Finite Group Databasebeta

← #26214084

Subgroups — #26214084

Order 12032.

1598 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689603C211yesmaximal, characteristic
2C2×#15689122C211yesmaximal, characteristic
3C2×#15689117C211yesmaximal, characteristic
4C2×#15689079C211yesmaximal, characteristic, complemented
5#20243221C212yesmaximal, complemented
6#20243216C212yesmaximal, complemented
7C4×#13093947C212yesmaximal, complemented
8C4×#13093944C212yesmaximal, complemented
9#20243109C211yesmaximal, characteristic, complemented
10C2×#15688885C211yesmaximal, characteristic, complemented
11C47×#2274C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C22×#10625329C2211yescharacteristic
14C23×C47⋊C8C2211yescharacteristic
15#15689603—24no—
16#15689603C418yescomplemented
17C2×C4×Dic94C2212yes—
18C2×C4×D188C2212yes—
19C2×(C22⋊D47)⋊C4—22no—
20C2×#13093982—21no—
21C22×C4⋊Dic47C2211yescharacteristic
22C2×C4⋊C4×D47—21no—
23#15689122C418yescomplemented
24#15689117C418yescomplemented
25C2×#13093947C2211yescharacteristic
26C2×#13093947C2212yes—
27C2×#13093947C2211yescharacteristic
28C2×#13093944C2211yescharacteristic
29C2×#13093944C2212yes—
30C2×#13093944C2211yescharacteristic
31#15689079—24no—
32C2×#13093935C2212yes—
33C2×C4×C47⋊C8C2212yes—
34#15688885C2212yes—
35#15688885C2212yes—
36#15688885C2212yes—
37#15688885C2212yes—
38C2×#117×C47C2211yescharacteristic
39C2×#117×C47C2211yescharacteristic
40C22×C4⋊C4×C47C2211yescharacteristic
41C4×C4⋊C4×C47—21no—
42C4×C4⋊C4×C47—21no—
43C4×C4⋊C4×C47C2212yes—
44C4×C4⋊C4×C47C2212yes—
45C4×C22⋊C4×C47—22no—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350—22no—
49C2×#10625350C2×C414yes—
50C2×#10625350C2×C414yes—
51C2×#10625350C2×C414yes—
52C22×Dic94C2311yescharacteristic
53C22×D188C2311yescharacteristic
54C22×C22⋊D47—21no—
55C23×Dic47—21no—
56C22×C4×D47—21no—
57C22×C4×D47—22no—
58C2×#10625329—22no—
59C2×#10625329—22no—
60C2×#10625329C2×C414yescomplemented
61C22×C47⋊C8—22no—
62C22×C47⋊C8—22no—
63C22×C47⋊C8C2×C414yescomplemented
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2312yes—
66C22×C47⋊C8C2311yescharacteristic
67C4×Dic94—24no—
68C4×Dic94C2×C418yescomplemented
69C4×D188—24no—
70C4×D188C2×C418yescomplemented
71(C22⋊D47)⋊C4—216no—
72#13093982—28no—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47—22no—
75C2×C4⋊Dic47C2×C414yes—
76C2×C4⋊Dic47C2312yes—
77C2×Dic47⋊C4—22no—
78C4⋊C4×D47—28no—
79C2×(C2×D47)⋊C4—22no—
80C2×C4×Dic47—22no—
81#13093947C2×C414yescomplemented
82#13093947C2×C414yescomplemented
83#13093947C2×C418yescomplemented
84#13093944C2×C414yescomplemented
85#13093944C2×C414yescomplemented
86#13093944C2×C418yescomplemented
87#13093935—24no—
88C4×C47⋊C8—24no—
89C23×C4×C47—21no—
90C23×C4×C47C2311yescharacteristic
91#117×C47—22no—
92#117×C47C2×C414yes—
93#117×C47C2×C414yescomplemented
94#117×C47D414yes—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—22no—
97C2×C4⋊C4×C47C2×C414yescomplemented
98C2×C4⋊C4×C47D411yescharacteristic
99C2×C4⋊C4×C47D411yescharacteristic
100C2×C4⋊C4×C47C2312yes—
101C2×C4⋊C4×C47C2311yescharacteristic
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C22⋊C4×C47—21no—
104C2×C22⋊C4×C47D412yes—
105C2×C42×C47—21no—
106C2×C42×C47—22no—
107C2×C42×C47—22no—
108C2×C42×C47C2312yes—
109#95×C47—22no—
110#95×C47—22no—
111C24×C47C2×D411yescharacteristic
112C23×D47—21no—
113C22×C4×C47—21no—
114C22×C4×C47—21no—
115C22×C4×C47—22no—
116C22×C4×C47—24no—
117C22×C4×C47—81no—
118C22×C4×C47—81no—
119C22×C4×C47C22×C412yes—
120C22×C4×C47C22×C412yes—
121C22×C4×C47C22×C412yes—
122C22×C4×C47C2×D412yes—
123C22×C4×C47C2×D412yes—
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D412yes—
126C22×C4×C47#4112yes—
127C22×C4×C47#4112yes—
128C22×C4×C47C2411yescharacteristic
129C2×C22⋊D47—24no—
130C2×C22⋊D47—24no—
131C2×C22⋊D47—24no—
132C22×Dic47—21no—
133C22×Dic47—22no—
134C22×Dic47—24no—
135C22×Dic47—81no—
136C22×Dic47—81no—
137#10625350—28no—
138#10625350—28no—
139#10625350—28no—
140#10625350C42116yescomplemented
141C2×D188—21no—
142C2×D188—21no—
143C2×D188—21no—
144C2×D188C22×C412yes—
145C2×D188C22×C412yes—
146C2×D188C22×C412yes—
147C2×C4×D47—24no—
148C2×C4×D47—24no—
149C2×C4×D47—24no—
150C2×C4×D47—28no—
151C2×C4×D47—82no—
152C2×C4×D47—82no—
153C2×Dic94—21no—
154C2×Dic94—21no—
155C2×Dic94—21no—
156C2×Dic94C22×C412yes—
157C2×Dic94C22×C412yes—
158C2×Dic94C22×C412yes—
159C4⋊C4×C47—24no—
160C4⋊C4×C47—22no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47C22⋊C414yes—
164C4⋊C4×C47C22⋊C414yes—
165C4⋊C4×C47C22×C414yes—
166C4⋊C4×C47C22×C412yes—
167C4⋊C4×C47C22×C412yes—
168C22⋊C4×C47—24no—
169C22⋊C4×C47C22⋊C418yes—
170C42×C47—24no—
171C42×C47—82no—
172C42×C47—82no—
173C42×C47C22×C414yes—
174C42×C47C2×D414yes—
175(C2×D47)⋊C4—28no—
176C4⋊Dic47—24no—
177C4⋊Dic47—22no—
178C4⋊Dic47C22×C414yes—
179Dic47⋊C4—28no—
180C4×Dic47—28no—
181#10625329—28no—
182C2×C47⋊C8—28no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—22no—
189C2×C47⋊C8C22×C412yes—
190C2×C47⋊C8C22×C412yes—
191C2×C47⋊C8C22×C414yes—
192C23×C47—21no—
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47C2×C22⋊C412yes—
197C23×C47C4×D412yes—
198C23×C47C4×D412yes—
199C23×C47C22×D411yescharacteristic
200C22×D47—22no—
201C22×D47—22no—
202C22×D47—22no—
203C22×D47—41no—
204C22×D47—41no—
205C22×D47—41no—
206C22×D47—41no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—24no—
214C2×C4×C47—41no—
215C2×C4×C47—42no—
216C2×C4×C47—41no—
217C2×C4×C47—42no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—84no—
221C2×C4×C47C2×C4214yes—
222C2×C4×C47C2×C22⋊C414yes—
223C2×C4×C47C2×C22⋊C414yes—
224C2×C4×C47C2×C22⋊C412yes—
225C2×C4×C47#11714yes—
226C2×C4×C47C4×D412yes—
227C2×C4×C47C4×D412yes—
228C2×C4×C47C4×D414yes—
229C2×C4×C47C4×D414yes—
230C2×C4×C47C23×C411yescharacteristic
231C2×C4×C47C23×C411yescharacteristic
232C2×C4×C47C23×C411yescharacteristic
233C2×C4×C47C22×D411yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C22⋊D47—216no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—22no—
240C2×Dic47—24no—
241C2×Dic47—42no—
242C2×Dic47—42no—
243C2×Dic47—84no—
244D188—22no—
245D188—22no—
246D188—22no—
247D188C2×C4214yes—
248C4×D47—216no—
249C4×D47—88no—
250Dic94—22no—
251Dic94—22no—
252Dic94—22no—
253Dic94C2×C4214yes—
254C47⋊C8—22no—
255C47⋊C8—22no—
256C47⋊C8—24no—
257#20415—471nomaximal, Sylow
258C22×C47—22no—
259C22×C47—22no—
260C22×C47—22no—
261C22×C47—22no—
262C22×C47—22no—
263C22×C47—22no—
264C22×C47C4×C22⋊C414yes—
265C22×C47C22×C22⋊C411yescharacteristic
266C22×C47C2×C4×D411yescharacteristic
267C22×C47C2×C4×D411yescharacteristic
268C22×C47#57611yescharacteristic
269C22×C47#57711yescharacteristic
270C22×C47#57711yescharacteristic
271C22×C47#57811yescharacteristic
272C2×D47—24no—
273C2×D47—42no—
274C2×D47—42no—
275C2×D47—42no—
276C2×D47—42no—
277C2×D47—42no—
278C2×D47—42no—
279C4×C47—24no—
280C4×C47—24no—
281C4×C47—44no—
282C4×C47—42no—
283C4×C47—42no—
284C4×C47C4×C22⋊C414yes—
285C4×C47C22×C4211yescharacteristic
286C4×C47C2×#11711yescharacteristic
287C4×C47C2×C4×D411yescharacteristic
288C4×C47C2×C4×D411yescharacteristic
289Dic47—24no—
290Dic47—44no—
291C2×#520—471no—
292C2×#417—471no—
293C2×#416—471no—
294C2×#405—471no—
295#2274D4711yescharacteristic, complemented
296#1763—472no—
297#1762—472no—
298C4×#103—472no—
299C4×#102—472no—
300#1736—471no—
301C2×#340—471no—
302C2×C47—24no—
303C2×C47C2×C4×C22⋊C411yescharacteristic
304C2×C47#289711yescharacteristic
305C2×C47#289811yescharacteristic
306C2×C47#294611yescharacteristic
307C2×C47#294711yescharacteristic
308C2×C47#295111yescharacteristic
309C2×C47#295211yescharacteristic
310D47—44no—
311C22×#41—471no—
312C22×M16—471no—
313C23×C8—471no—
314#520—478no—
315#520—944no—
316C2×C4×Q8—472no—
317C2×C4×D4—472no—
318C2×C4×D4—942no—
319C2×#117—941no—
320C2×#117C2×D4711yescharacteristic
321C2×#117C2×D4711yescharacteristic
322C22×C4⋊C4—471no—
323C22×C4⋊C4—941no—
324C22×C4⋊C4C2×D4711yescharacteristic
325#417—478no—
326#416—478no—
327C2×#103—471no—
328C2×#103—471no—
329C2×#103—472no—
330C2×#102—471no—
331C2×#102—471no—
332C2×#102—472no—
333#405—944no—
334C2×#97—472no—
335C2×C4×C8—472no—
336C4×C4⋊C4—21no—
337C4×C4⋊C4—21no—
338C4×C4⋊C4C2×D4712yes—
339C4×C4⋊C4C2×D4712yes—
340C4×C22⋊C4—22no—
341#340—472no—
342#340—472no—
343#340—472no—
344#340—472no—
345C47#2041511yescharacteristic, complemented, Sylow
346C2×#41—474no—
347C2×#41—474no—
348C2×#41—474no—
349C2×#41—942no—
350C2×#41—942no—
351C2×#41—942no—
352C22×Q8—471no—
353C22×D4—471no—
354C22×D4—941no—
355C23×C4—21no—
356C23×C4—941no—
357C23×C4—941no—
358C23×C4—942no—
359C23×C4C22×D4711yescharacteristic
360C2×M16—474no—
361C2×M16—942no—
362C2×M16—942no—
363C22×C8—474no—
364C22×C8—471no—
365C22×C8—471no—
366C22×C8—472no—
367C22×C8—942no—
368C22×C8—942no—
369C4×Q8—478no—
370C4×Q8—944no—
371C4×D4—478no—
372C4×D4—9416no—
373C4×D4—944no—
374#117—22no—
375#117—948no—
376#117C4×D4714yes—
377#117C4×D4714yescomplemented
378#117D18814yes—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—474no—
382C2×C4⋊C4—472no—
383C2×C4⋊C4—942no—
384C2×C4⋊C4—941no—
385C2×C4⋊C4—942no—
386C2×C4⋊C4—948no—
387C2×C4⋊C4C4×D4714yescomplemented
388C2×C4⋊C4C22⋊D4711yescharacteristic
389C2×C4⋊C4C22⋊D4711yescharacteristic
390C2×C4⋊C4C22×D4712yes—
391C2×C4⋊C4C22×D4711yescharacteristic
392C2×C4⋊C4C22×D4711yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—942no—
395C2×C22⋊C4C22⋊D4712yes—
396C2×C42—21no—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42—942no—
400C2×C42C22×D4712yes—
401#103—474no—
402#103—474no—
403#103—478no—
404#102—474no—
405#102—474no—
406#102—478no—
407#97—944no—
408C4×C8—944no—
409#95—22no—
410#95—22no—
411C24—941no—
412C24C2×C22⋊D4711yescharacteristic
413#41—4716no—
414#41—948no—
415#41—948no—
416#41—948no—
417C2×Q8—472no—
418C2×Q8—472no—
419C2×Q8—472no—
420C2×Q8—941no—
421C2×Q8—941no—
422C2×Q8—941no—
423C2×D4—472no—
424C2×D4—472no—
425C2×D4—472no—
426C2×D4—944no—
427C2×D4—944no—
428C2×D4—944no—
429C2×D4—941no—
430C2×D4—941no—
431C2×D4—941no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—941no—
439C22×C4—944no—
440C22×C4—944no—
441C22×C4—944no—
442C22×C4—944no—
443C22×C4—942no—
444C22×C4—948no—
445C22×C4—3762no—
446C22×C4—3762no—
447C22×C4—3761no—
448C22×C4—3761no—
449C22×C4C2×C4×D4712yes—
450C22×C4C2×C4×D4712yes—
451C22×C4C2×C4×D4712yes—
452C22×C4C2×D18812yes—
453C22×C4#1062535012yes—
454C22×C4#1062535012yes—
455C22×C4C2×C22⋊D4712yes—
456C22×C4C2×C22⋊D4712yes—
457C22×C4C2×C22⋊D4711yescharacteristic
458C22×C4C23×D4711yescharacteristic
459M16—948no—
460C2×C8—472no—
461C2×C8—472no—
462C2×C8—474no—
463C2×C8—948no—
464C2×C8—941no—
465C2×C8—942no—
466C2×C8—941no—
467C2×C8—941no—
468C2×C8—941no—
469C2×C8—942no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—474no—
475C4⋊C4—948no—
476C4⋊C4—944no—
477C4⋊C4—942no—
478C4⋊C4(C2×D47)⋊C414yes—
479C4⋊C4(C2×D47)⋊C414yes—
480C4⋊C4C2×C4×D4714yes—
481C4⋊C4C2×C4×D4712yes—
482C4⋊C4C2×C4×D4712yes—
483C22⋊C4—24no—
484C22⋊C4—948no—
485C22⋊C4(C2×D47)⋊C418yes—
486C42—24no—
487C42—82no—
488C42—82no—
489C42—948no—
490C42C2×C4×D4714yes—
491C42C2×D18814yes—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—942no—
497C23—942no—
498C23—942no—
499C23—1881no—
500C23—1881no—
501C23—1881no—
502C23—1881no—
503C23C2×(C2×D47)⋊C412yes—
504C23C4×C22⋊D4712yes—
505C23C4×C22⋊D4712yes—
506C23C22×C22⋊D4711yescharacteristic
507Q8—474no—
508Q8—942no—
509Q8—942no—
510Q8—942no—
511D4—474no—
512D4—9416no—
513D4—942no—
514D4—942no—
515D4—942no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—84no—
530C2×C4—942no—
531C2×C4—942no—
532C2×C4—942no—
533C2×C4—9416no—
534C2×C4—944no—
535C2×C4—3768no—
536C2×C4—3764no—
537C2×C4—1882no—
538C2×C4—1882no—
539C2×C4C42×D4714yes—
540C2×C4C2×(C2×D47)⋊C414yes—
541C2×C4C2×(C2×D47)⋊C414yes—
542C2×C4C2×(C2×D47)⋊C412yes—
543C2×C4#1309398114yes—
544C2×C4C4×C22⋊D4712yes—
545C2×C4C4×C22⋊D4712yes—
546C2×C4C4×D18814yes—
547C2×C4C4×D18814yes—
548C2×C4C22×C4×D4711yescharacteristic
549C2×C4C22×C4×D4711yescharacteristic
550C2×C4C22×C4×D4711yescharacteristic
551C2×C4C22×D18811yescharacteristic
552C2×C4C2×#1062535011yescharacteristic
553C2×C4C2×#1062535011yescharacteristic
554C8—942no—
555C8—942no—
556C8—944no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—944no—
564C22—1882no—
565C22—1882no—
566C22—1882no—
567C22—1882no—
568C22—1882no—
569C22—1882no—
570C22C4×(C2×D47)⋊C414yes—
571C22C22×(C2×D47)⋊C411yescharacteristic
572C22C2×C4×C22⋊D4711yescharacteristic
573C22C2×C4×C22⋊D4711yescharacteristic
574C22#1568992611yescharacteristic
575C22#1568993111yescharacteristic
576C22#1568993411yescharacteristic
577C22#1568993811yescharacteristic
578C4—24no—
579C4—24no—
580C4—44no—
581C4—42no—
582C4—42no—
583C4—944no—
584C4—1884no—
585C4C4×(C2×D47)⋊C414yes—
586C4C2×C42×D4711yescharacteristic
587C4C2×#1309398111yescharacteristic
588C4C2×C4×D18811yescharacteristic
589C4C2×C4×D18811yescharacteristic
590C2—24no—
591C2—1884no—
592C2C2×C4×(C2×D47)⋊C411yescharacteristic
593C2#2025191111yescharacteristic
594C2#2025191811yescharacteristic
595C2#2025222011yescharacteristic
596C2#2025223111yescharacteristic
597C2#2025226111yescharacteristic
598C2#2025226911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC47×#2274C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenter#2274D47
hypocenterC47#20415
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2274

Complements

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

NH ≅ G/Ntype
4. C2×#15689079C2semidirect
5. #20243221C2semidirect
6. #20243216C2semidirect
7. C4×#13093947C2semidirect
8. C4×#13093944C2semidirect
9. #20243109C2semidirect
10. C2×#15688885C2semidirect
11. C47×#2274C2semidirect
16. #15689603C4semidirect
23. #15689122C4semidirect
24. #15689117C4semidirect
60. C2×#10625329C2×C4semidirect
63. C22×C47⋊C8C2×C4semidirect
68. C4×Dic94C2×C4semidirect
70. C4×D188C2×C4semidirect
81. #13093947C2×C4semidirect
82. #13093947C2×C4semidirect
83. #13093947C2×C4semidirect
84. #13093944C2×C4semidirect
85. #13093944C2×C4semidirect
86. #13093944C2×C4semidirect
93. #117×C47C2×C4semidirect
97. C2×C4⋊C4×C47C2×C4semidirect
140. #10625350C42semidirect
295. #2274D47semidirect
345. C47#20415semidirect
377. #117C4×D47semidirect
387. C2×C4⋊C4C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896032
C2×#156891222
C2×#156891172
C2×#156890792
#202432212
#202432162
C4×#130939472
C4×#130939442
#202431092
C2×#156888852
C47×#22742
#2041547