Finite Group Databasebeta

← #26225964

Subgroups — #26225964

Order 12032.

1166 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689654C211yesmaximal, characteristic
2C2×#15689173C211yesmaximal, characteristic, complemented
3C2×#15688988C211yesmaximal, characteristic, complemented
4#20244351C212yesmaximal, complemented
5#20244198C212yesmaximal, complemented
6#20243706C211yesmaximal, characteristic
7#20243695C211yesmaximal, characteristic, complemented
8#20243693C211yesmaximal, characteristic, complemented
9#20243402C212yesmaximal, complemented
10#20243396C212yesmaximal, complemented
11C47×#1874C211yesmaximal, characteristic
12#15689654—24no—
13#15689654C418yescomplemented
14C2×#13094017—21no—
15C2×#13094015C2211yescharacteristic
16C2×#13094011C2211yescharacteristic
17C2×#13094005—21no—
18C2×#13094003C2211yescharacteristic
19C2×#13093989C2211yescharacteristic
20C2×D4×Dic47—21no—
21C2×D4×Dic47C2211yescharacteristic
22C2×#13093980—21no—
23C22×C4⋊Dic47C2211yescharacteristic
24C22×C4×Dic47C2211yescharacteristic
25#15689173—24no—
26C2×Dic47⋊C8C2212yes—
27C2×#13093949C2212yes—
28C2×#13093947C2212yes—
29C2×#13093945C2212yes—
30C2×C22⋊(C47⋊C8)C2211yescharacteristic
31C2×C8×Dic47C2212yes—
32#15688988—21no—
33#15688988—21no—
34#15688988C2212yes—
35#15688988C2212yes—
36C2×C47⋊#95—21no—
37C2×C47⋊#95—21no—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×#121×C47C2211yescharacteristic
42C22×C4⋊C4×C47C2211yescharacteristic
43C2×#102×C47C2212yes—
44C2×C22⋊C8×C47C2211yescharacteristic
45C47×#340C2212yes—
46C22×Dic94C2311yescharacteristic
47C23×Dic47—21no—
48C23×Dic47—21no—
49C23×Dic47D411yescharacteristic
50C23×Dic47D411yescharacteristic
51C22×C47⋊C8C2312yes—
52#13094017—28no—
53#13094015—22no—
54#13094015C2×C414yescomplemented
55#13094011—22no—
56#13094011C2×C414yescomplemented
57#13094005—28no—
58#13094003—22no—
59#13094003C2×C414yes—
60#13093989—22no—
61#13093989C2×C414yescomplemented
62D4×Dic47—28no—
63D4×Dic47—22no—
64D4×Dic47C2×C414yescomplemented
65#13093980—28no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—42no—
69C2×C4⋊Dic47C2312yes—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×C22⋊Dic47—21no—
75C2×C22⋊Dic47—21no—
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47—21no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47C2×C414yes—
80C2×C4×Dic47D411yescharacteristic
81C2×C4×Dic47D411yescharacteristic
82C2×C4×Dic47D414yes—
83C2×C4×Dic47C2311yescharacteristic
84C2×C4×Dic47C2311yescharacteristic
85Dic47⋊C8—24no—
86#13093949—42no—
87#13093947—42no—
88#13093945—42no—
89C22⋊(C47⋊C8)—22no—
90C8×Dic47—24no—
91C47⋊#95—22no—
92C47⋊#95—22no—
93C47⋊#95—22no—
94C47⋊#95—22no—
95C22×D4×C47—21no—
96C22×D4×C47C2311yescharacteristic
97C23×C4×C47—21no—
98C23×C4×C47C2311yescharacteristic
99C22×C8×C47C2312yes—
100#121×C47—22no—
101#121×C47C2×C414yes—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47—42no—
104C2×C4⋊C4×C47C2312yes—
105C2×C4⋊C4×C47C2311yescharacteristic
106C2×C22⋊C4×C47—21no—
107#102×C47—42no—
108C22⋊C8×C47—22no—
109C24×C47—21no—
110C24×C47C2×D411yescharacteristic
111C2×D4×C47—24no—
112C2×D4×C47—21no—
113C2×D4×C47—22no—
114C2×D4×C47—22no—
115C2×D4×C47—42no—
116C2×D4×C47—42no—
117C2×D4×C47C22×C412yes—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—21no—
121C22×C4×C47—22no—
122C22×C4×C47—42no—
123C22×C4×C47—42no—
124C22×C4×C47—42no—
125C22×C4×C47C22×C412yes—
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D412yes—
128C22×C4×C47C2411yescharacteristic
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—21no—
132C22×Dic47—21no—
133C22×Dic47—24no—
134C22×Dic47—22no—
135C22×Dic47—41no—
136C22×Dic47—41no—
137C22×Dic47—42no—
138C22×Dic47—42no—
139C22×Dic47—42no—
140C22×Dic47—42no—
141C22×Dic47—42no—
142C22×Dic47C22⋊C414yes—
143C22×Dic47C22⋊C414yes—
144C22×Dic47C2×D411yescharacteristic
145C22×Dic47C2×D411yescharacteristic
146C22×Dic47C2×D411yescharacteristic
147C22×Dic47C2×D411yescharacteristic
148C2×Dic94—22no—
149C2×Dic94—22no—
150C2×Dic94—21no—
151C2×Dic94C22×C412yes—
152C2×C8×C47—22no—
153C2×C8×C47—22no—
154C2×C8×C47—22no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47—22no—
158C4⋊C4×C47—42no—
159C4⋊C4×C47C22×C412yes—
160C22⋊C4×C47—24no—
161C22⋊Dic47—24no—
162C22⋊Dic47—24no—
163C22⋊Dic47—24no—
164C22⋊Dic47—24no—
165C4⋊Dic47—21no—
166C4⋊Dic47—24no—
167C4⋊Dic47—22no—
168C4⋊Dic47—22no—
169C4⋊Dic47—42no—
170C4⋊Dic47C22×C412yes—
171Dic47⋊C4—24no—
172Dic47⋊C4—24no—
173Dic47⋊C4—24no—
174C4×Dic47—24no—
175C4×Dic47C22⋊C414yes—
176C4×Dic47C22⋊C414yes—
177C4×Dic47C22×C412yes—
178C4×Dic47C22×C412yes—
179C4×Dic47C2×D412yes—
180C4×Dic47C2×D412yes—
181C2×C47⋊C8—22no—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—22no—
184C23×C47—21no—
185C23×C47—21no—
186C23×C47—21no—
187C23×C47—21no—
188C23×C47—22no—
189C23×C47—41no—
190C23×C47—41no—
191C23×C47—41no—
192C23×C47—41no—
193C23×C47—41no—
194C23×C47—41no—
195C23×C47C2×C22⋊C412yes—
196C23×C47#13512yes—
197C23×C47#13512yes—
198C23×C47C22×D411yescharacteristic
199D4×C47—24no—
200D4×C47—48no—
201D4×C47—42no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—24no—
205C2×C4×C47—24no—
206C2×C4×C47—22no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—84no—
213C2×C4×C47C2×C22⋊C412yes—
214C2×C4×C47C23×C411yescharacteristic
215C2×C4×C47C22×D411yescharacteristic
216C2×C4×C47H(5,𝔽2)12yes—
217C2×C4×C472−1+412yes—
218C2×Dic47—21no—
219C2×Dic47—21no—
220C2×Dic47—21no—
221C2×Dic47—21no—
222C2×Dic47—21no—
223C2×Dic47—21no—
224C2×Dic47—21no—
225C2×Dic47—21no—
226C2×Dic47—28no—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—44no—
230C2×Dic47—48no—
231C2×Dic47—41no—
232C2×Dic47—41no—
233C2×Dic47—41no—
234C2×Dic47—41no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47—84no—
239C2×Dic47C2×C22⋊C412yes—
240C2×Dic47C2×C22⋊C412yes—
241C2×Dic47C2×C22⋊C412yes—
242C2×Dic47C2×C22⋊C412yes—
243Dic94—24no—
244Dic94—42no—
245C8×C47—42no—
246C47⋊C8—42no—
247#21861—471nomaximal, Sylow
248C22×C47—22no—
249C22×C47—22no—
250C22×C47—22no—
251C22×C47—22no—
252C22×C47—22no—
253C22×C47—22no—
254C22×C47—42no—
255C22×C47—42no—
256C22×C47—42no—
257C22×C47—81no—
258C22×C47—81no—
259C22×C47—81no—
260C22×C47—81no—
261C22×C47#41614yes—
262C22×C47C22×C22⋊C411yescharacteristic
263C22×C47#53511yescharacteristic
264C22×C47#53611yescharacteristic
265C22×C47C2×#13511yescharacteristic
266C22×C47C2×#13511yescharacteristic
267C22×C47#57511yescharacteristic
268C22×C47#57511yescharacteristic
269C4×C47—22no—
270C4×C47—42no—
271C4×C47—82no—
272C4×C47#52012yes—
273C4×C47#53712yes—
274Dic47—22no—
275Dic47—22no—
276Dic47—22no—
277Dic47—22no—
278Dic47—42no—
279Dic47—42no—
280Dic47—82no—
281C2×#529—471no—
282C2×#431—471no—
283C2×#382—471no—
284#1991—472no—
285#1958—472no—
286#1874—471no—
287#1874D4711yescharacteristic
288#1872—471no—
289#1872—471no—
290#1805—472no—
291#1804—472no—
292C2×C47—24no—
293C2×C47—82no—
294C2×C47#231111yescharacteristic
295C2×C47C2×#41611yescharacteristic
296C2×C47#289611yescharacteristic
297C2×C47#321111yescharacteristic
298C2×C47#321111yescharacteristic
299C2×C47#323211yescharacteristic
300C2×C47#323211yescharacteristic
301#529—478no—
302#529—944no—
303C2×#126—941no—
304C2×#125—471no—
305C2×#124—471no—
306C2×#123—941no—
307C2×C22⋊Q8—471no—
308C2×#121C2×D4711yescharacteristic
309C2×C4×Q8—471no—
310C2×C4×D4—471no—
311C2×C4×D4—941no—
312C2×#117—941no—
313C22×C4⋊C4—471no—
314C22×C4⋊C4C2×D4711yescharacteristic
315C22×C42—471no—
316#431—944no—
317C2×C4⋊C8—472no—
318C2×#103—472no—
319C2×#103—472no—
320C2×#102—472no—
321C2×#102C2×D4712yes—
322C2×C22⋊C8—471no—
323C2×C22⋊C8C2×D4711yescharacteristic
324C2×C4×C8—472no—
325#382—472no—
326#382—472no—
327#382—941no—
328#382—941no—
329C2×#95—941no—
330C2×#95—941no—
331#340—472no—
332#340—472no—
333#340—472no—
334#340C2×D4712yes—
335C47#2186111yescharacteristic, complemented, Sylow
336C22×Q8—471no—
337C22×D4—21no—
338C22×D4C22×D4711yescharacteristic
339C23×C4—21no—
340C23×C4—471no—
341C23×C4—471no—
342C23×C4—941no—
343C23×C4—941no—
344C23×C4C22×D4711yescharacteristic
345C22×C8—472no—
346C22×C8C22×D4712yes—
347#126—948no—
348#125—474no—
349#125—942no—
350#124—474no—
351#124—942no—
352#123—948no—
353C22⋊Q8—474no—
354C22⋊Q8—942no—
355#121—22no—
356#121C4×D4714yes—
357C4×Q8—474no—
358C4×Q8—942no—
359C4×D4—474no—
360C4×D4—942no—
361C4×D4—948no—
362#117—948no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—42no—
365C2×C4⋊C4—471no—
366C2×C4⋊C4—472no—
367C2×C4⋊C4—941no—
368C2×C4⋊C4—941no—
369C2×C4⋊C4—941no—
370C2×C4⋊C4—941no—
371C2×C4⋊C4—941no—
372C2×C4⋊C4—1882no—
373C2×C4⋊C4C22×D4712yes—
374C2×C4⋊C4C22×D4711yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—941no—
377C2×C22⋊C4—941no—
378C2×C22⋊C4—941no—
379C2×C22⋊C4—941no—
380C2×C42—474no—
381C2×C42—471no—
382C2×C42—471no—
383C2×C42—474no—
384C2×C42—471no—
385C2×C42—471no—
386C2×C42—941no—
387C4⋊C8—944no—
388#103—1882no—
389#103—1882no—
390#102—42no—
391#102—1882no—
392C22⋊C8—22no—
393C22⋊C8—942no—
394C4×C8—944no—
395#95—942no—
396#95—942no—
397#95—942no—
398#95—942no—
399C24—21no—
400C24D4×D4711yescharacteristic
401C2×Q8—472no—
402C2×Q8—941no—
403C2×Q8—942no—
404C2×Q8—942no—
405C2×D4—24no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4C2×C4×D4712yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—474no—
420C22×C4—471no—
421C22×C4—471no—
422C22×C4—471no—
423C22×C4—471no—
424C22×C4—474no—
425C22×C4—944no—
426C22×C4—942no—
427C22×C4—942no—
428C22×C4—941no—
429C22×C4—941no—
430C22×C4—942no—
431C22×C4—1881no—
432C22×C4—1881no—
433C22×C4—1882no—
434C22×C4—1882no—
435C22×C4—1882no—
436C22×C4—1882no—
437C22×C4—1882no—
438C22×C4C2×C4×D4712yes—
439C22×C4D4×D4711yescharacteristic
440C22×C4D4×D4712yes—
441C22×C4C23×D4711yescharacteristic
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—942no—
446C2×C8—942no—
447C2×C8—942no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—42no—
452C4⋊C4—472no—
453C4⋊C4—944no—
454C4⋊C4—944no—
455C4⋊C4—944no—
456C4⋊C4—944no—
457C4⋊C4—941no—
458C4⋊C4—942no—
459C4⋊C4—942no—
460C4⋊C4—1882no—
461C4⋊C4C2×C4×D4712yes—
462C22⋊C4—24no—
463C22⋊C4—944no—
464C22⋊C4—944no—
465C22⋊C4—944no—
466C22⋊C4—944no—
467C42—474no—
468C42—474no—
469C42—472no—
470C42—472no—
471C42—472no—
472C42—472no—
473C42—944no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23C22⋊C4×D4712yes—
486C23#1309404312yes—
487C23#1309404512yes—
488C23C2×D4×D4711yescharacteristic
489Q8—944no—
490Q8—1882no—
491D4—24no—
492D4—48no—
493D4—42no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—84no—
505C2×C4—472no—
506C2×C4—472no—
507C2×C4—472no—
508C2×C4—472no—
509C2×C4—948no—
510C2×C4—941no—
511C2×C4—941no—
512C2×C4—941no—
513C2×C4—941no—
514C2×C4—941no—
515C2×C4—941no—
516C2×C4—941no—
517C2×C4—941no—
518C2×C4—942no—
519C2×C4—942no—
520C2×C4—3764no—
521C2×C4—1884no—
522C2×C4—1888no—
523C2×C4—1882no—
524C2×C4—1882no—
525C2×C4—1882no—
526C2×C4—1881no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4C22⋊C4×D4712yes—
531C2×C4C22×C4×D4711yescharacteristic
532C2×C4C2×D4×D4711yescharacteristic
533C2×C4#1309406912yes—
534C2×C4#1309407112yes—
535C8—42no—
536C8—1882no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22#1568911614yes—
551C22C2×C22⋊C4×D4711yescharacteristic
552C22#1568968811yescharacteristic
553C22#1568969511yescharacteristic
554C22C2×#1309404311yescharacteristic
555C22C2×#1309404511yescharacteristic
556C22#1568991911yescharacteristic
557C22#1568992211yescharacteristic
558C4—22no—
559C4—42no—
560C4—82no—
561C4—942no—
562C4—942no—
563C4—942no—
564C4—942no—
565C4—3762no—
566C4—1882no—
567C4—1882no—
568C4#1568960212yes—
569C4#1568970012yes—
570C2—24no—
571C2—82no—
572C2#2024597511yescharacteristic
573C2C2×#1568911611yescharacteristic
574C2#2025190411yescharacteristic
575C2#2025435011yescharacteristic
576C2#2025435211yescharacteristic
577C2#2025455311yescharacteristic
578C2#2025455511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1874C2
socleC23×C47C22×D4
prime radicalC2×#121×C47C22
cosocleC22×C4×C47C24
hypercenter#1874D47
hypocenterC47#21861
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
2. C2×#15689173C2semidirect
3. C2×#15688988C2semidirect
4. #20244351C2semidirect
5. #20244198C2semidirect
7. #20243695C2semidirect
8. #20243693C2semidirect
9. #20243402C2semidirect
10. #20243396C2semidirect
13. #15689654C4semidirect
54. #13094015C2×C4semidirect
56. #13094011C2×C4semidirect
61. #13093989C2×C4semidirect
64. D4×Dic47C2×C4semidirect
335. C47#21861semidirect

Maximal subgroups

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

subgroupindex
C2×#156896542
C2×#156891732
C2×#156889882
#202443512
#202441982
#202437062
#202436952
#202436932
#202434022
#202433962
C47×#18742
#2186147