Finite Group Databasebeta

← #26220618

Subgroups — #26220618

Order 12032.

874 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
1C2×#15689224C211yesmaximal, characteristic
2#20245863C211yesmaximal, characteristic, complemented
3#20245850C211yesmaximal, characteristic
4#20244271C211yesmaximal, characteristic
5#20244124C211yesmaximal, characteristic, complemented
6#20244121C211yesmaximal, characteristic, complemented
7#20244082C211yesmaximal, characteristic
8#20244075C211yesmaximal, characteristic, complemented
9#20244074C211yesmaximal, characteristic
10#20244067C211yesmaximal, characteristic, complemented
11#20243288C211yesmaximal, characteristic, complemented
12#20243220C211yesmaximal, characteristic, complemented
13#20243215C211yesmaximal, characteristic
14C4×#13093934C211yesmaximal, characteristic, complemented
15C47×#1968C211yesmaximal, characteristic
16C2×Q8×Dic47C2211yescharacteristic
17C2×D4×Dic47C2211yescharacteristic
18C2×#13093980—21no—
19#15689224C418yescomplemented
20C2×#13093958C2211yescharacteristic
21C2×#13093958C2211yescharacteristic
22C2×#13093948C2211yescharacteristic
23C2×#13093948C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093934C2211yescharacteristic
27C2×#13093934C2211yescharacteristic
28C2×#13093934C2211yescharacteristic
29C2×#13093934C2211yescharacteristic
30C2×C4×C47⋊C8C2211yescharacteristic
31#15689030C2211yescharacteristic
32#15689026C2211yescharacteristic
33#15689021C2211yescharacteristic
34#15689021C2211yescharacteristic
35#15689019C2211yescharacteristic
36#15689019C2211yescharacteristic
37#15689017—21no—
38#15689011—21no—
39#15689008—21no—
40#15689006—21no—
41C4⋊C4×Dic47C2211yescharacteristic
42C4⋊C4×Dic47C2211yescharacteristic
43#15688974—21no—
44C22⋊C4×Dic47—21no—
45#15688968—21no—
46C42×Dic47C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C22×SD16×C47C2211yescharacteristic
54C2×#106×C47C2211yescharacteristic
55C2×#103×C47C2211yescharacteristic
56C2×#102×C47C2211yescharacteristic
57C2×C4×C8×C47C2211yescharacteristic
58C47×#391C2211yescharacteristic
59C47×#390C2211yescharacteristic
60C23×Dic47—21no—
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63Q8×Dic47—22no—
64Q8×Dic47C2×C414yes—
65D4×Dic47—22no—
66D4×Dic47C2×C414yes—
67#13093980—44no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×C22⋊Dic47—21no—
78C2×C22⋊Dic47—21no—
79C2×C22⋊Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47D411yescharacteristic
85C2×C4×Dic47D411yescharacteristic
86C2×C4×Dic47D411yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88C2×C4×Dic47C2311yescharacteristic
89#13093958—22no—
90#13093958C2×C414yes—
91#13093948—22no—
92#13093948C2×C414yes—
93#13093945—22no—
94#13093945C2×C414yes—
95#13093934—22no—
96#13093934—22no—
97#13093934—22no—
98#13093934C2×C414yes—
99C4×C47⋊C8—22no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C22×Q8×C47C2311yescharacteristic
109C22×D4×C47C2311yescharacteristic
110C2×SD16×C47—22no—
111C2×SD16×C47—22no—
112C2×SD16×C47C2×C414yes—
113C22×C8×C47C2311yescharacteristic
114C22×C8×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C22⋊C4×C47—21no—
118C2×C22⋊C4×C47—21no—
119C2×C42×C47C2311yescharacteristic
120#106×C47—22no—
121#103×C47—22no—
122#102×C47—22no—
123C4×C8×C47—22no—
124#95×C47—21no—
125#95×C47—21no—
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×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2411yescharacteristic
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47—24no—
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—42no—
151C22×Dic47—42no—
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×C47#4112yes—
161C2×C8×C47#4112yes—
162C2×C8×C47#4112yes—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C22⋊C4×C47—42no—
168C22⋊C4×C47—42no—
169C42×C47—21no—
170C42×C47—21no—
171C22⋊Dic47—24no—
172C22⋊Dic47—42no—
173C22⋊Dic47—42no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—24no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47C22×C412yes—
182C4⋊Dic47C22×C412yes—
183Dic47⋊C4—42no—
184Dic47⋊C4—42no—
185Dic47⋊C4—42no—
186Dic47⋊C4—42no—
187C4×Dic47—22no—
188C4×Dic47—22no—
189C4×Dic47—21no—
190C4×Dic47—21no—
191C4×Dic47—24no—
192C4×Dic47—41no—
193C4×Dic47—41no—
194C4×Dic47—42no—
195C4×Dic47—42no—
196C4×Dic47C22×C412yes—
197C4×Dic47#4112yes—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8C22×C412yes—
203C2×C47⋊C8#4112yes—
204C23×C47—22no—
205C23×C47—41no—
206C23×C47—41no—
207C23×C47—41no—
208C23×C47—41no—
209C23×C47—41no—
210C23×C47—41no—
211C23×C47C22×D411yescharacteristic
212Q8×C47—22no—
213Q8×C47—22no—
214Q8×C47—41no—
215Q8×C47—41no—
216D4×C47—22no—
217D4×C47—22no—
218D4×C47—41no—
219D4×C47—41no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—22no—
225C2×C4×C47—41no—
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×C47—41no—
233C2×C4×C47AGL(1,ℤ/8)12yes—
234C2×C4×C47#13712yes—
235C2×C4×C47C23×C411yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×C4×C47C2×#4111yescharacteristic
239C2×C4×C47C2×#4111yescharacteristic
240C2×C4×C47C2×#4111yescharacteristic
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—22no—
246C2×Dic47—22no—
247C2×Dic47—21no—
248C2×Dic47—21no—
249C2×Dic47—21no—
250C2×Dic47—21no—
251C2×Dic47—21no—
252C2×Dic47—21no—
253C2×Dic47—48no—
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×Dic47C4×D412yes—
265C2×Dic47C4×D412yes—
266C8×C47—22no—
267C8×C47#11714yescomplemented
268C47⋊C8—22no—
269C47⋊C8—22no—
270#21288—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×C4×D411yescharacteristic
279C22×C47#53211yescharacteristic
280C22×C47#53211yescharacteristic
281C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C47C2×#13711yescharacteristic
283C22×C47#57511yescharacteristic
284C22×C47#57511yescharacteristic
285C4×C47—22no—
286C4×C47—22no—
287C4×C47—42no—
288C4×C47—81no—
289C4×C47—81no—
290C4×C47C2×#11711yescharacteristic
291C4×C47C4×#4111yescharacteristic
292C4×C47#52911yescharacteristic
293C4×C47#52911yescharacteristic
294Dic47—22no—
295Dic47—22no—
296Dic47—41no—
297Dic47—41no—
298Dic47—42no—
299Dic47—42no—
300Dic47—81no—
301Dic47—81no—
302C2×#440—471no—
303#2295—471no—
304#2293—471no—
305#1973—471no—
306#1968D4711yescharacteristic
307#1943—471no—
308#1942—471no—
309#1936—471no—
310#1935—471no—
311#1934—471no—
312#1933—471no—
313#1776—471no—
314#1763—471no—
315#1762—471no—
316C4×#97—471no—
317C2×C47—82no—
318C2×C47#229311yescharacteristic
319C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
320C2×C47C4×#13711yescharacteristic
321C2×C47#318011yescharacteristic
322C2×C47#318111yescharacteristic
323C2×C47#318211yescharacteristic
324C2×C47#318311yescharacteristic
325C22×SD16C2×D4711yescharacteristic
326C2×C4×Q8—471no—
327C2×C4×D4—471no—
328C2×#117—941no—
329#440—478no—
330C2×#107—471no—
331C2×#107—471no—
332C2×#106C2×D4711yescharacteristic
333C2×#103—471no—
334C2×#103—471no—
335C2×#103C2×D4711yescharacteristic
336C2×#102—471no—
337C2×#102—471no—
338C2×#102C2×D4711yescharacteristic
339C2×#97—471no—
340C2×#97—471no—
341C2×#97—471no—
342C2×#97—471no—
343C2×C4×C8—471no—
344C2×C4×C8C2×D4711yescharacteristic
345#391—471no—
346#391C2×D4711yescharacteristic
347#390—471no—
348#390C2×D4711yescharacteristic
349#389—471no—
350#389—471no—
351#389—471no—
352#389—471no—
353#388—941no—
354#388—941no—
355#387—941no—
356#387—941no—
357C4×C4⋊C4—471no—
358C4×C4⋊C4—471no—
359C4×C4⋊C4—941no—
360C4×C22⋊C4—941no—
361#376—941no—
362C43—471no—
363#340—471no—
364#340—471no—
365#340—471no—
366#340—471no—
367#340—471no—
368#340—471no—
369C47#2128811yescharacteristic, complemented, Sylow
370C22×Q8C22×D4711yescharacteristic
371C22×D4C22×D4711yescharacteristic
372C23×C4—941no—
373C2×SD16—22no—
374C2×SD16—22no—
375C2×SD16C4×D4714yes—
376C22×C8—471no—
377C22×C8—471no—
378C22×C8C22×D4711yescharacteristic
379C22×C8C22×D4711yescharacteristic
380C4×Q8—474no—
381C4×Q8—942no—
382C4×D4—474no—
383C4×D4—942no—
384#117—1884no—
385C2×C4⋊C4—471no—
386C2×C4⋊C4—471no—
387C2×C4⋊C4—471no—
388C2×C4⋊C4—471no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4C22×D4711yescharacteristic
395C2×C4⋊C4C22×D4711yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—941no—
399C2×C22⋊C4—941no—
400C2×C22⋊C4—941no—
401C2×C42—471no—
402C2×C42—471no—
403C2×C42—471no—
404C2×C42—471no—
405C2×C42—471no—
406C2×C42—471no—
407C2×C42—941no—
408C2×C42—941no—
409C2×C42—941no—
410C2×C42C22×D4711yescharacteristic
411#107—474no—
412#107—942no—
413#106—22no—
414#103—22no—
415#103—474no—
416#103—942no—
417#102—22no—
418#102—474no—
419#102—942no—
420#97—474no—
421#97—942no—
422#97—942no—
423#97—942no—
424C4×C8—22no—
425C4×C8—942no—
426#95—21no—
427#95—21no—
428#95—941no—
429#95—941no—
430#95—941no—
431#95—941no—
432#95—941no—
433#95—941no—
434#95—941no—
435#95—941no—
436C24—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D4712yes—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C2×C4×D4712yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—471no—
453C22×C4—471no—
454C22×C4—471no—
455C22×C4—471no—
456C22×C4—944no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—941no—
460C22×C4—941no—
461C22×C4—1882no—
462C22×C4—1882no—
463C22×C4D4×D4711yescharacteristic
464C22×C4D4×D4711yescharacteristic
465C22×C4C23×D4711yescharacteristic
466SD16—28no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—472no—
470C2×C8—472no—
471C2×C8—941no—
472C2×C8—941no—
473C2×C8—941no—
474C2×C8—941no—
475C2×C8C2×C4×D4712yes—
476C2×C8#1062535212yes—
477C2×C8#1062535212yes—
478C2×C8#1062535212yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—472no—
484C4⋊C4—472no—
485C4⋊C4—944no—
486C4⋊C4—941no—
487C4⋊C4—941no—
488C4⋊C4—941no—
489C4⋊C4—941no—
490C4⋊C4—941no—
491C4⋊C4—941no—
492C4⋊C4—1882no—
493C4⋊C4—1882no—
494C4⋊C4—1882no—
495C4⋊C4—1882no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—944no—
499C22⋊C4—1882no—
500C22⋊C4—1882no—
501C42—21no—
502C42—21no—
503C42—472no—
504C42—472no—
505C42—944no—
506C42—942no—
507C42—942no—
508C42—941no—
509C42—941no—
510C42—1881no—
511C42—1881no—
512C42—1882no—
513C42—1882no—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D4711yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—472no—
544C2×C4—472no—
545C2×C4—941no—
546C2×C4—941no—
547C2×C4—941no—
548C2×C4—941no—
549C2×C4—942no—
550C2×C4—942no—
551C2×C4—941no—
552C2×C4—941no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—941no—
556C2×C4—941no—
557C2×C4—1888no—
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—1881no—
567C2×C4—1881no—
568C2×C4#1309405112yes—
569C2×C4#1309405312yes—
570C2×C4C22×C4×D4711yescharacteristic
571C2×C4#41×D4711yescharacteristic
572C2×C4#41×D4711yescharacteristic
573C2×C4C2×#1062535211yescharacteristic
574C2×C4C2×#1062535211yescharacteristic
575C2×C4C2×#1062535211yescharacteristic
576C8—22no—
577C8—942no—
578C8—942no—
579C8#1309398014yescomplemented
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C4×D4×D4711yescharacteristic
588C22#1568966811yescharacteristic
589C22#1568967111yescharacteristic
590C22C2×#1309405111yescharacteristic
591C22C2×#1309405311yescharacteristic
592C22#1568992011yescharacteristic
593C22#1568992011yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—942no—
600C4—942no—
601C4—3761no—
602C4—3761no—
603C4—1882no—
604C4—1882no—
605C4—1881no—
606C4—1881no—
607C4C2×#1309398011yescharacteristic
608C4C4×#1062535211yescharacteristic
609C4#1568965211yescharacteristic
610C4#1568965411yescharacteristic
611C2—82no—
612C2#2024585011yescharacteristic
613C2C4×#1309405111yescharacteristic
614C2C4×#1309405311yescharacteristic
615C2#2025407911yescharacteristic
616C2#2025409011yescharacteristic
617C2#2025410411yescharacteristic
618C2#2025411411yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1968C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#1968D47
hypocenterC47#21288
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
2. #20245863C2semidirect
5. #20244124C2semidirect
6. #20244121C2semidirect
8. #20244075C2semidirect
10. #20244067C2semidirect
11. #20243288C2semidirect
12. #20243220C2semidirect
14. C4×#13093934C2semidirect
19. #15689224C4semidirect
267. C8×C47#117semidirect
369. C47#21288semidirect
579. C8#13093980semidirect

Maximal subgroups

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

subgroupindex
C2×#156892242
#202458632
#202458502
#202442712
#202441242
#202441212
#202440822
#202440752
#202440742
#202440672
#202432882
#202432202
#202432152
C4×#130939342
C47×#19682
#2128847