Finite Group Databasebeta

← #26223535

Subgroups — #26223535

Order 12032.

900 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689441C211yesmaximal, characteristic
2#20246081C211yesmaximal, characteristic
3#20244015C211yesmaximal, characteristic, complemented
4#20244008C211yesmaximal, characteristic, complemented
5#20243973C211yesmaximal, characteristic, complemented
6#20243966C211yesmaximal, characteristic, complemented
7#20243710C211yesmaximal, characteristic
8#20243587C211yesmaximal, characteristic, complemented
9#20243474C211yesmaximal, characteristic, complemented
10#20243286C211yesmaximal, characteristic, complemented
11C4×#13093955C211yesmaximal, characteristic, complemented
12#20243216C211yesmaximal, characteristic, complemented
13C4×#13093944C211yesmaximal, characteristic, complemented
14#20243154C211yesmaximal, characteristic, complemented
15C47×#2274C211yesmaximal, characteristic, complemented
16C2×#13093998C2211yescharacteristic
17C2×C4×D188C2211yescharacteristic
18#15689441C418yescomplemented
19C2×#13093957C2211yescharacteristic
20C2×#13093955C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093944C2211yescharacteristic
24C2×#13093944C2211yescharacteristic
25C2×#13093944C2211yescharacteristic
26C2×#13093944C2211yescharacteristic
27C2×C22⋊(C47⋊C8)C2211yescharacteristic
28C2×C22⋊(C47⋊C8)C2211yescharacteristic
29C2×#13093935C2211yescharacteristic
30C2×C4×C47⋊C8C2211yescharacteristic
31#15689025C2211yescharacteristic
32#15689023—21no—
33#15689021C2211yescharacteristic
34#15689013—21no—
35#15688999—21no—
36#15688994—21no—
37#15688983C2211yescharacteristic
38C4×C4⋊Dic47C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688878C2211yescharacteristic
48#15688878C2211yescharacteristic
49C2×#117×C47C2211yescharacteristic
50C2×#117×C47C2211yescharacteristic
51C22×C4⋊C4×C47C2211yescharacteristic
52C4×C4⋊C4×C47—21no—
53C4×C4⋊C4×C47—21no—
54C4×C4⋊C4×C47C2211yescharacteristic
55C4×C4⋊C4×C47C2211yescharacteristic
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C4×C22⋊C4×C47—21no—
59C4×C22⋊C4×C47—21no—
60C22×D188C2311yescharacteristic
61C22×C22⋊D47—21no—
62C22×C4×D47—21no—
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67#13093998—22no—
68#13093998C2×C414yes—
69C4×D188—22no—
70C4×D188—22no—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×Dic47⋊C4—21no—
75C2×C22⋊Dic47—21no—
76C2×(C2×D47)⋊C4—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×C4×Dic47—21no—
81#13093957—22no—
82#13093955—22no—
83#13093955C2×C414yescomplemented
84#13093955C2×C414yescomplemented
85#13093944—22no—
86#13093944—22no—
87#13093944C2×C414yes—
88#13093944C2×C414yes—
89C22⋊(C47⋊C8)—22no—
90C22⋊(C47⋊C8)C2×C414yescomplemented
91#13093935—22no—
92C4×C47⋊C8—22no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C23×C4×C47—21no—
97C23×C4×C47C2311yescharacteristic
98#117×C47—22no—
99#117×C47—22no—
100#117×C47—22no—
101#117×C47—22no—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47—22no—
104C2×C4⋊C4×C47C2×C414yescomplemented
105C2×C4⋊C4×C47D411yescharacteristic
106C2×C4⋊C4×C47D411yescharacteristic
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C22⋊C4×C47—21no—
112C2×C22⋊C4×C47D411yescharacteristic
113C2×C22⋊C4×C47D411yescharacteristic
114C2×C42×C47—21no—
115C2×C42×C47—21no—
116C2×C42×C47—21no—
117C2×C42×C47—21no—
118C2×C42×C47—21no—
119C2×C42×C47C2311yescharacteristic
120C2×C42×C47C2311yescharacteristic
121#95×C47—21no—
122#95×C47—21no—
123#95×C47—21no—
124#95×C47—21no—
125C24×C47C2×D411yescharacteristic
126C23×D47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—24no—
136C22×C4×C47—81no—
137C22×C4×C47—81no—
138C22×C4×C47C22×C412yes—
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47#4112yes—
145C22×C4×C47C2411yescharacteristic
146C2×C22⋊D47—24no—
147C2×C22⋊D47—81no—
148C2×C22⋊D47—81no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47—21no—
152C2×D188—21no—
153C2×D188—21no—
154C2×D188—21no—
155C2×D188—21no—
156C2×D188—21no—
157C2×D188C22×C412yes—
158C2×C4×D47—42no—
159C2×C4×D47—81no—
160C2×C4×D47—81no—
161C4⋊C4×C47—24no—
162C4⋊C4×C47—22no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47C22×C412yes—
169C4⋊C4×C47C22×C412yes—
170C4⋊C4×C47#4112yes—
171C4⋊C4×C47#4112yes—
172C22⋊C4×C47—24no—
173C22⋊C4×C47—22no—
174C22⋊C4×C47—22no—
175C42×C47—24no—
176C42×C47—21no—
177C42×C47—21no—
178C42×C47—81no—
179C42×C47—81no—
180C42×C47—81no—
181C42×C47—81no—
182C42×C47#4112yes—
183C42×C47#4112yes—
184C22⋊Dic47—81no—
185(C2×D47)⋊C4—24no—
186(C2×D47)⋊C4—42no—
187(C2×D47)⋊C4—81no—
188(C2×D47)⋊C4—81no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—21no—
193C4⋊Dic47—21no—
194C4⋊Dic47C22×C412yes—
195Dic47⋊C4—81no—
196C4×Dic47—81no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8C22×C412yes—
208C2×C47⋊C8C22×C412yes—
209C23×C47—21no—
210C23×C47—21no—
211C23×C47—21no—
212C23×C47—21no—
213C23×C47—21no—
214C23×C47—21no—
215C23×C47C4×D412yes—
216C23×C47C22×D411yescharacteristic
217C22×D47—22no—
218C22×D47—41no—
219C22×D47—41no—
220C22×D47—41no—
221C22×D47—41no—
222C22×D47—41no—
223C22×D47—41no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—22no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—84no—
253C2×C4×C47C4×D412yes—
254C2×C4×C47#12312yes—
255C2×C4×C47#12312yes—
256C2×C4×C47C23×C411yescharacteristic
257C2×C4×C47C2×#4111yescharacteristic
258C2×C4×C47C2×#4111yescharacteristic
259C2×C4×C47C2×#4111yescharacteristic
260C2×C4×C47C2×#4111yescharacteristic
261C2×C4×C47C2×#4111yescharacteristic
262C22⋊D47—84no—
263C2×Dic47—22no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272D188—22no—
273D188—22no—
274D188—41no—
275D188—41no—
276C4×D47—84no—
277C47⋊C8—22no—
278C47⋊C8—22no—
279C47⋊C8—41no—
280C47⋊C8—41no—
281#21584—471nomaximal, Sylow
282C22×C47—22no—
283C22×C47—22no—
284C22×C47—22no—
285C22×C47—41no—
286C22×C47—41no—
287C22×C47—41no—
288C22×C47—41no—
289C22×C47C2×C4×D411yescharacteristic
290C22×C47C2×#12311yescharacteristic
291C22×C47#52511yescharacteristic
292C22×C47#57611yescharacteristic
293C22×C47#57711yescharacteristic
294C22×C47#57711yescharacteristic
295C22×C47#57711yescharacteristic
296C2×D47—42no—
297C2×D47—42no—
298C2×D47—42no—
299C2×D47—81no—
300C2×D47—81no—
301C2×D47—81no—
302C2×D47—81no—
303C4×C47—22no—
304C4×C47—22no—
305C4×C47—22no—
306C4×C47—41no—
307C4×C47—41no—
308C4×C47—42no—
309C4×C47—42no—
310C4×C47—42no—
311C4×C47—42no—
312C4×C47C4×#4111yescharacteristic
313C4×C47C4×#4111yescharacteristic
314C4×C47#52911yescharacteristic
315C4×C47#52911yescharacteristic
316Dic47—42no—
317Dic47—81no—
318Dic47—81no—
319C2×#481—471no—
320#2324—471no—
321#2274D4711yescharacteristic, complemented
322#1926—471no—
323#1925—471no—
324#1920—471no—
325#1919—471no—
326#1875—471no—
327#1854—471no—
328#1823—471no—
329#1776—471no—
330C4×#106—471no—
331#1762—471no—
332C4×#102—471no—
333#1750—471no—
334C2×C47—42no—
335C2×C47C4×#12311yescharacteristic
336C2×C47#294611yescharacteristic
337C2×C47#294911yescharacteristic
338C2×C47#309611yescharacteristic
339C2×C47#309911yescharacteristic
340C2×C47#311911yescharacteristic
341C2×C47#312511yescharacteristic
342D47—82no—
343C2×#121—471no—
344C2×C4×D4—471no—
345C2×#117C2×D4711yescharacteristic
346C2×#117C2×D4711yescharacteristic
347C22×C4⋊C4C2×D4711yescharacteristic
348#481—478no—
349C2×#107—471no—
350C2×#106—471no—
351C2×#106—471no—
352C2×#106—471no—
353C2×#102—471no—
354C2×#102—471no—
355C2×#102—471no—
356C2×#102—471no—
357C2×C22⋊C8—471no—
358C2×C22⋊C8—471no—
359C2×#97—471no—
360C2×C4×C8—471no—
361#390—471no—
362#390—941no—
363#389—471no—
364#388—941no—
365#386—941no—
366#385—941no—
367#380—471no—
368C4×C4⋊C4—21no—
369C4×C4⋊C4—21no—
370C4×C4⋊C4—471no—
371C4×C4⋊C4C2×D4711yescharacteristic
372C4×C4⋊C4C2×D4711yescharacteristic
373C4×C4⋊C4C2×D4711yescharacteristic
374C4×C4⋊C4C2×D4711yescharacteristic
375C4×C22⋊C4—21no—
376C4×C22⋊C4—21no—
377#340—471no—
378#340—471no—
379#340—471no—
380#340—471no—
381#340—471no—
382#340—471no—
383#340—471no—
384#340—471no—
385#336—471no—
386#336—471no—
387C47#2158411yescharacteristic, complemented, Sylow
388C22×D4—471no—
389C22×D4—941no—
390C23×C4—21no—
391C23×C4—941no—
392C23×C4C22×D4711yescharacteristic
393C22×C8—471no—
394C22×C8—471no—
395C22×C8—471no—
396C22×C8—471no—
397#121—474no—
398#121—942no—
399C4×D4—942no—
400C4×D4—942no—
401#117—22no—
402#117—22no—
403#117—22no—
404#117—22no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—471no—
408C2×C4⋊C4—471no—
409C2×C4⋊C4—471no—
410C2×C4⋊C4—941no—
411C2×C4⋊C4C4×D4714yescomplemented
412C2×C4⋊C4C22⋊D4711yescharacteristic
413C2×C4⋊C4C22⋊D4711yescharacteristic
414C2×C4⋊C4C22×D4711yescharacteristic
415C2×C4⋊C4C22×D4711yescharacteristic
416C2×C4⋊C4C22×D4711yescharacteristic
417C2×C4⋊C4C22×D4711yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—941no—
420C2×C22⋊C4—941no—
421C2×C22⋊C4—941no—
422C2×C22⋊C4—941no—
423C2×C22⋊C4—941no—
424C2×C22⋊C4C22⋊D4711yescharacteristic
425C2×C22⋊C4C22⋊D4711yescharacteristic
426C2×C42—21no—
427C2×C42—21no—
428C2×C42—21no—
429C2×C42—21no—
430C2×C42—21no—
431C2×C42—941no—
432C2×C42C22×D4711yescharacteristic
433C2×C42C22×D4711yescharacteristic
434#107—942no—
435#106—474no—
436#106—474no—
437#106—942no—
438#102—474no—
439#102—474no—
440#102—942no—
441#102—942no—
442C22⋊C8—474no—
443C22⋊C8—942no—
444#97—942no—
445C4×C8—942no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—941no—
451#95—941no—
452#95—941no—
453C24—941no—
454C24C2×C22⋊D4711yescharacteristic
455C2×D4—472no—
456C2×D4—944no—
457C2×D4—941no—
458C2×D4—941no—
459C2×D4—941no—
460C2×D4—941no—
461C2×D4—941no—
462C2×D4—3761no—
463C2×D4—3761no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—941no—
476C22×C4—941no—
477C22×C4—941no—
478C22×C4—3761no—
479C22×C4—3761no—
480C22×C4—1882no—
481C22×C4C2×C4×D4712yes—
482C22×C4#1062535012yes—
483C22×C4C2×C22⋊D4711yescharacteristic
484C22×C4C2×C22⋊D4711yescharacteristic
485C22×C4C2×C22⋊D4711yescharacteristic
486C22×C4C2×C22⋊D4711yescharacteristic
487C22×C4C2×C22⋊D4711yescharacteristic
488C22×C4C23×D4711yescharacteristic
489C2×C8—472no—
490C2×C8—472no—
491C2×C8—941no—
492C2×C8—941no—
493C2×C8—941no—
494C2×C8—941no—
495C2×C8—941no—
496C2×C8—941no—
497C2×C8—941no—
498C2×C8—941no—
499C2×C8—941no—
500C2×C8—941no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—472no—
509C4⋊C4—941no—
510C4⋊C4—941no—
511C4⋊C4—941no—
512C4⋊C4—941no—
513C4⋊C4—941no—
514C4⋊C4—3761no—
515C4⋊C4C2×C4×D4712yes—
516C4⋊C4C2×C4×D4712yes—
517C4⋊C4#1062535012yes—
518C4⋊C4#1062535012yes—
519C22⋊C4—24no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—944no—
523C22⋊C4—3761no—
524C22⋊C4—3761no—
525C22⋊C4—3761no—
526C22⋊C4—1882no—
527C42—24no—
528C42—21no—
529C42—21no—
530C42—81no—
531C42—81no—
532C42—81no—
533C42—81no—
534C42—3761no—
535C42#1062535012yes—
536C42#1062535012yes—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—942no—
544C23—1881no—
545C23—1881no—
546C23—1881no—
547C23—1881no—
548C23—1881no—
549C23—1881no—
550C23C4×C22⋊D4712yes—
551C23C22×C22⋊D4711yescharacteristic
552D4—942no—
553D4—942no—
554D4—3764no—
555D4—1881no—
556D4—1881no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—84no—
586C2×C4—942no—
587C2×C4—3764no—
588C2×C4—1881no—
589C2×C4—1881no—
590C2×C4—1881no—
591C2×C4—1881no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4—1881no—
595C2×C4—1881no—
596C2×C4C4×C22⋊D4712yes—
597C2×C4#1309400712yes—
598C2×C4#1309400712yes—
599C2×C4C22×C4×D4711yescharacteristic
600C2×C4C2×#1062535011yescharacteristic
601C2×C4C2×#1062535011yescharacteristic
602C2×C4C2×#1062535011yescharacteristic
603C2×C4C2×#1062535011yescharacteristic
604C2×C4C2×#1062535011yescharacteristic
605C8—942no—
606C8—942no—
607C8—1881no—
608C8—1881no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—3761no—
617C22—3761no—
618C22—3761no—
619C22—3761no—
620C22—1882no—
621C22—1882no—
622C22—1882no—
623C22C2×C4×C22⋊D4711yescharacteristic
624C22C2×#1309400711yescharacteristic
625C22#1568963411yescharacteristic
626C22#1568992611yescharacteristic
627C22#1568993111yescharacteristic
628C22#1568993111yescharacteristic
629C22#1568993411yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—41no—
637C4—41no—
638C4—42no—
639C4—3761no—
640C4—3761no—
641C4—1882no—
642C4C4×#1062535011yescharacteristic
643C4C4×#1062535011yescharacteristic
644C4#1568965711yescharacteristic
645C4#1568965711yescharacteristic
646C2—42no—
647C2—3762no—
648C2C4×#1309400711yescharacteristic
649C2#2025222011yescharacteristic
650C2#2025224711yescharacteristic
651C2#2025334611yescharacteristic
652C2#2025337711yescharacteristic
653C2#2025358711yescharacteristic
654C2#2025366511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2274C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
cosocleC22×C4×C47C24
hypercenter#2274D47
hypocenterC47#21584
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2274

Complements

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

NH ≅ G/Ntype
3. #20244015C2semidirect
4. #20244008C2semidirect
5. #20243973C2semidirect
6. #20243966C2semidirect
8. #20243587C2semidirect
9. #20243474C2semidirect
10. #20243286C2semidirect
11. C4×#13093955C2semidirect
12. #20243216C2semidirect
13. C4×#13093944C2semidirect
14. #20243154C2semidirect
15. C47×#2274C2semidirect
18. #15689441C4semidirect
83. #13093955C2×C4semidirect
84. #13093955C2×C4semidirect
90. C22⋊(C47⋊C8)C2×C4semidirect
104. C2×C4⋊C4×C47C2×C4semidirect
321. #2274D47semidirect
387. C47#21584semidirect
411. C2×C4⋊C4C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156894412
#202460812
#202440152
#202440082
#202439732
#202439662
#202437102
#202435872
#202434742
#202432862
C4×#130939552
#202432162
C4×#130939442
#202431542
C47×#22742
#2158447