Finite Group Databasebeta

← #26217539

Subgroups — #26217539

Order 12032.

1970 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
1C2×#15689690C211yesmaximal, characteristic
2C2×#15689604C211yesmaximal, characteristic, complemented
3C2×#15689128C211yesmaximal, characteristic, complemented
4C2×#15689089C211yesmaximal, characteristic, complemented
5#20243707C214yesmaximal, complemented
6#20243702C214yesmaximal, complemented
7#20243107C211yesmaximal, characteristic, complemented
8#20243105C211yesmaximal, characteristic, complemented
9C22×#102×C47C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12#15689690—24no—
13#15689690C418yescomplemented
14C2×#13094020—21no—
15C2×#13094011—21no—
16C2×#13093995C2214yes—
17C2×#13093993—22no—
18#15689604—24no—
19#15689604—24no—
20C2×#13093989—21no—
21C2×#13093986—21no—
22C2×C4×C22⋊D47—22no—
23C2×#13093983C2211yescharacteristic
24C2×#13093983C2211yescharacteristic
25C2×#13093982C2212yescomplemented
26C2×C4⋊C4×D47C2212yescomplemented
27#15689128—24no—
28C2×#13093945C2212yes—
29C2×#13093945C2212yes—
30#15689089—24no—
31C2×(C2×D47)⋊C8C2214yescomplemented
32#15688886C2212yescomplemented
33#15688886C2212yescomplemented
34#15688885C2214yescomplemented
35C23×D4×C47C2211yescharacteristic
36C23×C8×C47C2211yescharacteristic
37C22×C4⋊C4×C47C2211yescharacteristic
38C2×#102×C47—24no—
39C2×#102×C47C2214yes—
40C2×#10625350—22no—
41C2×#10625350—22no—
42C2×#10625350—22no—
43C2×#10625350—22no—
44C2×#10625350C2×C414yes—
45C2×#10625350D414yes—
46C22×Dic94D411yescharacteristic
47C22×D188D411yescharacteristic
48C22×C22⋊D47—24no—
49C22×C22⋊D47D412yes—
50C22×C4×D47—22no—
51C22×C4×D47C2312yes—
52C2×#10625329—22no—
53C2×#10625329—22no—
54C2×#10625329—22no—
55C22×C47⋊C8C2312yes—
56#13094020—28no—
57#13094011—28no—
58#13093995—28no—
59#13093995C2×C4116yescomplemented
60#13093993—216no—
61#13093989—28no—
62#13093986—28no—
63C4×C22⋊D47—216no—
64#13093983—22no—
65#13093983—22no—
66#13093983C2×C414yescomplemented
67#13093983D414yescomplemented
68#13093982—24no—
69#13093982—24no—
70C2×C4⋊Dic47C2312yes—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—22no—
74C4⋊C4×D47—24no—
75C4⋊C4×D47—24no—
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47—21no—
78C2×C22⋊Dic47—24no—
79C2×(C2×D47)⋊C4—22no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—22no—
83#13093945—24no—
84#13093945—24no—
85(C2×D47)⋊C8—28no—
86C25×C47—21no—
87C22×D4×C47—22no—
88C22×D4×C47—22no—
89C22×D4×C47—21no—
90C22×D4×C47—22no—
91C22×D4×C47—22no—
92C22×D4×C47—22no—
93C22×D4×C47C2×C414yescomplemented
94C22×D4×C47C2312yes—
95C23×C4×C47—21no—
96C23×C4×C47C2311yescharacteristic
97C22×C8×C47—22no—
98C22×C8×C47—22no—
99C22×C8×C47—22no—
100C22×C8×C47C2312yes—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47—22no—
103C2×C4⋊C4×C47D414yescomplemented
104C2×C4⋊C4×C47C2312yes—
105#102×C47—28no—
106C24×C47—22no—
107C24×C47—24no—
108C24×C47—42no—
109C24×C47—42no—
110C24×C47—82no—
111C24×C47—82no—
112C24×C47C2×D411yescharacteristic
113C23×D47C2×D412yes—
114C2×D4×C47—24no—
115C2×D4×C47—28no—
116C2×D4×C47—28no—
117C2×D4×C47—22no—
118C2×D4×C47—22no—
119C2×D4×C47—22no—
120C2×D4×C47—22no—
121C2×D4×C47—22no—
122C2×D4×C47—42no—
123C2×D4×C47—44no—
124C2×D4×C47—44no—
125C2×D4×C47—42no—
126C2×D4×C47C22×C414yes—
127C22×C4×C47—22no—
128C22×C4×C47—24no—
129C22×C4×C47—82no—
130C22×C4×C47C22×C412yes—
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D412yes—
133C22×C4×C47C2×D412yes—
134C22×C4×C47C2×D412yes—
135C22×C4×C47#4112yes—
136C22×C4×C47#4112yes—
137C22×C4×C47C2411yescharacteristic
138C2×C22⋊D47—24no—
139C2×C22⋊D47—24no—
140C2×C22⋊D47—216no—
141C2×C22⋊D47—84no—
142C2×C22⋊D47—84no—
143C2×C22⋊D47C22⋊C418yes—
144C22×Dic47—21no—
145C22×Dic47—22no—
146C22×Dic47—21no—
147C22×Dic47C2×D412yes—
148#10625350—28no—
149#10625350—28no—
150#10625350—28no—
151#10625350—28no—
152C2×D188—22no—
153C2×D188—22no—
154C2×D188C22⋊C414yes—
155C2×C4×D47—22no—
156C2×C4×D47—22no—
157C2×C4×D47—22no—
158C2×C4×D47—22no—
159C2×C4×D47—28no—
160C2×C4×D47—84no—
161C2×C4×D47C22×C414yes—
162C2×C4×D47C2×D414yes—
163C2×Dic94—22no—
164C2×Dic94—22no—
165C2×Dic94C22⋊C414yes—
166C2×C8×C47—22no—
167C2×C8×C47—22no—
168C2×C8×C47—22no—
169C2×C8×C47—44no—
170C4⋊C4×C47—24no—
171C4⋊C4×C47—22no—
172C4⋊C4×C47C2×D414yes—
173C22⋊Dic47—24no—
174C22⋊Dic47—24no—
175C22⋊Dic47—216no—
176(C2×D47)⋊C4—28no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—22no—
180C4⋊Dic47C22×C414yes—
181C4⋊Dic47C2×D412yes—
182C4⋊Dic47C2×D412yes—
183Dic47⋊C4—28no—
184C4×Dic47—28no—
185C4×Dic47—24no—
186C4×Dic47—24no—
187#10625329—44no—
188C2×C47⋊C8—22no—
189C2×C47⋊C8—22no—
190C2×C47⋊C8—22no—
191C23×C47—22no—
192C23×C47—22no—
193C23×C47—24no—
194C23×C47—42no—
195C23×C47—42no—
196C23×C47—44no—
197C23×C47—44no—
198C23×C47—48no—
199C23×C47—88no—
200C23×C47—88no—
201C23×C47—84no—
202C23×C47—84no—
203C23×C47C2×C22⋊C412yes—
204C23×C47#12012yes—
205C23×C47#12312yes—
206C23×C47C22×D411yescharacteristic
207C22×D47—22no—
208C22×D47—22no—
209C22×D47—24no—
210C22×D47—24no—
211C22×D47C2×C22⋊C414yes—
212D4×C47—24no—
213D4×C47—24no—
214D4×C47—42no—
215D4×C47—42no—
216D4×C47—48no—
217C2×C4×C47—24no—
218C2×C4×C47—44no—
219C2×C4×C47—84no—
220C2×C4×C47C2×C22⋊C412yes—
221C2×C4×C47C4×D414yes—
222C2×C4×C47C4×D414yes—
223C2×C4×C47#12012yes—
224C2×C4×C47#12114yes—
225C2×C4×C47#12114yes—
226C2×C4×C47#12312yes—
227C2×C4×C47C23×C411yescharacteristic
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C47C22×D411yescharacteristic
230C2×C4×C47C22×D411yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C22⋊D47—216no—
234C22⋊D47—816no—
235C2×Dic47—22no—
236C2×Dic47—22no—
237C2×Dic47—24no—
238C2×Dic47—22no—
239C2×Dic47—22no—
240C2×Dic47—42no—
241C2×Dic47—42no—
242C2×Dic47—42no—
243C2×Dic47—42no—
244C2×Dic47C2×C22⋊C414yes—
245D188—28no—
246C4×D47—24no—
247C4×D47—24no—
248C4×D47—24no—
249C4×D47—24no—
250C4×D47—88no—
251Dic94—28no—
252C8×C47—42no—
253C47⋊C8—42no—
254#20914—471nomaximal, Sylow
255C22×C47—24no—
256C22×C47—24no—
257C22×C47—22no—
258C22×C47—22no—
259C22×C47—48no—
260C22×C47—44no—
261C22×C47—816no—
262C22×C47—84no—
263C22×C47—82no—
264C22×C47—82no—
265C22×C47#38614yes—
266C22×C47C22×C22⋊C411yescharacteristic
267C22×C47C2×#12011yescharacteristic
268C22×C47C2×#12311yescharacteristic
269C22×C47#57612yes—
270C22×C47#57712yes—
271C2×D47—28no—
272C2×D47—24no—
273C2×D47—44no—
274C2×D47—42no—
275C2×D47—42no—
276C4×C47—44no—
277C4×C47#38614yes—
278C4×C47C2×C4×D411yescharacteristic
279C4×C47C2×C4×D411yescharacteristic
280C4×C47C2×#12111yescharacteristic
281C4×C47C2×#12111yescharacteristic
282Dic47—24no—
283Dic47—44no—
284Dic47—42no—
285Dic47—42no—
286C2×#535—471no—
287C2×#520—471no—
288C2×#418—471no—
289C22×#102D4711yescharacteristic, complemented
290C2×#408—471no—
291#1875—474no—
292#1874—474no—
293#1736—471no—
294#1735—471no—
295C2×C47—24no—
296C2×C47—84no—
297C2×C47C2×#38611yescharacteristic
298C2×C47#289712yes—
299C2×C47#300811yescharacteristic
300C2×C47#301011yescharacteristic
301C2×C47#309612yes—
302D47—44no—
303C22×#41—471no—
304C23×D4C2×D4711yescharacteristic
305C22×M16—471no—
306C23×C8C2×D4711yescharacteristic
307#535—478no—
308#535—944no—
309C2×#127—941no—
310C2×#124—941no—
311C2×#121—474no—
312C2×#120—942no—
313#520—944no—
314#520—944no—
315C2×C4×Q8—941no—
316C2×C4×D4—941no—
317C2×C4×D4—942no—
318C2×#117—471no—
319C2×#117—471no—
320C2×#117—472no—
321C22×C4⋊C4—472no—
322C22×C4⋊C4C2×D4711yescharacteristic
323#418—944no—
324C2×#102—24no—
325C2×#102—472no—
326C2×#102—472no—
327C2×#102C2×D4714yes—
328#408—944no—
329C2×C22⋊C8—474no—
330#340—472no—
331#340—472no—
332#340—474no—
333C47#2091411yescharacteristic, complemented, Sylow
334C25—21no—
335C2×#41—474no—
336C2×#41—474no—
337C2×#41—942no—
338C2×#41—942no—
339C2×#41—942no—
340C2×#41—942no—
341C22×Q8—471no—
342C22×D4—22no—
343C22×D4—22no—
344C22×D4—21no—
345C22×D4—22no—
346C22×D4—22no—
347C22×D4—22no—
348C22×D4—472no—
349C22×D4—471no—
350C22×D4—944no—
351C22×D4C4×D4714yescomplemented
352C22×D4C22×D4712yes—
353C23×C4—21no—
354C23×C4—472no—
355C23×C4—942no—
356C23×C4C22×D4711yescharacteristic
357C2×M16—942no—
358C2×M16—942no—
359C2×M16—942no—
360C22×C8—22no—
361C22×C8—22no—
362C22×C8—22no—
363C22×C8—472no—
364C22×C8C22×D4712yes—
365#127—948no—
366#124—948no—
367#121—4716no—
368#121—948no—
369#120—9416no—
370C4×Q8—948no—
371C4×D4—948no—
372C4×D4—9416no—
373#117—474no—
374#117—474no—
375#117—944no—
376#117—942no—
377#117—942no—
378#117—944no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—471no—
382C2×C4⋊C4—471no—
383C2×C4⋊C4—472no—
384C2×C4⋊C4—944no—
385C2×C4⋊C4—942no—
386C2×C4⋊C4—944no—
387C2×C4⋊C4C22⋊D4714yescomplemented
388C2×C4⋊C4C22×D4712yes—
389C2×C22⋊C4—942no—
390C2×C22⋊C4—944no—
391C2×C22⋊C4—941no—
392C2×C22⋊C4—941no—
393C2×C42—942no—
394C2×C42—941no—
395C2×C42—941no—
396#102—28no—
397#102—944no—
398#102—944no—
399C22⋊C8—948no—
400C24—22no—
401C24—24no—
402C24—42no—
403C24—42no—
404C24—82no—
405C24—82no—
406C24—472no—
407C24D4×D4711yescharacteristic
408#41—948no—
409#41—948no—
410#41—948no—
411#41—948no—
412C2×Q8—474no—
413C2×Q8—942no—
414C2×Q8—942no—
415C2×D4—24no—
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—478no—
428C2×D4—474no—
429C2×D4—944no—
430C2×D4—944no—
431C2×D4—9416no—
432C2×D4—942no—
433C2×D4—942no—
434C2×D4—3764no—
435C2×D4—3764no—
436C2×D4C2×C4×D4714yes—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—82no—
440C22×C4—472no—
441C22×C4—474no—
442C22×C4—474no—
443C22×C4—948no—
444C22×C4—942no—
445C22×C4—942no—
446C22×C4—942no—
447C22×C4—942no—
448C22×C4—941no—
449C22×C4—942no—
450C22×C4—941no—
451C22×C4—3764no—
452C22×C4C2×C4×D4712yes—
453C22×C4#1062535012yes—
454C22×C4D4×D4711yescharacteristic
455C22×C4D4×D4712yes—
456C22×C4D4×D4712yes—
457C22×C4#1062535212yes—
458C22×C4C2×C22⋊D4712yes—
459C22×C4C23×D4711yescharacteristic
460M16—1884no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—44no—
465C2×C8—942no—
466C2×C8—942no—
467C2×C8—942no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—474no—
471C4⋊C4—472no—
472C4⋊C4—472no—
473C4⋊C4—948no—
474C4⋊C4—942no—
475C4⋊C4—941no—
476C4⋊C4—941no—
477C4⋊C4C2×C22⋊D4714yes—
478C22⋊C4—948no—
479C22⋊C4—9416no—
480C22⋊C4—944no—
481C22⋊C4—944no—
482C42—948no—
483C42—944no—
484C42—944no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—48no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—44no—
493C23—88no—
494C23—88no—
495C23—84no—
496C23—84no—
497C23—474no—
498C23—942no—
499C23—942no—
500C23—944no—
501C23—944no—
502C23C22⋊C4×D4712yes—
503C23#1309399212yes—
504C23#1309400612yes—
505C23C2×D4×D4711yescharacteristic
506Q8—948no—
507D4—24no—
508D4—24no—
509D4—48no—
510D4—42no—
511D4—42no—
512D4—9416no—
513D4—948no—
514D4—37616no—
515C2×C4—24no—
516C2×C4—44no—
517C2×C4—84no—
518C2×C4—474no—
519C2×C4—942no—
520C2×C4—942no—
521C2×C4—944no—
522C2×C4—944no—
523C2×C4—944no—
524C2×C4—944no—
525C2×C4—944no—
526C2×C4—942no—
527C2×C4—942no—
528C2×C4—3768no—
529C2×C4—1882no—
530C2×C4—1882no—
531C2×C4—1882no—
532C2×C4—1882no—
533C2×C4C22⋊C4×D4712yes—
534C2×C4(C22⋊D47)⋊C414yes—
535C2×C4C4×C22⋊D4714yes—
536C2×C4#1309399212yes—
537C2×C4#1309399514yes—
538C2×C4#1309399714yes—
539C2×C4#1309400612yes—
540C2×C4C22×C4×D4711yescharacteristic
541C2×C4C2×D4×D4711yescharacteristic
542C2×C4C2×D4×D4711yescharacteristic
543C2×C4C22×C22⋊D4711yescharacteristic
544C2×C4C2×#1062535211yescharacteristic
545C2×C4C2×#1062535011yescharacteristic
546C8—42no—
547C8—1882no—
548C22—24no—
549C22—24no—
550C22—22no—
551C22—22no—
552C22—48no—
553C22—44no—
554C22—816no—
555C22—84no—
556C22—82no—
557C22—82no—
558C22—948no—
559C22—944no—
560C22—1884no—
561C22—1882no—
562C22—1882no—
563C22#1568899914yes—
564C22C2×C22⋊C4×D4711yescharacteristic
565C22C2×#1309399211yescharacteristic
566C22C2×#1309400611yescharacteristic
567C22#1568992412yes—
568C22#1568992912yes—
569C4—44no—
570C4—944no—
571C4—1884no—
572C4—1882no—
573C4—1882no—
574C4#1568899914yes—
575C4C2×(C22⋊D47)⋊C411yescharacteristic
576C4C2×C4×C22⋊D4711yescharacteristic
577C4C2×#1309399511yescharacteristic
578C4C2×#1309399711yescharacteristic
579C2—24no—
580C2—84no—
581C2—1884no—
582C2C2×#1568899911yescharacteristic
583C2#2025190912yes—
584C2#2025265211yescharacteristic
585C2#2025267011yescharacteristic
586C2#2025334212yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#102×C47C2
socleC23×C47C22×D4
prime radicalC2×#15689690C2
cosocleC22×C4×C47C24
hypercenterC22×#102D47
hypocenterC47#20914
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
2. C2×#15689604C2semidirect
3. C2×#15689128C2semidirect
4. C2×#15689089C2semidirect
5. #20243707C2semidirect
6. #20243702C2semidirect
7. #20243107C2semidirect
8. #20243105C2semidirect
9. C22×#102×C47C2semidirect
13. #15689690C4semidirect
25. C2×#13093982C22semidirect
26. C2×C4⋊C4×D47C22semidirect
31. C2×(C2×D47)⋊C8C22semidirect
32. #15688886C22semidirect
33. #15688886C22semidirect
34. #15688885C22semidirect
59. #13093995C2×C4semidirect
66. #13093983C2×C4semidirect
67. #13093983D4semidirect
93. C22×D4×C47C2×C4semidirect
103. C2×C4⋊C4×C47D4semidirect
289. C22×#102D47semidirect
333. C47#20914semidirect
351. C22×D4C4×D47semidirect
387. C2×C4⋊C4C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896902
C2×#156896042
C2×#156891282
C2×#156890892
#202437072
#202437022
#202431072
#202431052
C22×#102×C472
#2091447