Finite Group Databasebeta

← #26223449

Subgroups — #26223449

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×#15689461C211yesmaximal, characteristic
2C22⋊Q8×Dic47C211yesmaximal, characteristic
3#117×Dic47C211yesmaximal, characteristic
4#20244032C211yesmaximal, characteristic, complemented
5#20244025C211yesmaximal, characteristic, complemented
6#20243969C211yesmaximal, characteristic, complemented
7#20243962C211yesmaximal, characteristic, complemented
8#20243587C211yesmaximal, characteristic
9#20243484C211yesmaximal, characteristic
10#20243280C211yesmaximal, characteristic, complemented
11#20243276C211yesmaximal, characteristic, complemented
12#103×Dic47C211yesmaximal, characteristic, complemented
13#103×Dic47C211yesmaximal, characteristic, complemented
14C22⋊C8×Dic47C211yesmaximal, characteristic
15C47×#1873C211yesmaximal, characteristic
16C2×Q8×Dic47C2211yescharacteristic
17C2×#13093983C2211yescharacteristic
18C22×C4×Dic47C2211yescharacteristic
19#15689461C418yescomplemented
20C2×#13093957C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093948C2211yescharacteristic
25C2×#13093948C2211yescharacteristic
26C2×C22⋊(C47⋊C8)C2211yescharacteristic
27C2×C8×Dic47C2211yescharacteristic
28C2×C8×Dic47C2211yescharacteristic
29#15689030C2211yescharacteristic
30#15689022C2211yescharacteristic
31#15689009—21no—
32#15688998—21no—
33#15688983C2211yescharacteristic
34C4⋊C4×Dic47—21no—
35C4⋊C4×Dic47C2211yescharacteristic
36C4⋊C4×Dic47C2211yescharacteristic
37C4⋊C4×Dic47C2211yescharacteristic
38C4⋊C4×Dic47C2211yescharacteristic
39C4⋊C4×Dic47C2211yescharacteristic
40C22⋊C4×Dic47—21no—
41C22⋊C4×Dic47—21no—
42C22⋊C4×Dic47—21no—
43#15688969—21no—
44C42×Dic47—21no—
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×C22⋊Q8×C47C2211yescharacteristic
54C2×#117×C47C2211yescharacteristic
55C2×#103×C47C2211yescharacteristic
56C2×#103×C47C2211yescharacteristic
57C2×C22⋊C8×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C23×Dic47D411yescharacteristic
61C23×Dic47D411yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64Q8×Dic47—22no—
65Q8×Dic47—22no—
66#13093983—22no—
67#13093983C2×C414yes—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C22⋊Dic47—21no—
75C2×C22⋊Dic47—21no—
76C2×C4×Dic47—21no—
77C2×C4×Dic47—22no—
78C2×C4×Dic47—22no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47D411yescharacteristic
86C2×C4×Dic47D411yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88C2×C4×Dic47C2311yescharacteristic
89#13093957—22no—
90#13093957C2×C414yescomplemented
91#13093955—22no—
92#13093955C2×C414yescomplemented
93#13093948C2×C414yescomplemented
94#13093948C2×C414yescomplemented
95C22⋊(C47⋊C8)C2×C414yes—
96C8×Dic47—22no—
97C8×Dic47—22no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C22×Q8×C47C2311yescharacteristic
108C23×C4×C47C2311yescharacteristic
109C22×C8×C47C2311yescharacteristic
110C22×C8×C47C2311yescharacteristic
111C22⋊Q8×C47—22no—
112C22⋊Q8×C47C2×C414yes—
113#117×C47—22no—
114#117×C47—22no—
115C2×C4⋊C4×C47—21no—
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C22⋊C4×C47—21no—
120C2×C22⋊C4×C47—21no—
121C2×C42×C47—21no—
122#103×C47—22no—
123#103×C47—22no—
124C22⋊C8×C47—22no—
125C24×C47C2×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×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47C22×C412yes—
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47#4112yes—
143C22×C4×C47C2411yescharacteristic
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—41no—
154C22×Dic47C2×D411yescharacteristic
155C22×Dic47C2×D411yescharacteristic
156C22×Dic47C2×D411yescharacteristic
157C22×Dic47C2×D411yescharacteristic
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47—21no—
164C4⋊C4×C47—24no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47C22×C412yes—
169C4⋊C4×C47#4112yes—
170C4⋊C4×C47#4112yes—
171C22⋊C4×C47—24no—
172C22⋊C4×C47—24no—
173C42×C47—24no—
174C22⋊Dic47—24no—
175C22⋊Dic47—42no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47—42no—
183C4⋊Dic47—42no—
184C4⋊Dic47C22×C412yes—
185C4⋊Dic47C22×C412yes—
186C4×Dic47—24no—
187C4×Dic47—21no—
188C4×Dic47—21no—
189C4×Dic47—22no—
190C4×Dic47—22no—
191C4×Dic47—42no—
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⋊C8C22×C412yes—
204C2×C47⋊C8C22×C412yes—
205C23×C47—21no—
206C23×C47—21no—
207C23×C47—21no—
208C23×C47—21no—
209C23×C47—21no—
210C23×C47—21no—
211C23×C47C4×D412yes—
212C23×C47C22×D411yescharacteristic
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—22no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
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—41no—
235C2×C4×C47C4×D412yes—
236C2×C4×C47C23×C411yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×C4×C47C2×#4111yescharacteristic
239C2×C4×C47C2×#4111yescharacteristic
240C2×C4×C47C2×#4111yescharacteristic
241C2×C4×C47C2×#4111yescharacteristic
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—22no—
247C2×Dic47—22no—
248C2×Dic47—21no—
249C2×Dic47—21no—
250C2×Dic47—21no—
251C2×Dic47—44no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—44no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—41no—
263C2×Dic47#12312yes—
264C2×Dic47#12312yes—
265C2×Dic47#13512yes—
266C2×Dic47#13512yes—
267C2×Dic47#13712yes—
268C8×C47—41no—
269C8×C47—41no—
270C47⋊C8—22no—
271C47⋊C8—22no—
272C4×#485—471nomaximal, Sylow
273C22×C47—22no—
274C22×C47—22no—
275C22×C47—22no—
276C22×C47—41no—
277C22×C47—41no—
278C22×C47—41no—
279C22×C47—41no—
280C22×C47C2×C4×D411yescharacteristic
281C22×C47C2×#12311yescharacteristic
282C22×C47#52511yescharacteristic
283C22×C47C2×#13511yescharacteristic
284C22×C47C2×#13511yescharacteristic
285C22×C47C2×#13711yescharacteristic
286C22×C47#57511yescharacteristic
287C4×C47—22no—
288C4×C47—42no—
289C4×C47—42no—
290C4×C47—42no—
291C4×C47—42no—
292C4×C47C4×#4111yescharacteristic
293C4×C47C4×#4111yescharacteristic
294C4×C47#52911yescharacteristic
295C4×C47#52911yescharacteristic
296Dic47—22no—
297Dic47—22no—
298Dic47—41no—
299Dic47—42no—
300Dic47—42no—
301Dic47—81no—
302Dic47—81no—
303Dic47#48514yescomplemented
304C2×#485—471no—
305C4×C22⋊Q8—471no—
306C4×#117—471no—
307#1928—471no—
308#1927—471no—
309#1919—471no—
310#1918—471no—
311#1873D4711yescharacteristic
312#1854—471no—
313#1825—471no—
314C4×#107—471no—
315C4×#106—471no—
316C4×#103—471no—
317C4×#103—471no—
318C4×C22⋊C8—471no—
319C2×C47—42no—
320C2×C47C4×#12311yescharacteristic
321C2×C47C4×#13511yescharacteristic
322C2×C47C4×#13711yescharacteristic
323C2×C47C2×#48511yescharacteristic
324C2×C47#309111yescharacteristic
325C2×C47#310211yescharacteristic
326C2×C47#310711yescharacteristic
327C2×C22⋊Q8C2×D4711yescharacteristic
328C2×C4×Q8—471no—
329C2×#117—471no—
330C2×#117C2×D4711yescharacteristic
331C22×C42—471no—
332#485—478no—
333C2×#107—471no—
334C2×#107—471no—
335C2×#106—471no—
336C2×#106—471no—
337C2×#103—471no—
338C2×#103—471no—
339C2×#103C2×D4711yescharacteristic
340C2×#103C2×D4711yescharacteristic
341C2×C22⋊C8—471no—
342C2×C22⋊C8C2×D4711yescharacteristic
343C2×C4×C8—471no—
344C2×C4×C8—471no—
345#391—471no—
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—471no—
355C4×C4⋊C4—941no—
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#340C2×D4711yescharacteristic
368#340C2×D4711yescharacteristic
369#336—471no—
370#336—471no—
371C47C4×#48511yescharacteristic, complemented, Sylow
372C22×Q8C22×D4711yescharacteristic
373C23×C4—471no—
374C23×C4—471no—
375C23×C4C22×D4711yescharacteristic
376C22×C8—471no—
377C22×C8—471no—
378C22×C8C22×D4711yescharacteristic
379C22×C8C22×D4711yescharacteristic
380C22⋊Q8—22no—
381C22⋊Q8C4×D4714yes—
382C4×Q8—942no—
383C4×Q8—942no—
384#117—22no—
385#117—22no—
386#117—474no—
387#117—942no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—471no—
390C2×C4⋊C4—471no—
391C2×C4⋊C4—471no—
392C2×C4⋊C4—471no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4—941no—
395C2×C4⋊C4C22×D4711yescharacteristic
396C2×C4⋊C4C22×D4711yescharacteristic
397C2×C4⋊C4C22×D4711yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—941no—
401C2×C22⋊C4—941no—
402C2×C42—21no—
403C2×C42—471no—
404C2×C42—471no—
405C2×C42—471no—
406C2×C42—471no—
407C2×C42—942no—
408C2×C42—942no—
409C2×C42—941no—
410C2×C42—941no—
411C2×C42—941no—
412C2×C42—941no—
413C2×C42—941no—
414C2×C42—941no—
415C2×C42—941no—
416#107—474no—
417#107—942no—
418#106—474no—
419#106—942no—
420#103—22no—
421#103—22no—
422#103—474no—
423#103—474no—
424C22⋊C8—22no—
425C22⋊C8—474no—
426C4×C8—942no—
427C4×C8—942no—
428#95—941no—
429#95—941no—
430#95—941no—
431#95—941no—
432#95—941no—
433#95—941no—
434#95—941no—
435#95—941no—
436#95—941no—
437C24D4×D4711yescharacteristic
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—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—471no—
453C22×C4—471no—
454C22×C4—471no—
455C22×C4—471no—
456C22×C4—942no—
457C22×C4—942no—
458C22×C4—942no—
459C22×C4—942no—
460C22×C4—942no—
461C22×C4—941no—
462C22×C4—941no—
463C22×C4—941no—
464C22×C4—941no—
465C22×C4—1881no—
466C22×C4C2×C4×D4712yes—
467C22×C4D4×D4711yescharacteristic
468C22×C4#1062535212yes—
469C22×C4C23×D4711yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—472no—
477C2×C8—472no—
478C2×C8—941no—
479C2×C8—941no—
480C2×C8—941no—
481C2×C8—941no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—472no—
487C4⋊C4—472no—
488C4⋊C4—941no—
489C4⋊C4—941no—
490C4⋊C4—941no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4—941no—
494C4⋊C4—1882no—
495C4⋊C4—1882no—
496C4⋊C4C2×C4×D4712yes—
497C4⋊C4#1062535212yes—
498C4⋊C4#1062535212yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—944no—
502C22⋊C4—1882no—
503C42—24no—
504C42—472no—
505C42—472no—
506C42—942no—
507C42—942no—
508C42—944no—
509C42—941no—
510C42—941no—
511C42—1882no—
512C42—1882no—
513C42—1882no—
514C42—1882no—
515C42—1882no—
516C42—1882no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23(C22⋊D47)⋊C412yes—
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—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—472no—
548C2×C4—472no—
549C2×C4—472no—
550C2×C4—472no—
551C2×C4—472no—
552C2×C4—941no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—941no—
556C2×C4—942no—
557C2×C4—942no—
558C2×C4—941no—
559C2×C4—941no—
560C2×C4—941no—
561C2×C4—1884no—
562C2×C4—1884no—
563C2×C4—1881no—
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(C22⋊D47)⋊C412yes—
574C2×C4C22×C4×D4711yescharacteristic
575C2×C4#41×D4711yescharacteristic
576C2×C4#41×D4711yescharacteristic
577C2×C4C2×#1062535211yescharacteristic
578C2×C4C2×#1062535211yescharacteristic
579C2×C4C2×#1062535211yescharacteristic
580C8—41no—
581C8—41no—
582C8—942no—
583C8—942no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×(C22⋊D47)⋊C411yescharacteristic
592C22#123×D4711yescharacteristic
593C22#1568962911yescharacteristic
594C22#135×D4711yescharacteristic
595C22#135×D4711yescharacteristic
596C22#137×D4711yescharacteristic
597C22#1568992211yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—474no—
604C4—942no—
605C4—942no—
606C4—3761no—
607C4—3761no—
608C4—1882no—
609C4—1881no—
610C4—1882no—
611C4C4×#1062535211yescharacteristic
612C4C4×#1062535211yescharacteristic
613C4#1568965411yescharacteristic
614C4#1568965411yescharacteristic
615C2—42no—
616C2#2024591311yescharacteristic
617C2#2025215611yescharacteristic
618C2#2025220311yescharacteristic
619C2#485×D4711yescharacteristic
620C2#2025330611yescharacteristic
621C2#2025340711yescharacteristic
622C2#2025345211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1873C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#1873D47
hypocenterC47C4×#485
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
4. #20244032C2semidirect
5. #20244025C2semidirect
6. #20243969C2semidirect
7. #20243962C2semidirect
10. #20243280C2semidirect
11. #20243276C2semidirect
12. #103×Dic47C2semidirect
13. #103×Dic47C2semidirect
19. #15689461C4semidirect
90. #13093957C2×C4semidirect
92. #13093955C2×C4semidirect
93. #13093948C2×C4semidirect
94. #13093948C2×C4semidirect
303. Dic47#485semidirect
371. C47C4×#485semidirect

Maximal subgroups

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

subgroupindex
C2×#156894612
C22⋊Q8×Dic472
#117×Dic472
#202440322
#202440252
#202439692
#202439622
#202435872
#202434842
#202432802
#202432762
#103×Dic472
C22⋊C8×Dic472
C47×#18732
C4×#48547