Finite Group Databasebeta

← #26216826

Subgroups — #26216826

Order 12032.

886 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M16×Dic47C211yesmaximal, characteristic
2#20245790C211yesmaximal, characteristic
3#20245783C211yesmaximal, characteristic
4#20243555C211yesmaximal, characteristic, complemented
5#20243553C211yesmaximal, characteristic, complemented
6#20243520C211yesmaximal, characteristic, complemented
7#20243520C211yesmaximal, characteristic, complemented
8#20243421C211yesmaximal, characteristic
9#20243419C211yesmaximal, characteristic
10#20243419C211yesmaximal, characteristic
11#107×Dic47C211yesmaximal, characteristic, complemented
12#20243280C211yesmaximal, characteristic, complemented
13#106×Dic47C211yesmaximal, characteristic, complemented
14#20243276C211yesmaximal, characteristic, complemented
15C47×#1809C211yesmaximal, characteristic
16C22×#10625329C2211yescharacteristic
17C2×#13093980—21no—
18C22×C4×Dic47C2211yescharacteristic
19C2×#13093958C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
22C2×#13093956C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25M16×Dic47C418yescomplemented
26C2×#13093934C2211yescharacteristic
27C2×#13093934C2211yescharacteristic
28C2×C8×Dic47C2211yescharacteristic
29C2×C8×Dic47C2211yescharacteristic
30#15688992C2211yescharacteristic
31#15688992C2211yescharacteristic
32#15688990—21no—
33#15688988C2211yescharacteristic
34#15688988C2211yescharacteristic
35#15688987—21no—
36#15688984—21no—
37#15688983C2211yescharacteristic
38#15688983C2211yescharacteristic
39#15688981—21no—
40C4⋊C4×Dic47C2211yescharacteristic
41C4⋊C4×Dic47C2211yescharacteristic
42C4⋊C4×Dic47C2211yescharacteristic
43C4⋊C4×Dic47C2211yescharacteristic
44#15688974—21no—
45C22⋊C4×Dic47—21no—
46#15688968—21no—
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C22×M16×C47C2211yescharacteristic
54C2×#107×C47C2211yescharacteristic
55C2×#106×C47C2211yescharacteristic
56C47×#380C2211yescharacteristic
57C47×#380C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C23×Dic47D411yescharacteristic
61C23×Dic47D411yescharacteristic
62C2×#10625329—22no—
63C2×#10625329—22no—
64C2×#10625329C2×C414yes—
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67#13093980—44no—
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47—21no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47C2×C414yes—
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47D411yescharacteristic
85C2×C4×Dic47Q814yes—
86C2×C4×Dic47C2311yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88#13093958—22no—
89#13093957—22no—
90#13093957—22no—
91#13093956—22no—
92#13093955—22no—
93#13093955—22no—
94#13093934C2×C414yes—
95#13093934C2×C414yes—
96C8×Dic47C2×C414yes—
97C8×Dic47C2×C414yes—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C23×C4×C47C2311yescharacteristic
109C2×M16×C47—22no—
110C2×M16×C47—22no—
111C2×M16×C47C2×C414yes—
112C22×C8×C47C2311yescharacteristic
113C22×C8×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C22⋊C4×C47—21no—
119C2×C22⋊C4×C47—21no—
120#107×C47—22no—
121#106×C47—22no—
122#95×C47—21no—
123#95×C47—21no—
124C24×C47C2×D411yescharacteristic
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×C47—21no—
133C22×C4×C47C22×C412yes—
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×Q812yes—
136C22×C4×C47C2411yescharacteristic
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—41no—
144C22×Dic47—41no—
145C22×Dic47C4⋊C414yes—
146C22×Dic47C4⋊C414yes—
147C22×Dic47C2×D411yescharacteristic
148C22×Dic47C2×D411yescharacteristic
149C22×Dic47C2×D411yescharacteristic
150C22×Dic47C2×D411yescharacteristic
151M16×C47—28no—
152C2×C8×C47—21no—
153C2×C8×C47—21no—
154C2×C8×C47C22×C412yes—
155C2×C8×C47C22×C412yes—
156C2×C8×C47#4112yes—
157C2×C8×C47#4112yes—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C22⋊C4×C47—42no—
167C22⋊C4×C47—42no—
168C22⋊Dic47—42no—
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—
178Dic47⋊C4—42no—
179Dic47⋊C4—42no—
180Dic47⋊C4—42no—
181Dic47⋊C4—42no—
182C4×Dic47—42no—
183C4×Dic47—42no—
184C4×Dic47—42no—
185C4×Dic47—42no—
186C4×Dic47C4⋊C414yes—
187C4×Dic47C4⋊C414yes—
188C4×Dic47C22×C412yes—
189C4×Dic47C22×C412yes—
190C4×Dic47C2×Q812yes—
191C4×Dic47C2×Q812yes—
192#10625329—28no—
193C2×C47⋊C8—21no—
194C2×C47⋊C8—21no—
195C2×C47⋊C8C22×C412yes—
196C2×C47⋊C8C22×C412yes—
197C2×C47⋊C8#4112yes—
198C2×C47⋊C8#4112yes—
199C23×C47—21no—
200C23×C47—21no—
201C23×C47—21no—
202C23×C47—21no—
203C23×C47—21no—
204C23×C47—21no—
205C23×C47C2×C4⋊C412yes—
206C23×C47C22×D411yescharacteristic
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—41no—
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×C47C2×C4⋊C412yes—
226C2×C4×C47C23×C411yescharacteristic
227C2×C4×C47C22×Q811yescharacteristic
228C2×C4×C47C2×#4111yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—21no—
238C2×Dic47—28no—
239C2×Dic47—44no—
240C2×Dic47—41no—
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×Dic47C2×C4⋊C412yes—
253C2×Dic47C2×C4⋊C412yes—
254C2×Dic47C2×C4⋊C412yes—
255C2×Dic47C2×C4⋊C412yes—
256C2×Dic47AGL(1,ℤ/8)12yes—
257C2×Dic47#13712yes—
258C8×C47—22no—
259C8×C47#11714yescomplemented
260C47⋊C8—22no—
261C47⋊C8#11714yescomplemented
262#20819—471nomaximal, Sylow
263C22×C47—22no—
264C22×C47—22no—
265C22×C47—22no—
266C22×C47—41no—
267C22×C47—41no—
268C22×C47—41no—
269C22×C47—41no—
270C22×C47C22×C4⋊C411yescharacteristic
271C22×C47#53211yescharacteristic
272C22×C47#53211yescharacteristic
273C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C47C2×#13711yescharacteristic
275C22×C47#57511yescharacteristic
276C22×C47#57511yescharacteristic
277C4×C47—22no—
278C4×C47—81no—
279C4×C47—81no—
280C4×C47—81no—
281C4×C47—81no—
282C4×C47C2×#11711yescharacteristic
283C4×C47C2×#11711yescharacteristic
284C4×C47#53311yescharacteristic
285C4×C47#53311yescharacteristic
286Dic47—22no—
287Dic47—22no—
288Dic47—22no—
289Dic47—81no—
290Dic47—81no—
291Dic47—81no—
292Dic47—81no—
293Dic47#42814yes—
294C2×C4×M16—471no—
295#2285—471no—
296#2283—471no—
297#1847—471no—
298#1847—471no—
299#1836—471no—
300#1836—471no—
301#1809—471no—
302#1809—471no—
303#1809—471no—
304#1809D4711yescharacteristic
305C4×#107—471no—
306C4×#107—471no—
307C4×#106—471no—
308C4×#106—471no—
309C2×C47—42no—
310C2×C47#228511yescharacteristic
311C2×C47C2×#42811yescharacteristic
312C2×C47#291111yescharacteristic
313C2×C47#317111yescharacteristic
314C2×C47#317111yescharacteristic
315C2×C47#317211yescharacteristic
316C2×C47#317211yescharacteristic
317C22×M16—471no—
318C22×M16C2×D4711yescharacteristic
319C2×#117—941no—
320C22×C42—471no—
321C2×#107—471no—
322C2×#107—471no—
323C2×#107—471no—
324C2×#107C2×D4711yescharacteristic
325C2×#106—471no—
326C2×#106—471no—
327C2×#106—471no—
328C2×#106C2×D4711yescharacteristic
329C4×M16—478no—
330C2×#97—471no—
331C2×#97—471no—
332C2×C4×C8—471no—
333C2×C4×C8—471no—
334#384—471no—
335#384—471no—
336#383—941no—
337#382—471no—
338#382—471no—
339#382—941no—
340#381—941no—
341#380—471no—
342#380—471no—
343#380—941no—
344#380C2×D4711yescharacteristic
345#380C2×D4711yescharacteristic
346C4×C4⋊C4—471no—
347C4×C4⋊C4—471no—
348C4×C4⋊C4—471no—
349C4×C4⋊C4—471no—
350C4×C4⋊C4—941no—
351C4×C22⋊C4—941no—
352#376—941no—
353#340—471no—
354#340—471no—
355#340—471no—
356#340—471no—
357#340—471no—
358#340—471no—
359#340C2×D4711yescharacteristic
360#340C2×D4711yescharacteristic
361C47#2081911yescharacteristic, complemented, Sylow
362C23×C4—471no—
363C23×C4—471no—
364C23×C4C22×D4711yescharacteristic
365C2×M16—22no—
366C2×M16—22no—
367C2×M16—474no—
368C2×M16—942no—
369C2×M16—942no—
370C2×M16C4×D4714yes—
371C22×C8—471no—
372C22×C8—471no—
373C22×C8C22×D4711yescharacteristic
374C22×C8C22×D4711yescharacteristic
375#117—1884no—
376C2×C4⋊C4—471no—
377C2×C4⋊C4—471no—
378C2×C4⋊C4—471no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—941no—
381C2×C4⋊C4—941no—
382C2×C4⋊C4—941no—
383C2×C4⋊C4—941no—
384C2×C4⋊C4C22×D4711yescharacteristic
385C2×C4⋊C4C22×D4711yescharacteristic
386C2×C4⋊C4C22×D4711yescharacteristic
387C2×C4⋊C4C22×D4711yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—941no—
391C2×C22⋊C4—941no—
392C2×C42—474no—
393C2×C42—471no—
394C2×C42—471no—
395C2×C42—474no—
396C2×C42—471no—
397C2×C42—471no—
398C2×C42—941no—
399C2×C42—941no—
400C2×C42—941no—
401C2×C42—941no—
402#107—22no—
403#107—942no—
404#107—942no—
405#107—942no—
406#106—22no—
407#106—942no—
408#106—942no—
409#106—942no—
410#97—474no—
411#97—474no—
412C4×C8—474no—
413C4×C8—474no—
414#95—21no—
415#95—21no—
416#95—941no—
417#95—941no—
418#95—941no—
419#95—941no—
420#95—941no—
421#95—941no—
422#95—941no—
423#95—941no—
424#95—941no—
425#95—941no—
426C24D4×D4711yescharacteristic
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—471no—
436C22×C4—474no—
437C22×C4—471no—
438C22×C4—471no—
439C22×C4—471no—
440C22×C4—474no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—941no—
444C22×C4—941no—
445C22×C4—941no—
446C22×C4—941no—
447C22×C4—1881no—
448C22×C4—1881no—
449C22×C4C2×C4×D4712yes—
450C22×C4D4×D4711yescharacteristic
451C22×C4Q8×D4712yes—
452C22×C4C23×D4711yescharacteristic
453M16—28no—
454M16—948no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—472no—
458C2×C8—472no—
459C2×C8—472no—
460C2×C8—472no—
461C2×C8—941no—
462C2×C8—941no—
463C2×C8C2×C4×D4712yes—
464C2×C8C2×C4×D4712yes—
465C2×C8#1062535212yes—
466C2×C8#1062535212yes—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—941no—
476C4⋊C4—941no—
477C4⋊C4—941no—
478C4⋊C4—941no—
479C4⋊C4—941no—
480C4⋊C4—941no—
481C4⋊C4—941no—
482C4⋊C4—941no—
483C4⋊C4—1882no—
484C4⋊C4—1882no—
485C4⋊C4—1882no—
486C4⋊C4—1882no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—1882no—
490C22⋊C4—1882no—
491C42—474no—
492C42—474no—
493C42—472no—
494C42—472no—
495C42—472no—
496C42—472no—
497C42—1882no—
498C42—1882no—
499C42—1882no—
500C42—1882no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23C4⋊C4×D4712yes—
508C23C2×D4×D4711yescharacteristic
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—41no—
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—472no—
528C2×C4—472no—
529C2×C4—472no—
530C2×C4—472no—
531C2×C4—472no—
532C2×C4—472no—
533C2×C4—948no—
534C2×C4—941no—
535C2×C4—941no—
536C2×C4—941no—
537C2×C4—941no—
538C2×C4—941no—
539C2×C4—941no—
540C2×C4—1884no—
541C2×C4—1881no—
542C2×C4—1881no—
543C2×C4—1881no—
544C2×C4—1881no—
545C2×C4—1881no—
546C2×C4—1881no—
547C2×C4—1881no—
548C2×C4—1881no—
549C2×C4—1881no—
550C2×C4—1881no—
551C2×C4—1881no—
552C2×C4—1881no—
553C2×C4C4⋊C4×D4712yes—
554C2×C4C22×C4×D4711yescharacteristic
555C2×C4C2×Q8×D4711yescharacteristic
556C2×C4#41×D4711yescharacteristic
557C2×C4#41×D4711yescharacteristic
558C2×C4C2×#1062535211yescharacteristic
559C2×C4C2×#1062535211yescharacteristic
560C8—22no—
561C8—474no—
562C8—942no—
563C8#1309398014yescomplemented
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22C2×C4⋊C4×D4711yescharacteristic
572C22#1568966811yescharacteristic
573C22#1568966811yescharacteristic
574C22AGL(1,ℤ/8)×D4711yescharacteristic
575C22#137×D4711yescharacteristic
576C22#1568991911yescharacteristic
577C22#1568992211yescharacteristic
578C4—22no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—474no—
584C4—942no—
585C4—942no—
586C4—942no—
587C4—3761no—
588C4—3761no—
589C4—3761no—
590C4—3761no—
591C4#117×D4711yescharacteristic
592C4C2×#1309398011yescharacteristic
593C4#1568967511yescharacteristic
594C4#1568967511yescharacteristic
595C2—42no—
596C2#2024579011yescharacteristic
597C2#428×D4711yescharacteristic
598C2#2025197511yescharacteristic
599C2#2025397611yescharacteristic
600C2#2025397811yescharacteristic
601C2#2025398711yescharacteristic
602C2#2025398911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1809C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#1809D47
hypocenterC47#20819
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
4. #20243555C2semidirect
5. #20243553C2semidirect
6. #20243520C2semidirect
7. #20243520C2semidirect
11. #107×Dic47C2semidirect
12. #20243280C2semidirect
13. #106×Dic47C2semidirect
14. #20243276C2semidirect
25. M16×Dic47C4semidirect
259. C8×C47#117semidirect
261. C47⋊C8#117semidirect
361. C47#20819semidirect
563. C8#13093980semidirect

Maximal subgroups

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

subgroupindex
C2×M16×Dic472
#202457902
#202457832
#202435552
#202435532
#202435202
#202434212
#202434192
#107×Dic472
#202432802
#106×Dic472
#202432762
C47×#18092
#2081947