Finite Group Databasebeta

← #26138846

Subgroups — #26138846

Order 12032.

1846 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689016C212yesmaximal
2#20241641C212yesmaximal, complemented
3#20241637C212yesmaximal
4#20241620C212yesmaximal, complemented
5#20241618C211yesmaximal, characteristic, complemented
6#20241615C211yesmaximal, characteristic, complemented
7#20241615C211yesmaximal, characteristic, complemented
8#20241611C211yesmaximal, characteristic
9#20241611C211yesmaximal, characteristic
10C4×C47⋊#95C211yesmaximal, characteristic, complemented
11C22×#117×C47C211yesmaximal, characteristic
12C22×C4⋊Dic47C2212yes—
13C22×C22⋊Dic47C2212yes—
14C22×C22⋊Dic47C2212yes—
15C22×C4×Dic47C2212yes—
16#15689016—28no—
17C2×C47⋊#95C2212yes—
18C2×C47⋊#95C2212yes—
19C2×C47⋊#95C2211yescharacteristic
20C2×C47⋊#95C2211yescharacteristic
21C2×C47⋊#95C2212yes—
22C2×C47⋊#95C2211yescharacteristic
23C2×C47⋊#95C2211yescharacteristic
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C24×C4×C47C2211yescharacteristic
30C2×#117×C47—28no—
31C2×#117×C47C418yescomplemented
32C22×C4⋊C4×C47C2211yescharacteristic
33C22×C4⋊C4×C47C2211yescharacteristic
34C22×C22⋊C4×C47C2211yescharacteristic
35C22×C22⋊C4×C47C2211yescharacteristic
36C22×C42×C47C2211yescharacteristic
37C22×C42×C47C2211yescharacteristic
38C23×Dic47C2312yes—
39C23×Dic47C2312yes—
40C23×Dic47C2312yes—
41C23×Dic47C2312yes—
42C2×C4⋊Dic47—24no—
43C2×C4⋊Dic47—24no—
44C2×C4⋊Dic47—42no—
45C2×C22⋊Dic47—24no—
46C2×C22⋊Dic47—24no—
47C2×C22⋊Dic47—24no—
48C2×C22⋊Dic47—24no—
49C2×C22⋊Dic47—42no—
50C2×C22⋊Dic47—42no—
51C2×C4×Dic47—24no—
52C2×C4×Dic47—24no—
53C2×C4×Dic47—42no—
54C47⋊#95—24no—
55C47⋊#95—24no—
56C47⋊#95—22no—
57C47⋊#95—22no—
58C47⋊#95—24no—
59C47⋊#95—24no—
60C47⋊#95—22no—
61C47⋊#95—22no—
62C47⋊#95—24no—
63C47⋊#95—24no—
64C47⋊#95—42no—
65C47⋊#95—42no—
66C25×C47C2311yescharacteristic
67C23×C4×C47—24no—
68C23×C4×C47—22no—
69C23×C4×C47—22no—
70C23×C4×C47—24no—
71C23×C4×C47C2×C414yes—
72C23×C4×C47C2311yescharacteristic
73C23×C4×C47C2311yescharacteristic
74C23×C4×C47C2311yescharacteristic
75C23×C4×C47C2311yescharacteristic
76C23×C4×C47C2311yescharacteristic
77C23×C4×C47C2311yescharacteristic
78#117×C47—232no—
79C2×C4⋊C4×C47—24no—
80C2×C4⋊C4×C47—24no—
81C2×C4⋊C4×C47C2×C414yes—
82C2×C4⋊C4×C47C2×C414yes—
83C2×C22⋊C4×C47—24no—
84C2×C22⋊C4×C47C2×C414yes—
85C2×C22⋊C4×C47C2×C414yes—
86C2×C22⋊C4×C47D418yes—
87C2×C42×C47—24no—
88C2×C42×C47—24no—
89C2×C42×C47C2×C414yes—
90C2×C42×C47C2×C414yes—
91C24×C47—21no—
92C24×C47—21no—
93C24×C47—22no—
94C24×C47—22no—
95C24×C47—22no—
96C24×C47—21no—
97C24×C47—21no—
98C24×C47—21no—
99C24×C47—21no—
100C24×C47C22×C412yes—
101C24×C47C2×D414yes—
102C24×C47C2411yescharacteristic
103C22×C4×C47—216no—
104C22×C4×C47—216no—
105C22×C4×C47—28no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—21no—
109C22×C4×C47—21no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—21no—
114C22×C4×C47—21no—
115C22×C4×C47—21no—
116C22×C4×C47—21no—
117C22×C4×C47—28no—
118C22×C4×C47—28no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47C22×C412yes—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C22×C412yes—
130C22×C4×C47C22×C412yes—
131C22×C4×C47C22×C412yes—
132C22×C4×C47C22×C412yes—
133C22×C4×C47C2×D414yes—
134C22×C4×C47C2×D414yes—
135C22×C4×C47#4112yes—
136C22×C4×C47#4112yes—
137C22×C4×C47#4114yes—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47—22no—
162C22×Dic47—22no—
163C22×Dic47—22no—
164C22×Dic47#4114yes—
165C22×Dic47#4114yes—
166C4⋊C4×C47—28no—
167C4⋊C4×C47—28no—
168C22⋊C4×C47—28no—
169C22⋊C4×C47C22⋊C4116yes—
170C42×C47—28no—
171C42×C47—28no—
172C22⋊Dic47—48no—
173C22⋊Dic47—48no—
174C4⋊Dic47—48no—
175C4×Dic47—48no—
176C23×C47—22no—
177C23×C47—22no—
178C23×C47—22no—
179C23×C47—22no—
180C23×C47—22no—
181C23×C47—22no—
182C23×C47—24no—
183C23×C47—24no—
184C23×C47—24no—
185C23×C47—24no—
186C23×C47—24no—
187C23×C47—24no—
188C23×C47—42no—
189C23×C47—42no—
190C23×C47—41no—
191C23×C47—41no—
192C23×C47—41no—
193C23×C47—41no—
194C23×C47—42no—
195C23×C47—42no—
196C23×C47—42no—
197C23×C47—42no—
198C23×C47C2×C22⋊C414yes—
199C23×C47C23×C411yescharacteristic
200C23×C47C22×D412yes—
201C23×C47C2×#4112yes—
202C23×C47C2×#4112yes—
203C23×C47C2×#4111yescharacteristic
204C23×C47C2×#4111yescharacteristic
205C23×C47C2×#4112yes—
206C23×C47H(5,𝔽2)11yescharacteristic
207C23×C47H(5,𝔽2)11yescharacteristic
208C23×C472−1+411yescharacteristic
209C23×C472−1+411yescharacteristic
210C2×C4×C47—232no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—24no—
220C2×C4×C47—24no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—24no—
225C2×C4×C47—24no—
226C2×C4×C47—24no—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—42no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47C2×C22⋊C414yes—
238C2×C4×C47C2×C22⋊C414yes—
239C2×C4×C47#11714yes—
240C2×C4×C47#11714yes—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—24no—
244C2×Dic47—24no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—42no—
248C2×Dic47—42no—
249C2×Dic47—42no—
250C2×Dic47—42no—
251C2×Dic47—42no—
252C2×Dic47—42no—
253C2×Dic47—42no—
254C2×Dic47—42no—
255C2×Dic47—42no—
256C2×Dic47—42no—
257C2×Dic47—42no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47—42no—
263C2×Dic47—42no—
264C2×Dic47—42no—
265C2×Dic47—42no—
266C2×Dic47—42no—
267C2×Dic47—42no—
268C2×Dic47—42no—
269#13943—471nomaximal, Sylow
270C22×C47—24no—
271C22×C47—24no—
272C22×C47—24no—
273C22×C47—42no—
274C22×C47—42no—
275C22×C47—42no—
276C22×C47—44no—
277C22×C47—44no—
278C22×C47—44no—
279C22×C47—44no—
280C22×C47—42no—
281C22×C47C22×C22⋊C411yescharacteristic
282C22×C47C2×#11711yescharacteristic
283C22×C47C2×#11711yescharacteristic
284C22×C47C4×#4111yescharacteristic
285C22×C47C4×#4111yescharacteristic
286C22×C47#52011yescharacteristic
287C22×C47#52011yescharacteristic
288C22×C47#52512yes—
289C22×C47#52512yes—
290C22×C47#52912yes—
291C22×C47#52912yes—
292C22×C47#54011yescharacteristic
293C22×C47#54111yescharacteristic
294C22×C47#54211yescharacteristic
295C22×C47#54211yescharacteristic
296C22×C47#54612yes—
297C22×C47#54612yes—
298C22×C47#54712yes—
299C22×C47#54712yes—
300C22×C47#55212yes—
301C22×C47#55312yes—
302C22×C47#55512yes—
303C22×C47#55612yes—
304C4×C47—24no—
305C4×C47—24no—
306C4×C47—24no—
307C4×C47—24no—
308C4×C47—42no—
309C4×C47—42no—
310Dic47—44no—
311Dic47—44no—
312Dic47—82no—
313Dic47—82no—
314C22×#117D4711yescharacteristic
315C2×#388—472no—
316#1441—472no—
317#1440—472no—
318#1436—472no—
319C4⋊#95—471no—
320#1434—471no—
321#1434—471no—
322C22⋊#95—471no—
323C22⋊#95—471no—
324C4×#95—471no—
325C2×C47—44no—
326C2×C47#228011yescharacteristic
327C2×C47#229811yescharacteristic
328C2×C47#231411yescharacteristic
329C2×C47#233911yescharacteristic
330C2×C47#233911yescharacteristic
331C2×C47#234611yescharacteristic
332C2×C47#234711yescharacteristic
333C2×C47#247012yes—
334C2×C47#247412yes—
335C2×C47#247612yes—
336C2×C47#247812yes—
337C24×C4C2×D4711yescharacteristic
338C2×#117—28no—
339C2×#117Dic4718yescomplemented
340C22×C4⋊C4—472no—
341C22×C4⋊C4C2×D4711yescharacteristic
342C22×C4⋊C4C2×D4711yescharacteristic
343C22×C22⋊C4—472no—
344C22×C22⋊C4—472no—
345C22×C22⋊C4C2×D4711yescharacteristic
346C22×C22⋊C4C2×D4711yescharacteristic
347C22×C42—472no—
348C22×C42C2×D4711yescharacteristic
349C22×C42C2×D4711yescharacteristic
350#388—948no—
351C2×#95—471no—
352C2×#95—471no—
353C2×#95—472no—
354C2×#95—471no—
355C2×#95—471no—
356C2×#95—472no—
357C2×#95—472no—
358C2×#95—472no—
359C2×#95—472no—
360C2×#95—472no—
361C2×#95—472no—
362C2×#95—472no—
363C47#1394311yescharacteristic, complemented, Sylow
364C25C22×D4711yescharacteristic
365C23×C4—24no—
366C23×C4—22no—
367C23×C4—22no—
368C23×C4—24no—
369C23×C4—472no—
370C23×C4—472no—
371C23×C4—472no—
372C23×C4—472no—
373C23×C4C2×Dic4714yes—
374C23×C4C22×D4711yescharacteristic
375C23×C4C22×D4711yescharacteristic
376C23×C4C22×D4711yescharacteristic
377C23×C4C22×D4711yescharacteristic
378C23×C4C22×D4711yescharacteristic
379C23×C4C22×D4711yescharacteristic
380#117—232no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—944no—
384C2×C4⋊C4—944no—
385C2×C4⋊C4—1882no—
386C2×C4⋊C4C2×Dic4714yes—
387C2×C4⋊C4C2×Dic4714yes—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—944no—
390C2×C22⋊C4—944no—
391C2×C22⋊C4—944no—
392C2×C22⋊C4—944no—
393C2×C22⋊C4—1882no—
394C2×C22⋊C4—1882no—
395C2×C22⋊C4C2×Dic4714yes—
396C2×C22⋊C4C2×Dic4714yes—
397C2×C22⋊C4C22⋊D4718yes—
398C2×C42—24no—
399C2×C42—24no—
400C2×C42—944no—
401C2×C42—944no—
402C2×C42—1882no—
403C2×C42C2×Dic4714yes—
404C2×C42C2×Dic4714yes—
405#95—944no—
406#95—944no—
407#95—942no—
408#95—942no—
409#95—944no—
410#95—944no—
411#95—942no—
412#95—942no—
413#95—944no—
414#95—944no—
415#95—1882no—
416#95—1882no—
417C24—22no—
418C24—22no—
419C24—21no—
420C24—21no—
421C24—22no—
422C24—21no—
423C24—21no—
424C24—21no—
425C24—21no—
426C24C22×Dic4712yes—
427C24C2×C22⋊D4714yes—
428C24C23×D4711yescharacteristic
429C22×C4—216no—
430C22×C4—216no—
431C22×C4—28no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—474no—
454C22×C4—474no—
455C22×C4—942no—
456C22×C4—942no—
457C22×C4—942no—
458C22×C4—942no—
459C22×C4—942no—
460C22×C4—942no—
461C22×C4—942no—
462C22×C4—942no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—942no—
466C22×C4—942no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—942no—
472C22×C4—942no—
473C22×C4—942no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4—942no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4#1062535012yes—
482C22×C4#1062535012yes—
483C22×C4#1062535014yes—
484C22×C4C22×Dic4712yes—
485C22×C4C22×Dic4712yes—
486C22×C4C22×Dic4712yes—
487C22×C4C22×Dic4712yes—
488C22×C4C22×Dic4712yes—
489C22×C4C22×Dic4712yes—
490C22×C4C2×C22⋊D4714yes—
491C22×C4C2×C22⋊D4714yes—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—1888no—
495C22⋊C4—28no—
496C22⋊C4—1888no—
497C22⋊C4—1888no—
498C22⋊C4C22⋊Dic47116yes—
499C42—28no—
500C42—28no—
501C42—1888no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23C2×C22⋊Dic4714yes—
525C23C23×Dic4711yescharacteristic
526C23C22×C22⋊D4712yes—
527C23#41×D4712yes—
528C23#41×D4712yes—
529C23C2×#1062535011yescharacteristic
530C23C2×#1062535011yescharacteristic
531C23C2×#1062535012yes—
532C23#1309407011yescharacteristic
533C23#1309407011yescharacteristic
534C23#1309407111yescharacteristic
535C23#1309407111yescharacteristic
536C2×C4—232no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—944no—
564C2×C4—944no—
565C2×C4—944no—
566C2×C4—944no—
567C2×C4—944no—
568C2×C4—944no—
569C2×C4—1882no—
570C2×C4—1882no—
571C2×C4—1882no—
572C2×C4—1882no—
573C2×C4—1882no—
574C2×C4—1882no—
575C2×C4—1882no—
576C2×C4—1882no—
577C2×C4—1882no—
578C2×C4—1882no—
579C2×C4—1882no—
580C2×C4—1882no—
581C2×C4—1882no—
582C2×C4—1882no—
583C2×C4—1882no—
584C2×C4—1882no—
585C2×C4—1882no—
586C2×C4—1882no—
587C2×C4—1882no—
588C2×C4—1882no—
589C2×C4—1882no—
590C2×C4—1882no—
591C2×C4C2×C22⋊Dic4714yes—
592C2×C4C2×C22⋊Dic4714yes—
593C2×C4#1309398314yes—
594C2×C4#1309398314yes—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—42no—
599C22—42no—
600C22—44no—
601C22—44no—
602C22—42no—
603C22—44no—
604C22—42no—
605C22—44no—
606C22C22×C22⋊Dic4711yescharacteristic
607C22C2×#1309398311yescharacteristic
608C22C2×#1309398311yescharacteristic
609C22#41×Dic4711yescharacteristic
610C22#41×Dic4711yescharacteristic
611C22#1568960611yescharacteristic
612C22#1568960611yescharacteristic
613C22#1568963412yes—
614C22#1568963412yes—
615C22#1568965112yes—
616C22#1568965112yes—
617C22#1568972811yescharacteristic
618C22#1568973511yescharacteristic
619C22#1568974411yescharacteristic
620C22#1568974411yescharacteristic
621C22#1568976112yes—
622C22#1568976112yes—
623C22#1568977212yes—
624C22#1568977212yes—
625C22#1568981212yes—
626C22#1568981712yes—
627C22#1568983612yes—
628C22#1568984112yes—
629C4—24no—
630C4—24no—
631C4—24no—
632C4—24no—
633C4—42no—
634C4—42no—
635C4—3762no—
636C4—3762no—
637C4—1884no—
638C4—1884no—
639C2—44no—
640C2#2024577111yescharacteristic
641C2#2024588611yescharacteristic
642C2#2024599711yescharacteristic
643C2#2024621811yescharacteristic
644C2#2024621811yescharacteristic
645C2#2024627311yescharacteristic
646C2#2024628111yescharacteristic
647C2#2024747812yes—
648C2#2024753812yes—
649C2#2024756812yes—
650C2#2024759812yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC22×#117×C47C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenterC22×#117D47
hypocenterC47#13943
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. #20241641C2semidirect
4. #20241620C2semidirect
5. #20241618C2semidirect
6. #20241615C2semidirect
7. #20241615C2semidirect
10. C4×C47⋊#95C2semidirect
31. C2×#117×C47C4semidirect
339. C2×#117Dic47semidirect
363. C47#13943semidirect

Maximal subgroups

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

subgroupindex
C2×#156890162
#202416412
#202416372
#202416202
#202416182
#202416152
#202416112
C4×C47⋊#952
C22×#117×C472
#1394347