Finite Group Databasebeta

← #26224052

Subgroups — #26224052

Order 12032.

876 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689411C211yesmaximal, characteristic
2#20246132C211yesmaximal, characteristic, complemented
3#20245749C211yesmaximal, characteristic
4#20244338C211yesmaximal, characteristic, complemented
5#20244159C211yesmaximal, characteristic, complemented
6#20243976C211yesmaximal, characteristic, complemented
7#20243965C211yesmaximal, characteristic, complemented
8#20243647C211yesmaximal, characteristic
9#20243637C211yesmaximal, characteristic
10#20243396C211yesmaximal, characteristic
11#20243285C211yesmaximal, characteristic, complemented
12C4×#13093955C211yesmaximal, characteristic, complemented
13#20243248C211yesmaximal, characteristic, complemented
14C4×#13093945C211yesmaximal, characteristic
15C47×#1762C211yesmaximal, characteristic
16C2×#13094023C2211yescharacteristic
17C2×D4×Dic47C2211yescharacteristic
18C2×D4×Dic47C2211yescharacteristic
19#15689411C418yescomplemented
20C2×#13093958C2211yescharacteristic
21C2×#13093956C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×Dic47⋊C8C2211yescharacteristic
25C2×Dic47⋊C8C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×#13093945C2211yescharacteristic
28C2×C4×C47⋊C8C2211yescharacteristic
29#15689029—21no—
30#15689027—21no—
31#15689022C2211yescharacteristic
32#15689019C2211yescharacteristic
33#15689010—21no—
34#15689007—21no—
35#15688988C2211yescharacteristic
36C4⋊C4×Dic47C2211yescharacteristic
37C4⋊C4×Dic47C2211yescharacteristic
38C4×C4⋊Dic47C2211yescharacteristic
39C4×C4⋊Dic47C2211yescharacteristic
40C22⋊C4×Dic47—21no—
41C22⋊C4×Dic47—21no—
42C4×C22⋊Dic47—21no—
43#15688969C2211yescharacteristic
44C2×C47⋊#95—21no—
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C2×#102×C47C2211yescharacteristic
55C2×#102×C47C2211yescharacteristic
56C2×#97×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×Dic94—21no—
61C23×Dic47—21no—
62C23×Dic47—21no—
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65#13094023—22no—
66#13094023C2×C414yescomplemented
67D4×Dic47—22no—
68D4×Dic47—22no—
69D4×Dic47—22no—
70D4×Dic47C2×C414yes—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×C22⋊Dic47—21no—
79C2×C22⋊Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47C2311yescharacteristic
85C2×C4×Dic47C2311yescharacteristic
86#13093958—22no—
87#13093956C2×C414yescomplemented
88#13093955—22no—
89#13093955C2×C414yescomplemented
90Dic47⋊C8—22no—
91Dic47⋊C8C2×C414yescomplemented
92#13093945—22no—
93#13093945C2×C414yes—
94C4×C47⋊C8—22no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95D411yescharacteristic
104C47⋊#95D411yescharacteristic
105C47⋊#95D411yescharacteristic
106C47⋊#95D411yescharacteristic
107C22×D4×C47C2311yescharacteristic
108C23×C4×C47—21no—
109C22×C8×C47C2311yescharacteristic
110C22×C8×C47C2311yescharacteristic
111C4×D4×C47—22no—
112C4×D4×C47Q814yescomplemented
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C22⋊C4×C47—21no—
117C2×C42×C47—21no—
118C2×C42×C47C2311yescharacteristic
119#102×C47—22no—
120#102×C47C2×C414yes—
121#97×C47—22no—
122#95×C47—21no—
123C24×C47—21no—
124C2×D4×C47—21no—
125C2×D4×C47—21no—
126C2×D4×C47—21no—
127C2×D4×C47C22×C412yes—
128C2×D4×C47C2×Q812yes—
129C2×D4×C47#4112yes—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—24no—
133C22×C4×C47—21no—
134C22×C4×C47—81no—
135C22×C4×C47—81no—
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2411yescharacteristic
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—24no—
143C22×Dic47—24no—
144C22×Dic47—21no—
145C22×Dic47—81no—
146C22×Dic47—81no—
147C22×Dic47—81no—
148C22×Dic47—81no—
149C22×Dic47C2×D411yescharacteristic
150C22×Dic47C2×D411yescharacteristic
151C22×Dic47C2×D411yescharacteristic
152C22×Dic47C2×D411yescharacteristic
153C2×Dic94—24no—
154C2×Dic94—81no—
155C2×Dic94—81no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47C22×C412yes—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47C22×C412yes—
167C4⋊C4×C47C2×Q812yes—
168C22⋊C4×C47—24no—
169C42×C47—81no—
170C42×C47C2×Q812yes—
171C42×C47#4112yes—
172C22⋊Dic47—24no—
173C22⋊Dic47—24no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—81no—
180C4⋊Dic47C22×C412yes—
181C4⋊Dic47C22×C412yes—
182C4⋊Dic47#4112yes—
183Dic47⋊C4—24no—
184Dic47⋊C4—81no—
185C4×Dic47—21no—
186C4×Dic47—21no—
187C4×Dic47—81no—
188C4×Dic47—81no—
189C4×Dic47—81no—
190C4×Dic47—81no—
191C4×Dic47C22×C412yes—
192C4×Dic47#4112yes—
193C2×C47⋊C8—21no—
194C2×C47⋊C8—21no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8C22×C412yes—
199C23×C47—22no—
200C23×C47—22no—
201C23×C47—22no—
202C23×C47—41no—
203C23×C47—41no—
204C23×C47—41no—
205C23×C47—41no—
206C23×C47C22×D411yescharacteristic
207D4×C47—22no—
208D4×C47—22no—
209D4×C47—22no—
210D4×C47C4×Q814yes—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—41no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—84no—
225C2×C4×C47C22⋊Q812yes—
226C2×C4×C47C22⋊Q812yes—
227C2×C4×C47C23×C411yescharacteristic
228C2×C4×C47C22×Q811yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—22no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—84no—
256C2×Dic47—84no—
257C2×Dic47C4×D412yes—
258C2×Dic47C4×D412yes—
259Dic94—84no—
260C8×C47—22no—
261C8×C47—41no—
262C47⋊C8—22no—
263C47⋊C8—41no—
264#21623—471nomaximal, Sylow
265C22×C47—24no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—42no—
269C22×C47—42no—
270C22×C47—42no—
271C22×C47—42no—
272C22×C47C2×C4×D411yescharacteristic
273C22×C47C2×C22⋊Q811yescharacteristic
274C22×C47#52511yescharacteristic
275C22×C47#57611yescharacteristic
276C22×C47#57711yescharacteristic
277C22×C47#57711yescharacteristic
278C22×C47#57711yescharacteristic
279C4×C47—22no—
280C4×C47—22no—
281C4×C47—42no—
282C4×C47—42no—
283C4×C47—81no—
284C4×C47C2×C4×Q811yescharacteristic
285C4×C47C4×#4111yescharacteristic
286C4×C47#52911yescharacteristic
287C4×C47#53311yescharacteristic
288Dic47—22no—
289Dic47—22no—
290Dic47—41no—
291Dic47—42no—
292Dic47—41no—
293Dic47—42no—
294Dic47—42no—
295Dic47—81no—
296C2×#476—471no—
297#2331—471no—
298#2275—471no—
299#1988—471no—
300#1949—471no—
301#1920—471no—
302#1919—471no—
303#1864—471no—
304#1862—471no—
305#1804—471no—
306#1776—471no—
307C4×#106—471no—
308#1768—471no—
309#1762D4711yescharacteristic
310C4×#102—471no—
311C2×C47—44no—
312C2×C47C4×C22⋊Q811yescharacteristic
313C2×C47#294611yescharacteristic
314C2×C47#294911yescharacteristic
315C2×C47#308111yescharacteristic
316C2×C47#308411yescharacteristic
317C2×C47#311811yescharacteristic
318C2×C47#312511yescharacteristic
319C2×C4⋊Q8—471no—
320C2×C4×D4—471no—
321C2×C4×D4—471no—
322C2×C4×D4C2×D4711yescharacteristic
323#476—478no—
324C2×#107—471no—
325C2×#106—471no—
326C2×#106—471no—
327C2×#106—471no—
328C2×C4⋊C8—471no—
329C2×C4⋊C8—471no—
330C2×#102—471no—
331C2×#102—471no—
332C2×#102C2×D4711yescharacteristic
333C2×#102C2×D4711yescharacteristic
334C2×#97C2×D4711yescharacteristic
335C2×C4×C8—471no—
336#391—941no—
337#391—941no—
338#389—471no—
339#389—471no—
340#387—941no—
341#387—941no—
342#382—471no—
343C4×C4⋊C4—471no—
344C4×C4⋊C4—471no—
345C4×C4⋊C4—471no—
346C4×C4⋊C4—471no—
347C4×C4⋊C4C2×D4711yescharacteristic
348C4×C22⋊C4—941no—
349C4×C22⋊C4—941no—
350C4×C22⋊C4—941no—
351#376—471no—
352C2×#95—941no—
353#340—471no—
354#340—471no—
355#340—471no—
356#340—471no—
357#340—471no—
358#340—471no—
359#340C2×D4711yescharacteristic
360#340C2×D4711yescharacteristic
361#336—471no—
362#336—471no—
363C47#2162311yescharacteristic, complemented, Sylow
364C22×Q8—941no—
365C22×D4C22×D4711yescharacteristic
366C23×C4—21no—
367C23×C4—941no—
368C23×C4—941no—
369C22×C8—471no—
370C22×C8—471no—
371C22×C8C22×D4711yescharacteristic
372C22×C8C22×D4711yescharacteristic
373C4⋊Q8—474no—
374C4⋊Q8—942no—
375C4×D4—22no—
376C4×D4—474no—
377C4×D4—942no—
378C4×D4—942no—
379C4×D4—942no—
380C4×D4Dic9414yescomplemented
381C2×C4⋊C4—471no—
382C2×C4⋊C4—471no—
383C2×C4⋊C4—471no—
384C2×C4⋊C4—471no—
385C2×C4⋊C4—941no—
386C2×C4⋊C4—941no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4C22×D4711yescharacteristic
389C2×C4⋊C4C22×D4711yescharacteristic
390C2×C4⋊C4C22×D4711yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—941no—
393C2×C22⋊C4—941no—
394C2×C42—21no—
395C2×C42—471no—
396C2×C42—471no—
397C2×C42—941no—
398C2×C42—941no—
399C2×C42—941no—
400C2×C42—941no—
401C2×C42C22×D4711yescharacteristic
402#107—942no—
403#106—474no—
404#106—474no—
405#106—942no—
406C4⋊C8—474no—
407C4⋊C8—942no—
408#102—22no—
409#102—474no—
410#102—942no—
411#102C4×D4714yes—
412#97—22no—
413C4×C8—942no—
414#95—21no—
415#95—471no—
416#95—471no—
417#95—471no—
418#95—471no—
419#95—941no—
420#95—941no—
421#95—941no—
422#95—941no—
423#95—941no—
424#95—941no—
425#95—941no—
426#95—941no—
427C24—21no—
428C2×Q8—944no—
429C2×Q8—3761no—
430C2×Q8—3761no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C2×Dic9412yes—
435C2×D4C2×C4×D4712yes—
436C2×D4#1062535012yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—471no—
444C22×C4—471no—
445C22×C4—471no—
446C22×C4—471no—
447C22×C4—944no—
448C22×C4—944no—
449C22×C4—941no—
450C22×C4—941no—
451C22×C4—941no—
452C22×C4—941no—
453C22×C4—3761no—
454C22×C4—3761no—
455C22×C4—3761no—
456C22×C4—3761no—
457C22×C4D4×D4711yescharacteristic
458C22×C4D4×D4711yescharacteristic
459C22×C4C23×D4711yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—472no—
466C2×C8—941no—
467C2×C8—941no—
468C2×C8—941no—
469C2×C8—941no—
470C2×C8—941no—
471C2×C8C2×C4×D4712yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—472no—
477C4⋊C4—472no—
478C4⋊C4—472no—
479C4⋊C4—944no—
480C4⋊C4—941no—
481C4⋊C4—941no—
482C4⋊C4—941no—
483C4⋊C4—941no—
484C4⋊C4—941no—
485C4⋊C4—3761no—
486C4⋊C4—3761no—
487C4⋊C4C2×Dic9412yes—
488C4⋊C4C2×C4×D4712yes—
489C22⋊C4—24no—
490C22⋊C4—944no—
491C22⋊C4—944no—
492C42—81no—
493C42—472no—
494C42—472no—
495C42—941no—
496C42—941no—
497C42—3761no—
498C42—3761no—
499C42—3761no—
500C42—3761no—
501C42C2×Dic9412yes—
502C42#1062535212yes—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C2×D4×D4711yescharacteristic
511Q8—3764no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4C4×Dic9414yes—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—84no—
530C2×C4—472no—
531C2×C4—472no—
532C2×C4—941no—
533C2×C4—941no—
534C2×C4—941no—
535C2×C4—941no—
536C2×C4—942no—
537C2×C4—942no—
538C2×C4—941no—
539C2×C4—941no—
540C2×C4—942no—
541C2×C4—941no—
542C2×C4—941no—
543C2×C4—941no—
544C2×C4—941no—
545C2×C4—3764no—
546C2×C4—3764no—
547C2×C4—1881no—
548C2×C4—1881no—
549C2×C4—1881no—
550C2×C4—1881no—
551C2×C4—1881no—
552C2×C4—1881no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4—1881no—
556C2×C4#1309400112yes—
557C2×C4#1309400112yes—
558C2×C4C22×C4×D4711yescharacteristic
559C2×C4C22×Dic9411yescharacteristic
560C2×C4#41×D4711yescharacteristic
561C2×C4#41×D4711yescharacteristic
562C2×C4C2×#1062535211yescharacteristic
563C2×C4C2×#1062535011yescharacteristic
564C8—22no—
565C8—41no—
566C8—942no—
567C8—1881no—
568C22—24no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22C4×D4×D4711yescharacteristic
576C22C2×#1309400111yescharacteristic
577C22#1568963211yescharacteristic
578C22#1568992711yescharacteristic
579C22#1568993211yescharacteristic
580C22#1568993311yescharacteristic
581C22#1568993311yescharacteristic
582C4—22no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—81no—
587C4—942no—
588C4—942no—
589C4—3761no—
590C4—1882no—
591C4—1882no—
592C4—1881no—
593C4—1882no—
594C4—1881no—
595C4C2×C4×Dic9411yescharacteristic
596C4C4×#1062535211yescharacteristic
597C4#1568965311yescharacteristic
598C4#1568967611yescharacteristic
599C2—44no—
600C2C4×#1309400111yescharacteristic
601C2#2025221711yescharacteristic
602C2#2025224411yescharacteristic
603C2#2025322111yescharacteristic
604C2#2025325411yescharacteristic
605C2#2025357611yescharacteristic
606C2#2025366911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1762C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#1762D47
hypocenterC47#21623
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. #20246132C2semidirect
4. #20244338C2semidirect
5. #20244159C2semidirect
6. #20243976C2semidirect
7. #20243965C2semidirect
11. #20243285C2semidirect
12. C4×#13093955C2semidirect
13. #20243248C2semidirect
19. #15689411C4semidirect
66. #13094023C2×C4semidirect
87. #13093956C2×C4semidirect
89. #13093955C2×C4semidirect
91. Dic47⋊C8C2×C4semidirect
112. C4×D4×C47Q8semidirect
363. C47#21623semidirect
380. C4×D4Dic94semidirect

Maximal subgroups

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

subgroupindex
C2×#156894112
#202461322
#202457492
#202443382
#202441592
#202439762
#202439652
#202436472
#202436372
#202433962
#202432852
C4×#130939552
#202432482
C4×#130939452
C47×#17622
#2162347