Finite Group Databasebeta

← #26139037

Subgroups — #26139037

Order 12032.

1862 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688980C211yesmaximal, characteristic
2C2×#15688970C211yesmaximal, characteristic
3C2×#95×D47C211yesmaximal, characteristic
4#20241647C211yesmaximal, characteristic, complemented
5#20241645C211yesmaximal, characteristic, complemented
6#20241644C211yesmaximal, characteristic, complemented
7#20241613C212yesmaximal, complemented
8#20241612C212yesmaximal, complemented
9(C2×D47)⋊#95C212yesmaximal
10(C2×D47)⋊#95C212yesmaximal
11C47×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D47C2211yescharacteristic
13C23×C4×D47C2211yescharacteristic
14C23×C4×D47C2211yescharacteristic
15C22×C4⋊Dic47C2211yescharacteristic
16C22×(C2×D47)⋊C4C2212yes—
17C22×(C2×D47)⋊C4C2212yes—
18C22×C4×Dic47C2212yes—
19#15688980—24no—
20#15688970C418yescomplemented
21C2×C47⋊#95C2212yes—
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2211yescharacteristic
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2211yescharacteristic
27C2×C47⋊#95C2211yescharacteristic
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2211yescharacteristic
32#95×D47—24no—
33C22×C4⋊C4×C47C2211yescharacteristic
34C2×#95×C47C2212yes—
35C2×#95×C47C2211yescharacteristic
36C2×#95×C47C2212yes—
37C2×#95×C47C2211yescharacteristic
38C24×D47C2311yescharacteristic
39C23×Dic47C2312yes—
40C23×Dic47C2312yes—
41C23×Dic47C2311yescharacteristic
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2311yescharacteristic
44C22×C4×D47—22no—
45C22×C4×D47—22no—
46C22×C4×D47—22no—
47C22×C4×D47—22no—
48C22×C4×D47—22no—
49C22×C4×D47—22no—
50C22×C4×D47—24no—
51C22×C4×D47—24no—
52C22×C4×D47—42no—
53C22×C4×D47—42no—
54C22×C4×D47—42no—
55C22×C4×D47—42no—
56C22×C4×D47C2×C414yes—
57C22×C4×D47D414yes—
58C22×C4×D47Q814yes—
59C2×C4⋊Dic47—22no—
60C2×C4⋊Dic47—42no—
61C2×(C2×D47)⋊C4—24no—
62C2×(C2×D47)⋊C4—24no—
63C2×(C2×D47)⋊C4—24no—
64C2×(C2×D47)⋊C4—24no—
65C2×(C2×D47)⋊C4—24no—
66C2×(C2×D47)⋊C4C2×C418yes—
67C2×C4×Dic47—42no—
68C2×C4×Dic47—42no—
69C2×C4×Dic47C2×C418yescomplemented
70C47⋊#95—22no—
71C47⋊#95—24no—
72C47⋊#95—22no—
73C47⋊#95—22no—
74C47⋊#95—24no—
75C47⋊#95—24no—
76C47⋊#95—24no—
77C47⋊#95—42no—
78C47⋊#95—42no—
79C47⋊#95—42no—
80C47⋊#95C2×C414yescomplemented
81C23×C4×C47C2312yes—
82C23×C4×C47C2312yes—
83C23×C4×C47C2311yescharacteristic
84C23×C4×C47C2311yescharacteristic
85C23×C4×C47C2311yescharacteristic
86C2×C4⋊C4×C47—22no—
87C2×C4⋊C4×C47—42no—
88#95×C47—22no—
89#95×C47—24no—
90#95×C47—42no—
91#95×C47C2×C414yescomplemented
92C24×C47C2411yescharacteristic
93C23×D47—21no—
94C23×D47—21no—
95C23×D47—21no—
96C23×D47—21no—
97C23×D47—22no—
98C23×D47—22no—
99C23×D47C22×C412yes—
100C23×D47C2×D412yes—
101C23×D47C2×Q812yes—
102C23×D47#4114yes—
103C23×D47#4114yes—
104C22×C4×C47—22no—
105C22×C4×C47—22no—
106C22×C4×C47—21no—
107C22×C4×C47—21no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—21no—
112C22×C4×C47—21no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47C22×C414yes—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C2×D412yes—
130C22×C4×C47C2×Q812yes—
131C22×C4×C47#4112yes—
132C22×C4×C47#4112yes—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—21no—
136C22×Dic47—21no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—21no—
142C22×Dic47—21no—
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×Dic47C22×C414yes—
157C22×Dic47C22×C412yes—
158C22×Dic47C2×D412yes—
159C22×Dic47C2×Q812yes—
160C22×Dic47#4112yes—
161C22×Dic47#4112yes—
162C2×C4×D47—28no—
163C2×C4×D47—28no—
164C2×C4×D47—48no—
165C2×C4×D47—44no—
166C2×C4×D47—44no—
167C2×C4×D47—48no—
168C2×C4×D47—48no—
169C2×C4×D47—48no—
170C2×C4×D47—48no—
171C2×C4×D47—48no—
172C2×C4×D47—44no—
173C2×C4×D47—44no—
174C2×C4×D47—44no—
175C2×C4×D47—44no—
176C2×C4×D47C4⋊C4116yes—
177C4⋊C4×C47—44no—
178(C2×D47)⋊C4—216no—
179(C2×D47)⋊C4—48no—
180C4⋊Dic47—44no—
181C4×Dic47—48no—
182C23×C47C23×C411yescharacteristic
183C23×C47C22×D411yescharacteristic
184C23×C47C22×Q811yescharacteristic
185C23×C47C2×#4112yes—
186C23×C47C2×#4112yes—
187C23×C47C2×#4111yescharacteristic
188C23×C47C2×#4111yescharacteristic
189C23×C47C2×#4111yescharacteristic
190C23×C47C2×#4111yescharacteristic
191C23×C47H(5,𝔽2)12yes—
192C23×C47H(5,𝔽2)12yes—
193C22×D47—22no—
194C22×D47—22no—
195C22×D47—24no—
196C22×D47—22no—
197C22×D47—22no—
198C22×D47—22no—
199C22×D47—22no—
200C22×D47—22no—
201C22×D47—22no—
202C22×D47—24no—
203C22×D47—24no—
204C22×D47—24no—
205C22×D47—24no—
206C22×D47—24no—
207C22×D47—24no—
208C22×D47—24no—
209C22×D47—24no—
210C22×D47—24no—
211C22×D47C2×C4⋊C414yes—
212C22×D47#11718yes—
213C22×D47#12414yes—
214C22×D47#12414yes—
215C22×D47#12518yes—
216C2×C4×C47—24no—
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—22no—
224C2×C4×C47—22no—
225C2×C4×C47—24no—
226C2×C4×C47—24no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47—42no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47—42no—
238C2×C4×C47—42no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47—42no—
243C2×C4×C47—42no—
244C2×C4×C47C2×C4⋊C414yes—
245C2×Dic47—24no—
246C2×Dic47—22no—
247C2×Dic47—22no—
248C2×Dic47—24no—
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—24no—
252C2×Dic47—22no—
253C2×Dic47—22no—
254C2×Dic47—24no—
255C2×Dic47—24no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
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—
269C2×Dic47—42no—
270C2×Dic47—42no—
271C2×Dic47—42no—
272C2×Dic47—42no—
273C2×Dic47C2×C4⋊C414yes—
274C4×D47—416no—
275C4×D47—416no—
276C4×D47—416no—
277#13966—471nomaximal, Sylow
278C22×C47C22×C4⋊C411yescharacteristic
279C22×C47C2×#11712yes—
280C22×C47C4×#4111yescharacteristic
281C22×C47C4×#4111yescharacteristic
282C22×C47#51812yes—
283C22×C47#52511yescharacteristic
284C22×C47#52511yescharacteristic
285C22×C47C2×#12411yescharacteristic
286C22×C47C2×#12411yescharacteristic
287C22×C47C2×#12512yes—
288C22×C47#53311yescharacteristic
289C22×C47#53311yescharacteristic
290C22×C47#53411yescharacteristic
291C22×C47#53411yescharacteristic
292C22×C47#54312yes—
293C22×C47#55112yes—
294C22×C47#55112yes—
295C22×C47#55112yes—
296C22×C47#55112yes—
297C22×C47#55312yes—
298C22×C47#55312yes—
299C22×C47#55312yes—
300C22×C47#55312yes—
301C2×D47—24no—
302C2×D47—24no—
303C2×D47—28no—
304C2×D47—28no—
305C2×D47—24no—
306C2×D47—24no—
307C2×D47—24no—
308C2×D47—28no—
309C2×D47—28no—
310C2×D47—24no—
311C2×D47#38218yes—
312C4×C47—24no—
313C4×C47—42no—
314C4×C47—42no—
315C4×C47—44no—
316C4×C47—82no—
317Dic47—24no—
318Dic47—42no—
319Dic47—42no—
320Dic47—44no—
321Dic47—82no—
322C2×#380—471no—
323C2×C4×C22⋊C4—471no—
324C22×#95—471no—
325#1442—471no—
326#1442—471no—
327#1442—471no—
328#1442D4711yescharacteristic, complemented
329#1434—472no—
330#1434—472no—
331C22⋊#95—472no—
332C22⋊#95—472no—
333C2×C47C2×#38211yescharacteristic
334C2×C47#229611yescharacteristic
335C2×C47#231511yescharacteristic
336C2×C47#235312yes—
337C2×C47#235312yes—
338C2×C47#256111yescharacteristic
339C2×C47#256111yescharacteristic
340C2×C47#256912yes—
341C2×C47#256912yes—
342C2×C47#259111yescharacteristic
343C2×C47#259111yescharacteristic
344D47—28no—
345C24×C4—471no—
346C24×C4—471no—
347C24×C4—471no—
348C22×C4⋊C4—471no—
349C22×C4⋊C4C2×D4711yescharacteristic
350C22×C22⋊C4—472no—
351C22×C22⋊C4—472no—
352C22×C42—472no—
353#380—944no—
354C4×C22⋊C4—478no—
355C2×#95—471no—
356C2×#95—472no—
357C2×#95—472no—
358C2×#95—472no—
359C2×#95—471no—
360C2×#95—471no—
361C2×#95—472no—
362C2×#95—471no—
363C2×#95—472no—
364C2×#95—472no—
365C2×#95—472no—
366C2×#95—944no—
367C2×#95C2×D4712yes—
368C2×#95C2×D4711yescharacteristic
369C2×#95C2×D4712yes—
370C2×#95C2×D4711yescharacteristic
371C47#1396611yescharacteristic, complemented, Sylow
372C25—471no—
373C23×C4—471no—
374C23×C4—474no—
375C23×C4—474no—
376C23×C4—474no—
377C23×C4—472no—
378C23×C4—471no—
379C23×C4—471no—
380C23×C4—472no—
381C23×C4—942no—
382C23×C4—942no—
383C23×C4—942no—
384C23×C4—942no—
385C23×C4—944no—
386C23×C4—944no—
387C23×C4—942no—
388C23×C4—942no—
389C23×C4—1882no—
390C23×C4—1882no—
391C23×C4—1882no—
392C23×C4—1882no—
393C23×C4C22×D4712yes—
394C23×C4C22×D4712yes—
395C23×C4C22×D4711yescharacteristic
396C23×C4C22×D4711yescharacteristic
397C23×C4C22×D4711yescharacteristic
398C2×C4⋊C4—22no—
399C2×C4⋊C4—42no—
400C2×C4⋊C4—942no—
401C2×C4⋊C4—1882no—
402C2×C22⋊C4—478no—
403C2×C22⋊C4—944no—
404C2×C22⋊C4—944no—
405C2×C22⋊C4—944no—
406C2×C22⋊C4—944no—
407C2×C22⋊C4—944no—
408C2×C42—478no—
409C2×C42—1882no—
410C2×C42—1882no—
411#95—22no—
412#95—24no—
413#95—42no—
414#95—474no—
415#95—944no—
416#95—944no—
417#95—942no—
418#95—942no—
419#95—944no—
420#95—942no—
421#95—944no—
422#95—1882no—
423#95—1882no—
424#95—1882no—
425#95C4×D4714yescomplemented
426C24—472no—
427C24—472no—
428C24—472no—
429C24—474no—
430C24—474no—
431C24—941no—
432C24—941no—
433C24—941no—
434C24—941no—
435C24—942no—
436C24—942no—
437C24C23×D4711yescharacteristic
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—472no—
462C22×C4—472no—
463C22×C4—472no—
464C22×C4—4716no—
465C22×C4—474no—
466C22×C4—472no—
467C22×C4—472no—
468C22×C4—948no—
469C22×C4—948no—
470C22×C4—942no—
471C22×C4—942no—
472C22×C4—941no—
473C22×C4—941no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4—942no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4—941no—
480C22×C4—942no—
481C22×C4—941no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4—942no—
487C22×C4—942no—
488C22×C4—942no—
489C22×C4—942no—
490C22×C4—942no—
491C22×C4—942no—
492C22×C4—942no—
493C22×C4—1888no—
494C22×C4—1888no—
495C22×C4—1884no—
496C22×C4—1884no—
497C22×C4—1888no—
498C22×C4—1888no—
499C22×C4—1888no—
500C22×C4—1888no—
501C22×C4—1884no—
502C22×C4—1884no—
503C22×C4—1884no—
504C22×C4—1884no—
505C22×C4C2×C4×D4714yes—
506C22×C4C2×C4×D4712yes—
507C22×C4D4×D4712yes—
508C22×C4#1062535212yes—
509C22×C4Q8×D4712yes—
510C22×C4#1062535412yes—
511C4⋊C4—44no—
512C4⋊C4—1884no—
513C22⋊C4—9416no—
514C22⋊C4—1888no—
515C42—1888no—
516C23—474no—
517C23—478no—
518C23—474no—
519C23—474no—
520C23—478no—
521C23—944no—
522C23—944no—
523C23—944no—
524C23—944no—
525C23—944no—
526C23—944no—
527C23—944no—
528C23—944no—
529C23—944no—
530C23—942no—
531C23—942no—
532C23—944no—
533C23—942no—
534C23—942no—
535C23—942no—
536C23—942no—
537C23—942no—
538C23—942no—
539C23C22×C4×D4711yescharacteristic
540C23C2×D4×D4711yescharacteristic
541C23C2×Q8×D4711yescharacteristic
542C23#41×D4712yes—
543C23#41×D4711yescharacteristic
544C23#41×D4711yescharacteristic
545C23#41×D4712yes—
546C23C2×#1062535211yescharacteristic
547C23C2×#1062535411yescharacteristic
548C23#1309406912yes—
549C23#1309407012yes—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—474no—
579C2×C4—944no—
580C2×C4—944no—
581C2×C4—942no—
582C2×C4—942no—
583C2×C4—944no—
584C2×C4—944no—
585C2×C4—944no—
586C2×C4—944no—
587C2×C4—944no—
588C2×C4—942no—
589C2×C4—942no—
590C2×C4—18816no—
591C2×C4—18816no—
592C2×C4—18816no—
593C2×C4—1882no—
594C2×C4—1882no—
595C2×C4—1882no—
596C2×C4—1882no—
597C2×C4—1882no—
598C2×C4—1882no—
599C2×C4—1882no—
600C2×C4—1882no—
601C2×C4—1882no—
602C2×C4—1882no—
603C2×C4—1882no—
604C2×C4—1882no—
605C2×C4—1882no—
606C2×C4—1881no—
607C2×C4—1881no—
608C2×C4—1881no—
609C2×C4—1881no—
610C2×C4C4⋊C4×D4714yes—
611C22—478no—
612C22—948no—
613C22—944no—
614C22—948no—
615C22—944no—
616C22—948no—
617C22—948no—
618C22—944no—
619C22—944no—
620C22—944no—
621C22—944no—
622C22C2×C4⋊C4×D4711yescharacteristic
623C22#117×D4712yes—
624C22C4×#1062535211yescharacteristic
625C22C4×#1062535411yescharacteristic
626C22#1568959512yes—
627C22#1568963311yescharacteristic
628C22#1568963611yescharacteristic
629C22#124×D4711yescharacteristic
630C22#124×D4711yescharacteristic
631C22#125×D4712yes—
632C22#1568967811yescharacteristic
633C22#1568967911yescharacteristic
634C22#1568968311yescharacteristic
635C22#1568968411yescharacteristic
636C22#1568974812yes—
637C22#1568979612yes—
638C22#1568979812yes—
639C22#1568980012yes—
640C22#1568980312yes—
641C22#1568981812yes—
642C22#1568982012yes—
643C22#1568982412yes—
644C22#1568982612yes—
645C4—24no—
646C4—44no—
647C4—42no—
648C4—42no—
649C4—82no—
650C4—944no—
651C4—3762no—
652C4—1884no—
653C4—1882no—
654C4—1882no—
655C2—948no—
656C2#382×D4711yescharacteristic
657C2#2024586911yescharacteristic
658C2#2024599911yescharacteristic
659C2#2024632312yes—
660C2#2024632712yes—
661C2#2024854011yescharacteristic
662C2#2024854211yescharacteristic
663C2#2024860612yes—
664C2#2024860812yes—
665C2#2024886211yescharacteristic
666C2#2024886411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1442C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenter#1442D47
hypocenterC47#13966
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
4. #20241647C2semidirect
5. #20241645C2semidirect
6. #20241644C2semidirect
7. #20241613C2semidirect
8. #20241612C2semidirect
11. C47×#1442C2semidirect
20. #15688970C4semidirect
69. C2×C4×Dic47C2×C4semidirect
80. C47⋊#95C2×C4semidirect
91. #95×C47C2×C4semidirect
328. #1442D47semidirect
371. C47#13966semidirect
425. #95C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156889802
C2×#156889702
C2×#95×D472
#202416472
#202416452
#202416442
#202416132
#202416122
(C2×D47)⋊#952
C47×#14422
#1396647