Finite Group Databasebeta

← #26226807

Subgroups — #26226807

Order 12032.

834 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689175C211yesmaximal, characteristic
2#20245784C211yesmaximal, characteristic
3#382×D47C211yesmaximal, characteristic
4#20244158C211yesmaximal, characteristic, complemented
5#20244156C211yesmaximal, characteristic, complemented
6#20244154C211yesmaximal, characteristic, complemented
7#20243582C211yesmaximal, characteristic
8#20243579C211yesmaximal, characteristic
9#20243577C211yesmaximal, characteristic
10#20243574C211yesmaximal, characteristic
11#20243551C211yesmaximal, characteristic, complemented
12#20243549C211yesmaximal, characteristic, complemented
13#20243517C211yesmaximal, characteristic, complemented
14#20243517C211yesmaximal, characteristic, complemented
15C47×#1949C211yesmaximal, characteristic, complemented
16C2×C4⋊C4×D47C2211yescharacteristic
17C2×C4⋊C4×D47C2211yescharacteristic
18C2×C42×D47C2211yescharacteristic
19#15689175C418yescomplemented
20C2×#13093958C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
22C2×#13093957C2211yescharacteristic
23C2×#13093956C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×(C2×D47)⋊C8C2211yescharacteristic
28C2×(C2×D47)⋊C8C2211yescharacteristic
29C2×C8×Dic47C2211yescharacteristic
30C2×C8×Dic47C2211yescharacteristic
31#15688990—21no—
32#15688989C2211yescharacteristic
33#15688988C2211yescharacteristic
34#15688988C2211yescharacteristic
35#15688988C2211yescharacteristic
36#15688988C2211yescharacteristic
37#15688987—21no—
38#15688987—21no—
39#15688985—21no—
40#15688980C2211yescharacteristic
41#15688980C2211yescharacteristic
42C4×C4⋊Dic47C2211yescharacteristic
43C4×(C2×D47)⋊C4—21no—
44#15688969—21no—
45#95×D47—21no—
46#95×D47—21no—
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C2×#107×C47C2211yescharacteristic
54C2×#106×C47C2211yescharacteristic
55C2×C4⋊C8×C47C2211yescharacteristic
56C47×#382C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×C4×D47—21no—
61C22×C4×D47—21no—
62C22×C4×D47D411yescharacteristic
63C22×C4×D47D411yescharacteristic
64C22×C4×D47C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C4⋊C4×D47—41no—
74C4⋊C4×D47—41no—
75C4⋊C4×D47—41no—
76C4⋊C4×D47—41no—
77C2×(C2×D47)⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47D411yescharacteristic
81C2×C4×Dic47D411yescharacteristic
82C2×C4×Dic47C2311yescharacteristic
83C42×D47C2×C414yes—
84C42×D47Q814yes—
85#13093958—41no—
86#13093957—41no—
87#13093957—41no—
88#13093956—41no—
89#13093955—41no—
90#13093955—41no—
91C4⋊(C47⋊C8)C2×C414yescomplemented
92(C2×D47)⋊C8C2×C414yes—
93(C2×D47)⋊C8C2×C414yes—
94C8×Dic47C2×C414yescomplemented
95C8×Dic47C2×C414yescomplemented
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C22×C8×C47C2311yescharacteristic
110C22×C8×C47C2311yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C42×C47—21no—
116C2×C42×C47C2311yescharacteristic
117#107×C47—41no—
118#106×C47—41no—
119C4⋊C8×C47C2×C414yescomplemented
120#95×C47—21no—
121#95×C47—21no—
122#95×C47—21no—
123C23×D47C2×D411yescharacteristic
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47C2×D411yescharacteristic
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2411yescharacteristic
131C22×Dic47—21no—
132C22×Dic47—21no—
133C22×Dic47—21no—
134C22×Dic47—21no—
135C22×Dic47C2×D411yescharacteristic
136C22×Dic47C2×D411yescharacteristic
137C22×Dic47C2×D411yescharacteristic
138C2×C4×D47—22no—
139C2×C4×D47—22no—
140C2×C4×D47—21no—
141C2×C4×D47—21no—
142C2×C4×D47—21no—
143C2×C4×D47—21no—
144C2×C4×D47—41no—
145C2×C4×D47—41no—
146C2×C4×D47—42no—
147C2×C4×D47—42no—
148C2×C4×D47—42no—
149C2×C4×D47—42no—
150C2×C4×D47—42no—
151C2×C4×D47—42no—
152C2×C4×D47C4⋊C414yes—
153C2×C4×D47C4⋊C414yes—
154C2×C4×D47C22×C412yes—
155C2×C4×D47C2×Q812yes—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47C22×C412yes—
161C2×C8×C47C22×C412yes—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C42×C47—42no—
171C42×C47C22×C412yes—
172C42×C47C2×Q812yes—
173(C2×D47)⋊C4—42no—
174(C2×D47)⋊C4—42no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183Dic47⋊C4—42no—
184Dic47⋊C4—42no—
185C4×Dic47—42no—
186C4×Dic47C4⋊C414yes—
187C4×Dic47C4⋊C414yes—
188C4×Dic47C22×C412yes—
189C4×Dic47C2×Q812yes—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—21no—
192C2×C47⋊C8—21no—
193C2×C47⋊C8—21no—
194C2×C47⋊C8C22×C412yes—
195C2×C47⋊C8C22×C412yes—
196C23×C47C22×D411yescharacteristic
197C22×D47—21no—
198C22×D47—21no—
199C22×D47—21no—
200C22×D47—21no—
201C22×D47C2×C4⋊C412yes—
202C22×D47#13512yes—
203C22×D47#13512yes—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—41no—
209C2×C4×C47—41no—
210C2×C4×C47—41no—
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×C47C2×C4⋊C412yes—
221C2×C4×C47C2×C4⋊C412yes—
222C2×C4×C47C23×C411yescharacteristic
223C2×C4×C47C22×Q811yescharacteristic
224C2×C4×C47H(5,𝔽2)11yescharacteristic
225C2×C4×C47H(5,𝔽2)11yescharacteristic
226C2×C4×C472−1+411yescharacteristic
227C2×C4×C472−1+411yescharacteristic
228C2×Dic47—21no—
229C2×Dic47—21no—
230C2×Dic47—21no—
231C2×Dic47—21no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—41no—
235C2×Dic47—41no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47C2×C4⋊C412yes—
247C2×Dic47C2×C4⋊C412yes—
248C2×Dic47C2×C4⋊C412yes—
249C4×D47—28no—
250C4×D47—22no—
251C4×D47—22no—
252C4×D47—22no—
253C4×D47—22no—
254C4×D47—44no—
255C4×D47—84no—
256C4×D47—84no—
257C8×C47—22no—
258C8×C47—22no—
259C47⋊C8—22no—
260C47⋊C8—22no—
261#21983—471nomaximal, Sylow
262C22×C47C22×C4⋊C411yescharacteristic
263C22×C47#53511yescharacteristic
264C22×C47#53611yescharacteristic
265C22×C47C2×#13511yescharacteristic
266C22×C47C2×#13511yescharacteristic
267C22×C47#57511yescharacteristic
268C22×C47#57511yescharacteristic
269C2×D47—22no—
270C2×D47—22no—
271C2×D47—22no—
272C2×D47—22no—
273C2×D47—22no—
274C2×D47—22no—
275C2×D47#42714yes—
276C4×C47—22no—
277C4×C47—22no—
278C4×C47—81no—
279C4×C47—81no—
280C4×C47—81no—
281C4×C47—81no—
282C4×C47#52011yescharacteristic
283C4×C47#52011yescharacteristic
284C4×C47#54411yescharacteristic
285C4×C47#54411yescharacteristic
286Dic47—22no—
287Dic47—22no—
288Dic47—22no—
289Dic47—81no—
290Dic47—81no—
291Dic47—81no—
292Dic47—81no—
293C2×#431—471no—
294#2283—471no—
295C2×#382—471no—
296#1949—471no—
297#1949—471no—
298#1949—471no—
299#1949D4711yescharacteristic, complemented
300#1853—471no—
301#1853—471no—
302#1852—471no—
303#1852—471no—
304#1846—471no—
305#1846—471no—
306#1835—471no—
307#1835—471no—
308C2×C47#231711yescharacteristic
309C2×C47C2×#42711yescharacteristic
310C2×C47#291111yescharacteristic
311C2×C47#321111yescharacteristic
312C2×C47#321111yescharacteristic
313C2×C47#323211yescharacteristic
314C2×C47#323211yescharacteristic
315D47—24no—
316C22×C4⋊C4—471no—
317C22×C4⋊C4—471no—
318C22×C42—471no—
319#431—478no—
320C2×#107—471no—
321C2×#107—471no—
322C2×#107—471no—
323C2×#107C2×D4711yescharacteristic
324C2×#106—471no—
325C2×#106—471no—
326C2×#106—471no—
327C2×#106C2×D4711yescharacteristic
328C2×C4⋊C8—471no—
329C2×C4⋊C8C2×D4711yescharacteristic
330C2×C22⋊C8—471no—
331C2×C22⋊C8—471no—
332C2×C4×C8—471no—
333C2×C4×C8—471no—
334#383—941no—
335#382—471no—
336#382—471no—
337#382—471no—
338#382—471no—
339#382—471no—
340#382—941no—
341#382—941no—
342#382C2×D4711yescharacteristic
343#381—941no—
344#380—471no—
345#380—471no—
346C4×C4⋊C4—471no—
347C4×C4⋊C4C2×D4711yescharacteristic
348C4×C22⋊C4—941no—
349#376—941no—
350C2×#95—941no—
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
360C47#2198311yescharacteristic, complemented, Sylow
361C23×C4—471no—
362C23×C4—471no—
363C23×C4—471no—
364C23×C4—941no—
365C23×C4—941no—
366C22×C8—471no—
367C22×C8—471no—
368C22×C8C22×D4711yescharacteristic
369C22×C8C22×D4711yescharacteristic
370C2×C4⋊C4—471no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—471no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4—1881no—
377C2×C4⋊C4—1881no—
378C2×C4⋊C4—1881no—
379C2×C4⋊C4—1881no—
380C2×C4⋊C4C22×D4711yescharacteristic
381C2×C4⋊C4C22×D4711yescharacteristic
382C2×C4⋊C4C22×D4711yescharacteristic
383C2×C4⋊C4C22×D4711yescharacteristic
384C2×C22⋊C4—941no—
385C2×C22⋊C4—941no—
386C2×C42—21no—
387C2×C42—471no—
388C2×C42—471no—
389C2×C42—471no—
390C2×C42—474no—
391C2×C42—474no—
392C2×C42—941no—
393C2×C42C22×D4711yescharacteristic
394#107—41no—
395#107—1881no—
396#107—1881no—
397#107—1881no—
398#106—41no—
399#106—1881no—
400#106—1881no—
401#106—1881no—
402C4⋊C8—474no—
403C4⋊C8C4×D4714yescomplemented
404C22⋊C8—474no—
405C22⋊C8—474no—
406C4×C8—474no—
407C4×C8—474no—
408#95—21no—
409#95—21no—
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—
423#95—941no—
424C24—471no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—471no—
430C22×C4—471no—
431C22×C4—471no—
432C22×C4—474no—
433C22×C4—472no—
434C22×C4—472no—
435C22×C4—474no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—941no—
439C22×C4—941no—
440C22×C4—941no—
441C22×C4—941no—
442C22×C4—941no—
443C22×C4—941no—
444C22×C4—941no—
445C22×C4—941no—
446C22×C4—1881no—
447C22×C4—1881no—
448C22×C4—1882no—
449C22×C4—1882no—
450C22×C4—1882no—
451C22×C4—1882no—
452C22×C4—1882no—
453C22×C4—1882no—
454C22×C4D4×D4711yescharacteristic
455C22×C4D4×D4711yescharacteristic
456C22×C4C23×D4711yescharacteristic
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—472no—
462C2×C8—472no—
463C2×C8—941no—
464C2×C8—941no—
465C2×C8—941no—
466C2×C8—941no—
467C2×C8C2×C4×D4712yes—
468C2×C8C2×C4×D4712yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—941no—
478C4⋊C4—941no—
479C4⋊C4—941no—
480C4⋊C4—941no—
481C4⋊C4—941no—
482C4⋊C4—941no—
483C4⋊C4—941no—
484C4⋊C4—941no—
485C4⋊C4—1882no—
486C4⋊C4—1882no—
487C22⋊C4—1882no—
488C22⋊C4—1882no—
489C42—42no—
490C42—474no—
491C42—474no—
492C42—472no—
493C42—472no—
494C42—1882no—
495C42C2×C4×D4712yes—
496C42Q8×D4712yes—
497C23—472no—
498C23—472no—
499C23—472no—
500C23—941no—
501C23—941no—
502C23—941no—
503C23—941no—
504C23C2×D4×D4711yescharacteristic
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—472no—
522C2×C4—472no—
523C2×C4—472no—
524C2×C4—948no—
525C2×C4—941no—
526C2×C4—941no—
527C2×C4—941no—
528C2×C4—941no—
529C2×C4—941no—
530C2×C4—941no—
531C2×C4—942no—
532C2×C4—942no—
533C2×C4—942no—
534C2×C4—942no—
535C2×C4—3764no—
536C2×C4—3764no—
537C2×C4—1884no—
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×C4C4⋊C4×D4712yes—
551C2×C4C4⋊C4×D4712yes—
552C2×C4C22×C4×D4711yescharacteristic
553C2×C4C2×Q8×D4711yescharacteristic
554C2×C4#1309406911yescharacteristic
555C2×C4#1309407011yescharacteristic
556C2×C4#1309407111yescharacteristic
557C2×C4#1309407211yescharacteristic
558C8—22no—
559C8—22no—
560C8—942no—
561C8—942no—
562C22—474no—
563C22—942no—
564C22—942no—
565C22—942no—
566C22—942no—
567C22—942no—
568C22—942no—
569C22C2×C4⋊C4×D4711yescharacteristic
570C22#1568968911yescharacteristic
571C22#1568969611yescharacteristic
572C22#135×D4711yescharacteristic
573C22#135×D4711yescharacteristic
574C22#1568991911yescharacteristic
575C22#1568992211yescharacteristic
576C4—22no—
577C4—22no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—942no—
583C4—942no—
584C4—942no—
585C4—3761no—
586C4—3761no—
587C4—3761no—
588C4—3761no—
589C4#1568960211yescharacteristic
590C4#1568960511yescharacteristic
591C4#1568975011yescharacteristic
592C4#1568975311yescharacteristic
593C2—944no—
594C2#2024601011yescharacteristic
595C2#427×D4711yescharacteristic
596C2#2025197511yescharacteristic
597C2#2025435111yescharacteristic
598C2#2025435711yescharacteristic
599C2#2025455411yescharacteristic
600C2#2025456011yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1949C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1949D47
hypocenterC47#21983
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. #20244158C2semidirect
5. #20244156C2semidirect
6. #20244154C2semidirect
11. #20243551C2semidirect
12. #20243549C2semidirect
13. #20243517C2semidirect
14. #20243517C2semidirect
15. C47×#1949C2semidirect
19. #15689175C4semidirect
91. C4⋊(C47⋊C8)C2×C4semidirect
94. C8×Dic47C2×C4semidirect
95. C8×Dic47C2×C4semidirect
119. C4⋊C8×C47C2×C4semidirect
299. #1949D47semidirect
360. C47#21983semidirect
403. C4⋊C8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156891752
#202457842
#382×D472
#202441582
#202441562
#202441542
#202435822
#202435792
#202435772
#202435742
#202435512
#202435492
#202435172
C47×#19492
#2198347