Finite Group Databasebeta

← #26219339

Subgroups — #26219339

Order 12032.

922 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689221C211yesmaximal, characteristic
2#20245852C211yesmaximal, characteristic
3#20244015C211yesmaximal, characteristic
4#20244008C211yesmaximal, characteristic
5#20243980C211yesmaximal, characteristic, complemented
6#20243959C211yesmaximal, characteristic, complemented
7#20243743C211yesmaximal, characteristic
8#20243743C211yesmaximal, characteristic
9#20243656C211yesmaximal, characteristic, complemented
10#20243642C211yesmaximal, characteristic, complemented
11#20243286C211yesmaximal, characteristic, complemented
12#20243218C211yesmaximal, characteristic, complemented
13#20243216C211yesmaximal, characteristic
14#20243137C211yesmaximal, characteristic, complemented
15C42×Q8×C47C211yesmaximal, characteristic, complemented
16C22×#10625336C2211yescharacteristic
17C2×C4×D188C2211yescharacteristic
18C2×#13093981—21no—
19#15689221C418yescomplemented
20C2×#13093957C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093948C2211yescharacteristic
24C2×#13093948C2211yescharacteristic
25C2×#13093948C2211yescharacteristic
26C2×#13093944C2211yescharacteristic
27C2×#13093944C2211yescharacteristic
28C2×#13093944C2211yescharacteristic
29C2×#13093935C2211yescharacteristic
30C2×#13093935C2211yescharacteristic
31#15689025C2211yescharacteristic
32#15689025C2211yescharacteristic
33#15689021C2211yescharacteristic
34#15689021C2211yescharacteristic
35#15689014—21no—
36#15689014—21no—
37C4×C4⋊Dic47C2211yescharacteristic
38C4×(C2×D47)⋊C4—21no—
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688878C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48#15688878C2211yescharacteristic
49C2×C4×Q8×C47C2211yescharacteristic
50C2×C4×Q8×C47C2211yescharacteristic
51C2×C4×Q8×C47C2211yescharacteristic
52C4×C4⋊C4×C47—21no—
53C4×C4⋊C4×C47—21no—
54C4×C4⋊C4×C47—21no—
55C4×C4⋊C4×C47C2211yescharacteristic
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C43×C47—21no—
59C43×C47C2211yescharacteristic
60C22×D188C2311yescharacteristic
61C22×C4×D47—21no—
62C2×#10625336—22no—
63C2×#10625336—22no—
64C2×#10625336C2×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#13093957C2×C414yes—
85#13093955—22no—
86#13093948—22no—
87#13093948—22no—
88#13093948C2×C414yes—
89#13093944—22no—
90#13093944—22no—
91#13093944C2×C414yes—
92#13093935—22no—
93#13093935C2×C414yes—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C22×Q8×C47C2311yescharacteristic
98C4×Q8×C47—22no—
99C4×Q8×C47—22no—
100C4×Q8×C47—22no—
101C4×Q8×C47—22no—
102C4×Q8×C47—22no—
103C4×Q8×C47C2×C414yes—
104C2×C4⋊C4×C47—21no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C42×C47—21no—
111C2×C42×C47—21no—
112C2×C42×C47—21no—
113C2×C42×C47—21no—
114C2×C42×C47—21no—
115C2×C42×C47—21no—
116C2×C42×C47D411yescharacteristic
117C2×C42×C47D411yescharacteristic
118C2×C42×C47D411yescharacteristic
119C2×C42×C47D411yescharacteristic
120C2×C42×C47C2311yescharacteristic
121C2×C42×C47C2311yescharacteristic
122C2×C42×C47C2311yescharacteristic
123#95×C47—21no—
124#95×C47—21no—
125#95×C47—21no—
126C23×D47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47C22×C412yes—
131C2×Q8×C47#4112yes—
132C2×Q8×C47#4112yes—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2411yescharacteristic
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—21no—
147C2×D188—21no—
148C2×D188—21no—
149C2×D188—21no—
150C2×D188—21no—
151C2×D188—21no—
152C2×D188C22×C412yes—
153C2×C4×D47—24no—
154C2×C4×D47—42no—
155C2×C4×D47—42no—
156C4⋊C4×C47—24no—
157C4⋊C4×C47—24no—
158C4⋊C4×C47—24no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47C22×C412yes—
163C4⋊C4×C47#4112yes—
164C4⋊C4×C47#4112yes—
165C42×C47—24no—
166C42×C47—24no—
167C42×C47—24no—
168C42×C47—22no—
169C42×C47—22no—
170C42×C47—21no—
171C42×C47—21no—
172C42×C47—21no—
173C42×C47—21no—
174C42×C47—41no—
175C42×C47—41no—
176C42×C47—42no—
177C42×C47—42no—
178C42×C47—42no—
179C42×C47C22×C412yes—
180C42×C47#4112yes—
181#10625336—28no—
182(C2×D47)⋊C4—24no—
183(C2×D47)⋊C4—42no—
184(C2×D47)⋊C4—42no—
185(C2×D47)⋊C4—42no—
186(C2×D47)⋊C4—42no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47C22×C412yes—
193Dic47⋊C4—42no—
194Dic47⋊C4—42no—
195C4×Dic47—42no—
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—21no—
206C2×C47⋊C8C22×C412yes—
207C2×C47⋊C8C22×C412yes—
208C23×C47C22×D411yescharacteristic
209C22×D47—22no—
210C22×D47—41no—
211C22×D47—41no—
212C22×D47—41no—
213C22×D47—41no—
214C22×D47—41no—
215C22×D47—41no—
216Q8×C47—22no—
217Q8×C47—22no—
218Q8×C47—22no—
219Q8×C47#11714yescomplemented
220C2×C4×C47—22no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—22no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—22no—
235C2×C4×C47—22no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47C4×D412yes—
247C2×C4×C47C4×D412yes—
248C2×C4×C47AGL(1,ℤ/8)12yes—
249C2×C4×C47#13712yes—
250C2×C4×C47C23×C411yescharacteristic
251C2×C4×C47C2×#4111yescharacteristic
252C2×C4×C47C2×#4111yescharacteristic
253C2×C4×C47C2×#4111yescharacteristic
254C2×C4×C47C2×#4111yescharacteristic
255C2×C4×C47C2×#4111yescharacteristic
256C2×Dic47—22no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265D188—22no—
266D188—22no—
267D188—41no—
268D188—41no—
269C4×D47—48no—
270C47⋊C8—22no—
271C47⋊C8—22no—
272C47⋊C8—41no—
273C47⋊C8—41no—
274#21124—471nomaximal, Sylow
275C22×C47C2×C4×D411yescharacteristic
276C22×C47#52511yescharacteristic
277C22×C47#52511yescharacteristic
278C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C47C2×#13711yescharacteristic
280C22×C47#57511yescharacteristic
281C22×C47#57511yescharacteristic
282C2×D47—42no—
283C2×D47—42no—
284C2×D47—42no—
285C2×D47—81no—
286C2×D47—81no—
287C2×D47—81no—
288C2×D47—81no—
289C4×C47—22no—
290C4×C47—22no—
291C4×C47—22no—
292C4×C47—22no—
293C4×C47—24no—
294C4×C47—41no—
295C4×C47—42no—
296C4×C47—41no—
297C4×C47—42no—
298C4×C47—42no—
299C4×C47C2×#11711yescharacteristic
300C4×C47C4×#4111yescharacteristic
301C4×C47#52911yescharacteristic
302C4×C47#52911yescharacteristic
303Dic47—42no—
304Dic47—81no—
305Dic47—81no—
306C2×#440—471no—
307#2293—471no—
308C42×Q8D4711yescharacteristic, complemented
309#1926—471no—
310#1925—471no—
311#1921—471no—
312#1918—471no—
313#1880—471no—
314#1880—471no—
315#1865—471no—
316#1863—471no—
317#1776—471no—
318#1763—471no—
319#1762—471no—
320#1745—471no—
321C2×C47#229011yescharacteristic
322C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
323C2×C47C4×#13711yescharacteristic
324C2×C47#311111yescharacteristic
325C2×C47#311211yescharacteristic
326C2×C47#311511yescharacteristic
327C2×C47#311711yescharacteristic
328D47—82no—
329C22×SD16—471no—
330C2×C4×Q8C2×D4711yescharacteristic
331C2×C4×Q8C2×D4711yescharacteristic
332C2×C4×Q8C2×D4711yescharacteristic
333C2×C4×D4—471no—
334C2×#117—941no—
335#440—478no—
336C2×#107—471no—
337C2×#107—471no—
338C2×#106—471no—
339C2×#103—471no—
340C2×#103—471no—
341C2×#103—471no—
342C2×#102—471no—
343C2×#102—471no—
344C2×#102—471no—
345C2×#97—471no—
346C2×#97—471no—
347#390—471no—
348#390—471no—
349#389—471no—
350#389—471no—
351#388—941no—
352#388—941no—
353C4×C4⋊C4—21no—
354C4×C4⋊C4—21no—
355C4×C4⋊C4—21no—
356C4×C4⋊C4—471no—
357C4×C4⋊C4C2×D4711yescharacteristic
358C4×C4⋊C4C2×D4711yescharacteristic
359C4×C4⋊C4C2×D4711yescharacteristic
360C4×C22⋊C4—941no—
361C43—21no—
362C43C2×D4711yescharacteristic
363#340—471no—
364#340—471no—
365#340—471no—
366#340—471no—
367#340—471no—
368#340—471no—
369#336—471no—
370#336—471no—
371#336—471no—
372#336—471no—
373C47#2112411yescharacteristic, complemented, Sylow
374C22×Q8C22×D4711yescharacteristic
375C22×D4—471no—
376C23×C4—941no—
377C2×SD16—474no—
378C2×SD16—942no—
379C2×SD16—942no—
380C22×C8—471no—
381C22×C8—471no—
382C22×C8—471no—
383C22×C8—471no—
384C4×Q8—22no—
385C4×Q8—22no—
386C4×Q8—22no—
387C4×Q8—22no—
388C4×Q8—22no—
389C4×Q8C4×D4714yes—
390C4×D4—474no—
391C4×D4—942no—
392#117—1884no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—471no—
397C2×C4⋊C4—471no—
398C2×C4⋊C4—471no—
399C2×C4⋊C4—941no—
400C2×C4⋊C4—941no—
401C2×C4⋊C4C22×D4711yescharacteristic
402C2×C4⋊C4C22×D4711yescharacteristic
403C2×C4⋊C4C22×D4711yescharacteristic
404C2×C22⋊C4—941no—
405C2×C22⋊C4—941no—
406C2×C22⋊C4—941no—
407C2×C22⋊C4—941no—
408C2×C22⋊C4—941no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—21no—
414C2×C42—21no—
415C2×C42—941no—
416C2×C42C22⋊D4711yescharacteristic
417C2×C42C22⋊D4711yescharacteristic
418C2×C42C22⋊D4711yescharacteristic
419C2×C42C22⋊D4711yescharacteristic
420C2×C42C22×D4711yescharacteristic
421C2×C42C22×D4711yescharacteristic
422C2×C42C22×D4711yescharacteristic
423#107—474no—
424#107—942no—
425#106—942no—
426#103—474no—
427#103—942no—
428#103—942no—
429#102—474no—
430#102—942no—
431#102—942no—
432#97—474no—
433#97—942no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—941no—
438#95—941no—
439#95—941no—
440C24—941no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×C4×D4712yes—
445C2×Q8#1062535012yes—
446C2×Q8#1062535012yes—
447C2×D4—472no—
448C2×D4—941no—
449C2×D4—941no—
450C2×D4—941no—
451C2×D4—941no—
452C2×D4—941no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—941no—
458C22×C4—944no—
459C22×C4—941no—
460C22×C4—941no—
461C22×C4—1882no—
462C22×C4—1882no—
463C22×C4C2×C22⋊D4711yescharacteristic
464C22×C4C2×C22⋊D4711yescharacteristic
465C22×C4C2×C22⋊D4711yescharacteristic
466C22×C4C2×C22⋊D4711yescharacteristic
467C22×C4C2×C22⋊D4711yescharacteristic
468C22×C4C2×C22⋊D4711yescharacteristic
469C22×C4C23×D4711yescharacteristic
470SD16—948no—
471C2×C8—472no—
472C2×C8—472no—
473C2×C8—941no—
474C2×C8—941no—
475C2×C8—941no—
476C2×C8—941no—
477C2×C8—941no—
478C2×C8—941no—
479C2×C8—941no—
480C2×C8—941no—
481C2×C8—941no—
482C2×C8—941no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—472no—
490C4⋊C4—941no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4—941no—
494C4⋊C4—941no—
495C4⋊C4—1882no—
496C4⋊C4—1882no—
497C4⋊C4C2×C4×D4712yes—
498C4⋊C4#1062535012yes—
499C4⋊C4#1062535012yes—
500C22⋊C4—944no—
501C22⋊C4—1882no—
502C22⋊C4—1882no—
503C22⋊C4—1882no—
504C22⋊C4—1882no—
505C42—24no—
506C42—24no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—41no—
515C42—41no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—1882no—
520C42C2×C4×D4712yes—
521C42#1062535012yes—
522C23—942no—
523C23—1881no—
524C23—1881no—
525C23—1881no—
526C23—1881no—
527C23—1881no—
528C23—1881no—
529C23C22×C22⋊D4711yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8#1309398114yescomplemented
534D4—942no—
535D4—942no—
536D4—1881no—
537D4—1881no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—942no—
565C2×C4—1888no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4—1881no—
571C2×C4—1881no—
572C2×C4—1881no—
573C2×C4—1881no—
574C2×C4C4×C22⋊D4712yes—
575C2×C4C4×C22⋊D4712yes—
576C2×C4#1309404912yes—
577C2×C4#1309405512yes—
578C2×C4C22×C4×D4711yescharacteristic
579C2×C4C2×#1062535011yescharacteristic
580C2×C4C2×#1062535011yescharacteristic
581C2×C4C2×#1062535011yescharacteristic
582C2×C4C2×#1062535011yescharacteristic
583C2×C4C2×#1062535011yescharacteristic
584C8—942no—
585C8—942no—
586C8—1881no—
587C8—1881no—
588C22—3761no—
589C22—3761no—
590C22—3761no—
591C22—3761no—
592C22—1882no—
593C22—1882no—
594C22—1882no—
595C22C2×C4×C22⋊D4711yescharacteristic
596C22#1568963411yescharacteristic
597C22#1568963411yescharacteristic
598C22C2×#1309404911yescharacteristic
599C22C2×#1309405511yescharacteristic
600C22#1568992111yescharacteristic
601C22#1568992111yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—24no—
606C4—22no—
607C4—42no—
608C4—41no—
609C4—42no—
610C4—41no—
611C4—42no—
612C4—3761no—
613C4—3761no—
614C4—1882no—
615C4C2×#1309398111yescharacteristic
616C4C4×#1062535011yescharacteristic
617C4#1568965711yescharacteristic
618C4#1568965711yescharacteristic
619C2—3762no—
620C2#2024583111yescharacteristic
621C2C4×#1309404911yescharacteristic
622C2C4×#1309405511yescharacteristic
623C2#2025349711yescharacteristic
624C2#2025351011yescharacteristic
625C2#2025353911yescharacteristic
626C2#2025355911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC42×Q8×C47C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenterC42×Q8D47
hypocenterC47#21124
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
5. #20243980C2semidirect
6. #20243959C2semidirect
9. #20243656C2semidirect
10. #20243642C2semidirect
11. #20243286C2semidirect
12. #20243218C2semidirect
14. #20243137C2semidirect
15. C42×Q8×C47C2semidirect
19. #15689221C4semidirect
219. Q8×C47#117semidirect
308. C42×Q8D47semidirect
373. C47#21124semidirect
533. Q8#13093981semidirect

Maximal subgroups

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

subgroupindex
C2×#156892212
#202458522
#202440152
#202440082
#202439802
#202439592
#202437432
#202436562
#202436422
#202432862
#202432182
#202432162
#202431372
C42×Q8×C472
#2112447