Finite Group Databasebeta

← #26216094

Subgroups — #26216094

Order 12032.

1374 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689604C211yesmaximal, characteristic
2C2×#15689089C211yesmaximal, characteristic
3#20245981C211yesmaximal, characteristic
4#20243482C212yesmaximal, complemented
5#20243481C212yesmaximal, complemented
6#20243477C212yesmaximal, complemented
7#20243473C212yesmaximal, complemented
8#20243107C211yesmaximal, characteristic, complemented
9#20243107C211yesmaximal, characteristic, complemented
10#20243105C211yesmaximal, characteristic, complemented
11C2×C47×#340C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C22×#10625329C2211yescharacteristic
14#15689604—24no—
15#15689604C418yescomplemented
16C2×#13093989—21no—
17C2×#13093986—21no—
18C2×C4×C22⋊D47—22no—
19C2×#13093983C2211yescharacteristic
20C2×#13093983C2211yescharacteristic
21C2×#13093982C2212yes—
22C2×C4⋊C4×D47C2212yes—
23#15689089—24no—
24C2×(C2×D47)⋊C8C2212yes—
25C2×(C2×D47)⋊C8C2212yes—
26#15689028—21no—
27#15689024—21no—
28#15689002—22no—
29#15688980C2212yes—
30#15688980C2212yes—
31#15688886C2211yescharacteristic
32#15688886C2211yescharacteristic
33#15688886C2211yescharacteristic
34#15688886C2211yescharacteristic
35#15688885C2212yes—
36#15688885C2212yes—
37#15688885C2212yes—
38#15688885C2212yes—
39C23×C8×C47C2211yescharacteristic
40C22×C4⋊C4×C47C2211yescharacteristic
41C22×C4⋊C4×C47C2211yescharacteristic
42C47×#340C2212yes—
43C47×#340C2212yes—
44C2×#10625350—22no—
45C2×#10625350—22no—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350C2×C414yes—
49C2×#10625350D414yes—
50C22×Dic94D411yescharacteristic
51C22×D188D411yescharacteristic
52C22×C22⋊D47D412yes—
53C22×C4×D47—22no—
54C22×C4×D47C2312yes—
55C2×#10625329—22no—
56C2×#10625329—22no—
57C2×#10625329—22no—
58C22×C47⋊C8C2312yes—
59#13093989—28no—
60#13093986—28no—
61C4×C22⋊D47—216no—
62#13093983—22no—
63#13093983—22no—
64#13093983—22no—
65#13093983C2×C414yes—
66#13093982—24no—
67#13093982C2×C418yescomplemented
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×Dic47⋊C4—22no—
73C4⋊C4×D47—24no—
74C4⋊C4×D47C2×C418yescomplemented
75C2×C22⋊Dic47—21no—
76C2×C22⋊Dic47—21no—
77C2×(C2×D47)⋊C4—22no—
78C2×(C2×D47)⋊C4—22no—
79C2×(C2×D47)⋊C4—22no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—22no—
82C2×C4×Dic47—21no—
83(C2×D47)⋊C8—24no—
84(C2×D47)⋊C8—24no—
85C47⋊#95—22no—
86C47⋊#95—22no—
87C23×C4×C47—21no—
88C23×C4×C47—21no—
89C23×C4×C47C2311yescharacteristic
90C22×C8×C47—22no—
91C22×C8×C47—22no—
92C22×C8×C47—22no—
93C22×C8×C47C2311yescharacteristic
94C22×C8×C47C2311yescharacteristic
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—22no—
97C2×C4⋊C4×C47—22no—
98C2×C4⋊C4×C47—22no—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47C2×C414yes—
101C2×C4⋊C4×C47C2312yes—
102C2×C4⋊C4×C47C2312yes—
103C24×C47C2×D411yescharacteristic
104C23×D47C2×D412yes—
105C22×C4×C47—22no—
106C22×C4×C47—24no—
107C22×C4×C47—22no—
108C22×C4×C47—42no—
109C22×C4×C47—81no—
110C22×C4×C47—81no—
111C22×C4×C47—81no—
112C22×C4×C47—81no—
113C22×C4×C47C22×C412yes—
114C22×C4×C47C2×D411yescharacteristic
115C22×C4×C47C2×D412yes—
116C22×C4×C47#4112yes—
117C22×C4×C47#4112yes—
118C22×C4×C47#4112yes—
119C22×C4×C47#4112yes—
120C22×C4×C47C2411yescharacteristic
121C2×C22⋊D47—24no—
122C2×C22⋊D47—24no—
123C2×C22⋊D47C22⋊C418yes—
124C22×Dic47—21no—
125C22×Dic47—21no—
126C22×Dic47—21no—
127C22×Dic47—21no—
128C22×Dic47C2×D412yes—
129#10625350—28no—
130#10625350—28no—
131#10625350—28no—
132#10625350—28no—
133C2×D188—22no—
134C2×D188—22no—
135C2×D188C22⋊C414yes—
136C2×C4×D47—22no—
137C2×C4×D47—22no—
138C2×C4×D47—22no—
139C2×C4×D47—22no—
140C2×C4×D47—28no—
141C2×C4×D47—82no—
142C2×C4×D47—82no—
143C2×C4×D47C22×C414yes—
144C2×C4×D47C2×D414yes—
145C2×Dic94—22no—
146C2×Dic94—22no—
147C2×Dic94C22⋊C414yes—
148C2×C8×C47—21no—
149C2×C8×C47—21no—
150C2×C8×C47—21no—
151C2×C8×C47—21no—
152C2×C8×C47—21no—
153C2×C8×C47—21no—
154C2×C8×C47—44no—
155C4⋊C4×C47—24no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47—22no—
158C4⋊C4×C47—22no—
159C4⋊C4×C47—42no—
160C4⋊C4×C47C22×C414yes—
161C22⋊Dic47—24no—
162C22⋊Dic47—42no—
163(C2×D47)⋊C4—28no—
164(C2×D47)⋊C4—82no—
165(C2×D47)⋊C4—82no—
166C4⋊Dic47—21no—
167C4⋊Dic47—21no—
168C4⋊Dic47—21no—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47C22×C412yes—
173C4⋊Dic47C22×C412yes—
174Dic47⋊C4—28no—
175C4×Dic47—24no—
176C4×Dic47—28no—
177C4×Dic47—42no—
178#10625329—44no—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8—22no—
182C23×C47—21no—
183C23×C47—21no—
184C23×C47—21no—
185C23×C47—21no—
186C23×C47C2×C22⋊C412yes—
187C23×C47#12312yes—
188C23×C47#12312yes—
189C23×C47C22×D411yescharacteristic
190C22×D47—22no—
191C22×D47—22no—
192C22×D47—22no—
193C22×D47—22no—
194C22×D47—22no—
195C22×D47—22no—
196C22×D47C2×C22⋊C414yes—
197C2×C4×C47—24no—
198C2×C4×C47—42no—
199C2×C4×C47—42no—
200C2×C4×C47—42no—
201C2×C4×C47—42no—
202C2×C4×C47—42no—
203C2×C4×C47—84no—
204C2×C4×C47—84no—
205C2×C4×C47C2×C22⋊C412yes—
206C2×C4×C47#11714yes—
207C2×C4×C47#11714yes—
208C2×C4×C47#12314yes—
209C2×C4×C47#12312yes—
210C2×C4×C47#12314yes—
211C2×C4×C47#12312yes—
212C2×C4×C47C23×C411yescharacteristic
213C2×C4×C47C22×D411yescharacteristic
214C2×C4×C47C2×#4111yescharacteristic
215C2×C4×C47C2×#4111yescharacteristic
216C2×C4×C47C2×#4111yescharacteristic
217C2×C4×C47C2×#4111yescharacteristic
218C22⋊D47—216no—
219C2×Dic47—22no—
220C2×Dic47—22no—
221C2×Dic47—22no—
222C2×Dic47—22no—
223C2×Dic47—41no—
224C2×Dic47—41no—
225C2×Dic47—41no—
226C2×Dic47—41no—
227C2×Dic47—41no—
228C2×Dic47—41no—
229C2×Dic47—41no—
230C2×Dic47—41no—
231C2×Dic47—41no—
232C2×Dic47—41no—
233C2×Dic47C2×C22⋊C414yes—
234D188—28no—
235C4×D47—24no—
236C4×D47—24no—
237C4×D47—24no—
238C4×D47—24no—
239C4×D47—88no—
240Dic94—28no—
241C8×C47—41no—
242C8×C47—41no—
243C47⋊C8—42no—
244#20703—471nomaximal, Sylow
245C22×C47—22no—
246C22×C47—22no—
247C22×C47—22no—
248C22×C47—22no—
249C22×C47—22no—
250C22×C47—22no—
251C22×C47#38114yes—
252C22×C47C22×C22⋊C411yescharacteristic
253C22×C47C2×#12311yescharacteristic
254C22×C47C2×#12311yescharacteristic
255C22×C47#57611yescharacteristic
256C22×C47#57711yescharacteristic
257C22×C47#57711yescharacteristic
258C22×C47#57811yescharacteristic
259C2×D47—24no—
260C2×D47—24no—
261C2×D47—24no—
262C2×D47—42no—
263C2×D47—42no—
264C2×D47—42no—
265C2×D47—42no—
266C4×C47—44no—
267C4×C47—82no—
268C4×C47#38114yes—
269C4×C47C2×#11711yescharacteristic
270C4×C47C2×#11711yescharacteristic
271C4×C47C2×#12311yescharacteristic
272C4×C47C2×#12311yescharacteristic
273Dic47—24no—
274Dic47—42no—
275Dic47—42no—
276Dic47—81no—
277Dic47—81no—
278C2×#520—471no—
279C2×#408—471no—
280#2311—471no—
281#1825—472no—
282#1825—472no—
283#1824—472no—
284#1823—472no—
285#1736—471no—
286#1736—471no—
287#1735—471no—
288C2×#340D4711yescharacteristic, complemented
289C2×C47—24no—
290C2×C47C2×#38111yescharacteristic
291C2×C47#289711yescharacteristic
292C2×C47#289811yescharacteristic
293C2×C47#309611yescharacteristic
294C2×C47#309711yescharacteristic
295C2×C47#309811yescharacteristic
296C2×C47#309911yescharacteristic
297D47—44no—
298C22×#41—471no—
299C22×M16—471no—
300C23×C8C2×D4711yescharacteristic
301#520—478no—
302#520—944no—
303C2×C4×Q8—941no—
304C2×C4×D4—941no—
305C2×C4×D4—942no—
306C2×#117—471no—
307C2×#117—471no—
308C2×#117—472no—
309C22×C4⋊C4—472no—
310C22×C4⋊C4C2×D4711yescharacteristic
311C22×C4⋊C4C2×D4711yescharacteristic
312#408—944no—
313C2×C22⋊C8—472no—
314C2×C22⋊C8—472no—
315#391—941no—
316#390—941no—
317#386—942no—
318#380—472no—
319#380—472no—
320#340—471no—
321#340—471no—
322#340—471no—
323#340—471no—
324#340—472no—
325#340—472no—
326#340—472no—
327#340—472no—
328#340C2×D4712yes—
329#340C2×D4712yes—
330C47#2070311yescharacteristic, complemented, Sylow
331C2×#41—474no—
332C2×#41—474no—
333C2×#41—942no—
334C2×#41—942no—
335C2×#41—942no—
336C2×#41—942no—
337C22×Q8—471no—
338C22×D4—472no—
339C22×D4—471no—
340C23×C4—21no—
341C23×C4—21no—
342C23×C4—472no—
343C23×C4—942no—
344C23×C4C22×D4711yescharacteristic
345C2×M16—942no—
346C2×M16—942no—
347C2×M16—942no—
348C22×C8—22no—
349C22×C8—22no—
350C22×C8—22no—
351C22×C8—472no—
352C22×C8C22×D4711yescharacteristic
353C22×C8C22×D4711yescharacteristic
354C4×Q8—948no—
355C4×D4—948no—
356C4×D4—9416no—
357#117—474no—
358#117—478no—
359#117—944no—
360#117—942no—
361#117—942no—
362#117—942no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—471no—
370C2×C4⋊C4—478no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—944no—
374C2×C4⋊C4—942no—
375C2×C4⋊C4C4×D4714yes—
376C2×C4⋊C4C22×D4712yes—
377C2×C4⋊C4C22×D4712yes—
378C2×C22⋊C4—942no—
379C2×C22⋊C4—941no—
380C2×C22⋊C4—942no—
381C2×C22⋊C4—942no—
382C2×C22⋊C4—941no—
383C2×C42—942no—
384C2×C42—941no—
385C2×C42—941no—
386C22⋊C8—944no—
387C22⋊C8—944no—
388#95—942no—
389#95—942no—
390C24—472no—
391C24D4×D4711yescharacteristic
392#41—948no—
393#41—948no—
394#41—948no—
395#41—948no—
396C2×Q8—474no—
397C2×Q8—942no—
398C2×Q8—942no—
399C2×D4—478no—
400C2×D4—474no—
401C2×D4—944no—
402C2×D4—944no—
403C2×D4—942no—
404C2×D4—942no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4—472no—
414C22×C4—474no—
415C22×C4—474no—
416C22×C4—948no—
417C22×C4—942no—
418C22×C4—942no—
419C22×C4—942no—
420C22×C4—942no—
421C22×C4—941no—
422C22×C4—941no—
423C22×C4—941no—
424C22×C4—941no—
425C22×C4—3762no—
426C22×C4—3762no—
427C22×C4C2×C4×D4712yes—
428C22×C4#1062535012yes—
429C22×C4#1062535012yes—
430C22×C4D4×D4711yescharacteristic
431C22×C4D4×D4712yes—
432C22×C4#1062535212yes—
433C22×C4#1062535412yes—
434C22×C4C23×D4711yescharacteristic
435M16—1884no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—44no—
443C2×C8—942no—
444C2×C8—942no—
445C2×C8—942no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—472no—
452C4⋊C4—472no—
453C4⋊C4—948no—
454C4⋊C4—941no—
455C4⋊C4—941no—
456C4⋊C4—941no—
457C4⋊C4—941no—
458C4⋊C4—941no—
459C4⋊C4—941no—
460C4⋊C4C2×C4×D4714yes—
461C22⋊C4—948no—
462C22⋊C4—944no—
463C22⋊C4—3762no—
464C22⋊C4—3762no—
465C22⋊C4—1882no—
466C42—948no—
467C42—944no—
468C42—1882no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—474no—
474C23—942no—
475C23—942no—
476C23—942no—
477C23—942no—
478C23—942no—
479C23—942no—
480C23C22⋊C4×D4712yes—
481C23#1309400612yes—
482C23#1309400912yes—
483C23C2×D4×D4711yescharacteristic
484Q8—948no—
485D4—9416no—
486D4—948no—
487C2×C4—24no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—84no—
494C2×C4—84no—
495C2×C4—474no—
496C2×C4—942no—
497C2×C4—942no—
498C2×C4—944no—
499C2×C4—944no—
500C2×C4—944no—
501C2×C4—944no—
502C2×C4—942no—
503C2×C4—942no—
504C2×C4—3768no—
505C2×C4—1881no—
506C2×C4—1881no—
507C2×C4—1881no—
508C2×C4—1881no—
509C2×C4—1881no—
510C2×C4—1881no—
511C2×C4—1881no—
512C2×C4—1881no—
513C2×C4—1881no—
514C2×C4—1881no—
515C2×C4C22⋊C4×D4712yes—
516C2×C4#1309398114yes—
517C2×C4#1309398214yes—
518C2×C4#1309400612yes—
519C2×C4#1309400614yes—
520C2×C4#1309400912yes—
521C2×C4#1309400914yes—
522C2×C4C22×C4×D4711yescharacteristic
523C2×C4C2×D4×D4711yescharacteristic
524C2×C4C2×#1062535211yescharacteristic
525C2×C4C2×#1062535011yescharacteristic
526C2×C4C2×#1062535011yescharacteristic
527C2×C4C2×#1062535411yescharacteristic
528C8—41no—
529C8—41no—
530C8—1882no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—944no—
538C22—944no—
539C22—944no—
540C22—1882no—
541C22—1882no—
542C22—1882no—
543C22—1882no—
544C22#1568898514yes—
545C22C2×C22⋊C4×D4711yescharacteristic
546C22C2×#1309400611yescharacteristic
547C22C2×#1309400911yescharacteristic
548C22#1568992411yescharacteristic
549C22#1568992911yescharacteristic
550C22#1568992911yescharacteristic
551C22#1568993611yescharacteristic
552C4—44no—
553C4—82no—
554C4—944no—
555C4—3761no—
556C4—3761no—
557C4—1882no—
558C4—1882no—
559C4#1568898514yes—
560C4C2×#1309398111yescharacteristic
561C4C2×#1309398211yescharacteristic
562C4C2×#1309400611yescharacteristic
563C4C2×#1309400911yescharacteristic
564C2—24no—
565C2—1884no—
566C2C2×#1568898511yescharacteristic
567C2#2025190911yescharacteristic
568C2#2025191611yescharacteristic
569C2#2025334211yescharacteristic
570C2#2025335311yescharacteristic
571C2#2025337111yescharacteristic
572C2#2025338211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#340C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenterC2×#340D47
hypocenterC47#20703
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
4. #20243482C2semidirect
5. #20243481C2semidirect
6. #20243477C2semidirect
7. #20243473C2semidirect
8. #20243107C2semidirect
9. #20243107C2semidirect
10. #20243105C2semidirect
11. C2×C47×#340C2semidirect
15. #15689604C4semidirect
67. #13093982C2×C4semidirect
74. C4⋊C4×D47C2×C4semidirect
288. C2×#340D47semidirect
330. C47#20703semidirect

Maximal subgroups

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

subgroupindex
C2×#156896042
C2×#156890892
#202459812
#202434822
#202434812
#202434772
#202434732
#202431072
#202431052
C2×C47×#3402
#2070347