Finite Group Databasebeta

← #26236157

Subgroups — #26236157

Order 12032.

838 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246353C211yesmaximal, characteristic, complemented
2#20246337C211yesmaximal, characteristic
3#20244271C211yesmaximal, characteristic
4#20244244C211yesmaximal, characteristic
5#20244031C211yesmaximal, characteristic, complemented
6#20244024C211yesmaximal, characteristic, complemented
7#20244017C211yesmaximal, characteristic
8#20244010C211yesmaximal, characteristic
9#20243556C211yesmaximal, characteristic, complemented
10#20243398C211yesmaximal, characteristic
11#20243393C211yesmaximal, characteristic, complemented
12#20243287C211yesmaximal, characteristic, complemented
13#107×Dic47C211yesmaximal, characteristic, complemented
14#20243260C211yesmaximal, characteristic, complemented
15C2×C47×#440C211yesmaximal, characteristic
16C2×#13093983—21no—
17C2×#13093958C2211yescharacteristic
18C2×#13093958C2211yescharacteristic
19C2×#13093956C2211yescharacteristic
20C2×#13093956C2211yescharacteristic
21C2×C4⋊(C47⋊C8)C2211yescharacteristic
22C2×#13093948C2211yescharacteristic
23C2×#13093948C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093934C2211yescharacteristic
27C2×C8×Dic47C2211yescharacteristic
28#15689030C2211yescharacteristic
29#15689030C2211yescharacteristic
30#15689026C2211yescharacteristic
31#15689026C2211yescharacteristic
32#15689016—21no—
33#15689016—21no—
34#15689009—21no—
35#15689009—21no—
36#15688993C2211yescharacteristic
37#15688991—21no—
38#15688989C2211yescharacteristic
39#15688983—21no—
40C4⋊C4×Dic47C2211yescharacteristic
41C4⋊C4×Dic47C2211yescharacteristic
42C4⋊C4×Dic47C2211yescharacteristic
43C4⋊C4×Dic47C2211yescharacteristic
44C4×C4⋊Dic47C2211yescharacteristic
45#15688969—21no—
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688877C2211yescharacteristic
51#15688877C2211yescharacteristic
52C22×SD16×C47C2211yescharacteristic
53C2×C4×Q8×C47C2211yescharacteristic
54C2×C4×D4×C47C2211yescharacteristic
55C47×#440C418yescomplemented
56C2×#107×C47C2211yescharacteristic
57C2×#103×C47C2211yescharacteristic
58C2×#102×C47C2211yescharacteristic
59C2×#97×C47C2211yescharacteristic
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62#13093983—44no—
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47D411yescharacteristic
66C2×C4⋊Dic47D411yescharacteristic
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C22⋊Dic47—21no—
72C2×C22⋊Dic47—21no—
73C2×C22⋊Dic47—21no—
74C2×C22⋊Dic47—21no—
75C2×C4×Dic47—21no—
76C2×C4×Dic47—21no—
77C2×C4×Dic47—21no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47C2311yescharacteristic
80C2×C4×Dic47C2311yescharacteristic
81#13093958—22no—
82#13093958—41no—
83#13093956—22no—
84#13093956—41no—
85C4⋊(C47⋊C8)—22no—
86#13093948—41no—
87#13093948—41no—
88#13093945—41no—
89#13093945—41no—
90#13093934—22no—
91C8×Dic47—22no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95D411yescharacteristic
103C47⋊#95D411yescharacteristic
104C22×Q8×C47C2311yescharacteristic
105C22×D4×C47C2311yescharacteristic
106C23×C4×C47—21no—
107C2×SD16×C47—22no—
108C2×SD16×C47—22no—
109C2×SD16×C47C2×C414yes—
110C22×C8×C47C2311yescharacteristic
111C22×C8×C47C2311yescharacteristic
112C4×Q8×C47—22no—
113C4×Q8×C47C2×C414yes—
114C4×D4×C47—22no—
115C4×D4×C47C2×C414yes—
116C2×C4⋊C4×C47—21no—
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C22⋊C4×C47—21no—
120C2×C42×C47—21no—
121C2×C42×C47C2311yescharacteristic
122#107×C47C2×C414yes—
123#103×C47C2×C414yes—
124#102×C47C2×C414yes—
125#97×C47C2×C414yes—
126C24×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47—21no—
131C2×Q8×C47—21no—
132C2×Q8×C47C22×C412yes—
133C2×D4×C47—21no—
134C2×D4×C47—21no—
135C2×D4×C47—21no—
136C2×D4×C47—21no—
137C2×D4×C47—21no—
138C2×D4×C47C22×C412yes—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—24no—
143C22×C4×C47—42no—
144C22×C4×C47—42no—
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2411yescharacteristic
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47C2×D411yescharacteristic
153C22×Dic47C2×D411yescharacteristic
154C22×Dic47C2×D411yescharacteristic
155C22×Dic47C2×D411yescharacteristic
156SD16×C47—28no—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47C22×C412yes—
160C2×C8×C47C22×C412yes—
161C2×C8×C47#4112yes—
162C2×C8×C47#4112yes—
163C4⋊C4×C47—24no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47C22×C412yes—
167C4⋊C4×C47C22×C412yes—
168C22⋊C4×C47—24no—
169C42×C47—24no—
170C42×C47C22×C412yes—
171C42×C47#4112yes—
172C22⋊Dic47—42no—
173C22⋊Dic47—42no—
174C22⋊Dic47—81no—
175C22⋊Dic47—81no—
176C4⋊Dic47—22no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—41no—
186C4⋊Dic47—42no—
187C4⋊Dic47—81no—
188C4×Dic47—21no—
189C4×Dic47—21no—
190C4×Dic47—21no—
191C4×Dic47—21no—
192C4×Dic47—42no—
193C4×Dic47—42no—
194C4×Dic47—42no—
195C4×Dic47—42no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C23×C47—22no—
203C23×C47—41no—
204C23×C47—41no—
205C23×C47—41no—
206C23×C47—41no—
207C23×C47—41no—
208C23×C47—41no—
209C23×C47C22×D411yescharacteristic
210Q8×C47—22no—
211Q8×C47—22no—
212Q8×C47—41no—
213Q8×C47—41no—
214D4×C47—22no—
215D4×C47—22no—
216D4×C47—41no—
217D4×C47—41no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—48no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47C4×D412yes—
233C2×C4×C47C4×D412yes—
234C2×C4×C47C23×C411yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×C4×C47H(5,𝔽2)11yescharacteristic
239C2×C4×C472−1+411yescharacteristic
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—21no—
249C2×Dic47—21no—
250C2×Dic47—21no—
251C2×Dic47—41no—
252C2×Dic47—41no—
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×Dic47AGL(1,ℤ/8)12yes—
264C8×C47—22no—
265C8×C47#11714yescomplemented
266C47⋊C8—41no—
267C47⋊C8—41no—
268#22820—471nomaximal, Sylow
269C22×C47—42no—
270C22×C47—42no—
271C22×C47—42no—
272C22×C47—81no—
273C22×C47—81no—
274C22×C47—81no—
275C22×C47—81no—
276C22×C47C2×C4×D411yescharacteristic
277C22×C47#54711yescharacteristic
278C22×C47#55011yescharacteristic
279C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C47#57511yescharacteristic
281C22×C47#57711yescharacteristic
282C22×C47#57811yescharacteristic
283C4×C47—22no—
284C4×C47—22no—
285C4×C47—42no—
286C4×C47—42no—
287C4×C47—42no—
288C4×C47C2×#11711yescharacteristic
289C4×C47#52011yescharacteristic
290C4×C47#55511yescharacteristic
291C4×C47#55611yescharacteristic
292Dic47—22no—
293Dic47—41no—
294Dic47—41no—
295Dic47—41no—
296Dic47—81no—
297Dic47—81no—
298Dic47—81no—
299Dic47—81no—
300C2×#440D4711yescharacteristic
301#2356—471no—
302#2354—471no—
303#1973—471no—
304#1968—471no—
305#1928—471no—
306#1927—471no—
307#1926—471no—
308#1925—471no—
309#1847—471no—
310#1804—471no—
311#1803—471no—
312#1776—471no—
313C4×#107—471no—
314#1771—471no—
315C2×C47—82no—
316C2×C47#235411yescharacteristic
317C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
318C2×C47#294711yescharacteristic
319C2×C47#334311yescharacteristic
320C2×C47#334811yescharacteristic
321C2×C47#339711yescharacteristic
322C2×C47#340211yescharacteristic
323C22×SD16C2×D4711yescharacteristic
324C2×C4×Q8C2×D4711yescharacteristic
325C2×C4×D4C2×D4711yescharacteristic
326C2×#117—941no—
327#440Dic4718yescomplemented
328C2×#107—471no—
329C2×#107—471no—
330C2×#107C2×D4711yescharacteristic
331C2×#106—471no—
332C2×#106—471no—
333C2×C4⋊C8—471no—
334C2×#103—471no—
335C2×#103—471no—
336C2×#103C2×D4711yescharacteristic
337C2×#102—471no—
338C2×#102—471no—
339C2×#102C2×D4711yescharacteristic
340C2×#97—471no—
341C2×#97C2×D4711yescharacteristic
342C2×C4×C8—471no—
343#391—471no—
344#391—471no—
345#390—471no—
346#390—471no—
347#388—941no—
348#388—941no—
349#387—941no—
350#387—941no—
351#384—471no—
352#383—941no—
353#382—471no—
354#380—941no—
355C4×C4⋊C4—471no—
356C4×C4⋊C4—471no—
357C4×C4⋊C4—471no—
358C4×C4⋊C4—471no—
359C4×C4⋊C4—471no—
360#376—941no—
361#340—471no—
362#340—471no—
363#340—471no—
364#340—471no—
365#336—471no—
366#336—471no—
367C47#2282011yescharacteristic, complemented, Sylow
368C22×Q8C22×D4711yescharacteristic
369C22×D4C22×D4711yescharacteristic
370C23×C4—21no—
371C2×SD16—22no—
372C2×SD16—22no—
373C2×SD16C2×Dic4714yes—
374C22×C8—471no—
375C22×C8—471no—
376C22×C8C22×D4711yescharacteristic
377C22×C8C22×D4711yescharacteristic
378C4×Q8—22no—
379C4×Q8C2×Dic4714yes—
380C4×D4—22no—
381C4×D4C2×Dic4714yes—
382#117—1884no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—471no—
385C2×C4⋊C4—471no—
386C2×C4⋊C4—471no—
387C2×C4⋊C4—471no—
388C2×C4⋊C4—471no—
389C2×C4⋊C4—471no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4C22×D4711yescharacteristic
393C2×C4⋊C4C22×D4711yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—941no—
396C2×C22⋊C4—941no—
397C2×C22⋊C4—941no—
398C2×C22⋊C4—941no—
399C2×C42—21no—
400C2×C42—471no—
401C2×C42—471no—
402C2×C42—941no—
403C2×C42—941no—
404C2×C42—941no—
405C2×C42—941no—
406C2×C42C22×D4711yescharacteristic
407#107—942no—
408#107—1881no—
409#107C2×Dic4714yes—
410#106—942no—
411#106—1881no—
412C4⋊C8—942no—
413#103—1881no—
414#103—1881no—
415#103C2×Dic4714yes—
416#102—1881no—
417#102—1881no—
418#102C2×Dic4714yes—
419#97—942no—
420#97C2×Dic4714yes—
421C4×C8—942no—
422#95—471no—
423#95—471no—
424#95—941no—
425#95—941no—
426#95—941no—
427#95—941no—
428#95—941no—
429#95—941no—
430#95—941no—
431#95—941no—
432#95—941no—
433#95—941no—
434C24—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8C22×Dic4712yes—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4C22×Dic4712yes—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—471no—
454C22×C4—471no—
455C22×C4—471no—
456C22×C4—471no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—941no—
460C22×C4—941no—
461C22×C4D4×D4711yescharacteristic
462C22×C4D4×D4711yescharacteristic
463C22×C4C23×D4711yescharacteristic
464SD16—28no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—941no—
468C2×C8—941no—
469C2×C8—941no—
470C2×C8—941no—
471C2×C8—941no—
472C2×C8—941no—
473C2×C8#1062535012yes—
474C2×C8#1062535012yes—
475C2×C8C22×Dic4712yes—
476C2×C8C22×Dic4712yes—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—942no—
481C4⋊C4—941no—
482C4⋊C4—941no—
483C4⋊C4—941no—
484C4⋊C4—941no—
485C4⋊C4—941no—
486C4⋊C4—941no—
487C4⋊C4—941no—
488C4⋊C4—941no—
489C4⋊C4—3761no—
490C4⋊C4—1881no—
491C4⋊C4—1882no—
492C4⋊C4C22×Dic4712yes—
493C4⋊C4C22×Dic4712yes—
494C22⋊C4—24no—
495C22⋊C4—3761no—
496C22⋊C4—3761no—
497C22⋊C4—1882no—
498C22⋊C4—1882no—
499C42—24no—
500C42—941no—
501C42—941no—
502C42—941no—
503C42—941no—
504C42—1882no—
505C42—1882no—
506C42—1882no—
507C42—1882no—
508C42#1062535212yes—
509C42C22×Dic4712yes—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C2×D4×D4711yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522D4—22no—
523D4—22no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—48no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—472no—
541C2×C4—941no—
542C2×C4—941no—
543C2×C4—941no—
544C2×C4—941no—
545C2×C4—941no—
546C2×C4—941no—
547C2×C4—941no—
548C2×C4—941no—
549C2×C4—941no—
550C2×C4—941no—
551C2×C4—941no—
552C2×C4—1881no—
553C2×C4—1881no—
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×C4D4×Dic4712yes—
565C2×C4D4×Dic4712yes—
566C2×C4C23×Dic4711yescharacteristic
567C2×C4C2×#1062535211yescharacteristic
568C2×C4C2×#1062535011yescharacteristic
569C2×C4C2×#1062535011yescharacteristic
570C2×C4#1309407011yescharacteristic
571C2×C4#1309407111yescharacteristic
572C8—22no—
573C8—1881no—
574C8—1881no—
575C8#1309398314yescomplemented
576C22—42no—
577C22—42no—
578C22—42no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22C2×D4×Dic4711yescharacteristic
584C22#1568977811yescharacteristic
585C22#1568979411yescharacteristic
586C22AGL(1,ℤ/8)×D4711yescharacteristic
587C22#1568991911yescharacteristic
588C22#1568993211yescharacteristic
589C22#1568993911yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—942no—
596C4—3761no—
597C4—3761no—
598C4—3761no—
599C4—3761no—
600C4—1881no—
601C4—1881no—
602C4—1881no—
603C4C2×#1309398311yescharacteristic
604C4#1568960611yescharacteristic
605C4#1568983811yescharacteristic
606C4#1568984311yescharacteristic
607C2—82no—
608C2#2024633711yescharacteristic
609C2AGL(1,ℤ/8)×Dic4711yescharacteristic
610C2#2025223211yescharacteristic
611C2#2025583311yescharacteristic
612C2#2025590811yescharacteristic
613C2#2025640911yescharacteristic
614C2#2025646811yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#440C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenterC2×#440D47
hypocenterC47#22820
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. #20246353C2semidirect
5. #20244031C2semidirect
6. #20244024C2semidirect
9. #20243556C2semidirect
11. #20243393C2semidirect
12. #20243287C2semidirect
13. #107×Dic47C2semidirect
14. #20243260C2semidirect
55. C47×#440C4semidirect
265. C8×C47#117semidirect
327. #440Dic47semidirect
367. C47#22820semidirect
575. C8#13093983semidirect

Maximal subgroups

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

subgroupindex
#202463532
#202463372
#202442712
#202442442
#202440312
#202440242
#202440172
#202440102
#202435562
#202433982
#202433932
#202432872
#107×Dic472
#202432602
C2×C47×#4402
#2282047