Finite Group Databasebeta

← #26220482

Subgroups — #26220482

Order 12032.

898 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689208C211yesmaximal, characteristic
2#20245863C211yesmaximal, characteristic, complemented
3#20245846C211yesmaximal, characteristic
4#20244245C211yesmaximal, characteristic
5#20244239C211yesmaximal, characteristic
6#20244115C211yesmaximal, characteristic, complemented
7#20244113C211yesmaximal, characteristic, complemented
8#20244071C211yesmaximal, characteristic
9#20244069C211yesmaximal, characteristic
10#20244066C211yesmaximal, characteristic, complemented
11#20243276C211yesmaximal, characteristic, complemented
12#103×Dic47C211yesmaximal, characteristic, complemented
13#20243202C211yesmaximal, characteristic
14C4×C8×Dic47C211yesmaximal, characteristic, complemented
15C47×#1933C211yesmaximal, characteristic, complemented
16C22×#10625336C2211yescharacteristic
17C2×Q8×Dic47C2211yescharacteristic
18C2×#13093986C2211yescharacteristic
19C2×#13093982—21no—
20#15689208C418yescomplemented
21C2×#13093955C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093948C2211yescharacteristic
25C2×#13093946C2211yescharacteristic
26C2×#13093946C2211yescharacteristic
27C2×#13093944C2211yescharacteristic
28C2×C4×C47⋊C8C2211yescharacteristic
29C2×C8×Dic47C2211yescharacteristic
30C2×C8×Dic47C2211yescharacteristic
31C2×C8×Dic47C2211yescharacteristic
32C2×C8×Dic47C2211yescharacteristic
33#15689030C2211yescharacteristic
34#15689025C2211yescharacteristic
35#15689024C2211yescharacteristic
36#15689022C2211yescharacteristic
37#15689021C2211yescharacteristic
38#15689019C2211yescharacteristic
39#15689015—21no—
40#15689013—21no—
41#15689008—21no—
42#15689006—21no—
43C4⋊C4×Dic47C2211yescharacteristic
44C4⋊C4×Dic47C2211yescharacteristic
45#15688974—21no—
46#15688970—21no—
47#15688968—21no—
48C42×Dic47C2211yescharacteristic
49#15688886C2211yescharacteristic
50#15688886C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C2×#103×C47C2211yescharacteristic
54C2×#103×C47C2211yescharacteristic
55C2×C4×C8×C47C2211yescharacteristic
56C47×#391C2211yescharacteristic
57C47×#389C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×D188C2311yescharacteristic
61C22×C4×D47—21no—
62C2×#10625336—22no—
63C2×#10625336—22no—
64C2×#10625336C2×C414yes—
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67Q8×Dic47—22no—
68Q8×Dic47C2×C414yes—
69#13093986—22no—
70#13093986C2×C414yes—
71#13093982—44no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×(C2×D47)⋊C4—21no—
81C2×(C2×D47)⋊C4—21no—
82C2×(C2×D47)⋊C4—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47D411yescharacteristic
88C2×C4×Dic47D411yescharacteristic
89C2×C4×Dic47D411yescharacteristic
90C2×C4×Dic47D411yescharacteristic
91C2×C4×Dic47C2311yescharacteristic
92C2×C4×Dic47C2311yescharacteristic
93#13093955—22no—
94#13093955—22no—
95#13093955C2×C414yes—
96#13093948—22no—
97#13093946—22no—
98#13093946C2×C414yes—
99#13093944—22no—
100C4×C47⋊C8—22no—
101C8×Dic47—22no—
102C8×Dic47—22no—
103C8×Dic47—22no—
104C8×Dic47C2×C414yes—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C47⋊#95—21no—
110C47⋊#95—21no—
111C47⋊#95—21no—
112C22×Q8×C47C2311yescharacteristic
113C22×C8×C47C2311yescharacteristic
114C22×C8×C47C2311yescharacteristic
115C2×C4⋊C4×C47—21no—
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C42×C47C2311yescharacteristic
120#103×C47—22no—
121#103×C47C2×C414yes—
122C4×C8×C47—22no—
123#95×C47—21no—
124#95×C47—21no—
125#95×C47—21no—
126C23×D47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47—21no—
131C2×Q8×C47—21no—
132C2×Q8×C47C22×C412yes—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2411yescharacteristic
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47C2×D411yescharacteristic
144C22×Dic47C2×D411yescharacteristic
145C22×Dic47C2×D411yescharacteristic
146C22×Dic47C2×D411yescharacteristic
147C2×D188—21no—
148C2×D188—21no—
149C2×D188—21no—
150C2×D188—21no—
151C2×D188—21no—
152C2×D188C22×C412yes—
153C2×C4×D47—24no—
154C2×C4×D47—42no—
155C2×C4×D47—42no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47C22×C412yes—
161C2×C8×C47#4112yes—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—42no—
168C4⋊C4×C47C22×C412yes—
169C42×C47—21no—
170C42×C47—21no—
171#10625336—28no—
172(C2×D47)⋊C4—24no—
173(C2×D47)⋊C4—42no—
174(C2×D47)⋊C4—42no—
175(C2×D47)⋊C4—42no—
176(C2×D47)⋊C4—42no—
177C4⋊Dic47—21no—
178C4⋊Dic47—24no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47—42no—
184C4⋊Dic47C22×C412yes—
185Dic47⋊C4—42no—
186Dic47⋊C4—42no—
187C4×Dic47—22no—
188C4×Dic47—21no—
189C4×Dic47—21no—
190C4×Dic47—24no—
191C4×Dic47—22no—
192C4×Dic47—22no—
193C4×Dic47—42no—
194C4×Dic47—42no—
195C4×Dic47SD1614yescomplemented
196C4×Dic47C22×C412yes—
197C4×Dic47#4112yes—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8C22×C412yes—
201C2×C47⋊C8#4112yes—
202C2×C47⋊C8#4112yes—
203C2×C47⋊C8#4112yes—
204C23×C47C22×D411yescharacteristic
205C22×D47—22no—
206C22×D47—41no—
207C22×D47—41no—
208C22×D47—41no—
209C22×D47—41no—
210C22×D47—41no—
211C22×D47—41no—
212Q8×C47—22no—
213Q8×C47—22no—
214Q8×C47—41no—
215Q8×C47—41no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47C2×SD1612yes—
233C2×C4×C47#13512yes—
234C2×C4×C47C23×C411yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×C4×C47C2×#4111yescharacteristic
239C2×C4×C47C2×#4111yescharacteristic
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—22no—
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×Dic47C4×D412yes—
256C2×Dic47C4×D412yes—
257C2×Dic47C2×SD1612yes—
258C2×Dic47C2×SD1612yes—
259C2×Dic47#13512yes—
260C2×Dic47#13512yes—
261D188—22no—
262D188—22no—
263D188—41no—
264D188—41no—
265C4×D47—48no—
266C8×C47—22no—
267C8×C47—22no—
268C47⋊C8—22no—
269C47⋊C8#11714yescomplemented
270#21274—471nomaximal, Sylow
271C22×C47C2×C4×D411yescharacteristic
272C22×C47#53211yescharacteristic
273C22×C47#53211yescharacteristic
274C22×C47C22×SD1611yescharacteristic
275C22×C47C2×#13511yescharacteristic
276C22×C47C2×#13511yescharacteristic
277C22×C47C2×#13511yescharacteristic
278C2×D47—42no—
279C2×D47—42no—
280C2×D47—42no—
281C2×D47—81no—
282C2×D47—81no—
283C2×D47—81no—
284C2×D47—81no—
285C4×C47—22no—
286C4×C47—22no—
287C4×C47—42no—
288C4×C47—42no—
289C4×C47—81no—
290C4×C47—81no—
291C4×C47C2×#11711yescharacteristic
292C4×C47C4×#4111yescharacteristic
293C4×C47#52911yescharacteristic
294C4×C47#52911yescharacteristic
295Dic47—22no—
296Dic47—22no—
297Dic47—22no—
298Dic47—42no—
299Dic47—81no—
300Dic47—81no—
301Dic47C4×SD1614yes—
302C2×C4×SD16—471no—
303#2295—471no—
304#2293—471no—
305#1968—471no—
306#1968—471no—
307#1941—471no—
308#1941—471no—
309#1934—471no—
310#1934—471no—
311#1933—471no—
312#1933D4711yescharacteristic, complemented
313C4×#106—471no—
314C4×#103—471no—
315C4×#102—471no—
316C42×C8—471no—
317C2×C47#229311yescharacteristic
318C2×C47C2×C4×SD1611yescharacteristic
319C2×C47C4×#13511yescharacteristic
320C2×C47#316711yescharacteristic
321C2×C47#316711yescharacteristic
322C2×C47#316811yescharacteristic
323C2×C47#316811yescharacteristic
324D47—82no—
325C22×SD16—471no—
326C2×C4×Q8—471no—
327C2×C4×D4—471no—
328C2×#117—941no—
329C4×SD16—478no—
330C2×#106—471no—
331C2×#106—471no—
332C2×#106—471no—
333C2×#103—471no—
334C2×#103C2×D4711yescharacteristic
335C2×#103C2×D4711yescharacteristic
336C2×#102—471no—
337C2×#102—471no—
338C2×#102—471no—
339C2×C4×C8—471no—
340C2×C4×C8—471no—
341C2×C4×C8—471no—
342C2×C4×C8—471no—
343C2×C4×C8—471no—
344C2×C4×C8C2×D4711yescharacteristic
345#391—471no—
346#391C2×D4711yescharacteristic
347#390—471no—
348#390—471no—
349#389—471no—
350#389—471no—
351#389—471no—
352#389C2×D4711yescharacteristic
353#388—941no—
354#388—941no—
355#387—941no—
356#387—941no—
357C4×C4⋊C4—471no—
358C4×C4⋊C4—471no—
359C4×C4⋊C4—941no—
360C4×C22⋊C4—941no—
361#376—941no—
362C43—471no—
363#340—471no—
364#340—471no—
365#340—471no—
366#340—471no—
367#340C2×D4711yescharacteristic
368#340C2×D4711yescharacteristic
369C47#2127411yescharacteristic, complemented, Sylow
370C22×Q8C22×D4711yescharacteristic
371C22×D4—471no—
372C23×C4—941no—
373C2×SD16—474no—
374C2×SD16—942no—
375C2×SD16—942no—
376C22×C8—471no—
377C22×C8—471no—
378C22×C8C22×D4711yescharacteristic
379C22×C8C22×D4711yescharacteristic
380C4×Q8—474no—
381C4×Q8—942no—
382C4×D4—474no—
383C4×D4—942no—
384#117—1884no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—471no—
387C2×C4⋊C4—471no—
388C2×C4⋊C4—471no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4C22×D4711yescharacteristic
394C2×C4⋊C4C22×D4711yescharacteristic
395C2×C4⋊C4C22×D4711yescharacteristic
396C2×C22⋊C4—941no—
397C2×C22⋊C4—941no—
398C2×C22⋊C4—941no—
399C2×C22⋊C4—941no—
400C2×C22⋊C4—941no—
401C2×C42—471no—
402C2×C42—471no—
403C2×C42—471no—
404C2×C42—471no—
405C2×C42—471no—
406C2×C42—471no—
407C2×C42—941no—
408C2×C42—941no—
409C2×C42—941no—
410C2×C42C22×D4711yescharacteristic
411#106—474no—
412#106—942no—
413#106—942no—
414#103—22no—
415#103—942no—
416#103C4×D4714yes—
417#102—474no—
418#102—942no—
419#102—942no—
420C4×C8—22no—
421C4×C8—474no—
422C4×C8—942no—
423C4×C8—942no—
424C4×C8—942no—
425C4×C8—942no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—941no—
430#95—941no—
431#95—941no—
432#95—941no—
433#95—941no—
434#95—941no—
435#95—941no—
436C24—941no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D4712yes—
443C2×D4—472no—
444C2×D4—941no—
445C2×D4—941no—
446C2×D4—941no—
447C2×D4—941no—
448C2×D4—941no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—471no—
454C22×C4—471no—
455C22×C4—471no—
456C22×C4—471no—
457C22×C4—944no—
458C22×C4—941no—
459C22×C4—941no—
460C22×C4—941no—
461C22×C4—1882no—
462C22×C4—1882no—
463C22×C4D4×D4711yescharacteristic
464C22×C4D4×D4711yescharacteristic
465C22×C4C23×D4711yescharacteristic
466SD16—948no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—472no—
472C2×C8—472no—
473C2×C8—472no—
474C2×C8—472no—
475C2×C8—941no—
476C2×C8—941no—
477C2×C8C2×C4×D4712yes—
478C2×C8#1062535212yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—472no—
486C4⋊C4—944no—
487C4⋊C4—941no—
488C4⋊C4—941no—
489C4⋊C4—941no—
490C4⋊C4—941no—
491C4⋊C4—941no—
492C4⋊C4—1882no—
493C4⋊C4—1882no—
494C4⋊C4—1882no—
495C4⋊C4C2×C4×D4712yes—
496C22⋊C4—944no—
497C22⋊C4—1882no—
498C22⋊C4—1882no—
499C22⋊C4—1882no—
500C22⋊C4—1882no—
501C42—21no—
502C42—21no—
503C42—474no—
504C42—472no—
505C42—472no—
506C42—942no—
507C42—942no—
508C42—944no—
509C42—942no—
510C42—941no—
511C42—941no—
512C42—1882no—
513C42—1882no—
514C23—942no—
515C23—1881no—
516C23—1881no—
517C23—1881no—
518C23—1881no—
519C23—1881no—
520C23—1881no—
521C23C2×D4×D4711yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—942no—
527D4—942no—
528D4—1881no—
529D4—1881no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—472no—
547C2×C4—472no—
548C2×C4—472no—
549C2×C4—472no—
550C2×C4—472no—
551C2×C4—472no—
552C2×C4—941no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—941no—
556C2×C4—941no—
557C2×C4—941no—
558C2×C4—942no—
559C2×C4—1888no—
560C2×C4—1881no—
561C2×C4—1881no—
562C2×C4—1881no—
563C2×C4—1881no—
564C2×C4—1881no—
565C2×C4—1881no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4SD16×D4712yes—
569C2×C4#1309404712yes—
570C2×C4C22×C4×D4711yescharacteristic
571C2×C4#41×D4711yescharacteristic
572C2×C4#41×D4711yescharacteristic
573C2×C4#41×D4711yescharacteristic
574C2×C4#41×D4711yescharacteristic
575C2×C4C2×#1062535211yescharacteristic
576C8—22no—
577C8—22no—
578C8—474no—
579C8—942no—
580C22—3761no—
581C22—3761no—
582C22—3761no—
583C22—3761no—
584C22—1882no—
585C22—1882no—
586C22—1882no—
587C22C4×D4×D4711yescharacteristic
588C22#1568966811yescharacteristic
589C22#1568967211yescharacteristic
590C22C2×SD16×D4711yescharacteristic
591C22#135×D4711yescharacteristic
592C22#135×D4711yescharacteristic
593C22C2×#1309404711yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—42no—
598C4—81no—
599C4—81no—
600C4—474no—
601C4—942no—
602C4—942no—
603C4—942no—
604C4—3761no—
605C4—3761no—
606C4—1882no—
607C4#117×D4711yescharacteristic
608C4C4×#1062535211yescharacteristic
609C4#1568965011yescharacteristic
610C4#1568965611yescharacteristic
611C2—3762no—
612C2#2024584811yescharacteristic
613C2C4×SD16×D4711yescharacteristic
614C2C4×#1309404711yescharacteristic
615C2#2025394211yescharacteristic
616C2#2025395011yescharacteristic
617C2#2025395711yescharacteristic
618C2#2025395911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1933C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenter#1933D47
hypocenterC47#21274
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1933

Complements

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

NH ≅ G/Ntype
2. #20245863C2semidirect
6. #20244115C2semidirect
7. #20244113C2semidirect
10. #20244066C2semidirect
11. #20243276C2semidirect
12. #103×Dic47C2semidirect
14. C4×C8×Dic47C2semidirect
15. C47×#1933C2semidirect
20. #15689208C4semidirect
195. C4×Dic47SD16semidirect
269. C47⋊C8#117semidirect
312. #1933D47semidirect
369. C47#21274semidirect

Maximal subgroups

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

subgroupindex
C2×#156892082
#202458632
#202458462
#202442452
#202442392
#202441152
#202441132
#202440712
#202440692
#202440662
#202432762
#103×Dic472
#202432022
C4×C8×Dic472
C47×#19332
#2127447