Finite Group Databasebeta

← #26111844

Subgroups — #26111844

Order 12032.

1062 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
2C2×#15688857C212yesmaximal, complemented
3#20241611C211yesmaximal, characteristic, complemented
4#20241266C212yesmaximal, complemented
5C47×#2847C211yesmaximal, characteristic
6C23×C47⋊C8—21no—
7C22×C22⋊Dic47—21no—
8C2×C22⋊(C47⋊C8)—24no—
9C2×C22⋊(C47⋊C8)—21no—
10C2×C22⋊(C47⋊C8)C2212yes—
11C2×C47⋊#95—21no—
12C2×C47⋊#95C2212yes—
13#15688857—24no—
14C24×C4×C47C2211yescharacteristic
15C2×#121×C47C411yescharacteristic, complemented
16C2×#121×C47C411yescharacteristic, complemented
17C2×#121×C47C2211yescharacteristic
18C2×#120×C47—21no—
19C47×#386—21no—
20C47×#386—21no—
21C47×#386—21no—
22C47×#380C411yescharacteristic, complemented
23C47×#380C411yescharacteristic, complemented
24C47×#380C2211yescharacteristic
25C23×Dic47—22no—
26C22×C47⋊C8—24no—
27C22×C47⋊C8—22no—
28C22×C47⋊C8—42no—
29C22×C47⋊C8—42no—
30C2×C22⋊Dic47—24no—
31C2×C22⋊Dic47—42no—
32C2×C22⋊Dic47—42no—
33C22⋊(C47⋊C8)—24no—
34C22⋊(C47⋊C8)—22no—
35C22⋊(C47⋊C8)D414yescomplemented
36C47⋊#95—22no—
37C47⋊#95—22no—
38C47⋊#95—42no—
39C25×C47D411yescharacteristic
40C22×D4×C47—21no—
41C22×D4×C47—21no—
42C22×D4×C47—21no—
43C22×D4×C47C2×C411yescharacteristic
44C23×C4×C47—22no—
45C23×C4×C47—22no—
46C23×C4×C47—42no—
47C23×C4×C47—42no—
48C23×C4×C47C2×C411yescharacteristic
49C23×C4×C47C2×C411yescharacteristic
50C23×C4×C47D411yescharacteristic
51C23×C4×C47D411yescharacteristic
52C23×C4×C47D411yescharacteristic
53C23×C4×C47C2311yescharacteristic
54#121×C47—22no—
55#121×C47—22no—
56#121×C47—42no—
57#121×C47—42no—
58#120×C47—42no—
59#120×C47—42no—
60C2×C4⋊C4×C47C2×C411yescharacteristic
61C2×C4⋊C4×C47C2×C411yescharacteristic
62C2×C4⋊C4×C47C2×C411yescharacteristic
63C2×C22⋊C4×C47—21no—
64C2×C22⋊C4×C47—21no—
65C2×C22⋊C4×C47—21no—
66C2×C22⋊C4×C47—21no—
67C2×C22⋊C4×C47—21no—
68C2×C22⋊C4×C47—21no—
69#95×C47—21no—
70#95×C47—21no—
71#95×C47—21no—
72C24×C47—22no—
73C24×C47—22no—
74C24×C47—22no—
75C24×C47—22no—
76C24×C47—21no—
77C24×C47—41no—
78C24×C47—41no—
79C24×C47C22⋊C411yescharacteristic
80C24×C47C22⋊C411yescharacteristic
81C24×C47D812yes—
82C24×C47SD1612yes—
83C24×C47C2×D411yescharacteristic
84C2×D4×C47—22no—
85C2×D4×C47—41no—
86C2×D4×C47—41no—
87C2×D4×C47—42no—
88C2×D4×C47—42no—
89C2×D4×C47—42no—
90C2×D4×C47—42no—
91C2×D4×C47—42no—
92C2×D4×C47—42no—
93C2×D4×C47—42no—
94C2×D4×C47—41no—
95C2×D4×C47—41no—
96C2×D4×C47—42no—
97C22×C4×C47—22no—
98C22×C4×C47—22no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—21no—
108C22×C4×C47—21no—
109C22×C4×C47—21no—
110C22×C4×C47—21no—
111C22×C4×C47—21no—
112C22×C4×C47—41no—
113C22×C4×C47—41no—
114C22×C4×C47—41no—
115C22×C4×C47—41no—
116C22×C4×C47—41no—
117C22×C4×C47—41no—
118C22×C4×C47—41no—
119C22×C4×C47—48no—
120C22×C4×C47—48no—
121C22×C4×C47C22⋊C411yescharacteristic
122C22×C4×C47C22⋊C411yescharacteristic
123C22×C4×C47C22×C411yescharacteristic
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D412yes—
126C22×C4×C47#4112yes—
127C22×Dic47—24no—
128C22×Dic47—42no—
129C22×Dic47—42no—
130C22×Dic47—42no—
131C22×Dic47—42no—
132C22×Dic47—42no—
133C22×Dic47—42no—
134C4⋊C4×C47—22no—
135C4⋊C4×C47—41no—
136C4⋊C4×C47—41no—
137C22⋊C4×C47—42no—
138C22⋊C4×C47—42no—
139C22⋊C4×C47—42no—
140C22⋊C4×C47—41no—
141C22⋊C4×C47—41no—
142C22⋊C4×C47—42no—
143C22⋊C4×C47—41no—
144C22⋊C4×C47—41no—
145C22⋊Dic47—48no—
146C2×C47⋊C8—24no—
147C2×C47⋊C8—48no—
148C2×C47⋊C8—42no—
149C2×C47⋊C8—42no—
150C23×C47—21no—
151C23×C47—21no—
152C23×C47—21no—
153C23×C47—21no—
154C23×C47—21no—
155C23×C47—21no—
156C23×C47—21no—
157C23×C47—21no—
158C23×C47—24no—
159C23×C47—24no—
160C23×C47—21no—
161C23×C47—21no—
162C23×C47—21no—
163C23×C47—21no—
164C23×C47—21no—
165C23×C47—21no—
166C23×C47—24no—
167C23×C47—24no—
168C23×C47—42no—
169C23×C47—42no—
170C23×C47—44no—
171C23×C47—44no—
172C23×C47—44no—
173C23×C47—44no—
174C23×C47—41no—
175C23×C47—41no—
176C23×C47—41no—
177C23×C47—41no—
178C23×C47—41no—
179C23×C47—41no—
180C23×C47—41no—
181C23×C47#9912yes—
182C23×C47#10212yes—
183C23×C47#10212yes—
184C23×C47#10412yes—
185C23×C47C2×C22⋊C411yescharacteristic
186C23×C47#12112yes—
187C23×C47C2×D811yescharacteristic
188C23×C47C2×SD1611yescharacteristic
189C23×C47#13512yes—
190D4×C47—82no—
191D4×C47—84no—
192D4×C47—84no—
193D4×C47—82no—
194D4×C47—82no—
195C2×C4×C47—24no—
196C2×C4×C47—24no—
197C2×C4×C47—21no—
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—44no—
209C2×C4×C47—44no—
210C2×C4×C47—44no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
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#9912yes—
223C2×C4×C47C4×D412yes—
224C2×C4×C47#12011yescharacteristic
225C2×C4×C47#12311yescharacteristic
226C2×Dic47—44no—
227C2×Dic47—44no—
228C2×Dic47—44no—
229C2×Dic47—82no—
230C2×Dic47—82no—
231C2×Dic47—82no—
232C2×Dic47—82no—
233C47⋊C8—44no—
234#9113—471nomaximal, Sylow
235C22×C47—22no—
236C22×C47—22no—
237C22×C47—22no—
238C22×C47—22no—
239C22×C47—22no—
240C22×C47—21no—
241C22×C47—21no—
242C22×C47—22no—
243C22×C47—22no—
244C22×C47—22no—
245C22×C47—22no—
246C22×C47—21no—
247C22×C47—22no—
248C22×C47—21no—
249C22×C47—22no—
250C22×C47—22no—
251C22×C47—22no—
252C22×C47—48no—
253C22×C47—48no—
254C22×C47—41no—
255C22×C47—41no—
256C22×C47—41no—
257C22×C47—41no—
258C22×C47—81no—
259C22×C47—81no—
260C22×C47—81no—
261C22×C47—81no—
262C22×C47—81no—
263C22×C47—81no—
264C22×C47—81no—
265C22×C47#32714yescomplemented
266C22×C47#38611yescharacteristic
267C22×C47C2×#9911yescharacteristic
268C22×C47C2×#9911yescharacteristic
269C22×C47C2×#10211yescharacteristic
270C22×C47#41611yescharacteristic
271C22×C47C2×#10411yescharacteristic
272C22×C47#42111yescharacteristic
273C22×C47#46512yes—
274C22×C47#46612yes—
275C4×C47—21no—
276C4×C47—22no—
277C4×C47—21no—
278C4×C47—22no—
279C4×C47—41no—
280C4×C47—81no—
281C4×C47—81no—
282C4×C47—81no—
283Dic47—84no—
284C22×C22⋊C8—471no—
285#2847D4711yescharacteristic
286C2×#327—472no—
287C22⋊#95—471no—
288#1294—472no—
289C2×C47—22no—
290C2×C47—24no—
291C2×C47—22no—
292C2×C47—42no—
293C2×C47—161no—
294C2×C47C2×#32711yescharacteristic
295C2×C47#150111yescharacteristic
296C2×C47#151611yescharacteristic
297C2×C47#151911yescharacteristic
298C2×C47#187111yescharacteristic
299C2×C47#189311yescharacteristic
300C2×C47#189811yescharacteristic
301C24×C4C2×D4711yescharacteristic
302C23×C8—941no—
303C2×#121Dic4711yescharacteristic, complemented
304C2×#121Dic4711yescharacteristic, complemented
305C2×#121C2×D4711yescharacteristic
306C2×#120—21no—
307C22×C22⋊C4—941no—
308C2×C22⋊C8—472no—
309C2×C22⋊C8—944no—
310C2×C22⋊C8—941no—
311#386—21no—
312#386—21no—
313#386—21no—
314#380Dic4711yescharacteristic, complemented
315#380Dic4711yescharacteristic, complemented
316#380C2×D4711yescharacteristic
317C2×#95—472no—
318C2×#95—941no—
319#327—944no—
320C47#911311yescharacteristic, complemented, Sylow
321C25C22⋊D4711yescharacteristic
322C22×D4—21no—
323C22×D4—21no—
324C22×D4—21no—
325C22×D4C2×Dic4711yescharacteristic
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—42no—
329C23×C4—42no—
330C23×C4—942no—
331C23×C4C2×Dic4711yescharacteristic
332C23×C4C2×Dic4711yescharacteristic
333C23×C4C22⋊D4711yescharacteristic
334C23×C4C22⋊D4711yescharacteristic
335C23×C4C22⋊D4711yescharacteristic
336C23×C4C22×D4711yescharacteristic
337C22×C8—944no—
338C22×C8—942no—
339C22×C8—1882no—
340C22×C8—1882no—
341#121—22no—
342#121—22no—
343#121—42no—
344#121—42no—
345#120—42no—
346#120—42no—
347C2×C4⋊C4C2×Dic4711yescharacteristic
348C2×C4⋊C4C2×Dic4711yescharacteristic
349C2×C4⋊C4C2×Dic4711yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—944no—
357C2×C22⋊C4—1882no—
358C2×C22⋊C4—1882no—
359C22⋊C8—474no—
360C22⋊C8—944no—
361C22⋊C8—942no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—942no—
366#95—942no—
367#95—1882no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—21no—
373C24—41no—
374C24—41no—
375C24#1062533412yes—
376C24#1062533512yes—
377C24C22⋊Dic4711yescharacteristic
378C24C22⋊Dic4711yescharacteristic
379C24C2×C22⋊D4711yescharacteristic
380C2×D4—22no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—42no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—944no—
418C22×C4—1882no—
419C22×C4—1882no—
420C22×C4—1882no—
421C22×C4—1882no—
422C22×C4—1882no—
423C22×C4—1882no—
424C22×C4C22⋊Dic4711yescharacteristic
425C22×C4C22⋊Dic4711yescharacteristic
426C22×C4D4×D4712yes—
427C22×C4#1062535212yes—
428C22×C4C22×Dic4711yescharacteristic
429C22×C4C2×C22⋊D4711yescharacteristic
430C2×C8—944no—
431C2×C8—1888no—
432C2×C8—1882no—
433C2×C8—1882no—
434C4⋊C4—22no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—42no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—1888no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23#1309393912yes—
478C23#1309394512yes—
479C23#1309394512yes—
480C23#1309395112yes—
481C23C2×C22⋊Dic4711yescharacteristic
482C23#1309399612yes—
483C23C2×#1062533411yescharacteristic
484C23C2×#1062533511yescharacteristic
485C23#1309404412yes—
486D4—82no—
487D4—84no—
488D4—82no—
489D4—82no—
490D4—84no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
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—3762no—
519C2×C4—3762no—
520C2×C4—3762no—
521C2×C4—3762no—
522C2×C4—1884no—
523C2×C4—1884no—
524C2×C4—1884no—
525C2×C4#1309393912yes—
526C2×C4D4×Dic4712yes—
527C2×C4#1309399211yescharacteristic
528C2×C4#1309400511yescharacteristic
529C8—1884no—
530C22—22no—
531C22—22no—
532C22—21no—
533C22—22no—
534C22—21no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—21no—
539C22—21no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—48no—
548C22—48no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22#1568885714yescomplemented
561C22#1568900311yescharacteristic
562C22C2×#1309393911yescharacteristic
563C22C2×#1309393911yescharacteristic
564C22C2×#1309394511yescharacteristic
565C22#1568911911yescharacteristic
566C22C2×#1309395111yescharacteristic
567C22#1568914011yescharacteristic
568C22#1568935012yes—
569C22#1568935712yes—
570C4—21no—
571C4—22no—
572C4—21no—
573C4—22no—
574C4—41no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—3764no—
579C2—22no—
580C2—22no—
581C2—24no—
582C2—42no—
583C2—161no—
584C2C2×#1568885711yescharacteristic
585C2#2024187911yescharacteristic
586C2#2024194411yescharacteristic
587C2#2024195611yescharacteristic
588C2#2024369111yescharacteristic
589C2#2024381711yescharacteristic
590C2#2024384211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
Fitting, prime radicalC47×#2847C2
socleC23×C47C2×C22⋊C4
cosocleC23×C4×C47C23
hypercenter#2847D47
hypocenterC47#9113
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2847

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊(C47⋊C8)C2semidirect
2. C2×#15688857C2semidirect
3. #20241611C2semidirect
4. #20241266C2semidirect
15. C2×#121×C47C4semidirect
16. C2×#121×C47C4semidirect
22. C47×#380C4semidirect
23. C47×#380C4semidirect
35. C22⋊(C47⋊C8)D4semidirect
265. C22×C47#327semidirect
303. C2×#121Dic47semidirect
304. C2×#121Dic47semidirect
314. #380Dic47semidirect
315. #380Dic47semidirect
320. C47#9113semidirect
560. C22#15688857semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊(C47⋊C8)2
C2×#156888572
#202416112
#202412662
C47×#28472
#911347