Finite Group Databasebeta

← #26316393

Subgroups — #26316393

Order 12032.

776 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20249218C211yesmaximal, characteristic, complemented
2#20244946C211yesmaximal, characteristic, complemented
3#20244921C211yesmaximal, characteristic, complemented
4#20244726C211yesmaximal, characteristic, complemented
5#20244534C211yesmaximal, characteristic, complemented
6#20243973C211yesmaximal, characteristic, complemented
7#20243966C211yesmaximal, characteristic, complemented
8#20243827C211yesmaximal, characteristic
9#20243817C211yesmaximal, characteristic
10#20243649C211yesmaximal, characteristic, complemented
11#20243635C211yesmaximal, characteristic, complemented
12#20243419C211yesmaximal, characteristic, complemented
13#20243411C211yesmaximal, characteristic, complemented
14#20243085C211yesmaximal, characteristic, complemented
15C47×#2324C211yesmaximal, characteristic
16C22×#10625329C2211yescharacteristic
17C23×C47⋊C8C2211yescharacteristic
18C2×#13094016C2211yescharacteristic
19C2×#13093957C2211yescharacteristic
20C2×#13093955C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093945C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27#15689056—21no—
28#15689056—21no—
29#15689046—21no—
30#15689046—21no—
31#15689021C2211yescharacteristic
32#15689021C2211yescharacteristic
33#15688993C2211yescharacteristic
34#15688983C2211yescharacteristic
35#15688983C2211yescharacteristic
36#15688885C2211yescharacteristic
37#15688885C2211yescharacteristic
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688878C2211yescharacteristic
45#15688878C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C2×#121×C47C2211yescharacteristic
49C2×C4×D4×C47C2211yescharacteristic
50C2×#117×C47C2211yescharacteristic
51C47×#390—21no—
52C47×#390C2211yescharacteristic
53C47×#389C2211yescharacteristic
54C47×#388—21no—
55C47×#386—21no—
56C47×#385—21no—
57C47×#380C2211yescharacteristic
58C4×C4⋊C4×C47C2211yescharacteristic
59C2×#10625329—22no—
60C2×#10625329—22no—
61C2×#10625329—41no—
62C22×C47⋊C8—22no—
63C22×C47⋊C8—22no—
64C22×C47⋊C8—41no—
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69#13094016—41no—
70#13094016—41no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47C2311yescharacteristic
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×C22⋊Dic47—21no—
82C2×C22⋊Dic47—21no—
83#13093957—41no—
84#13093955—22no—
85#13093955—22no—
86#13093955—41no—
87#13093945—22no—
88#13093945—22no—
89#13093945—41no—
90#13093945—41no—
91C47⋊#95—21no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C22×D4×C47—21no—
96C22×D4×C47C2311yescharacteristic
97C23×C4×C47—21no—
98C23×C4×C47C2311yescharacteristic
99#121×C47—22no—
100#121×C47Q814yes—
101C4×D4×C47—41no—
102C4×D4×C47—41no—
103#117×C47—41no—
104#117×C47—41no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47D411yescharacteristic
107C2×C4⋊C4×C47D411yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C22⋊C4×C47—21no—
112C2×C22⋊C4×C47—21no—
113C2×C22⋊C4×C47—21no—
114C2×C22⋊C4×C47—21no—
115C2×C22⋊C4×C47—21no—
116C2×C22⋊C4×C47D411yescharacteristic
117C2×C22⋊C4×C47D411yescharacteristic
118C2×C42×C47—21no—
119C2×C42×C47C2311yescharacteristic
120C2×C42×C47C2311yescharacteristic
121#95×C47—21no—
122#95×C47—21no—
123#95×C47—21no—
124C24×C47—21no—
125C24×C47C2×D411yescharacteristic
126C2×D4×C47—24no—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47—81no—
133C2×D4×C47—81no—
134C2×D4×C47C2×Q812yes—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—42no—
143C22×C4×C47—81no—
144C22×C4×C47—81no—
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×Q812yes—
151C22×C4×C47#4112yes—
152C22×C4×C47C2411yescharacteristic
153C22×Dic47—21no—
154C22×Dic47—21no—
155C22×Dic47—21no—
156C22×Dic47—21no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—41no—
163C4⋊C4×C47—41no—
164C4⋊C4×C47—81no—
165C4⋊C4×C47C2×Q812yes—
166C22⋊C4×C47—24no—
167C22⋊C4×C47—41no—
168C22⋊C4×C47—41no—
169C22⋊C4×C47—42no—
170C22⋊C4×C47—81no—
171C22⋊C4×C47—81no—
172C22⋊C4×C47—81no—
173C42×C47—21no—
174C42×C47—21no—
175C42×C47—21no—
176C42×C47—21no—
177C42×C47—81no—
178C22⋊Dic47—81no—
179C22⋊Dic47—81no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—21no—
188C4⋊Dic47—81no—
189C4⋊Dic47—81no—
190Dic47⋊C4—42no—
191Dic47⋊C4—42no—
192Dic47⋊C4—81no—
193Dic47⋊C4—81no—
194#10625329—44no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—44no—
206C2×C47⋊C8#4112yes—
207C2×C47⋊C8#4112yes—
208C23×C47—21no—
209C23×C47—21no—
210C23×C47—21no—
211C23×C47—21no—
212C23×C47—21no—
213C23×C47—21no—
214C23×C47—22no—
215C23×C47—41no—
216C23×C47—41no—
217C23×C47—41no—
218C23×C47—41no—
219C23×C47—41no—
220C23×C47—41no—
221C23×C47C22⋊Q812yes—
222C23×C47C22×D411yescharacteristic
223D4×C47—22no—
224D4×C47—22no—
225D4×C47—41no—
226D4×C47—41no—
227D4×C47—84no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—21no—
246C2×C4×C47—21no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—84no—
256C2×C4×C47C22⋊Q812yes—
257C2×C4×C47C22×Q811yescharacteristic
258C2×C4×C47C2×#4111yescharacteristic
259C2×C4×C47C2×#4111yescharacteristic
260C2×C4×C47C2×#4111yescharacteristic
261C2×C4×C47H(5,𝔽2)11yescharacteristic
262C2×C4×C47H(5,𝔽2)11yescharacteristic
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C47⋊C8—22no—
276C47⋊C8—22no—
277C47⋊C8—41no—
278C47⋊C8—41no—
279#29168—471nomaximal, Sylow
280C22×C47—22no—
281C22×C47—22no—
282C22×C47—22no—
283C22×C47—41no—
284C22×C47—41no—
285C22×C47—41no—
286C22×C47—41no—
287C22×C47—42no—
288C22×C47—42no—
289C22×C47—42no—
290C22×C47—81no—
291C22×C47—81no—
292C22×C47—81no—
293C22×C47—81no—
294C22×C47C2×C22⋊Q811yescharacteristic
295C22×C47#53211yescharacteristic
296C22×C47#53411yescharacteristic
297C22×C47#57611yescharacteristic
298C22×C47#57711yescharacteristic
299C22×C47#57711yescharacteristic
300C22×C47#57711yescharacteristic
301C4×C47—22no—
302C4×C47—42no—
303C4×C47—41no—
304C4×C47—41no—
305C4×C47—41no—
306C4×C47—41no—
307C4×C47—81no—
308C4×C47—81no—
309C4×C47#55111yescharacteristic
310C4×C47#55111yescharacteristic
311C4×C47#55411yescharacteristic
312C4×C47#55411yescharacteristic
313Dic47—81no—
314Dic47—81no—
315Dic47—81no—
316Dic47—81no—
317#2629—471no—
318#2324D4711yescharacteristic
319#2099—471no—
320#2094—471no—
321#2059—471no—
322#2025—471no—
323#1920—471no—
324#1919—471no—
325#1895—471no—
326#1893—471no—
327#1864—471no—
328#1862—471no—
329#1809—471no—
330#1807—471no—
331#1729—471no—
332C2×C47—42no—
333C2×C47—82no—
334C2×C47#256411yescharacteristic
335C2×C47#308111yescharacteristic
336C2×C47#308411yescharacteristic
337C2×C47#317411yescharacteristic
338C2×C47#317911yescharacteristic
339C2×C47#319811yescharacteristic
340C2×C47#320411yescharacteristic
341C22×M16—471no—
342C23×C8—471no—
343C2×#125—471no—
344C2×#121C2×D4711yescharacteristic
345C2×C4×D4C2×D4711yescharacteristic
346C2×#117C2×D4711yescharacteristic
347C2×#107—471no—
348C2×#106—471no—
349C2×#106—471no—
350C2×#106—471no—
351C2×#102—471no—
352C2×#102—471no—
353C2×#102—471no—
354C2×#102—471no—
355#398—941no—
356#398—941no—
357#396—941no—
358#396—941no—
359#390—21no—
360#390C2×D4711yescharacteristic
361#389—471no—
362#389—471no—
363#389C2×D4711yescharacteristic
364#388—21no—
365#386—21no—
366#385—21no—
367#384—471no—
368#380—471no—
369#380—471no—
370#380C2×D4711yescharacteristic
371C4×C4⋊C4C2×D4711yescharacteristic
372#340—471no—
373#340—471no—
374#340—471no—
375#340—471no—
376#340—471no—
377#340—471no—
378#340—471no—
379#340—471no—
380#336—471no—
381#336—471no—
382#336—471no—
383#336—471no—
384C47#2916811yescharacteristic, complemented, Sylow
385C22×D4—21no—
386C22×D4C22×D4711yescharacteristic
387C23×C4—21no—
388C23×C4C22×D4711yescharacteristic
389C2×M16—942no—
390C2×M16—942no—
391C2×M16—1881no—
392C22×C8—471no—
393C22×C8—471no—
394C22×C8—471no—
395C22×C8—471no—
396C22×C8—942no—
397C22×C8—942no—
398C22×C8—1881no—
399#125—1881no—
400#125—1881no—
401#121—22no—
402#121Dic9414yes—
403C4×D4—41no—
404C4×D4—41no—
405#117—41no—
406#117—41no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—471no—
409C2×C4⋊C4—471no—
410C2×C4⋊C4—471no—
411C2×C4⋊C4—471no—
412C2×C4⋊C4—941no—
413C2×C4⋊C4—941no—
414C2×C4⋊C4—941no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4—941no—
417C2×C4⋊C4—941no—
418C2×C4⋊C4C22⋊D4711yescharacteristic
419C2×C4⋊C4C22⋊D4711yescharacteristic
420C2×C4⋊C4C22×D4711yescharacteristic
421C2×C4⋊C4C22×D4711yescharacteristic
422C2×C4⋊C4C22×D4711yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—941no—
429C2×C22⋊C4—941no—
430C2×C22⋊C4C22⋊D4711yescharacteristic
431C2×C22⋊C4C22⋊D4711yescharacteristic
432C2×C42—21no—
433C2×C42C22×D4711yescharacteristic
434C2×C42C22×D4711yescharacteristic
435#107—1881no—
436#106—942no—
437#106—942no—
438#106—1881no—
439#102—942no—
440#102—942no—
441#102—1881no—
442#102—1881no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—941no—
447#95—941no—
448#95—941no—
449#95—941no—
450C24—21no—
451C24C2×C22⋊D4711yescharacteristic
452C2×D4—24no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—81no—
459C2×D4—81no—
460C2×D4C2×Dic9412yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—42no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4—941no—
472C22×C4—941no—
473C22×C4—941no—
474C22×C4—941no—
475C22×C4C2×Dic9412yes—
476C22×C4#1062535012yes—
477C22×C4C2×C22⋊D4711yescharacteristic
478C22×C4C2×C22⋊D4711yescharacteristic
479C22×C4C2×C22⋊D4711yescharacteristic
480C22×C4C2×C22⋊D4711yescharacteristic
481C22×C4C2×C22⋊D4711yescharacteristic
482C22×C4C23×D4711yescharacteristic
483M16—1884no—
484C2×C8—472no—
485C2×C8—472no—
486C2×C8—941no—
487C2×C8—941no—
488C2×C8—941no—
489C2×C8—941no—
490C2×C8—941no—
491C2×C8—941no—
492C2×C8—941no—
493C2×C8—941no—
494C2×C8—941no—
495C2×C8—941no—
496C2×C8—1884no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—81no—
505C4⋊C4—941no—
506C4⋊C4—941no—
507C4⋊C4—941no—
508C4⋊C4—941no—
509C4⋊C4—941no—
510C4⋊C4—941no—
511C4⋊C4—941no—
512C4⋊C4—941no—
513C4⋊C4—3761no—
514C4⋊C4—3761no—
515C4⋊C4—3761no—
516C4⋊C4—3761no—
517C4⋊C4—1882no—
518C4⋊C4—1882no—
519C4⋊C4C2×Dic9412yes—
520C22⋊C4—24no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—42no—
524C22⋊C4—81no—
525C22⋊C4—81no—
526C22⋊C4—81no—
527C22⋊C4—3761no—
528C22⋊C4—3761no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—81no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—22no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23#1309400312yes—
548C23C22×C22⋊D4711yescharacteristic
549D4—22no—
550D4—22no—
551D4—41no—
552D4—41no—
553D4—84no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—84no—
582C2×C4—1881no—
583C2×C4—1881no—
584C2×C4—1881no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4—1881no—
589C2×C4—1881no—
590C2×C4—1881no—
591C2×C4—1881no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4#1309400312yes—
595C2×C4C22×Dic9411yescharacteristic
596C2×C4#41×D4711yescharacteristic
597C2×C4#41×D4711yescharacteristic
598C2×C4C2×#1062535011yescharacteristic
599C2×C4#1309406911yescharacteristic
600C2×C4#1309406911yescharacteristic
601C8—942no—
602C8—942no—
603C8—1881no—
604C8—1881no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22C2×#1309400311yescharacteristic
620C22#1568966911yescharacteristic
621C22#1568968211yescharacteristic
622C22#1568992611yescharacteristic
623C22#1568993111yescharacteristic
624C22#1568993411yescharacteristic
625C22#1568993411yescharacteristic
626C4—22no—
627C4—41no—
628C4—41no—
629C4—42no—
630C4—41no—
631C4—41no—
632C4—81no—
633C4—81no—
634C4—3761no—
635C4—3761no—
636C4—3761no—
637C4—3761no—
638C4#1568980211yescharacteristic
639C4#1568980211yescharacteristic
640C4#1568983211yescharacteristic
641C4#1568983211yescharacteristic
642C2—42no—
643C2—82no—
644C2#2024856411yescharacteristic
645C2#2025321911yescharacteristic
646C2#2025325011yescharacteristic
647C2#2025401011yescharacteristic
648C2#2025406511yescharacteristic
649C2#2025424011yescharacteristic
650C2#2025429411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2324C2
socleC23×C47C22×D4
prime radicalC2×#121×C47C22
hypercenter#2324D47
hypocenterC47#29168
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2324

Complements

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

NH ≅ G/Ntype
1. #20249218C2semidirect
2. #20244946C2semidirect
3. #20244921C2semidirect
4. #20244726C2semidirect
5. #20244534C2semidirect
6. #20243973C2semidirect
7. #20243966C2semidirect
10. #20243649C2semidirect
11. #20243635C2semidirect
12. #20243419C2semidirect
13. #20243411C2semidirect
14. #20243085C2semidirect
384. C47#29168semidirect

Maximal subgroups

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

subgroupindex
#202492182
#202449462
#202449212
#202447262
#202445342
#202439732
#202439662
#202438272
#202438172
#202436492
#202436352
#202434192
#202434112
#202430852
C47×#23242
#2916847