Finite Group Databasebeta

← #26246331

Subgroups — #26246331

Order 12032.

752 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
1C4×#13094000C211yesmaximal, characteristic
2#20245784C211yesmaximal, characteristic
3#20244960C211yesmaximal, characteristic, complemented
4#20244934C211yesmaximal, characteristic, complemented
5#20244875C211yesmaximal, characteristic, complemented
6#20244827C211yesmaximal, characteristic
7#20244813C211yesmaximal, characteristic
8#20244809C211yesmaximal, characteristic, complemented
9#20244590C211yesmaximal, characteristic
10#20244582C211yesmaximal, characteristic
11C4×#13093958C211yesmaximal, characteristic, complemented
12C4×#13093955C211yesmaximal, characteristic, complemented
13C4×#13093948C211yesmaximal, characteristic, complemented
14C4×(C2×D47)⋊C8C211yesmaximal, characteristic
15C4×#103×C47C211yesmaximal, characteristic, complemented
16C2×#13094000C2211yescharacteristic
17C2×C42×D47C2211yescharacteristic
18C2×#13093958C2211yescharacteristic
19C2×#13093958C2211yescharacteristic
20C2×#13093955C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×#13093948C2211yescharacteristic
23C2×#13093948C2211yescharacteristic
24C2×(C2×D47)⋊C8C2211yescharacteristic
25C2×(C2×D47)⋊C8C2211yescharacteristic
26C2×C4×C47⋊C8C2211yescharacteristic
27#15689030C2211yescharacteristic
28#15689019C2211yescharacteristic
29#15689004—21no—
30#15688994—21no—
31#15688990—21no—
32#15688988C2211yescharacteristic
33#15688988C2211yescharacteristic
34#15688985—21no—
35#15688980C2211yescharacteristic
36#15688980C2211yescharacteristic
37#15688980C2211yescharacteristic
38C4×Dic47⋊C4—21no—
39C4×C4⋊Dic47C2211yescharacteristic
40C4×C4⋊Dic47C2211yescharacteristic
41C4×(C2×D47)⋊C4—21no—
42C4×(C2×D47)⋊C4—21no—
43#15688969—21no—
44#15688886C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688877C2211yescharacteristic
51#15688877C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C2×#103×C47C2211yescharacteristic
54C2×#103×C47C2211yescharacteristic
55C2×C4×C8×C47C2211yescharacteristic
56C4×C4⋊C4×C47C2211yescharacteristic
57C47×#340C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C22×C4×D47D411yescharacteristic
60C22×C4×D47D411yescharacteristic
61C22×C4×D47C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64#13094000—22no—
65#13094000—22no—
66C2×C4⋊Dic47C2311yescharacteristic
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×Dic47⋊C4—21no—
71C2×Dic47⋊C4—21no—
72C2×(C2×D47)⋊C4—21no—
73C2×(C2×D47)⋊C4—21no—
74C2×(C2×D47)⋊C4—21no—
75C2×(C2×D47)⋊C4—21no—
76C2×C4×Dic47—21no—
77C2×C4×Dic47—21no—
78C2×C4×Dic47D411yescharacteristic
79C2×C4×Dic47D411yescharacteristic
80C2×C4×Dic47C2311yescharacteristic
81C42×D47—22no—
82C42×D47—22no—
83#13093958—22no—
84#13093958—22no—
85#13093955—22no—
86#13093955—22no—
87#13093948—22no—
88#13093948—22no—
89(C2×D47)⋊C8—22no—
90(C2×D47)⋊C8—22no—
91C4×C47⋊C8—22no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—21no—
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⋊#95—21no—
104C22×Q8×C47C2311yescharacteristic
105C22×C8×C47C2311yescharacteristic
106C22×C8×C47C2311yescharacteristic
107C4×Q8×C47—22no—
108C4×Q8×C47—22no—
109C2×C4⋊C4×C47—21no—
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C42×C47—21no—
114C2×C42×C47—21no—
115C2×C42×C47C2311yescharacteristic
116#103×C47—22no—
117#103×C47—22no—
118C4×C8×C47—22no—
119#95×C47—21no—
120C23×D47C2×D411yescharacteristic
121C2×Q8×C47—21no—
122C2×Q8×C47—21no—
123C2×Q8×C47—21no—
124C2×Q8×C47—21no—
125C2×Q8×C47—21no—
126C2×Q8×C47#4112yes—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2411yescharacteristic
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47—21no—
137C22×Dic47—21no—
138C22×Dic47C2×D411yescharacteristic
139C22×Dic47C2×D411yescharacteristic
140C22×Dic47C2×D411yescharacteristic
141C2×C4×D47—22no—
142C2×C4×D47—22no—
143C2×C4×D47—22no—
144C2×C4×D47—22no—
145C2×C4×D47—22no—
146C2×C4×D47—21no—
147C2×C4×D47—21no—
148C2×C4×D47—21no—
149C2×C4×D47—21no—
150C2×C4×D47—21no—
151C2×C4×D47—21no—
152C2×C4×D47—41no—
153C2×C8×C47—21no—
154C2×C8×C47—21no—
155C2×C8×C47—21no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47#4112yes—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—42no—
166C42×C47—42no—
167C42×C47—42no—
168C42×C47#4112yes—
169C42×C47#4112yes—
170(C2×D47)⋊C4—24no—
171(C2×D47)⋊C4—42no—
172(C2×D47)⋊C4—42no—
173(C2×D47)⋊C4—42no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47#4112yes—
182Dic47⋊C4—24no—
183Dic47⋊C4—42no—
184C4×Dic47—22no—
185C4×Dic47—22no—
186C4×Dic47—21no—
187C4×Dic47—21no—
188C4×Dic47—42no—
189C4×Dic47—42no—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—21no—
192C2×C47⋊C8—21no—
193C2×C47⋊C8—21no—
194C2×C47⋊C8—21no—
195C2×C47⋊C8#4112yes—
196C23×C47C22×D411yescharacteristic
197C22×D47—21no—
198C22×D47—21no—
199C22×D47—21no—
200C22×D47—21no—
201C22×D47—21no—
202C22×D47—21no—
203C22×D47#13512yes—
204Q8×C47—22no—
205Q8×C47—22no—
206Q8×C47—41no—
207Q8×C47—41no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—22no—
211C2×C4×C47—41no—
212C2×C4×C47—41no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—41no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47#12312yes—
223C2×C4×C47#12312yes—
224C2×C4×C47#13512yes—
225C2×C4×C47#13712yes—
226C2×C4×C47C2×#4111yescharacteristic
227C2×C4×C47C2×#4111yescharacteristic
228C2×C4×C47C2×#4111yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—22no—
237C2×Dic47—21no—
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C4×D47—44no—
254C4×D47—41no—
255C4×D47—41no—
256C4×D47—41no—
257C4×D47—41no—
258C4×D47—44no—
259C8×C47—22no—
260C8×C47—41no—
261C47⋊C8—22no—
262C47⋊C8—41no—
263#23759—471nomaximal, Sylow
264C22×C47C2×#12311yescharacteristic
265C22×C47#52511yescharacteristic
266C22×C47#53211yescharacteristic
267C22×C47C2×#13511yescharacteristic
268C22×C47C2×#13511yescharacteristic
269C22×C47C2×#13711yescharacteristic
270C22×C47#57511yescharacteristic
271C2×D47—22no—
272C2×D47—22no—
273C2×D47—22no—
274C2×D47—41no—
275C2×D47—41no—
276C2×D47—41no—
277C2×D47—41no—
278C4×C47—22no—
279C4×C47—42no—
280C4×C47—81no—
281C4×C47—81no—
282C4×C47—81no—
283C4×C47#48514yes—
284C4×C47#52911yescharacteristic
285C4×C47#52911yescharacteristic
286C4×C47#52911yescharacteristic
287C4×C47#52911yescharacteristic
288Dic47—41no—
289Dic47—41no—
290Dic47—42no—
291Dic47—41no—
292Dic47—81no—
293Dic47—81no—
294Dic47—81no—
295C4×C22⋊Q8—471no—
296#2283—471no—
297#2101—471no—
298#2097—471no—
299#2085—471no—
300#2077—471no—
301#2075—471no—
302#2074—471no—
303#2036—471no—
304#2035—471no—
305C4×#107—471no—
306C4×#106—471no—
307C4×#103—471no—
308C4×#103D4711yescharacteristic, complemented
309C4×C22⋊C8—471no—
310C2×C47#239511yescharacteristic
311C2×C47C2×#48511yescharacteristic
312C2×C47#309111yescharacteristic
313C2×C47#310211yescharacteristic
314C2×C47#310811yescharacteristic
315C2×C47#316811yescharacteristic
316C2×C47#317211yescharacteristic
317D47—42no—
318C2×C22⋊Q8—471no—
319C2×C4×Q8C2×D4711yescharacteristic
320C22×C42—471no—
321C2×#107—471no—
322C2×#107—471no—
323C2×#106—471no—
324C2×#106—471no—
325C2×#103—471no—
326C2×#103—471no—
327C2×#103C2×D4711yescharacteristic
328C2×#103C2×D4711yescharacteristic
329C2×C22⋊C8—471no—
330C2×C22⋊C8—471no—
331C2×C4×C8—471no—
332C2×C4×C8C2×D4711yescharacteristic
333#391—471no—
334#389—471no—
335#387—941no—
336#385—941no—
337#383—941no—
338#382—471no—
339#382—471no—
340#381—941no—
341#380—471no—
342#380—471no—
343#380—471no—
344C4×C4⋊C4—471no—
345C4×C4⋊C4—471no—
346C4×C4⋊C4—941no—
347C4×C4⋊C4C2×D4711yescharacteristic
348C4×C22⋊C4—941no—
349C4×C22⋊C4—941no—
350#376—941no—
351#340—471no—
352#340—471no—
353#340—471no—
354#340—471no—
355#340—471no—
356#340—471no—
357#340C2×D4711yescharacteristic
358#340C2×D4711yescharacteristic
359#336—471no—
360#336—471no—
361C47#2375911yescharacteristic, complemented, Sylow
362C22×Q8C22×D4711yescharacteristic
363C23×C4—471no—
364C23×C4—471no—
365C23×C4—471no—
366C22×C8—471no—
367C22×C8—471no—
368C22×C8C22×D4711yescharacteristic
369C22×C8C22×D4711yescharacteristic
370C22⋊Q8—942no—
371C22⋊Q8—942no—
372C4×Q8—22no—
373C4×Q8—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—471no—
376C2×C4⋊C4—471no—
377C2×C4⋊C4—471no—
378C2×C4⋊C4—471no—
379C2×C4⋊C4—941no—
380C2×C4⋊C4—941no—
381C2×C4⋊C4C22×D4711yescharacteristic
382C2×C4⋊C4C22×D4711yescharacteristic
383C2×C4⋊C4C22×D4711yescharacteristic
384C2×C22⋊C4—941no—
385C2×C22⋊C4—941no—
386C2×C22⋊C4—941no—
387C2×C22⋊C4—941no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—471no—
391C2×C42—471no—
392C2×C42—471no—
393C2×C42—942no—
394C2×C42—942no—
395C2×C42—941no—
396C2×C42—941no—
397C2×C42C22×D4711yescharacteristic
398#107—942no—
399#107—942no—
400#106—942no—
401#106—942no—
402#103—22no—
403#103—22no—
404#103—942no—
405#103—942no—
406C22⋊C8—942no—
407C22⋊C8—942no—
408C4×C8—22no—
409C4×C8—942no—
410#95—21no—
411#95—941no—
412#95—941no—
413#95—941no—
414#95—941no—
415#95—941no—
416#95—941no—
417#95—941no—
418#95—941no—
419#95—941no—
420#95—941no—
421#95—941no—
422#95—941no—
423C24—471no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8#1062535012yes—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—471no—
435C22×C4—471no—
436C22×C4—471no—
437C22×C4—942no—
438C22×C4—941no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—941no—
444C22×C4—941no—
445C22×C4—941no—
446C22×C4—941no—
447C22×C4—941no—
448C22×C4—941no—
449C22×C4—941no—
450C22×C4—941no—
451C22×C4—941no—
452C22×C4—1881no—
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×C8#1062535012yes—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—472no—
476C4⋊C4—944no—
477C4⋊C4—941no—
478C4⋊C4—941no—
479C4⋊C4—941no—
480C4⋊C4—941no—
481C4⋊C4—941no—
482C4⋊C4—941no—
483C4⋊C4—941no—
484C4⋊C4—1882no—
485C22⋊C4—944no—
486C22⋊C4—1882no—
487C22⋊C4—1882no—
488C22⋊C4—1882no—
489C42—42no—
490C42—42no—
491C42—942no—
492C42—942no—
493C42—941no—
494C42—941no—
495C42—1882no—
496C42—1882no—
497C42#1062535012yes—
498C42#1062535212yes—
499C23—472no—
500C23—941no—
501C23—941no—
502C23—941no—
503C23—941no—
504C23—941no—
505C23—941no—
506C23C2×D4×D4711yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—941no—
526C2×C4—941no—
527C2×C4—941no—
528C2×C4—942no—
529C2×C4—941no—
530C2×C4—941no—
531C2×C4—941no—
532C2×C4—941no—
533C2×C4—941no—
534C2×C4—941no—
535C2×C4—1884no—
536C2×C4—1881no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4—1881no—
540C2×C4—1881no—
541C2×C4—1881no—
542C2×C4—1881no—
543C2×C4—1881no—
544C2×C4—1881no—
545C2×C4—1881no—
546C2×C4—1881no—
547C2×C4—1881no—
548C2×C4—1881no—
549C2×C4—1881no—
550C2×C4—1881no—
551C2×C4—1884no—
552C2×C4#1309400612yes—
553C2×C4#1309400612yes—
554C2×C4#1309404712yes—
555C2×C4#1309405312yes—
556C2×C4#41×D4711yescharacteristic
557C2×C4#41×D4711yescharacteristic
558C2×C4C2×#1062535211yescharacteristic
559C2×C4C2×#1062535011yescharacteristic
560C2×C4C2×#1062535011yescharacteristic
561C2×C4C2×#1062535011yescharacteristic
562C8—22no—
563C8—41no—
564C8—942no—
565C8—1881no—
566C22—942no—
567C22—942no—
568C22—942no—
569C22—1881no—
570C22—1881no—
571C22—1881no—
572C22—1881no—
573C22C2×#1309400611yescharacteristic
574C22#1568963211yescharacteristic
575C22#1568967011yescharacteristic
576C22#135×D4711yescharacteristic
577C22C2×#1309404711yescharacteristic
578C22C2×#1309405311yescharacteristic
579C22#1568992011yescharacteristic
580C4—22no—
581C4—42no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—3761no—
586C4—3761no—
587C4—3761no—
588C4—1881no—
589C4—1881no—
590C4—1881no—
591C4—1882no—
592C4#1568946314yes—
593C4#1568965111yescharacteristic
594C4#1568965311yescharacteristic
595C4#1568965311yescharacteristic
596C4#1568965711yescharacteristic
597C2—1882no—
598C2#2024664911yescharacteristic
599C2C2×#1568946311yescharacteristic
600C2#2025329911yescharacteristic
601C2#2025341211yescharacteristic
602C2#2025346611yescharacteristic
603C2#2025396111yescharacteristic
604C2#2025399011yescharacteristic
subgroupG/H
centerC22×C4D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC4×#103×C47C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
hypercenterC4×#103D47
hypocenterC47#23759
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
3. #20244960C2semidirect
4. #20244934C2semidirect
5. #20244875C2semidirect
8. #20244809C2semidirect
11. C4×#13093958C2semidirect
12. C4×#13093955C2semidirect
13. C4×#13093948C2semidirect
15. C4×#103×C47C2semidirect
308. C4×#103D47semidirect
361. C47#23759semidirect

Maximal subgroups

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

subgroupindex
C4×#130940002
#202457842
#202449602
#202449342
#202448752
#202448272
#202448132
#202448092
#202445902
#202445822
C4×#130939582
C4×#130939552
C4×#130939482
C4×(C2×D47)⋊C82
C4×#103×C472
#2375947