Finite Group Databasebeta

← #26218680

Subgroups — #26218680

Order 12032.

1218 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689166C211yesmaximal, characteristic, complemented
2C2×#15689124C211yesmaximal, characteristic
3C2×#15689124C211yesmaximal, characteristic
4C2×C4×C4⋊Dic47C211yesmaximal, characteristic, complemented
5#20244026C212yesmaximal, complemented
6#20244019C212yesmaximal, complemented
7#20244012C212yesmaximal, complemented
8#20244005C212yesmaximal, complemented
9#20243109C211yesmaximal, characteristic, complemented
10#20243082C211yesmaximal, characteristic, complemented
11C47×#2297C211yesmaximal, characteristic
12C22×#10625329C2211yescharacteristic
13C22×#10625329C2211yescharacteristic
14C22×C4⋊Dic47C2211yescharacteristic
15C22×C4⋊Dic47C2211yescharacteristic
16C22×C4×Dic47—21no—
17#15689166—24no—
18C2×#13093957C2212yes—
19C2×#13093955C2212yes—
20#15689124—24no—
21#15689124C418yescomplemented
22C2×#13093948C2212yes—
23C2×#13093948C2212yes—
24C2×#13093945C2212yes—
25C2×#13093945C2212yes—
26C4×C4⋊Dic47—21no—
27C4×C4⋊Dic47—21no—
28C4×C4⋊Dic47C2212yes—
29C4×C4⋊Dic47C2212yes—
30C2×C47⋊#95—21no—
31#15688885C2212yes—
32#15688885C2212yes—
33#15688878C2212yes—
34#15688878C2212yes—
35C22×#41×C47C2211yescharacteristic
36C2×#117×C47C2211yescharacteristic
37C22×C42×C47C2211yescharacteristic
38C47×#391C2211yescharacteristic
39C47×#391C2211yescharacteristic
40C47×#390C2211yescharacteristic
41C47×#390C2211yescharacteristic
42C47×#386—21no—
43C47×#386—21no—
44C4×C22⋊C4×C47—21no—
45C4×C22⋊C4×C47—21no—
46C23×Dic47—21no—
47C23×Dic47—21no—
48C2×#10625329—22no—
49C2×#10625329—22no—
50C2×#10625329—22no—
51C2×#10625329—22no—
52C2×#10625329—22no—
53C2×#10625329C2×C414yes—
54C22×C47⋊C8C2312yes—
55C22×C47⋊C8C2312yes—
56C2×C4⋊Dic47—21no—
57C2×C4⋊Dic47—21no—
58C2×C4⋊Dic47—22no—
59C2×C4⋊Dic47—22no—
60C2×C4⋊Dic47C2×C414yes—
61C2×C4⋊Dic47D414yes—
62C2×C4⋊Dic47C2312yes—
63C2×C4⋊Dic47C2312yes—
64C2×C4×Dic47—22no—
65C2×C4×Dic47—22no—
66C2×C4×Dic47—44no—
67#13093957—24no—
68#13093955—24no—
69#13093948—24no—
70#13093948C2×C418yescomplemented
71#13093945—24no—
72#13093945C2×C418yescomplemented
73C47⋊#95—22no—
74C47⋊#95—22no—
75C2×#41×C47—22no—
76C2×#41×C47—22no—
77C2×#41×C47—22no—
78C2×#41×C47—22no—
79C2×#41×C47—22no—
80C2×#41×C47C2×C414yes—
81C22×Q8×C47C2311yescharacteristic
82C22×D4×C47—21no—
83C22×D4×C47C2311yescharacteristic
84C23×C4×C47—21no—
85C23×C4×C47D411yescharacteristic
86C23×C4×C47D411yescharacteristic
87C23×C4×C47C2311yescharacteristic
88#117×C47—22no—
89#117×C47—22no—
90C2×C4⋊C4×C47C2311yescharacteristic
91C2×C4⋊C4×C47C2311yescharacteristic
92C2×C22⋊C4×C47—21no—
93C2×C22⋊C4×C47—21no—
94C2×C22⋊C4×C47—21no—
95C2×C22⋊C4×C47—21no—
96C2×C22⋊C4×C47—21no—
97C2×C42×C47—22no—
98C2×C42×C47—22no—
99C2×C42×C47—21no—
100C2×C42×C47D411yescharacteristic
101C2×C42×C47D411yescharacteristic
102C2×C42×C47C2311yescharacteristic
103C2×C42×C47C2311yescharacteristic
104#95×C47—21no—
105#95×C47—21no—
106#95×C47—21no—
107#95×C47—21no—
108C24×C47—21no—
109C24×C47C2×D411yescharacteristic
110#41×C47—28no—
111#41×C47—28no—
112#41×C47—44no—
113#41×C47—44no—
114C2×Q8×C47—21no—
115C2×Q8×C47—21no—
116C2×Q8×C47—21no—
117C2×Q8×C47—21no—
118C2×Q8×C47—21no—
119C2×Q8×C47C22×C412yes—
120C2×D4×C47—24no—
121C2×D4×C47—21no—
122C2×D4×C47—21no—
123C2×D4×C47—21no—
124C2×D4×C47—21no—
125C2×D4×C47—21no—
126C2×D4×C47—42no—
127C2×D4×C47—42no—
128C2×D4×C47C22×C412yes—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—21no—
134C22×C4×C47—24no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—41no—
138C22×C4×C47—41no—
139C22×C4×C47—42no—
140C22×C4×C47—42no—
141C22×C4×C47C22×C412yes—
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D412yes—
148C22×C4×C47#4112yes—
149C22×C4×C47#4112yes—
150C22×C4×C47#4112yes—
151C22×C4×C47#4112yes—
152C22×C4×C47C2411yescharacteristic
153C22×Dic47—24no—
154C22×Dic47—24no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—42no—
158C22×Dic47—42no—
159C22×Dic47—42no—
160C22×Dic47—42no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C22⋊C4×C47—42no—
166C22⋊C4×C47—42no—
167C22⋊C4×C47—42no—
168C22⋊C4×C47—42no—
169C22⋊C4×C47—42no—
170C42×C47—22no—
171C42×C47—22no—
172C42×C47—21no—
173C42×C47—21no—
174C42×C47—21no—
175C42×C47—21no—
176C42×C47—42no—
177C4⋊Dic47—24no—
178C4⋊Dic47—24no—
179C4⋊Dic47—22no—
180C4⋊Dic47—22no—
181C4⋊Dic47C22×C414yes—
182C4⋊Dic47C2×D414yes—
183C4×Dic47—44no—
184C4×Dic47—44no—
185#10625329—28no—
186#10625329—44no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—22no—
189C2×C47⋊C8—22no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—22no—
192C2×C47⋊C8C22×C414yes—
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47—21no—
197C23×C47—22no—
198C23×C47—41no—
199C23×C47—41no—
200C23×C47—41no—
201C23×C47—41no—
202C23×C47—41no—
203C23×C47—41no—
204C23×C47C4×D412yes—
205C23×C47#12112yes—
206C23×C47#12312yes—
207C23×C47C22×D411yescharacteristic
208Q8×C47—22no—
209Q8×C47—22no—
210Q8×C47—41no—
211Q8×C47—41no—
212D4×C47—22no—
213D4×C47—22no—
214D4×C47—48no—
215D4×C47—41no—
216D4×C47—41no—
217C2×C4×C47—28no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—22no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—44no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—48no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47#11714yes—
240C2×C4×C47C4×D414yes—
241C2×C4×C47C4×D412yes—
242C2×C4×C47#12112yes—
243C2×C4×C47#12312yes—
244C2×C4×C47#12414yes—
245C2×C4×C47#12614yes—
246C2×C4×C47AGL(1,ℤ/8)12yes—
247C2×C4×C47#13712yes—
248C2×C4×C47C23×C411yescharacteristic
249C2×C4×C47C22×D411yescharacteristic
250C2×C4×C47C2×#4111yescharacteristic
251C2×C4×C47C2×#4111yescharacteristic
252C2×C4×C47C2×#4111yescharacteristic
253C2×C4×C47C2×#4111yescharacteristic
254C2×Dic47—24no—
255C2×Dic47—24no—
256C2×Dic47—48no—
257C2×Dic47—48no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C47⋊C8—24no—
263C47⋊C8—42no—
264#21059—471nomaximal, Sylow
265C22×C47—22no—
266C22×C47—22no—
267C22×C47—22no—
268C22×C47—22no—
269C22×C47—22no—
270C22×C47—22no—
271C22×C47—42no—
272C22×C47—42no—
273C22×C47—42no—
274C22×C47—81no—
275C22×C47—81no—
276C22×C47—81no—
277C22×C47—81no—
278C22×C47#38814yes—
279C22×C47C2×C4×D411yescharacteristic
280C22×C47C2×#12111yescharacteristic
281C22×C47C2×#12311yescharacteristic
282C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C47C2×#13711yescharacteristic
284C22×C47#57511yescharacteristic
285C22×C47#57511yescharacteristic
286C4×C47—22no—
287C4×C47—22no—
288C4×C47—41no—
289C4×C47—41no—
290C4×C47—42no—
291C4×C47—81no—
292C4×C47—81no—
293C4×C47#38814yes—
294C4×C47C2×#11711yescharacteristic
295C4×C47C2×C4×D411yescharacteristic
296C4×C47C2×#12411yescharacteristic
297C4×C47C2×#12611yescharacteristic
298Dic47—44no—
299Dic47—44no—
300C2×#428—471no—
301C2×#417—471no—
302C2×#417—471no—
303#2297D4711yescharacteristic
304C2×C4×C4⋊C4—471no—
305#1928—472no—
306#1927—472no—
307#1926—472no—
308#1925—472no—
309#1736—471no—
310#1728—471no—
311C2×C47—24no—
312C2×C47—82no—
313C2×C47C2×#38811yescharacteristic
314C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
315C2×C47C4×#13711yescharacteristic
316C2×C47#303611yescharacteristic
317C2×C47#303711yescharacteristic
318C2×C47#309211yescharacteristic
319C2×C47#309311yescharacteristic
320C22×#41C2×D4711yescharacteristic
321C22×M16—471no—
322C22×M16—471no—
323C2×#117C2×D4711yescharacteristic
324C22×C4⋊C4—471no—
325C22×C4⋊C4—471no—
326C22×C42—941no—
327C22×C42C2×D4711yescharacteristic
328#428—944no—
329C2×#107—472no—
330C2×#106—472no—
331#417—478no—
332#417—944no—
333C2×#103—472no—
334C2×#103—472no—
335C2×#102—472no—
336C2×#102—472no—
337#391C2×D4711yescharacteristic
338#391C2×D4711yescharacteristic
339#390C2×D4711yescharacteristic
340#390C2×D4711yescharacteristic
341#386—21no—
342#386—21no—
343C4×C4⋊C4—472no—
344C4×C4⋊C4—472no—
345C4×C4⋊C4—941no—
346C4×C4⋊C4—941no—
347C4×C22⋊C4—21no—
348C4×C22⋊C4—21no—
349C2×#95—941no—
350#340—472no—
351#340—472no—
352#336—472no—
353#336—472no—
354C47#2105911yescharacteristic, complemented, Sylow
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41—22no—
360C2×#41C4×D4714yes—
361C22×Q8C22×D4711yescharacteristic
362C22×D4—21no—
363C22×D4C22×D4711yescharacteristic
364C23×C4—21no—
365C23×C4—941no—
366C23×C4—941no—
367C23×C4C22⋊D4711yescharacteristic
368C23×C4C22⋊D4711yescharacteristic
369C23×C4C22×D4711yescharacteristic
370C2×M16—474no—
371C2×M16—942no—
372C2×M16—942no—
373C2×M16—942no—
374C2×M16—942no—
375C2×M16—942no—
376C22×C8—472no—
377C22×C8—472no—
378#117—22no—
379#117—22no—
380C2×C4⋊C4—474no—
381C2×C4⋊C4—474no—
382C2×C4⋊C4—472no—
383C2×C4⋊C4—472no—
384C2×C4⋊C4—942no—
385C2×C4⋊C4—942no—
386C2×C4⋊C4—941no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4C22×D4711yescharacteristic
389C2×C4⋊C4C22×D4711yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—21no—
398C2×C42—942no—
399C2×C42—942no—
400C2×C42—1884no—
401C2×C42C22⋊D4711yescharacteristic
402C2×C42C22⋊D4711yescharacteristic
403C2×C42C22×D4711yescharacteristic
404C2×C42C22×D4711yescharacteristic
405#107—944no—
406#106—944no—
407#103—478no—
408#103—944no—
409#102—478no—
410#102—944no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—942no—
416#95—942no—
417C24—21no—
418C24C2×C22⋊D4711yescharacteristic
419#41—28no—
420#41—28no—
421#41—44no—
422#41—44no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8C2×C4×D4712yes—
429C2×D4—24no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4C2×C4×D4712yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—944no—
451C22×C4—944no—
452C22×C4—942no—
453C22×C4—942no—
454C22×C4—1882no—
455C22×C4—1882no—
456C22×C4—1882no—
457C22×C4—1882no—
458C22×C4C2×C4×D4712yes—
459C22×C4#1062535012yes—
460C22×C4D4×D4712yes—
461C22×C4#1062535212yes—
462C22×C4#1062535212yes—
463C22×C4#1062535212yes—
464C22×C4C2×C22⋊D4711yescharacteristic
465C22×C4C2×C22⋊D4711yescharacteristic
466C22×C4C2×C22⋊D4711yescharacteristic
467C22×C4C2×C22⋊D4711yescharacteristic
468C22×C4C2×C22⋊D4711yescharacteristic
469C22×C4C23×D4711yescharacteristic
470M16—948no—
471M16—1884no—
472C2×C8—474no—
473C2×C8—942no—
474C2×C8—942no—
475C2×C8—942no—
476C2×C8—942no—
477C2×C8—942no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—474no—
483C4⋊C4—474no—
484C4⋊C4—944no—
485C4⋊C4—944no—
486C4⋊C4—942no—
487C4⋊C4—942no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C42—22no—
494C42—22no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—42no—
500C42—1884no—
501C42—1884no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23C4×C22⋊D4712yes—
514C23#1309399512yes—
515C23#1309400512yes—
516C23C22×C22⋊D4711yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—28no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—48no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—944no—
549C2×C4—944no—
550C2×C4—1888no—
551C2×C4—1888no—
552C2×C4—1882no—
553C2×C4—1882no—
554C2×C4—1882no—
555C2×C4—1882no—
556C2×C4#1309398014yes—
557C2×C4(C22⋊D47)⋊C414yes—
558C2×C4C4×C22⋊D4712yes—
559C2×C4#1309399512yes—
560C2×C4#1309400512yes—
561C2×C4#1309401014yes—
562C2×C4#1309401714yes—
563C2×C4#1309404912yes—
564C2×C4#1309405412yes—
565C2×C4C22×C4×D4711yescharacteristic
566C2×C4C2×D4×D4711yescharacteristic
567C2×C4C2×#1062535211yescharacteristic
568C2×C4C2×#1062535211yescharacteristic
569C2×C4C2×#1062535211yescharacteristic
570C2×C4C2×#1062535011yescharacteristic
571C8—944no—
572C8—1882no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22#1568901114yes—
587C22C2×C4×C22⋊D4711yescharacteristic
588C22C2×#1309399511yescharacteristic
589C22C2×#1309400511yescharacteristic
590C22C2×#1309404911yescharacteristic
591C22C2×#1309405411yescharacteristic
592C22#1568992111yescharacteristic
593C22#1568992111yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—41no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4—1884no—
602C4—1884no—
603C4#1568901114yes—
604C4C2×#1309398011yescharacteristic
605C4C2×(C22⋊D47)⋊C411yescharacteristic
606C4C2×#1309401011yescharacteristic
607C4C2×#1309401711yescharacteristic
608C2—24no—
609C2—82no—
610C2C2×#1568901111yescharacteristic
611C2C4×#1309404911yescharacteristic
612C2C4×#1309405411yescharacteristic
613C2#2025284811yescharacteristic
614C2#2025285611yescharacteristic
615C2#2025330811yescharacteristic
616C2#2025331711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2297C2
socleC23×C47C22×D4
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenter#2297D47
hypocenterC47#21059
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
1. C2×#15689166C2semidirect
4. C2×C4×C4⋊Dic47C2semidirect
5. #20244026C2semidirect
6. #20244019C2semidirect
7. #20244012C2semidirect
8. #20244005C2semidirect
9. #20243109C2semidirect
10. #20243082C2semidirect
21. #15689124C4semidirect
70. #13093948C2×C4semidirect
72. #13093945C2×C4semidirect
354. C47#21059semidirect

Maximal subgroups

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

subgroupindex
C2×#156891662
C2×#156891242
C2×C4×C4⋊Dic472
#202440262
#202440192
#202440122
#202440052
#202431092
#202430822
C47×#22972
#2105947