Finite Group Databasebeta

← #26223992

Subgroups — #26223992

Order 12032.

868 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689421C211yesmaximal, characteristic
2#20246125C211yesmaximal, characteristic, complemented
3#20245749C211yesmaximal, characteristic
4#20244342C211yesmaximal, characteristic, complemented
5#20244150C211yesmaximal, characteristic, complemented
6#20243969C211yesmaximal, characteristic, complemented
7#20243958C211yesmaximal, characteristic, complemented
8#20243651C211yesmaximal, characteristic
9#20243647C211yesmaximal, characteristic
10#20243386C211yesmaximal, characteristic
11#20243285C211yesmaximal, characteristic, complemented
12C4×#13093955C211yesmaximal, characteristic, complemented
13#20243248C211yesmaximal, characteristic, complemented
14C4×#13093945C211yesmaximal, characteristic
15C47×#1762C211yesmaximal, characteristic
16C2×#13094015C2211yescharacteristic
17C2×D4×Dic47C2211yescharacteristic
18C2×D4×Dic47C2211yescharacteristic
19#15689421C418yescomplemented
20C2×#13093958C2211yescharacteristic
21C2×#13093956C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×Dic47⋊C8C2211yescharacteristic
25C2×Dic47⋊C8C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×#13093945C2211yescharacteristic
28C2×C4×C47⋊C8C2211yescharacteristic
29#15689022C2211yescharacteristic
30#15689021—21no—
31#15689020—21no—
32#15689019C2211yescharacteristic
33#15689007—21no—
34#15689006—21no—
35#15688992C2211yescharacteristic
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
60C23×Dic47—21no—
61C23×Dic47—21no—
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64#13094015—22no—
65#13094015C2×C414yescomplemented
66D4×Dic47—22no—
67D4×Dic47—22no—
68D4×Dic47—22no—
69D4×Dic47C2×C414yes—
70C2×C4⋊Dic47—21no—
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×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×C22⋊Dic47—21no—
81C2×C22⋊Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47C2311yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88#13093958C2×C414yescomplemented
89#13093956—22no—
90#13093955—22no—
91#13093955C2×C414yescomplemented
92Dic47⋊C8—22no—
93Dic47⋊C8C2×C414yescomplemented
94#13093945—22no—
95#13093945C2×C414yes—
96C4×C47⋊C8—22no—
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×C47Q814yes—
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×C8×C47—21no—
154C2×C8×C47—21no—
155C2×C8×C47—21no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47C22×C412yes—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47C22×C412yes—
164C4⋊C4×C47C2×Q812yes—
165C22⋊C4×C47—24no—
166C42×C47—81no—
167C42×C47C2×Q812yes—
168C42×C47#4112yes—
169C22⋊Dic47—24no—
170C22⋊Dic47—24no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—24no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—81no—
178C4⋊Dic47C22×C412yes—
179C4⋊Dic47C22×C412yes—
180C4⋊Dic47#4112yes—
181Dic47⋊C4—24no—
182Dic47⋊C4—81no—
183Dic47⋊C4—81no—
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—
259C8×C47—22no—
260C8×C47—41no—
261C47⋊C8—22no—
262C47⋊C8—41no—
263#21619—471nomaximal, Sylow
264C22×C47—24no—
265C22×C47—42no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—42no—
269C22×C47—42no—
270C22×C47—42no—
271C22×C47C2×C4×D411yescharacteristic
272C22×C47C2×C22⋊Q811yescharacteristic
273C22×C47#52511yescharacteristic
274C22×C47#57611yescharacteristic
275C22×C47#57711yescharacteristic
276C22×C47#57711yescharacteristic
277C22×C47#57711yescharacteristic
278C4×C47—22no—
279C4×C47—22no—
280C4×C47—42no—
281C4×C47—42no—
282C4×C47—81no—
283C4×C47C2×C4×Q811yescharacteristic
284C4×C47C4×#4111yescharacteristic
285C4×C47#52911yescharacteristic
286C4×C47#53311yescharacteristic
287Dic47—22no—
288Dic47—22no—
289Dic47—41no—
290Dic47—41no—
291Dic47—42no—
292Dic47—42no—
293Dic47—42no—
294Dic47—81no—
295C2×#478—471no—
296#2330—471no—
297#2275—471no—
298#1989—471no—
299#1948—471no—
300#1919—471no—
301#1918—471no—
302#1864—471no—
303#1864—471no—
304#1802—471no—
305#1776—471no—
306C4×#106—471no—
307#1768—471no—
308#1762D4711yescharacteristic
309C4×#102—471no—
310C2×C47—44no—
311C2×C47C4×C22⋊Q811yescharacteristic
312C2×C47#294611yescharacteristic
313C2×C47#295211yescharacteristic
314C2×C47#308111yescharacteristic
315C2×C47#308411yescharacteristic
316C2×C47#311911yescharacteristic
317C2×C47#312311yescharacteristic
318C2×#125—471no—
319C2×C4×D4—471no—
320C2×C4×D4—471no—
321C2×C4×D4C2×D4711yescharacteristic
322#478—478no—
323C2×#107—471no—
324C2×#106—471no—
325C2×#106—471no—
326C2×#106—471no—
327C2×C4⋊C8—471no—
328C2×C4⋊C8—471no—
329C2×#102—471no—
330C2×#102—471no—
331C2×#102C2×D4711yescharacteristic
332C2×#102C2×D4711yescharacteristic
333C2×#97C2×D4711yescharacteristic
334C2×C4×C8—471no—
335#389—471no—
336#389—471no—
337#389—941no—
338#389—941no—
339#387—941no—
340#387—941no—
341#384—471no—
342C4×C4⋊C4—471no—
343C4×C4⋊C4—471no—
344C4×C4⋊C4—471no—
345C4×C4⋊C4—471no—
346C4×C4⋊C4C2×D4711yescharacteristic
347C4×C22⋊C4—941no—
348C4×C22⋊C4—941no—
349C4×C22⋊C4—941no—
350#376—471no—
351C2×#95—941no—
352#340—471no—
353#340—471no—
354#340—471no—
355#340—471no—
356#340—471no—
357#340—471no—
358#340C2×D4711yescharacteristic
359#340C2×D4711yescharacteristic
360#336—471no—
361#336—471no—
362C47#2161911yescharacteristic, complemented, Sylow
363C22×D4C22×D4711yescharacteristic
364C23×C4—21no—
365C23×C4—941no—
366C23×C4—941no—
367C22×C8—471no—
368C22×C8—471no—
369C22×C8C22×D4711yescharacteristic
370C22×C8C22×D4711yescharacteristic
371#125—474no—
372#125—942no—
373C4×D4—22no—
374C4×D4—474no—
375C4×D4—942no—
376C4×D4—942no—
377C4×D4—942no—
378C4×D4Dic9414yes—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—471no—
381C2×C4⋊C4—471no—
382C2×C4⋊C4—471no—
383C2×C4⋊C4—941no—
384C2×C4⋊C4—941no—
385C2×C4⋊C4—941no—
386C2×C4⋊C4—941no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4C22×D4711yescharacteristic
390C2×C4⋊C4C22×D4711yescharacteristic
391C2×C4⋊C4C22×D4711yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—941no—
394C2×C22⋊C4—941no—
395C2×C42—21no—
396C2×C42—471no—
397C2×C42—471no—
398C2×C42—941no—
399C2×C42—941no—
400C2×C42—941no—
401C2×C42—941no—
402C2×C42C22×D4711yescharacteristic
403#107—474no—
404#106—474no—
405#106—942no—
406#106—942no—
407C4⋊C8—474no—
408C4⋊C8—942no—
409#102—22no—
410#102—474no—
411#102—942no—
412#102C4×D4714yes—
413#97—22no—
414C4×C8—942no—
415#95—21no—
416#95—471no—
417#95—471no—
418#95—471no—
419#95—471no—
420#95—941no—
421#95—941no—
422#95—941no—
423#95—941no—
424#95—941no—
425#95—941no—
426C24—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4C2×Dic9412yes—
431C2×D4C2×C4×D4712yes—
432C2×D4#1062535012yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—471no—
440C22×C4—471no—
441C22×C4—471no—
442C22×C4—471no—
443C22×C4—944no—
444C22×C4—944no—
445C22×C4—941no—
446C22×C4—941no—
447C22×C4—941no—
448C22×C4—941no—
449C22×C4—3761no—
450C22×C4—3761no—
451C22×C4—3761no—
452C22×C4—3761no—
453C22×C4D4×D4711yescharacteristic
454C22×C4D4×D4711yescharacteristic
455C22×C4C23×D4711yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—472no—
462C2×C8—941no—
463C2×C8—941no—
464C2×C8—941no—
465C2×C8—941no—
466C2×C8—941no—
467C2×C8C2×C4×D4712yes—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—472no—
473C4⋊C4—472no—
474C4⋊C4—472no—
475C4⋊C4—944no—
476C4⋊C4—944no—
477C4⋊C4—941no—
478C4⋊C4—941no—
479C4⋊C4—941no—
480C4⋊C4—941no—
481C4⋊C4—941no—
482C4⋊C4—3761no—
483C4⋊C4—3761no—
484C4⋊C4—3761no—
485C4⋊C4—3761no—
486C4⋊C4C2×Dic9412yes—
487C4⋊C4C2×C4×D4712yes—
488C22⋊C4—24no—
489C22⋊C4—944no—
490C22⋊C4—944no—
491C42—81no—
492C42—472no—
493C42—472no—
494C42—941no—
495C42—941no—
496C42—3761no—
497C42—3761no—
498C42—3761no—
499C42—3761no—
500C42C2×Dic9412yes—
501C42#1062535212yes—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C2×D4×D4711yescharacteristic
510D4—22no—
511D4—22no—
512D4—22no—
513D4C4×Dic9414yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—84no—
528C2×C4—472no—
529C2×C4—472no—
530C2×C4—941no—
531C2×C4—941no—
532C2×C4—941no—
533C2×C4—941no—
534C2×C4—942no—
535C2×C4—942no—
536C2×C4—941no—
537C2×C4—941no—
538C2×C4—942no—
539C2×C4—941no—
540C2×C4—941no—
541C2×C4—941no—
542C2×C4—941no—
543C2×C4—3764no—
544C2×C4—3764no—
545C2×C4—1881no—
546C2×C4—1881no—
547C2×C4—1881no—
548C2×C4—1881no—
549C2×C4—1881no—
550C2×C4—1881no—
551C2×C4—1881no—
552C2×C4—1881no—
553C2×C4—1881no—
554C2×C4#1309400112yes—
555C2×C4#1309400112yes—
556C2×C4C22×C4×D4711yescharacteristic
557C2×C4C22×Dic9411yescharacteristic
558C2×C4#41×D4711yescharacteristic
559C2×C4#41×D4711yescharacteristic
560C2×C4C2×#1062535211yescharacteristic
561C2×C4C2×#1062535011yescharacteristic
562C8—22no—
563C8—41no—
564C8—942no—
565C8—1881no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22C4×D4×D4711yescharacteristic
574C22C2×#1309400111yescharacteristic
575C22#1568963211yescharacteristic
576C22#1568992711yescharacteristic
577C22#1568993211yescharacteristic
578C22#1568993311yescharacteristic
579C22#1568993311yescharacteristic
580C4—22no—
581C4—22no—
582C4—42no—
583C4—42no—
584C4—81no—
585C4—942no—
586C4—942no—
587C4—3761no—
588C4—1882no—
589C4—1882no—
590C4—1881no—
591C4—1881no—
592C4—1882no—
593C4C2×C4×Dic9411yescharacteristic
594C4C4×#1062535211yescharacteristic
595C4#1568965311yescharacteristic
596C4#1568967611yescharacteristic
597C2—44no—
598C2C4×#1309400111yescharacteristic
599C2#2025221911yescharacteristic
600C2#2025227011yescharacteristic
601C2#2025322111yescharacteristic
602C2#2025325411yescharacteristic
603C2#2025359111yescharacteristic
604C2#2025363911yescharacteristic
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#21619
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. #20246125C2semidirect
4. #20244342C2semidirect
5. #20244150C2semidirect
6. #20243969C2semidirect
7. #20243958C2semidirect
11. #20243285C2semidirect
12. C4×#13093955C2semidirect
13. #20243248C2semidirect
19. #15689421C4semidirect
65. #13094015C2×C4semidirect
88. #13093958C2×C4semidirect
91. #13093955C2×C4semidirect
93. Dic47⋊C8C2×C4semidirect
362. C47#21619semidirect

Maximal subgroups

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

subgroupindex
C2×#156894212
#202461252
#202457492
#202443422
#202441502
#202439692
#202439582
#202436512
#202436472
#202433862
#202432852
C4×#130939552
#202432482
C4×#130939452
C47×#17622
#2161947