Finite Group Databasebeta

← #26220589

Subgroups — #26220589

Order 12032.

950 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689233C211yesmaximal, characteristic
2#20245852C211yesmaximal, characteristic, complemented
3#20244263C211yesmaximal, characteristic
4#20244253C211yesmaximal, characteristic
5#20244128C211yesmaximal, characteristic, complemented
6#20244113C211yesmaximal, characteristic, complemented
7#20244088C211yesmaximal, characteristic, complemented
8#20244084C211yesmaximal, characteristic, complemented
9#20244074C211yesmaximal, characteristic, complemented
10#20244070C211yesmaximal, characteristic, complemented
11#20243286C211yesmaximal, characteristic, complemented
12#20243216C211yesmaximal, characteristic, complemented
13#20243213C211yesmaximal, characteristic, complemented
14C4×#13093935C211yesmaximal, characteristic, complemented
15C47×#2293C211yesmaximal, characteristic, complemented
16C22×#10625334C2211yescharacteristic
17C2×C4×D188C2211yescharacteristic
18C2×#13093981—21no—
19#15689233C418yescomplemented
20C2×#13093957C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093945C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093944C2211yescharacteristic
27C2×#13093944C2211yescharacteristic
28C2×#13093944C2211yescharacteristic
29C2×#13093935C2211yescharacteristic
30C2×#13093935C2211yescharacteristic
31C2×#13093935C2211yescharacteristic
32C2×#13093935C2211yescharacteristic
33C2×C4×C47⋊C8C2211yescharacteristic
34C2×C4×C47⋊C8C2211yescharacteristic
35#15689025C2211yescharacteristic
36#15689025C2211yescharacteristic
37#15689021C2211yescharacteristic
38#15689021C2211yescharacteristic
39#15689014—21no—
40#15689014—21no—
41C4×C4⋊Dic47C2211yescharacteristic
42C4×(C2×D47)⋊C4—21no—
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49C2×C4×D4×C47C2211yescharacteristic
50C2×#117×C47—21no—
51C47×#390C2211yescharacteristic
52C47×#390C2211yescharacteristic
53C47×#389C2211yescharacteristic
54C47×#389C2211yescharacteristic
55C47×#388—21no—
56C47×#388—21no—
57C4×C4⋊C4×C47C2211yescharacteristic
58C4×C22⋊C4×C47—21no—
59C43×C47C2211yescharacteristic
60C22×D188C2311yescharacteristic
61C22×C4×D47—21no—
62C2×#10625334—22no—
63C2×#10625334—22no—
64C2×#10625334C2×C414yes—
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C4×D188—22no—
70C4×D188C2×C414yes—
71#13093981—44no—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×(C2×D47)⋊C4—21no—
81C2×(C2×D47)⋊C4—21no—
82C2×C4×Dic47—21no—
83#13093957—22no—
84#13093955—22no—
85#13093955C2×C414yes—
86#13093945—22no—
87#13093945—22no—
88#13093945C2×C414yes—
89#13093944—22no—
90#13093944—22no—
91#13093944C2×C414yes—
92#13093935—22no—
93#13093935—22no—
94#13093935—22no—
95#13093935C2×C414yes—
96C4×C47⋊C8—22no—
97C4×C47⋊C8—22no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C22×D4×C47C2311yescharacteristic
102C23×C4×C47—21no—
103C4×D4×C47—22no—
104C4×D4×C47C2×C414yes—
105#117×C47—44no—
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47—21no—
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×C42×C47—21no—
117C2×C42×C47D411yescharacteristic
118C2×C42×C47D411yescharacteristic
119C2×C42×C47D411yescharacteristic
120C2×C42×C47D411yescharacteristic
121C2×C42×C47C2311yescharacteristic
122C2×C42×C47C2311yescharacteristic
123C2×C42×C47C2311yescharacteristic
124#95×C47—21no—
125#95×C47—21no—
126#95×C47—21no—
127C24×C47—21no—
128C23×D47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47—21no—
133C2×D4×C47—21no—
134C2×D4×C47C22×C412yes—
135C22×C4×C47—21no—
136C22×C4×C47—24no—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—42no—
140C22×C4×C47—42no—
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2411yescharacteristic
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C2×D188—21no—
152C2×D188—21no—
153C2×D188—21no—
154C2×D188—21no—
155C2×D188—21no—
156C2×D188C22×C412yes—
157C2×C4×D47—24no—
158C2×C4×D47—42no—
159C2×C4×D47—42no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—42no—
166C4⋊C4×C47—42no—
167C4⋊C4×C47C22×C412yes—
168C22⋊C4×C47—24no—
169C22⋊C4×C47—42no—
170C22⋊C4×C47—42no—
171C22⋊C4×C47—42no—
172C22⋊C4×C47—42no—
173C42×C47—22no—
174C42×C47—22no—
175C42×C47—21no—
176C42×C47—21no—
177C42×C47—21no—
178C42×C47—21no—
179C42×C47—41no—
180C42×C47—41no—
181C42×C47—42no—
182C42×C47C22×C412yes—
183C42×C47#4112yes—
184#10625334—28no—
185(C2×D47)⋊C4—24no—
186(C2×D47)⋊C4—42no—
187(C2×D47)⋊C4—42no—
188(C2×D47)⋊C4—42no—
189(C2×D47)⋊C4—42no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—21no—
193C4⋊Dic47—21no—
194C4⋊Dic47—21no—
195C4⋊Dic47C22×C412yes—
196Dic47⋊C4—42no—
197Dic47⋊C4—42no—
198C4×Dic47—42no—
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⋊C8C22×C412yes—
206C2×C47⋊C8C22×C412yes—
207C2×C47⋊C8#4112yes—
208C2×C47⋊C8#4112yes—
209C2×C47⋊C8#4112yes—
210C2×C47⋊C8#4112yes—
211C23×C47—22no—
212C23×C47—41no—
213C23×C47—41no—
214C23×C47—41no—
215C23×C47—41no—
216C23×C47—41no—
217C23×C47—41no—
218C23×C47C22×D411yescharacteristic
219C22×D47—22no—
220C22×D47—41no—
221C22×D47—41no—
222C22×D47—41no—
223C22×D47—41no—
224C22×D47—41no—
225C22×D47—41no—
226D4×C47—22no—
227D4×C47—22no—
228D4×C47—41no—
229D4×C47—41no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—22no—
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—48no—
246C2×C4×C47—41no—
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×C47C4×D412yes—
255C2×C4×C47C4×D412yes—
256C2×C4×C47AGL(1,ℤ/8)12yes—
257C2×C4×C47AGL(1,ℤ/8)12yes—
258C2×C4×C47C23×C411yescharacteristic
259C2×C4×C47C2×#4111yescharacteristic
260C2×C4×C47C2×#4111yescharacteristic
261C2×C4×C47C2×#4111yescharacteristic
262C2×C4×C47C2×#4111yescharacteristic
263C2×C4×C47C2×#4111yescharacteristic
264C2×Dic47—22no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273D188—22no—
274D188—22no—
275D188—41no—
276D188—41no—
277C4×D47—48no—
278C47⋊C8—22no—
279C47⋊C8—22no—
280C47⋊C8—22no—
281C47⋊C8#11714yescomplemented
282#21286—471nomaximal, Sylow
283C22×C47—42no—
284C22×C47—42no—
285C22×C47—42no—
286C22×C47—81no—
287C22×C47—81no—
288C22×C47—81no—
289C22×C47—81no—
290C22×C47C2×C4×D411yescharacteristic
291C22×C47#53211yescharacteristic
292C22×C47#53211yescharacteristic
293C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
295C22×C47#57511yescharacteristic
296C22×C47#57511yescharacteristic
297C2×D47—42no—
298C2×D47—42no—
299C2×D47—42no—
300C2×D47—81no—
301C2×D47—81no—
302C2×D47—81no—
303C2×D47—81no—
304C4×C47—22no—
305C4×C47—22no—
306C4×C47—22no—
307C4×C47—22no—
308C4×C47—41no—
309C4×C47—41no—
310C4×C47—42no—
311C4×C47—81no—
312C4×C47—81no—
313C4×C47C2×#11711yescharacteristic
314C4×C47C4×#4111yescharacteristic
315C4×C47#52911yescharacteristic
316C4×C47#52911yescharacteristic
317Dic47—42no—
318Dic47—81no—
319Dic47—81no—
320C2×#442—471no—
321#2293—471no—
322#2293D4711yescharacteristic, complemented
323#1972—471no—
324#1970—471no—
325#1944—471no—
326#1941—471no—
327#1936—471no—
328#1936—471no—
329#1934—471no—
330#1934—471no—
331#1776—471no—
332#1762—471no—
333#1762—471no—
334C4×#97—471no—
335C2×C47—82no—
336C2×C47#229311yescharacteristic
337C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
338C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
339C2×C47#318011yescharacteristic
340C2×C47#318011yescharacteristic
341C2×C47#318311yescharacteristic
342C2×C47#318411yescharacteristic
343D47—82no—
344C22×D8—471no—
345C2×C4×D4—471no—
346C2×C4×D4C2×D4711yescharacteristic
347C2×#117—21no—
348C2×#117—941no—
349#442—478no—
350C2×#107—471no—
351C2×#106—471no—
352C2×#106—471no—
353C2×#102—471no—
354C2×#102—471no—
355C2×#102—471no—
356C2×#102—471no—
357C2×#102—471no—
358C2×#102—471no—
359C2×#97—471no—
360C2×#97—471no—
361C2×#97—471no—
362C2×#97—471no—
363C2×C4×C8—471no—
364C2×C4×C8—471no—
365#390—471no—
366#390—471no—
367#390C2×D4711yescharacteristic
368#390C2×D4711yescharacteristic
369#389—471no—
370#389—471no—
371#389C2×D4711yescharacteristic
372#389C2×D4711yescharacteristic
373#388—21no—
374#388—21no—
375#388—941no—
376#388—941no—
377C4×C4⋊C4—471no—
378C4×C4⋊C4C2×D4711yescharacteristic
379C4×C22⋊C4—21no—
380C4×C22⋊C4—941no—
381C43C2×D4711yescharacteristic
382#340—471no—
383#340—471no—
384#340—471no—
385#340—471no—
386#340—471no—
387#340—471no—
388C47#2128611yescharacteristic, complemented, Sylow
389C22×D4—471no—
390C22×D4C22×D4711yescharacteristic
391C23×C4—21no—
392C23×C4—941no—
393C2×D8—474no—
394C2×D8—942no—
395C2×D8—942no—
396C22×C8—471no—
397C22×C8—471no—
398C22×C8—471no—
399C22×C8—471no—
400C4×D4—22no—
401C4×D4—474no—
402C4×D4—942no—
403C4×D4C4×D4714yes—
404#117—44no—
405#117—1884no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—471no—
409C2×C4⋊C4—471no—
410C2×C4⋊C4—471no—
411C2×C4⋊C4—941no—
412C2×C4⋊C4—941no—
413C2×C4⋊C4C22×D4711yescharacteristic
414C2×C4⋊C4C22×D4711yescharacteristic
415C2×C4⋊C4C22×D4711yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—941no—
422C2×C22⋊C4—941no—
423C2×C22⋊C4—941no—
424C2×C22⋊C4—941no—
425C2×C22⋊C4—941no—
426C2×C42—21no—
427C2×C42—941no—
428C2×C42C22⋊D4711yescharacteristic
429C2×C42C22⋊D4711yescharacteristic
430C2×C42C22⋊D4711yescharacteristic
431C2×C42C22⋊D4711yescharacteristic
432C2×C42C22×D4711yescharacteristic
433C2×C42C22×D4711yescharacteristic
434C2×C42C22×D4711yescharacteristic
435#107—942no—
436#106—474no—
437#106—942no—
438#102—474no—
439#102—474no—
440#102—942no—
441#102—942no—
442#102—942no—
443#102—942no—
444#97—474no—
445#97—942no—
446#97—942no—
447#97—942no—
448C4×C8—942no—
449C4×C8—942no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—941no—
454#95—941no—
455#95—941no—
456C24—21no—
457C24—941no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—472no—
464C2×D4—941no—
465C2×D4—941no—
466C2×D4—941no—
467C2×D4—941no—
468C2×D4—941no—
469C2×D4C2×C4×D4712yes—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—941no—
477C22×C4—944no—
478C22×C4—941no—
479C22×C4—941no—
480C22×C4—1882no—
481C22×C4—1882no—
482C22×C4C2×C22⋊D4711yescharacteristic
483C22×C4C2×C22⋊D4711yescharacteristic
484C22×C4C2×C22⋊D4711yescharacteristic
485C22×C4C2×C22⋊D4711yescharacteristic
486C22×C4C2×C22⋊D4711yescharacteristic
487C22×C4C2×C22⋊D4711yescharacteristic
488C22×C4C23×D4711yescharacteristic
489D8—948no—
490C2×C8—472no—
491C2×C8—472no—
492C2×C8—472no—
493C2×C8—472no—
494C2×C8—472no—
495C2×C8—472no—
496C2×C8—941no—
497C2×C8—941no—
498C2×C8—941no—
499C2×C8—941no—
500C2×C8—941no—
501C2×C8—941no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—472no—
510C4⋊C4—941no—
511C4⋊C4—941no—
512C4⋊C4—941no—
513C4⋊C4—941no—
514C4⋊C4—941no—
515C4⋊C4—1882no—
516C4⋊C4—1882no—
517C4⋊C4C2×C4×D4712yes—
518C22⋊C4—24no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—944no—
524C22⋊C4—1882no—
525C22⋊C4—1882no—
526C22⋊C4—1882no—
527C22⋊C4—1882no—
528C42—22no—
529C42—22no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—41no—
535C42—41no—
536C42—42no—
537C42—1882no—
538C42C2×C4×D4712yes—
539C42#1062535012yes—
540C23—22no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—942no—
548C23—1881no—
549C23—1881no—
550C23—1881no—
551C23—1881no—
552C23—1881no—
553C23—1881no—
554C23C22×C22⋊D4711yescharacteristic
555D4—22no—
556D4—22no—
557D4—41no—
558D4—41no—
559D4—942no—
560D4—942no—
561D4—1881no—
562D4—1881no—
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—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—48no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—942no—
588C2×C4—1888no—
589C2×C4—1881no—
590C2×C4—1881no—
591C2×C4—1881no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4—1881no—
595C2×C4—1881no—
596C2×C4—1881no—
597C2×C4C4×C22⋊D4712yes—
598C2×C4C4×C22⋊D4712yes—
599C2×C4#1309404912yes—
600C2×C4#1309405012yes—
601C2×C4C22×C4×D4711yescharacteristic
602C2×C4#41×D4711yescharacteristic
603C2×C4#41×D4711yescharacteristic
604C2×C4#41×D4711yescharacteristic
605C2×C4#41×D4711yescharacteristic
606C2×C4C2×#1062535011yescharacteristic
607C8—474no—
608C8—942no—
609C8—942no—
610C8—942no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—3761no—
619C22—3761no—
620C22—3761no—
621C22—3761no—
622C22—1882no—
623C22—1882no—
624C22—1882no—
625C22C2×C4×C22⋊D4711yescharacteristic
626C22#1568966911yescharacteristic
627C22#1568966911yescharacteristic
628C22C2×#1309404911yescharacteristic
629C22C2×#1309405011yescharacteristic
630C22#1568992111yescharacteristic
631C22#1568992111yescharacteristic
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—42no—
637C4—41no—
638C4—41no—
639C4—81no—
640C4—81no—
641C4—3761no—
642C4—3761no—
643C4—1882no—
644C4#117×D4711yescharacteristic
645C4#1568959111yescharacteristic
646C4#1568965111yescharacteristic
647C4#1568965111yescharacteristic
648C2—82no—
649C2—3762no—
650C2#2024584711yescharacteristic
651C2C4×#1309404911yescharacteristic
652C2C4×#1309405011yescharacteristic
653C2#2025408011yescharacteristic
654C2#2025408711yescharacteristic
655C2#2025411111yescharacteristic
656C2#2025412211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2293C2
socleC23×C47C22×D4
prime radicalC22×#10625334C22
cosocleC22×C4×C47C24
hypercenter#2293D47
hypocenterC47#21286
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ #2293

Complements

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

NH ≅ G/Ntype
2. #20245852C2semidirect
5. #20244128C2semidirect
6. #20244113C2semidirect
7. #20244088C2semidirect
8. #20244084C2semidirect
9. #20244074C2semidirect
10. #20244070C2semidirect
11. #20243286C2semidirect
12. #20243216C2semidirect
13. #20243213C2semidirect
14. C4×#13093935C2semidirect
15. C47×#2293C2semidirect
19. #15689233C4semidirect
281. C47⋊C8#117semidirect
322. #2293D47semidirect
388. C47#21286semidirect

Maximal subgroups

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

subgroupindex
C2×#156892332
#202458522
#202442632
#202442532
#202441282
#202441132
#202440882
#202440842
#202440742
#202440702
#202432862
#202432162
#202432132
C4×#130939352
C47×#22932
#2128647