Finite Group Databasebeta

← #26223451

Subgroups — #26223451

Order 12032.

888 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689462C211yesmaximal, characteristic
2#20245926C211yesmaximal, characteristic
3C4×#13093982C211yesmaximal, characteristic
4#20244026C211yesmaximal, characteristic, complemented
5#20243967C211yesmaximal, characteristic, complemented
6#20243959C211yesmaximal, characteristic, complemented
7#20243697C211yesmaximal, characteristic
8#20243586C211yesmaximal, characteristic
9#20243481C211yesmaximal, characteristic
10#20243280C211yesmaximal, characteristic, complemented
11#106×Dic47C211yesmaximal, characteristic, complemented
12#103×Dic47C211yesmaximal, characteristic, complemented
13#103×Dic47C211yesmaximal, characteristic, complemented
14#20243145C211yesmaximal, characteristic
15C47×#1927C211yesmaximal, characteristic, complemented
16C2×#13094000C2211yescharacteristic
17C2×Q8×Dic47C2211yescharacteristic
18C2×#13093982C2211yescharacteristic
19C2×#13093982C2211yescharacteristic
20C2×C42×D47C2211yescharacteristic
21#15689462C418yescomplemented
22C2×#13093957C2211yescharacteristic
23C2×#13093957C2211yescharacteristic
24C2×#13093956C2211yescharacteristic
25C2×#13093948C2211yescharacteristic
26C2×#13093948C2211yescharacteristic
27C2×(C2×D47)⋊C8C2211yescharacteristic
28C2×(C2×D47)⋊C8C2211yescharacteristic
29C2×C8×Dic47C2211yescharacteristic
30C2×C8×Dic47C2211yescharacteristic
31#15689021C2211yescharacteristic
32#15689010—21no—
33#15688997—21no—
34#15688980C2211yescharacteristic
35C4⋊C4×Dic47C2211yescharacteristic
36C4⋊C4×Dic47C2211yescharacteristic
37C4⋊C4×Dic47C2211yescharacteristic
38C4×C4⋊Dic47C2211yescharacteristic
39#15688974—21no—
40C4×(C2×D47)⋊C4—21no—
41#15688970—21no—
42#15688970—21no—
43#15688968—21no—
44C42×Dic47—21no—
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688878C2211yescharacteristic
53C2×#106×C47C2211yescharacteristic
54C2×#103×C47C2211yescharacteristic
55C2×#103×C47C2211yescharacteristic
56C47×#391C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#336C2211yescharacteristic
60C22×C4×D47D411yescharacteristic
61C22×C4×D47D411yescharacteristic
62C22×C4×D47C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65#13094000—22no—
66#13094000C2×C414yes—
67Q8×Dic47—22no—
68Q8×Dic47—22no—
69#13093982—22no—
70#13093982—22no—
71#13093982—22no—
72#13093982C2×C414yes—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×(C2×D47)⋊C4—21no—
81C2×(C2×D47)⋊C4—21no—
82C2×(C2×D47)⋊C4—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47—21no—
90C2×C4×Dic47—21no—
91C2×C4×Dic47D411yescharacteristic
92C2×C4×Dic47D411yescharacteristic
93C2×C4×Dic47C2311yescharacteristic
94C42×D47—22no—
95C42×D47—22no—
96#13093957—22no—
97#13093957C2×C414yescomplemented
98#13093956C2×C414yescomplemented
99#13093948—22no—
100#13093948C2×C414yescomplemented
101(C2×D47)⋊C8—22no—
102(C2×D47)⋊C8C2×C414yes—
103C8×Dic47—22no—
104C8×Dic47—22no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C47⋊#95—21no—
110C47⋊#95—21no—
111C22×Q8×C47C2311yescharacteristic
112C22×C8×C47C2311yescharacteristic
113C22×C8×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C42×C47—21no—
118C2×C42×C47C2311yescharacteristic
119#106×C47—22no—
120#103×C47—22no—
121#103×C47C2×C414yescomplemented
122#95×C47—21no—
123#95×C47—21no—
124#95×C47—21no—
125C23×D47C2×D411yescharacteristic
126C2×Q8×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47—21no—
131C2×Q8×C47C22×C412yes—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2411yescharacteristic
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47C2×D411yescharacteristic
144C22×Dic47C2×D411yescharacteristic
145C22×Dic47C2×D411yescharacteristic
146C2×C4×D47—21no—
147C2×C4×D47—21no—
148C2×C4×D47—21no—
149C2×C4×D47—21no—
150C2×C4×D47—22no—
151C2×C4×D47—22no—
152C2×C4×D47—22no—
153C2×C4×D47—22no—
154C2×C4×D47—22no—
155C2×C4×D47—41no—
156C2×C4×D47C22×C412yes—
157C2×C4×D47#4112yes—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47C22×C412yes—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47C22×C412yes—
169C4⋊C4×C47#4112yes—
170C42×C47—21no—
171C42×C47—21no—
172C42×C47—42no—
173(C2×D47)⋊C4—24no—
174(C2×D47)⋊C4—24no—
175(C2×D47)⋊C4—24no—
176(C2×D47)⋊C4—42no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47—42no—
183C4⋊Dic47C22×C412yes—
184C4⋊Dic47C22×C412yes—
185C4⋊Dic47#4112yes—
186Dic47⋊C4—24no—
187Dic47⋊C4—42no—
188C4×Dic47—24no—
189C4×Dic47—24no—
190C4×Dic47—22no—
191C4×Dic47—22no—
192C4×Dic47—42no—
193C4×Dic47—42no—
194C4×Dic47—42no—
195C4×Dic47—42no—
196C4×Dic47—42no—
197C4×Dic47#4112yes—
198C4×Dic47#4112yes—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8C22×C412yes—
205C23×C47C22×D411yescharacteristic
206C22×D47—21no—
207C22×D47—21no—
208C22×D47—21no—
209C22×D47—21no—
210C22×D47—21no—
211C22×D47—21no—
212C22×D47C4×D412yes—
213Q8×C47—22no—
214Q8×C47—22no—
215Q8×C47—41no—
216Q8×C47—41no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—21no—
222C2×C4×C47—22no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47#13512yes—
235C2×C4×C47C23×C411yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×C4×C47C2×#4111yescharacteristic
239C2×C4×C47C2×#4111yescharacteristic
240C2×C4×C47C2×#4111yescharacteristic
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—22no—
246C2×Dic47—22no—
247C2×Dic47—22no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47C4×D412yes—
258C2×Dic47#12312yes—
259C2×Dic47#12312yes—
260C2×Dic47#13512yes—
261C2×Dic47#13712yes—
262C4×D47—22no—
263C4×D47—22no—
264C4×D47—22no—
265C4×D47—22no—
266C4×D47—44no—
267C4×D47—44no—
268C8×C47—22no—
269C8×C47—41no—
270C47⋊C8—22no—
271C47⋊C8—41no—
272C4×#485—471nomaximal, Sylow
273C22×C47C2×C4×D411yescharacteristic
274C22×C47C2×#12311yescharacteristic
275C22×C47#52511yescharacteristic
276C22×C47C2×#13511yescharacteristic
277C22×C47C2×#13511yescharacteristic
278C22×C47C2×#13711yescharacteristic
279C22×C47#57511yescharacteristic
280C2×D47—22no—
281C2×D47—22no—
282C2×D47—22no—
283C2×D47—41no—
284C2×D47—41no—
285C2×D47—41no—
286C2×D47—41no—
287C4×C47—22no—
288C4×C47—41no—
289C4×C47—42no—
290C4×C47—42no—
291C4×C47—42no—
292C4×C47—81no—
293C4×C47C4×#4111yescharacteristic
294C4×C47C4×#4111yescharacteristic
295C4×C47#52911yescharacteristic
296C4×C47#52911yescharacteristic
297Dic47—22no—
298Dic47—22no—
299Dic47—42no—
300Dic47—42no—
301Dic47—42no—
302Dic47—81no—
303Dic47#48514yescomplemented
304C2×#485—471no—
305C4×C22⋊Q8—471no—
306C4×#117—471no—
307#1928—471no—
308#1927D4711yescharacteristic, complemented
309#1919—471no—
310#1918—471no—
311#1873—471no—
312#1854—471no—
313#1825—471no—
314C4×#107—471no—
315C4×#106—471no—
316C4×#103—471no—
317C4×#103—471no—
318C4×C22⋊C8—471no—
319C2×C47C4×#12311yescharacteristic
320C2×C47C4×#13511yescharacteristic
321C2×C47C4×#13711yescharacteristic
322C2×C47C2×#48511yescharacteristic
323C2×C47#309111yescharacteristic
324C2×C47#310211yescharacteristic
325C2×C47#310711yescharacteristic
326D47—42no—
327C2×C22⋊Q8—471no—
328C2×C4×Q8—471no—
329C2×#117—471no—
330C2×#117—471no—
331C22×C42—471no—
332#485—478no—
333C2×#107—471no—
334C2×#107—471no—
335C2×#106—471no—
336C2×#106C2×D4711yescharacteristic
337C2×#103—471no—
338C2×#103—471no—
339C2×#103C2×D4711yescharacteristic
340C2×#103C2×D4711yescharacteristic
341C2×C22⋊C8—471no—
342C2×C22⋊C8—471no—
343C2×C4×C8—471no—
344C2×C4×C8—471no—
345#391C2×D4711yescharacteristic
346#389—471no—
347#387—941no—
348#385—941no—
349#380—471no—
350C4×C4⋊C4—471no—
351C4×C4⋊C4—471no—
352C4×C4⋊C4—471no—
353C4×C4⋊C4—471no—
354C4×C4⋊C4—941no—
355C4×C4⋊C4C2×D4711yescharacteristic
356C4×C22⋊C4—941no—
357C4×C22⋊C4—941no—
358C4×C22⋊C4—941no—
359#376—941no—
360C43—941no—
361#340—471no—
362#340—471no—
363#340—471no—
364#340—471no—
365#340—471no—
366#340—471no—
367#340—471no—
368#340C2×D4711yescharacteristic
369#336—471no—
370#336C2×D4711yescharacteristic
371C47C4×#48511yescharacteristic, complemented, Sylow
372C22×Q8C22×D4711yescharacteristic
373C23×C4—471no—
374C23×C4—471no—
375C23×C4—471no—
376C22×C8—471no—
377C22×C8—471no—
378C22×C8C22×D4711yescharacteristic
379C22×C8C22×D4711yescharacteristic
380C22⋊Q8—474no—
381C22⋊Q8—942no—
382C4×Q8—942no—
383C4×Q8—942no—
384#117—474no—
385#117—942no—
386#117—942no—
387#117—942no—
388C2×C4⋊C4—471no—
389C2×C4⋊C4—471no—
390C2×C4⋊C4—471no—
391C2×C4⋊C4—471no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4—941no—
395C2×C4⋊C4C22×D4711yescharacteristic
396C2×C4⋊C4C22×D4711yescharacteristic
397C2×C4⋊C4C22×D4711yescharacteristic
398C2×C22⋊C4—941no—
399C2×C22⋊C4—941no—
400C2×C22⋊C4—941no—
401C2×C22⋊C4—941no—
402C2×C42—21no—
403C2×C42—471no—
404C2×C42—471no—
405C2×C42—471no—
406C2×C42—942no—
407C2×C42—942no—
408C2×C42—941no—
409C2×C42—941no—
410C2×C42—941no—
411C2×C42—941no—
412C2×C42—941no—
413C2×C42—941no—
414C2×C42—941no—
415C2×C42C22×D4711yescharacteristic
416#107—474no—
417#107—942no—
418#106—22no—
419#106—474no—
420#103—22no—
421#103—474no—
422#103—942no—
423#103C4×D4714yescomplemented
424C22⋊C8—474no—
425C22⋊C8—942no—
426C4×C8—942no—
427C4×C8—942no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—941no—
432#95—941no—
433#95—941no—
434#95—941no—
435#95—941no—
436#95—941no—
437C24—471no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8C2×C4×D4712yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—471no—
449C22×C4—471no—
450C22×C4—471no—
451C22×C4—472no—
452C22×C4—472no—
453C22×C4—942no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—942no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—941no—
460C22×C4—941no—
461C22×C4—941no—
462C22×C4—941no—
463C22×C4—941no—
464C22×C4—942no—
465C22×C4—941no—
466C22×C4—1881no—
467C22×C4D4×D4711yescharacteristic
468C22×C4D4×D4711yescharacteristic
469C22×C4C23×D4711yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—472no—
476C2×C8—941no—
477C2×C8—941no—
478C2×C8—941no—
479C2×C8—941no—
480C2×C8—941no—
481C2×C8C2×C4×D4712yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—472no—
487C4⋊C4—472no—
488C4⋊C4—472no—
489C4⋊C4—944no—
490C4⋊C4—941no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4—941no—
494C4⋊C4—941no—
495C4⋊C4—1882no—
496C4⋊C4—1882no—
497C4⋊C4C2×C4×D4712yes—
498C4⋊C4#1062535212yes—
499C22⋊C4—944no—
500C22⋊C4—944no—
501C22⋊C4—944no—
502C22⋊C4—1882no—
503C42—21no—
504C42—21no—
505C42—42no—
506C42—472no—
507C42—472no—
508C42—942no—
509C42—942no—
510C42—944no—
511C42—944no—
512C42—1882no—
513C42—1882no—
514C42—1882no—
515C42—1882no—
516C42—1882no—
517C23—472no—
518C23—941no—
519C23—941no—
520C23—941no—
521C23—941no—
522C23—941no—
523C23—941no—
524C23C2×D4×D4711yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—472no—
547C2×C4—472no—
548C2×C4—472no—
549C2×C4—472no—
550C2×C4—472no—
551C2×C4—941no—
552C2×C4—941no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—942no—
556C2×C4—942no—
557C2×C4—942no—
558C2×C4—942no—
559C2×C4—942no—
560C2×C4—942no—
561C2×C4—942no—
562C2×C4—1884no—
563C2×C4—1884no—
564C2×C4—1881no—
565C2×C4—1881no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4—1881no—
571C2×C4—1881no—
572C2×C4—1881no—
573C2×C4#1309404512yes—
574C2×C4C22×C4×D4711yescharacteristic
575C2×C4#41×D4711yescharacteristic
576C2×C4#41×D4711yescharacteristic
577C2×C4#41×D4711yescharacteristic
578C2×C4#41×D4711yescharacteristic
579C2×C4C2×#1062535211yescharacteristic
580C8—22no—
581C8—41no—
582C8—942no—
583C8—1881no—
584C22—942no—
585C22—942no—
586C22—942no—
587C22—1881no—
588C22—1881no—
589C22—1881no—
590C22—1881no—
591C22C4×D4×D4711yescharacteristic
592C22#123×D4711yescharacteristic
593C22#1568963311yescharacteristic
594C22#135×D4711yescharacteristic
595C22C2×#1309404511yescharacteristic
596C22#137×D4711yescharacteristic
597C22#1568992211yescharacteristic
598C4—22no—
599C4—42no—
600C4—41no—
601C4—42no—
602C4—42no—
603C4—81no—
604C4—474no—
605C4—942no—
606C4—942no—
607C4—3761no—
608C4—1882no—
609C4—1882no—
610C4—1882no—
611C4C4×#1062535211yescharacteristic
612C4#1568959111yescharacteristic
613C4#1568965211yescharacteristic
614C4#1568965611yescharacteristic
615C2—1882no—
616C2#2024591511yescharacteristic
617C2C4×#1309404511yescharacteristic
618C2#2025220411yescharacteristic
619C2#485×D4711yescharacteristic
620C2#2025330211yescharacteristic
621C2#2025340511yescharacteristic
622C2#2025345611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1927C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1927D47
hypocenterC47C4×#485
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1927

Complements

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

NH ≅ G/Ntype
4. #20244026C2semidirect
5. #20243967C2semidirect
6. #20243959C2semidirect
10. #20243280C2semidirect
11. #106×Dic47C2semidirect
12. #103×Dic47C2semidirect
13. #103×Dic47C2semidirect
15. C47×#1927C2semidirect
21. #15689462C4semidirect
97. #13093957C2×C4semidirect
98. #13093956C2×C4semidirect
100. #13093948C2×C4semidirect
121. #103×C47C2×C4semidirect
303. Dic47#485semidirect
308. #1927D47semidirect
371. C47C4×#485semidirect
423. #103C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156894622
#202459262
C4×#130939822
#202440262
#202439672
#202439592
#202436972
#202435862
#202434812
#202432802
#106×Dic472
#103×Dic472
#202431452
C47×#19272
C4×#48547