Finite Group Databasebeta

← #26187636

Subgroups — #26187636

Order 12032.

606 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253419C211yesmaximal, characteristic
2#20252477C211yesmaximal, characteristic, complemented
3#20252454C211yesmaximal, characteristic
4#20242293C211yesmaximal, characteristic, complemented
5#20242275C211yesmaximal, characteristic, complemented
6#20242263C211yesmaximal, characteristic, complemented
7#20242248C211yesmaximal, characteristic, complemented
8#20242228C211yesmaximal, characteristic, complemented
9#20242227C211yesmaximal, characteristic, complemented
10#20242203C211yesmaximal, characteristic, complemented
11#20242185C211yesmaximal, characteristic, complemented
12#20242166C211yesmaximal, characteristic, complemented
13#20241858C211yesmaximal, characteristic
14#20241688C211yesmaximal, characteristic, complemented
15C47×#1479C211yesmaximal, characteristic, complemented
16#15689673C2211yescharacteristic
17#15689441C411yescharacteristic
18#15689441C411yescharacteristic
19#15689408C411yescharacteristic
20#15689408C411yescharacteristic
21#15689323C411yescharacteristic
22#15689323C411yescharacteristic
23#15689262C411yescharacteristic
24#15689262C411yescharacteristic
25#15689254—21no—
26#15689254—21no—
27#15689254C2211yescharacteristic
28#15689254C2211yescharacteristic
29Q8×C47⋊C8C2211yescharacteristic
30Q8×C47⋊C8C2211yescharacteristic
31C4×#10625336C2211yescharacteristic
32C4×#10625336C2211yescharacteristic
33C4×#10625336C2211yescharacteristic
34C4×#10625336C2211yescharacteristic
35#15689202—21no—
36#15689200C2211yescharacteristic
37#15689200C2211yescharacteristic
38C8×D188—21no—
39#15689188C2211yescharacteristic
40#15689188C2211yescharacteristic
41#15689180—21no—
42#15689180C2211yescharacteristic
43#15689176—21no—
44#15689176C2211yescharacteristic
45#15689176C2211yescharacteristic
46#15689088—21no—
47#15689088—21no—
48#15688875C2211yescharacteristic
49#15688875C2211yescharacteristic
50#15688873C2211yescharacteristic
51#15688873C2211yescharacteristic
52#15688852C2211yescharacteristic
53#15688852C2211yescharacteristic
54#15688852C2211yescharacteristic
55#15688852C2211yescharacteristic
56#15688845C2211yescharacteristic
57#15688845C2211yescharacteristic
58#15688844C2211yescharacteristic
59C8×C47⋊C8C2211yescharacteristic
60C2×C4×Q8×C47C2211yescharacteristic
61C47×#431C2211yescharacteristic
62C47×#431C2211yescharacteristic
63C47×Q8⋊C8C2211yescharacteristic
64C47×Q8⋊C8C2211yescharacteristic
65C47×Q8⋊C8C2211yescharacteristic
66C47×Q8⋊C8C2211yescharacteristic
67C2×#10625350—21no—
68C2×#10625336C2×C411yescharacteristic
69C2×#10625336C2×C411yescharacteristic
70C2×#10625336C2×C411yescharacteristic
71C2×#10625336C2×C411yescharacteristic
72C2×#10625324—21no—
73C2×#10625324—21no—
74C2×C8×D47—21no—
75C2×C8×D47—21no—
76#13094024C2×C411yescharacteristic
77#13094021C2×C411yescharacteristic
78#13094012—21no—
79#13093998—21no—
80#13093998C2×C411yescharacteristic
81#13093998C2×C411yescharacteristic
82C4×D188—21no—
83C4×D188C2311yescharacteristic
84C4×D188C2311yescharacteristic
85#13093955C2×C411yescharacteristic
86#13093955C2×C411yescharacteristic
87#13093955C2×C411yescharacteristic
88#13093955C2×C411yescharacteristic
89C4⋊(C47⋊C8)—21no—
90C4⋊(C47⋊C8)—21no—
91C4⋊(C47⋊C8)—21no—
92C4⋊(C47⋊C8)—21no—
93C4⋊(C47⋊C8)C2×C411yescharacteristic
94C4⋊(C47⋊C8)C2×C411yescharacteristic
95C4⋊(C47⋊C8)D411yescharacteristic
96C4⋊(C47⋊C8)D411yescharacteristic
97C4⋊(C47⋊C8)C2311yescharacteristic
98C4⋊(C47⋊C8)C2311yescharacteristic
99#13093948C2×C411yescharacteristic
100#13093948C2×C411yescharacteristic
101#13093948C2×C411yescharacteristic
102#13093948C2×C411yescharacteristic
103#13093944C2×C411yescharacteristic
104#13093944C2×C411yescharacteristic
105#13093944C2×C411yescharacteristic
106#13093944C2×C411yescharacteristic
107C22⋊(C47⋊C8)—21no—
108C22⋊(C47⋊C8)C2×C411yescharacteristic
109C22⋊(C47⋊C8)C2×C411yescharacteristic
110C22⋊(C47⋊C8)D411yescharacteristic
111C22⋊(C47⋊C8)D411yescharacteristic
112(C2×D47)⋊C8—21no—
113(C2×D47)⋊C8—21no—
114(C2×D47)⋊C8—21no—
115(C2×D47)⋊C8—21no—
116#13093935—21no—
117#13093935—21no—
118#13093935—21no—
119C4×C47⋊C8—21no—
120C4×C47⋊C8—21no—
121C4×C47⋊C8C2311yescharacteristic
122C4×C47⋊C8C2311yescharacteristic
123C4×C47⋊C8C2311yescharacteristic
124C4×C47⋊C8C2311yescharacteristic
125C22×Q8×C47C2×C411yescharacteristic
126C4×Q8×C47—21no—
127C4×Q8×C47—21no—
128C4×Q8×C47C2×C411yescharacteristic
129C4×Q8×C47C2×C411yescharacteristic
130C4×Q8×C47C2311yescharacteristic
131C4×Q8×C47C2311yescharacteristic
132C2×C4⋊C4×C47—21no—
133C2×C4⋊C4×C47C2×C411yescharacteristic
134C2×C42×C47—21no—
135C2×C42×C47C2311yescharacteristic
136C4⋊C8×C47—21no—
137C4⋊C8×C47C2311yescharacteristic
138C4⋊C8×C47C2311yescharacteristic
139C22⋊C8×C47—21no—
140C22⋊C8×C47—21no—
141C4×C8×C47—21no—
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C2×Q8×C47—21no—
145C2×Q8×C47—22no—
146C2×Q8×C47—41no—
147C2×Q8×C47C22×C411yescharacteristic
148C2×Q8×C47C22×C411yescharacteristic
149C22×C4×C47—21no—
150C22×C4×C47—21no—
151C22×C4×C47C22×C411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47#4111yescharacteristic
154C2×C22⋊D47—21no—
155C2×C22⋊D47—21no—
156#10625350—81no—
157#10625350—81no—
158C2×D188—21no—
159C2×D188—21no—
160C2×D188C22×C411yescharacteristic
161C2×D188C22×C411yescharacteristic
162C2×C4×D47—21no—
163C2×C4×D47—21no—
164C2×Dic94—21no—
165C2×C8×C47—21no—
166C2×C8×C47—21no—
167C2×C8×C47—21no—
168C2×C8×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47—21no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47C22×C411yescharacteristic
175C4⋊C4×C47C22×C411yescharacteristic
176C42×C47—21no—
177C42×C47—21no—
178C42×C47—21no—
179C42×C47—21no—
180C42×C47C22×C411yescharacteristic
181C42×C47C2×D411yescharacteristic
182C42×C47#4111yescharacteristic
183C42×C47C2411yescharacteristic
184#10625336—22no—
185#10625336—22no—
186#10625336—41no—
187#10625336—41no—
188(C2×D47)⋊C4—21no—
189(C2×D47)⋊C4—21no—
190(C2×D47)⋊C4—21no—
191(C2×D47)⋊C4—21no—
192C4⋊Dic47—21no—
193C4⋊Dic47C22×C411yescharacteristic
194C4⋊Dic47C22×C411yescharacteristic
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8C22×C411yescharacteristic
198C2×C47⋊C8C22×C411yescharacteristic
199C2×C47⋊C8C22×C411yescharacteristic
200C2×C47⋊C8C22×C411yescharacteristic
201C2×C47⋊C8C2×D411yescharacteristic
202C2×C47⋊C8C2×D411yescharacteristic
203C2×C47⋊C8C2×D411yescharacteristic
204C2×C47⋊C8C2×D411yescharacteristic
205#10625324—81no—
206#10625324—81no—
207C8×D47—81no—
208C8×D47—81no—
209C23×C47C4×D411yescharacteristic
210C22×D47—21no—
211C22×D47—21no—
212Q8×C47—21no—
213Q8×C47—21no—
214Q8×C47—21no—
215Q8×C47—42no—
216Q8×C47#13112yes—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—41no—
226C2×C4×C47—42no—
227C2×C4×C47—42no—
228C2×C4×C47C4×D411yescharacteristic
229C2×C4×C47C4×D411yescharacteristic
230C2×C4×C47C4×D411yescharacteristic
231C2×C4×C47C23×C411yescharacteristic
232C2×C4×C47C22×D411yescharacteristic
233C2×C4×C47C2×#4111yescharacteristic
234C22⋊D47—81no—
235C22⋊D47—81no—
236C2×Dic47—21no—
237C2×Dic47—21no—
238D188—21no—
239D188—21no—
240D188—21no—
241D188—21no—
242D188—42no—
243D188—81no—
244C4×D47—41no—
245C4×D47—41no—
246C4×D47—41no—
247C4×D47—41no—
248Dic94—81no—
249C8×C47—41no—
250C8×C47—41no—
251C8×C47—41no—
252C8×C47—41no—
253C47⋊C8—21no—
254C47⋊C8—21no—
255C47⋊C8—21no—
256C47⋊C8—21no—
257C47⋊C8—21no—
258C47⋊C8—21no—
259C47⋊C8—21no—
260C47⋊C8—21no—
261C47⋊C8—41no—
262C47⋊C8—41no—
263#18030—471nomaximal, Sylow
264C22×C47—21no—
265C22×C47—21no—
266C22×C47—21no—
267C22×C47C2×C4×D411yescharacteristic
268C2×D47—41no—
269C2×D47—41no—
270C2×D47—41no—
271C2×D47—41no—
272C2×D47—41no—
273C2×D47—41no—
274C4×C47—21no—
275C4×C47—22no—
276C4×C47—21no—
277C4×C47—21no—
278C4×C47—21no—
279C4×C47—41no—
280C4×C47—41no—
281C4×C47—41no—
282C4×C47C2×#13111yescharacteristic
283C4×C47#56811yescharacteristic
284C4×C47#57711yescharacteristic
285C4×C47#57711yescharacteristic
286Dic47—41no—
287Dic47—41no—
288#3103—471no—
289#2984—471no—
290#2979—471no—
291#1592—471no—
292#1590—471no—
293#1588—471no—
294#1585—471no—
295#1583—471no—
296#1582—471no—
297SD16⋊C8—471no—
298C8×SD16—471no—
299#1572—471no—
300#1496—471no—
301#1479D4711yescharacteristic, complemented
302#1454—471no—
303C2×C47—41no—
304C2×C47#293311yescharacteristic
305C2×C47#294911yescharacteristic
306C2×C47#295511yescharacteristic
307D47—81no—
308D47—81no—
309#532—471no—
310C2×C4×Q8C2×D4711yescharacteristic
311#481—471no—
312#481—471no—
313#475—471no—
314#475—471no—
315#460—471no—
316#460—471no—
317#448—471no—
318#448—471no—
319#446—471no—
320#446—471no—
321#446—941no—
322#446—941no—
323C8×Q8—471no—
324C8×Q8—471no—
325C4×SD16—471no—
326C4×SD16—471no—
327C4×SD16—471no—
328C4×SD16—471no—
329#436—471no—
330#436—471no—
331#436—941no—
332C8×D4—471no—
333C8×D4—471no—
334C8×D4—941no—
335#432—471no—
336#432—941no—
337#431—471no—
338#431—471no—
339#431—941no—
340#431C2×D4711yescharacteristic
341#431C2×D4711yescharacteristic
342#408—941no—
343#408—941no—
344#335—471no—
345#335—471no—
346#334—471no—
347#334—471no—
348Q8⋊C8C2×D4711yescharacteristic
349Q8⋊C8C2×D4711yescharacteristic
350Q8⋊C8C2×D4711yescharacteristic
351Q8⋊C8C2×D4711yescharacteristic
352D4⋊C8—471no—
353D4⋊C8—471no—
354D4⋊C8—471no—
355D4⋊C8—471no—
356#322—471no—
357#322—471no—
358#322—471no—
359C82—471no—
360C47#1803011yescharacteristic, complemented, Sylow
361C2×#41—941no—
362C22×Q8C4×D4711yescharacteristic
363C2×SD16—471no—
364C2×SD16—471no—
365C2×SD16—471no—
366C2×SD16—471no—
367C2×M16—941no—
368C2×M16—941no—
369C22×C8—941no—
370C22×C8—941no—
371C4⋊Q8—471no—
372#127—471no—
373#124—941no—
374#121—471no—
375#121—471no—
376#121—941no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8C4×D4711yescharacteristic
380C4×Q8C4×D4711yescharacteristic
381C4×Q8C22×D4711yescharacteristic
382C4×Q8C22×D4711yescharacteristic
383C4×D4—471no—
384C4×D4—471no—
385C4×D4—941no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4C4×D4711yescharacteristic
388C2×C42—21no—
389C2×C42C22×D4711yescharacteristic
390#106—471no—
391#106—471no—
392#106—471no—
393#106—471no—
394C4⋊C8—21no—
395C4⋊C8—471no—
396C4⋊C8—471no—
397C4⋊C8—471no—
398C4⋊C8—471no—
399C4⋊C8—471no—
400C4⋊C8—471no—
401C4⋊C8—941no—
402C4⋊C8—941no—
403C4⋊C8—941no—
404C4⋊C8—941no—
405C4⋊C8C22×D4711yescharacteristic
406C4⋊C8C22×D4711yescharacteristic
407#103—471no—
408#103—471no—
409#103—471no—
410#103—471no—
411#102—471no—
412#102—471no—
413#102—471no—
414#102—471no—
415C22⋊C8—21no—
416C22⋊C8—21no—
417C22⋊C8—471no—
418C22⋊C8—471no—
419C22⋊C8—471no—
420C22⋊C8—471no—
421C22⋊C8—941no—
422C22⋊C8—941no—
423C22⋊C8—941no—
424C22⋊C8—941no—
425C22⋊C8—941no—
426#97—941no—
427#97—941no—
428#97—941no—
429C4×C8—21no—
430C4×C8—471no—
431C4×C8—471no—
432C4×C8—471no—
433C4×C8—471no—
434C4×C8—941no—
435C4×C8—941no—
436C4×C8C22×D4711yescharacteristic
437C4×C8C22×D4711yescharacteristic
438#41—3761no—
439#41—3761no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—41no—
443C2×Q8—941no—
444C2×Q8C2×C4×D4711yescharacteristic
445C2×Q8C2×C4×D4711yescharacteristic
446C2×D4—471no—
447C2×D4—471no—
448C2×D4—941no—
449C2×D4—941no—
450C2×D4—941no—
451C2×D4—941no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—941no—
455C22×C4—941no—
456C22×C4C2×C4×D4711yescharacteristic
457C22×C4D4×D4711yescharacteristic
458C22×C4#1062535211yescharacteristic
459SD16—942no—
460SD16—942no—
461SD16—1881no—
462SD16—1881no—
463M16—3761no—
464M16—3761no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—471no—
470C2×C8—471no—
471C2×C8—471no—
472C2×C8—471no—
473C2×C8—471no—
474C2×C8—471no—
475C2×C8—471no—
476C2×C8—471no—
477C2×C8—941no—
478C2×C8—941no—
479C2×C8—3761no—
480C2×C8—3761no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—471no—
487C4⋊C4—471no—
488C4⋊C4—941no—
489C4⋊C4C2×C4×D4711yescharacteristic
490C4⋊C4C2×C4×D4711yescharacteristic
491C22⋊C4—941no—
492C22⋊C4—941no—
493C22⋊C4—941no—
494C22⋊C4—941no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42C2×C4×D4711yescharacteristic
500C42D4×D4711yescharacteristic
501C42#1062535211yescharacteristic
502C42C23×D4711yescharacteristic
503C23—941no—
504C23—941no—
505C23(C22⋊D47)⋊C411yescharacteristic
506Q8—21no—
507Q8—21no—
508Q8—21no—
509Q8—42no—
510Q8—3761no—
511Q8#1309403012yes—
512D4—941no—
513D4—941no—
514D4—941no—
515D4—941no—
516D4—3761no—
517D4—3761no—
518D4—3761no—
519D4—1882no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—941no—
532C2×C4—941no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4(C22⋊D47)⋊C411yescharacteristic
538C2×C4(C22⋊D47)⋊C411yescharacteristic
539C2×C4(C22⋊D47)⋊C411yescharacteristic
540C2×C4C22×C4×D4711yescharacteristic
541C2×C4C2×D4×D4711yescharacteristic
542C2×C4C2×#1062535211yescharacteristic
543C8—41no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—941no—
548C8—941no—
549C8—941no—
550C8—941no—
551C8—941no—
552C8—941no—
553C8—941no—
554C8—941no—
555C8—1881no—
556C8—1881no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—1881no—
561C22—1881no—
562C22—1881no—
563C22—1881no—
564C22—1881no—
565C22—1881no—
566C22C2×(C22⋊D47)⋊C411yescharacteristic
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—22no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—1881no—
576C4—1881no—
577C4C2×#1309403011yescharacteristic
578C4#1568988911yescharacteristic
579C4#1568993211yescharacteristic
580C4#1568993211yescharacteristic
581C2—41no—
582C2—3761no—
583C2—3761no—
584C2#2025210911yescharacteristic
585C2#2025224311yescharacteristic
586C2#2025228711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1479C2
socleC22×C47C2×C4×D4
prime radical#15689673C22
cosocleC42×C47C24
hypercenter#1479D47
hypocenterC47#18030
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
2. #20252477C2semidirect
4. #20242293C2semidirect
5. #20242275C2semidirect
6. #20242263C2semidirect
7. #20242248C2semidirect
8. #20242228C2semidirect
9. #20242227C2semidirect
10. #20242203C2semidirect
11. #20242185C2semidirect
12. #20242166C2semidirect
14. #20241688C2semidirect
15. C47×#1479C2semidirect
301. #1479D47semidirect
360. C47#18030semidirect

Maximal subgroups

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

subgroupindex
#202534192
#202524772
#202524542
#202422932
#202422752
#202422632
#202422482
#202422282
#202422272
#202422032
#202421852
#202421662
#202418582
#202416882
C47×#14792
#1803047