Finite Group Databasebeta

← #26224271

Subgroups — #26224271

Order 12032.

818 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689479C211yesmaximal, characteristic
2#20246128C211yesmaximal, characteristic, complemented
3#20246102C211yesmaximal, characteristic
4#20244355C211yesmaximal, characteristic, complemented
5#20244201C211yesmaximal, characteristic
6#20244085C211yesmaximal, characteristic
7#20244070C211yesmaximal, characteristic
8#20244067C211yesmaximal, characteristic, complemented
9#20243514C211yesmaximal, characteristic, complemented
10#20243220C211yesmaximal, characteristic, complemented
11#20243212C211yesmaximal, characteristic
12#103×Dic47C211yesmaximal, characteristic, complemented
13#20243202C211yesmaximal, characteristic
14#20243076C211yesmaximal, characteristic, complemented
15C47×#1935C211yesmaximal, characteristic, complemented
16C2×#13094015C2211yescharacteristic
17C2×#13094013C2211yescharacteristic
18C2×Q8×Dic47C2211yescharacteristic
19C2×#13093986C2211yescharacteristic
20#15689479C418yescomplemented
21C2×#13093948C2211yescharacteristic
22C2×#13093948C2211yescharacteristic
23C2×#13093946C2211yescharacteristic
24C2×#13093946C2211yescharacteristic
25C2×#13093944C2211yescharacteristic
26C2×#13093944C2211yescharacteristic
27C2×#13093934C2211yescharacteristic
28C2×#13093934C2211yescharacteristic
29C2×C4×C47⋊C8C2211yescharacteristic
30C2×C8×Dic47C2211yescharacteristic
31C2×C8×Dic47C2211yescharacteristic
32#15689025C2211yescharacteristic
33#15689021C2211yescharacteristic
34#15689019—21no—
35#15689018—21no—
36#15689012—21no—
37#15689012—21no—
38#15689008—21no—
39#15689005—21no—
40#15688997—21no—
41#15688997—21no—
42#15688992C2211yescharacteristic
43#15688992C2211yescharacteristic
44C4⋊C4×Dic47C2211yescharacteristic
45C4⋊C4×Dic47C2211yescharacteristic
46#15688969C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C2×#103×C47C2211yescharacteristic
54C2×#103×C47C2211yescharacteristic
55C2×#97×C47C2211yescharacteristic
56C47×#391C2211yescharacteristic
57C47×#389C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×D188C2311yescharacteristic
61C22×C4×D47—21no—
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64#13094015—22no—
65#13094015C2×C414yescomplemented
66#13094013—22no—
67#13094013C2×C414yes—
68Q8×Dic47—22no—
69Q8×Dic47—22no—
70#13093986—22no—
71#13093986—22no—
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×Dic47⋊C4—21no—
80C2×Dic47⋊C4—21no—
81C2×Dic47⋊C4—21no—
82C2×Dic47⋊C4—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C2×(C2×D47)⋊C4—21no—
87C2×(C2×D47)⋊C4—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47—21no—
90C2×C4×Dic47—21no—
91C2×C4×Dic47C2311yescharacteristic
92C2×C4×Dic47C2311yescharacteristic
93#13093948—22no—
94#13093948C2×C414yescomplemented
95#13093946—22no—
96#13093946C2×C414yes—
97#13093944—22no—
98#13093944C2×C414yes—
99#13093934—22no—
100#13093934—22no—
101C4×C47⋊C8—22no—
102C8×Dic47—22no—
103C8×Dic47C2×C414yescomplemented
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95D411yescharacteristic
108C47⋊#95D411yescharacteristic
109C47⋊#95D411yescharacteristic
110C47⋊#95D411yescharacteristic
111C22×Q8×C47C2311yescharacteristic
112C22×C8×C47C2311yescharacteristic
113C22×C8×C47C2311yescharacteristic
114C2×C4⋊C4×C47—21no—
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C42×C47C2311yescharacteristic
119#103×C47—22no—
120#103×C47C2×C414yescomplemented
121#97×C47—22no—
122#95×C47—21no—
123#95×C47—21no—
124#95×C47—21no—
125C23×D47—21no—
126C2×Q8×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47—21no—
131C2×Q8×C47C22×C412yes—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2411yescharacteristic
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47C2×D411yescharacteristic
143C22×Dic47C2×D411yescharacteristic
144C22×Dic47C2×D411yescharacteristic
145C22×Dic47C2×D411yescharacteristic
146C2×D188—21no—
147C2×D188—21no—
148C2×D188—21no—
149C2×D188—21no—
150C2×D188—21no—
151C2×D188C22×C412yes—
152C2×C4×D47—42no—
153C2×C4×D47—81no—
154C2×C4×D47—81no—
155C2×C8×C47—21no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47C22×C412yes—
160C2×C8×C47#4112yes—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—81no—
167C4⋊C4×C47C22×C412yes—
168C42×C47—21no—
169C42×C47—21no—
170(C2×D47)⋊C4—24no—
171(C2×D47)⋊C4—24no—
172(C2×D47)⋊C4—42no—
173(C2×D47)⋊C4—81no—
174(C2×D47)⋊C4—81no—
175C4⋊Dic47—21no—
176C4⋊Dic47—24no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—42no—
182C4⋊Dic47C22×C412yes—
183Dic47⋊C4—24no—
184Dic47⋊C4—81no—
185Dic47⋊C4—81no—
186Dic47⋊C4—81no—
187Dic47⋊C4—81no—
188Dic47⋊C4—81no—
189C4×Dic47—42no—
190C4×Dic47—81no—
191C4×Dic47—81no—
192C4×Dic47C22×C412yes—
193C4×Dic47#4112yes—
194C4×Dic47#4112yes—
195C4×Dic47#4112yes—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8C22×C412yes—
201C2×C47⋊C8#4112yes—
202C23×C47C22×D411yescharacteristic
203C22×D47—22no—
204C22×D47—41no—
205C22×D47—41no—
206C22×D47—41no—
207C22×D47—41no—
208C22×D47—41no—
209C22×D47—41no—
210Q8×C47—22no—
211Q8×C47—22no—
212Q8×C47—41no—
213Q8×C47—41no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
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×C47C23×C411yescharacteristic
233C2×C4×C47C2×#4111yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×Dic47—21no—
239C2×Dic47—21no—
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×Dic47#12412yes—
258C2×Dic47#12412yes—
259D188—22no—
260D188—22no—
261D188—41no—
262D188—41no—
263C4×D47—84no—
264C8×C47—22no—
265C8×C47—22no—
266C47⋊C8—22no—
267C47⋊C8—22no—
268#21643—471nomaximal, Sylow
269C22×C47C2×C4×D411yescharacteristic
270C22×C47C2×#12411yescharacteristic
271C22×C47#53211yescharacteristic
272C22×C47#57611yescharacteristic
273C22×C47#57711yescharacteristic
274C22×C47#57711yescharacteristic
275C22×C47#57811yescharacteristic
276C2×D47—42no—
277C2×D47—42no—
278C2×D47—42no—
279C2×D47—81no—
280C2×D47—81no—
281C2×D47—81no—
282C2×D47—81no—
283C4×C47—41no—
284C4×C47—41no—
285C4×C47—42no—
286C4×C47—42no—
287C4×C47—81no—
288C4×C47—81no—
289C4×C47C4×#4111yescharacteristic
290C4×C47C4×#4111yescharacteristic
291C4×C47#52911yescharacteristic
292C4×C47#52911yescharacteristic
293Dic47—22no—
294Dic47—22no—
295Dic47—22no—
296Dic47—22no—
297Dic47—42no—
298Dic47—81no—
299Dic47—81no—
300C2×#488—471no—
301#2330—471no—
302#2326—471no—
303#1992—471no—
304#1959—471no—
305#1936—471no—
306#1935D4711yescharacteristic, complemented
307#1934—471no—
308#1933—471no—
309#1834—471no—
310#1763—471no—
311#1762—471no—
312C4×#103—471no—
313C4×#102—471no—
314#1726—471no—
315C2×C47C4×#12411yescharacteristic
316C2×C47#295111yescharacteristic
317C2×C47#295211yescharacteristic
318C2×C47#313611yescharacteristic
319C2×C47#313711yescharacteristic
320C2×C47#317811yescharacteristic
321C2×C47#317911yescharacteristic
322D47—82no—
323C2×#125—471no—
324C2×#124—471no—
325C2×C4×Q8—471no—
326C2×C4×D4—471no—
327#488—478no—
328C2×#103—471no—
329C2×#103—471no—
330C2×#103C2×D4711yescharacteristic
331C2×#103C2×D4711yescharacteristic
332C2×#102—471no—
333C2×#102—471no—
334C2×#102—471no—
335C2×#102—471no—
336C2×#97—471no—
337C2×#97—471no—
338C2×#97C2×D4711yescharacteristic
339C2×C4×C8—471no—
340C2×C4×C8—471no—
341C2×C4×C8—471no—
342#391C2×D4711yescharacteristic
343#390—471no—
344#389—471no—
345#389—941no—
346#389—941no—
347#389C2×D4711yescharacteristic
348#388—941no—
349#388—941no—
350#387—941no—
351#387—941no—
352#385—941no—
353#385—941no—
354#384—471no—
355#384—471no—
356C4×C4⋊C4—471no—
357C4×C4⋊C4—471no—
358#376—471no—
359#340—471no—
360#340—471no—
361#340—471no—
362#340—471no—
363#340—471no—
364#340—471no—
365#340C2×D4711yescharacteristic
366#340C2×D4711yescharacteristic
367C47#2164311yescharacteristic, complemented, Sylow
368C22×Q8C22×D4711yescharacteristic
369C22×D4—471no—
370C23×C4—941no—
371C22×C8—471no—
372C22×C8—471no—
373C22×C8C22×D4711yescharacteristic
374C22×C8C22×D4711yescharacteristic
375#125—474no—
376#125—942no—
377#124—474no—
378#124—942no—
379C4×Q8—942no—
380C4×Q8—942no—
381C4×D4—942no—
382C4×D4—942no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—471no—
385C2×C4⋊C4—471no—
386C2×C4⋊C4—471no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4—941no—
395C2×C4⋊C4C22×D4711yescharacteristic
396C2×C4⋊C4C22×D4711yescharacteristic
397C2×C4⋊C4C22×D4711yescharacteristic
398C2×C22⋊C4—941no—
399C2×C22⋊C4—941no—
400C2×C22⋊C4—941no—
401C2×C22⋊C4—941no—
402C2×C22⋊C4—941no—
403C2×C42—471no—
404C2×C42—471no—
405C2×C42—941no—
406C2×C42—941no—
407C2×C42—941no—
408C2×C42C22×D4711yescharacteristic
409#103—22no—
410#103—474no—
411#103—942no—
412#103C4×D4714yescomplemented
413#102—474no—
414#102—474no—
415#102—942no—
416#102—942no—
417#97—22no—
418#97—942no—
419#97—942no—
420C4×C8—474no—
421C4×C8—942no—
422C4×C8—942no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—471no—
427#95—471no—
428#95—471no—
429#95—471no—
430#95—941no—
431#95—941no—
432#95—941no—
433C24—941no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8C2×C4×D4712yes—
440C2×D4—472no—
441C2×D4—941no—
442C2×D4—941no—
443C2×D4—941no—
444C2×D4—941no—
445C2×D4—941no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—471no—
451C22×C4—471no—
452C22×C4—471no—
453C22×C4—471no—
454C22×C4—941no—
455C22×C4—941no—
456C22×C4—941no—
457C22×C4—3761no—
458C22×C4—3761no—
459C22×C4—1882no—
460C22×C4D4×D4711yescharacteristic
461C22×C4D4×D4711yescharacteristic
462C22×C4C23×D4711yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—472no—
468C2×C8—472no—
469C2×C8—941no—
470C2×C8—941no—
471C2×C8—941no—
472C2×C8—941no—
473C2×C8C2×C4×D4712yes—
474C2×C8#1062535212yes—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—81no—
481C4⋊C4—472no—
482C4⋊C4—944no—
483C4⋊C4—944no—
484C4⋊C4—941no—
485C4⋊C4—941no—
486C4⋊C4—941no—
487C4⋊C4—941no—
488C4⋊C4—941no—
489C4⋊C4—3761no—
490C4⋊C4—3761no—
491C4⋊C4—3761no—
492C4⋊C4—3761no—
493C4⋊C4—3761no—
494C4⋊C4—1882no—
495C4⋊C4C2×C4×D4712yes—
496C22⋊C4—944no—
497C22⋊C4—944no—
498C22⋊C4—3761no—
499C22⋊C4—3761no—
500C22⋊C4—1882no—
501C42—21no—
502C42—21no—
503C42—472no—
504C42—472no—
505C42—472no—
506C42—472no—
507C42—3761no—
508C42—3761no—
509C42—1882no—
510C23—942no—
511C23—1881no—
512C23—1881no—
513C23—1881no—
514C23—1881no—
515C23—1881no—
516C23—1881no—
517C23C2×D4×D4711yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522D4—942no—
523D4—942no—
524D4—1881no—
525D4—1881no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
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—472no—
545C2×C4—472no—
546C2×C4—472no—
547C2×C4—472no—
548C2×C4—941no—
549C2×C4—941no—
550C2×C4—941no—
551C2×C4—941no—
552C2×C4—941no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—941no—
556C2×C4—942no—
557C2×C4—3764no—
558C2×C4—1881no—
559C2×C4—1881no—
560C2×C4—1881no—
561C2×C4—1881no—
562C2×C4—1881no—
563C2×C4—1881no—
564C2×C4—1881no—
565C2×C4—1881no—
566C2×C4C22×C4×D4711yescharacteristic
567C2×C4#41×D4711yescharacteristic
568C2×C4#41×D4711yescharacteristic
569C2×C4#41×D4711yescharacteristic
570C2×C4#41×D4711yescharacteristic
571C2×C4C2×#1062535211yescharacteristic
572C8—22no—
573C8—22no—
574C8—942no—
575C8—942no—
576C22—3761no—
577C22—3761no—
578C22—3761no—
579C22—3761no—
580C22—1882no—
581C22—1882no—
582C22—1882no—
583C22C4×D4×D4711yescharacteristic
584C22#124×D4711yescharacteristic
585C22#1568966811yescharacteristic
586C22#1568992711yescharacteristic
587C22#1568993211yescharacteristic
588C22#1568993211yescharacteristic
589C22#1568993911yescharacteristic
590C4—42no—
591C4—41no—
592C4—41no—
593C4—42no—
594C4—81no—
595C4—81no—
596C4—942no—
597C4—942no—
598C4—942no—
599C4—942no—
600C4—3761no—
601C4—3761no—
602C4—1882no—
603C4C4×#1062535211yescharacteristic
604C4#1568959111yescharacteristic
605C4#1568965011yescharacteristic
606C4#1568965611yescharacteristic
607C2—3762no—
608C2#2024592911yescharacteristic
609C2#2025226011yescharacteristic
610C2#2025226811yescharacteristic
611C2#2025373911yescharacteristic
612C2#2025375411yescharacteristic
613C2#2025405711yescharacteristic
614C2#2025406411yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1935C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenter#1935D47
hypocenterC47#21643
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1935

Complements

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

NH ≅ G/Ntype
2. #20246128C2semidirect
4. #20244355C2semidirect
8. #20244067C2semidirect
9. #20243514C2semidirect
10. #20243220C2semidirect
12. #103×Dic47C2semidirect
14. #20243076C2semidirect
15. C47×#1935C2semidirect
20. #15689479C4semidirect
65. #13094015C2×C4semidirect
94. #13093948C2×C4semidirect
103. C8×Dic47C2×C4semidirect
120. #103×C47C2×C4semidirect
306. #1935D47semidirect
367. C47#21643semidirect
412. #103C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156894792
#202461282
#202461022
#202443552
#202442012
#202440852
#202440702
#202440672
#202435142
#202432202
#202432122
#103×Dic472
#202432022
#202430762
C47×#19352
#2164347