Finite Group Databasebeta

← #26107174

Subgroups — #26107174

Order 12032.

804 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
1#121×Dic47C211yesmaximal, characteristic
2#20243152C211yesmaximal, characteristic, complemented
3C2×#15688857C211yesmaximal, characteristic
4#95×Dic47C211yesmaximal, characteristic, complemented
5#20241268C211yesmaximal, characteristic, complemented
6#20241267C211yesmaximal, characteristic, complemented
7C2×C47×#327C211yesmaximal, characteristic
8C2×D4×Dic47—21no—
9C2×D4×Dic47C411yescharacteristic, complemented
10C22×C4×Dic47—21no—
11C22×C4×Dic47C2211yescharacteristic
12C2×C22⋊(C47⋊C8)C2211yescharacteristic
13C2×#13093934—21no—
14#15689026C411yescharacteristic, complemented
15#15689022C411yescharacteristic, complemented
16#15689016—21no—
17#15689003—21no—
18#15688983C2211yescharacteristic
19C4⋊C4×Dic47C411yescharacteristic, complemented
20C22⋊C4×Dic47—21no—
21C2×C47⋊#95—21no—
22C2×C47⋊#95C2211yescharacteristic
23#15688877—21no—
24#15688857C414yescomplemented
25C2×#121×C47C2211yescharacteristic
26C2×C22⋊C8×C47C2211yescharacteristic
27C2×#95×C47C2211yescharacteristic
28C47×#327C414yescomplemented
29C23×Dic47—21no—
30C23×Dic47—21no—
31C23×Dic47D411yescharacteristic
32C23×Dic47D411yescharacteristic
33C22×C47⋊C8—21no—
34D4×Dic47—42no—
35D4×Dic47—42no—
36D4×Dic47—42no—
37C2×C4⋊Dic47—21no—
38C2×C4⋊Dic47C2×C411yescharacteristic
39C2×C4⋊Dic47C2×C411yescharacteristic
40C2×C22⋊Dic47—21no—
41C2×C22⋊Dic47—21no—
42C2×C22⋊Dic47—21no—
43C2×C22⋊Dic47—21no—
44C2×C4×Dic47—24no—
45C2×C4×Dic47—21no—
46C2×C4×Dic47—42no—
47C2×C4×Dic47—42no—
48C2×C4×Dic47—42no—
49C2×C4×Dic47C2×C411yescharacteristic
50C2×C4×Dic47C2×C411yescharacteristic
51C2×C4×Dic47D411yescharacteristic
52C2×C4×Dic47D411yescharacteristic
53C22⋊(C47⋊C8)—21no—
54C22⋊(C47⋊C8)C2×C412yescomplemented
55#13093934—42no—
56C47⋊#95—24no—
57C47⋊#95—21no—
58C47⋊#95—21no—
59C47⋊#95—21no—
60C47⋊#95—21no—
61C47⋊#95C2×C412yescomplemented
62C22×D4×C47—21no—
63C22×D4×C47C2×C411yescharacteristic
64C23×C4×C47—21no—
65C23×C4×C47C2311yescharacteristic
66C22×C8×C47—21no—
67#121×C47—21no—
68#121×C47—21no—
69#121×C47C2×C412yes—
70#121×C47C2×C412yes—
71C2×C4⋊C4×C47C2×C411yescharacteristic
72C2×C22⋊C4×C47—21no—
73C22⋊C8×C47—21no—
74C22⋊C8×C47C2×C412yescomplemented
75#95×C47—21no—
76#95×C47C2×C412yescomplemented
77C24×C47—21no—
78C24×C47C2×D411yescharacteristic
79C2×D4×C47—22no—
80C2×D4×C47—22no—
81C2×D4×C47—21no—
82C2×D4×C47—41no—
83C2×D4×C47—41no—
84C2×D4×C47—41no—
85C2×D4×C47—41no—
86C2×D4×C47—41no—
87C2×D4×C47—41no—
88C2×D4×C47C4212yescomplemented
89C22×C4×C47—22no—
90C22×C4×C47—22no—
91C22×C4×C47—21no—
92C22×C4×C47—22no—
93C22×C4×C47—41no—
94C22×C4×C47—41no—
95C22×C4×C47—41no—
96C22×C4×C47—41no—
97C22×C4×C47—41no—
98C22×C4×C47—41no—
99C22×C4×C47C22×C411yescharacteristic
100C22×C4×C47C22×C411yescharacteristic
101C22×C4×C47C22×C411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47#4111yescharacteristic
104C22×C4×C47#4111yescharacteristic
105C22×Dic47—21no—
106C22×Dic47—21no—
107C22×Dic47—22no—
108C22×Dic47—22no—
109C22×Dic47—22no—
110C22×Dic47—22no—
111C22×Dic47—22no—
112C22×Dic47—21no—
113C22×Dic47—21no—
114C22×Dic47—22no—
115C22×Dic47—21no—
116C22×Dic47—21no—
117C22×Dic47—41no—
118C22×Dic47—42no—
119C22×Dic47—42no—
120C22×Dic47—41no—
121C22×Dic47—41no—
122C22×Dic47—41no—
123C22×Dic47—41no—
124C22×Dic47—42no—
125C22×Dic47C22⋊C411yescharacteristic
126C22×Dic47C22⋊C411yescharacteristic
127C22×Dic47C22⋊C411yescharacteristic
128C22×Dic47C22⋊C411yescharacteristic
129C2×C8×C47—22no—
130C2×C8×C47—41no—
131C2×C8×C47—41no—
132C4⋊C4×C47—21no—
133C4⋊C4×C47C4212yescomplemented
134C22⋊C4×C47—22no—
135C22⋊C4×C47—22no—
136C22⋊Dic47—42no—
137C22⋊Dic47—41no—
138C22⋊Dic47—41no—
139C22⋊Dic47—42no—
140C22⋊Dic47—42no—
141C4⋊Dic47—42no—
142C4⋊Dic47—41no—
143C4⋊Dic47—41no—
144C4×Dic47—22no—
145C4×Dic47—22no—
146C4×Dic47—41no—
147C4×Dic47—41no—
148C4×Dic47—48no—
149C4×Dic47—42no—
150C2×C47⋊C8—22no—
151C2×C47⋊C8—41no—
152C2×C47⋊C8—41no—
153C23×C47—21no—
154C23×C47—21no—
155C23×C47—21no—
156C23×C47—21no—
157C23×C47—22no—
158C23×C47—41no—
159C23×C47—41no—
160C23×C47—41no—
161C23×C47—41no—
162C23×C47—41no—
163C23×C47—41no—
164C23×C47C2×C22⋊C411yescharacteristic
165C23×C47C4×D411yescharacteristic
166C23×C47C4×D411yescharacteristic
167C23×C47AGL(1,ℤ/8)11yescharacteristic
168C23×C47AGL(1,ℤ/8)11yescharacteristic
169C23×C47AGL(1,ℤ/8)11yescharacteristic
170C23×C47#13711yescharacteristic
171D4×C47—42no—
172D4×C47—42no—
173D4×C47—42no—
174D4×C47—42no—
175D4×C47—42no—
176D4×C47—42no—
177C2×C4×C47—21no—
178C2×C4×C47—21no—
179C2×C4×C47—21no—
180C2×C4×C47—21no—
181C2×C4×C47—24no—
182C2×C4×C47—22no—
183C2×C4×C47—41no—
184C2×C4×C47—41no—
185C2×C4×C47—41no—
186C2×C4×C47—41no—
187C2×C4×C47—42no—
188C2×C4×C47—42no—
189C2×C4×C47—42no—
190C2×C4×C47—42no—
191C2×C4×C47—42no—
192C2×C4×C47—41no—
193C2×C4×C47—41no—
194C2×C4×C47—42no—
195C2×C4×C47C2×C4211yescharacteristic
196C2×C4×C47#11711yescharacteristic
197C2×C4×C47C4×D411yescharacteristic
198C2×C4×C47C4×D411yescharacteristic
199C2×Dic47—24no—
200C2×Dic47—24no—
201C2×Dic47—21no—
202C2×Dic47—21no—
203C2×Dic47—21no—
204C2×Dic47—21no—
205C2×Dic47—21no—
206C2×Dic47—21no—
207C2×Dic47—24no—
208C2×Dic47—21no—
209C2×Dic47—21no—
210C2×Dic47—21no—
211C2×Dic47—42no—
212C2×Dic47—42no—
213C2×Dic47—42no—
214C2×Dic47—42no—
215C2×Dic47—42no—
216C2×Dic47—42no—
217C2×Dic47—42no—
218C2×Dic47—42no—
219C2×Dic47—41no—
220C2×Dic47—41no—
221C2×Dic47—41no—
222C2×Dic47—41no—
223C2×Dic47—41no—
224C2×Dic47—41no—
225C2×Dic47—84no—
226C2×Dic47#9912yes—
227C2×Dic47#10412yes—
228C2×Dic47#10412yes—
229C8×C47—42no—
230C47⋊C8—42no—
231#8218—471nomaximal, Sylow
232C22×C47—21no—
233C22×C47—21no—
234C22×C47—21no—
235C22×C47—21no—
236C22×C47—21no—
237C22×C47—21no—
238C22×C47—21no—
239C22×C47—21no—
240C22×C47—21no—
241C22×C47—21no—
242C22×C47—21no—
243C22×C47—21no—
244C22×C47—42no—
245C22×C47—42no—
246C22×C47—42no—
247C22×C47—81no—
248C22×C47—81no—
249C22×C47—81no—
250C22×C47—81no—
251C22×C47C4×C22⋊C411yescharacteristic
252C22×C47C2×#9911yescharacteristic
253C22×C47#41011yescharacteristic
254C22×C47#41711yescharacteristic
255C22×C47#41811yescharacteristic
256C22×C47C2×#10411yescharacteristic
257C22×C47C2×#10411yescharacteristic
258C22×C47#44011yescharacteristic
259C22×C47#44011yescharacteristic
260C22×C47#44211yescharacteristic
261C22×C47#44211yescharacteristic
262C4×C47—21no—
263C4×C47—21no—
264C4×C47—21no—
265C4×C47—21no—
266C4×C47—44no—
267C4×C47—42no—
268Dic47—22no—
269Dic47—22no—
270Dic47—22no—
271Dic47—44no—
272Dic47—41no—
273Dic47—81no—
274Dic47—81no—
275C4×#121—471no—
276#1749—471no—
277C2×#327—471no—
278C2×#327D4711yescharacteristic
279C4×#95—471no—
280#1294—471no—
281#1294—471no—
282C2×C47—21no—
283C2×C47—21no—
284C2×C47—21no—
285C2×C47—21no—
286C2×C47—82no—
287C2×C47#150411yescharacteristic
288C2×C47#150511yescharacteristic
289C2×C47#150711yescharacteristic
290C2×C47#150911yescharacteristic
291C2×C47C4×#9911yescharacteristic
292C2×C47C4×#10411yescharacteristic
293C2×C47#176211yescharacteristic
294C2×#121C2×D4711yescharacteristic
295C2×C4×D4—471no—
296C2×C4×D4—941no—
297C22×C42—471no—
298C22×C42—941no—
299C2×C22⋊C8—471no—
300C2×C22⋊C8C2×D4711yescharacteristic
301C2×#97—941no—
302#390—471no—
303#389—471no—
304#388—941no—
305#386—941no—
306#380—471no—
307C4×C4⋊C4—471no—
308C4×C22⋊C4—941no—
309C2×#95—471no—
310C2×#95—941no—
311C2×#95C2×D4711yescharacteristic
312#336—941no—
313#327—474no—
314#327Dic4714yescomplemented
315C47#821811yescharacteristic, complemented, Sylow
316C22×D4—21no—
317C22×D4C4×D4711yescharacteristic
318C23×C4—21no—
319C23×C4—471no—
320C23×C4—471no—
321C23×C4—941no—
322C23×C4—941no—
323C23×C4C22×D4711yescharacteristic
324C22×C8—21no—
325C22×C8—941no—
326#121—21no—
327#121—21no—
328#121C4×D4712yes—
329#121C2×Dic4712yes—
330C4×D4—1882no—
331C4×D4—1882no—
332C4×D4—1882no—
333C2×C4⋊C4—471no—
334C2×C4⋊C4—471no—
335C2×C4⋊C4—941no—
336C2×C4⋊C4C4×D4711yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—941no—
339C2×C22⋊C4—941no—
340C2×C22⋊C4—941no—
341C2×C22⋊C4—941no—
342C2×C42—471no—
343C2×C42—471no—
344C2×C42—471no—
345C2×C42—471no—
346C2×C42—941no—
347C2×C42—944no—
348C2×C42—1882no—
349C2×C42—1882no—
350C2×C42—1882no—
351C22⋊C8—21no—
352C22⋊C8—472no—
353C22⋊C8—941no—
354C22⋊C8C2×Dic4712yescomplemented
355#97—1882no—
356#95—21no—
357#95—472no—
358#95—941no—
359#95—941no—
360#95—944no—
361#95—941no—
362#95—941no—
363#95C2×Dic4712yescomplemented
364C24—21no—
365C24D4×D4711yescharacteristic
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4C4×Dic4712yescomplemented
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—471no—
387C22×C4—471no—
388C22×C4—471no—
389C22×C4—471no—
390C22×C4—942no—
391C22×C4—942no—
392C22×C4—942no—
393C22×C4—941no—
394C22×C4—941no—
395C22×C4—942no—
396C22×C4—942no—
397C22×C4—942no—
398C22×C4—941no—
399C22×C4—941no—
400C22×C4—941no—
401C22×C4—941no—
402C22×C4—1882no—
403C22×C4—1881no—
404C22×C4—1882no—
405C22×C4—1882no—
406C22×C4—1881no—
407C22×C4—1881no—
408C22×C4—1881no—
409C22×C4—1881no—
410C22×C4C2×C4×D4711yescharacteristic
411C22×C4C2×C4×D4711yescharacteristic
412C22×C4D4×D4711yescharacteristic
413C22×C4#1062535211yescharacteristic
414C22×C4#1062535211yescharacteristic
415C22×C4C22×Dic4711yescharacteristic
416C2×C8—22no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—942no—
420C2×C8—1881no—
421C2×C8—1881no—
422C4⋊C4—21no—
423C4⋊C4—1881no—
424C4⋊C4—1881no—
425C4⋊C4—1882no—
426C4⋊C4C4×Dic4712yescomplemented
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—1882no—
430C22⋊C4—1882no—
431C22⋊C4—1882no—
432C22⋊C4—1881no—
433C22⋊C4—1881no—
434C42—942no—
435C42—942no—
436C42—1881no—
437C42—1881no—
438C42—1888no—
439C42—1882no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—22no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23C22⋊C4×D4711yescharacteristic
452C23(C22⋊D47)⋊C411yescharacteristic
453C23D4×Dic4711yescharacteristic
454C23#1309404811yescharacteristic
455C23#1309404811yescharacteristic
456C23#1309405111yescharacteristic
457C23#1309405311yescharacteristic
458D4—42no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—42no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—472no—
483C2×C4—472no—
484C2×C4—472no—
485C2×C4—944no—
486C2×C4—944no—
487C2×C4—941no—
488C2×C4—941no—
489C2×C4—941no—
490C2×C4—941no—
491C2×C4—941no—
492C2×C4—941no—
493C2×C4—944no—
494C2×C4—941no—
495C2×C4—941no—
496C2×C4—941no—
497C2×C4—3764no—
498C2×C4—1882no—
499C2×C4—1882no—
500C2×C4—1881no—
501C2×C4—1881no—
502C2×C4—1881no—
503C2×C4—1881no—
504C2×C4—1881no—
505C2×C4—1881no—
506C2×C4—1882no—
507C2×C4—1882no—
508C2×C4—1882no—
509C2×C4—1882no—
510C2×C4—1882no—
511C2×C4—1882no—
512C2×C4C2×C4×Dic4711yescharacteristic
513C2×C4#1309398011yescharacteristic
514C2×C4(C22⋊D47)⋊C411yescharacteristic
515C2×C4D4×Dic4711yescharacteristic
516C8—42no—
517C8—1882no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22C22⋊C4×Dic4711yescharacteristic
538C22#99×D4711yescharacteristic
539C22#1568909511yescharacteristic
540C22#1568912111yescharacteristic
541C22#1568912611yescharacteristic
542C22#104×D4711yescharacteristic
543C22#104×D4711yescharacteristic
544C22#1568922011yescharacteristic
545C22#1568922411yescharacteristic
546C22#1568923211yescharacteristic
547C22#1568923511yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—44no—
553C4—42no—
554C4—942no—
555C4—942no—
556C4—942no—
557C4—3761no—
558C4—3761no—
559C4—1884no—
560C4—1881no—
561C2—21no—
562C2—21no—
563C2—21no—
564C2—21no—
565C2—82no—
566C2#2024188911yescharacteristic
567C2#2024189411yescharacteristic
568C2#2024190211yescharacteristic
569C2#2024191011yescharacteristic
570C2#99×Dic4711yescharacteristic
571C2#104×Dic4711yescharacteristic
572C2#2024321511yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC2×C4×C47C2×C42
FrattiniC23×C4C22×D47
FittingC2×C47×#327C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#121×C47C22
cosocleC23×C4×C47C23
hypercenterC2×#327D47
hypocenterC47#8218
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. #20243152C2semidirect
4. #95×Dic47C2semidirect
5. #20241268C2semidirect
6. #20241267C2semidirect
9. C2×D4×Dic47C4semidirect
14. #15689026C4semidirect
15. #15689022C4semidirect
19. C4⋊C4×Dic47C4semidirect
24. #15688857C4semidirect
28. C47×#327C4semidirect
54. C22⋊(C47⋊C8)C2×C4semidirect
61. C47⋊#95C2×C4semidirect
74. C22⋊C8×C47C2×C4semidirect
76. #95×C47C2×C4semidirect
88. C2×D4×C47C42semidirect
133. C4⋊C4×C47C42semidirect
314. #327Dic47semidirect
315. C47#8218semidirect
354. C22⋊C8C2×Dic47semidirect
363. #95C2×Dic47semidirect
375. C2×D4C4×Dic47semidirect
426. C4⋊C4C4×Dic47semidirect

Maximal subgroups

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

subgroupindex
#121×Dic472
#202431522
C2×#156888572
#95×Dic472
#202412682
#202412672
C2×C47×#3272
#821847