407011yescharacteristic
576C8—1881no—
577C8—1881no—
578C8—1881no—
579C8—1881no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C2×#1309400311yescharacteristic
588C22C2×#1309401111yescharacteristic
589C22#1568968711yescharacteristic
590C22#1568992611yescharacteristic
591C22#1568993111yescharacteristic
592C22#1568993411yescharacteristic
593C22#1568993811yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—42no—
601C4—81no—
602C4—81no—
603C4—3761no—
604C4—3761no—
605C4—3761no—
606C4—3761no—
607C4#1568983011yescharacteristic
608C4#1568983011yescharacteristic
609C4#1568983811yescharacteristic
610C4#1568983811yescharacteristic
611C2—82no—
612C2#2024856111yescharacteristic
613C2#2025323911yescharacteristic
614C2#2025325011yescharacteristic
615C2#2025374211yescharacteristic
616C2#2025375311yescharacteristic
617C2#2025447611yescharacteristic
618C2#2025449111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2326C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
hypercenter#2326D47
hypocenterC47#29151
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ #376 ◁ #2326

Complements

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

NH ≅ G/Ntype
1. #20249507C2semidirect
2. #20244539C2semidirect
4. #20244529C2semidirect
8. #20243523C2semidirect
9. #20243515C2semidirect
10. #20243403C2semidirect
12. #20243393C2semidirect
14. #20243245C2semidirect
99. #124×C47Q8semidirect
368. C47#29151semidirect
382. #124Dic94semidirect

Maximal subgroups

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

subgroupindex
#202495072
#202445392
#202445342
#202445292
#202445242
#202442052
#202442002
#202435232
#202435152
#202434032
#202433982
#202433932
#202433882
#202432452
C47×#23262
#2915147