Finite Group Databasebeta

← #26111834

Subgroups — #26111834

Order 12032.

984 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊(C47⋊C8)C211yesmaximal, characteristic
2C2×#15688860C212yesmaximal, complemented
3#20241611C211yesmaximal, characteristic
4#20241266C212yesmaximal, complemented
5C47×#2848C211yesmaximal, 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#15688860—24no—
14C24×C4×C47C2211yescharacteristic
15C2×C22⋊Q8×C47C411yescharacteristic, complemented
16C2×C22⋊Q8×C47C411yescharacteristic, complemented
17C2×C22⋊Q8×C47C2211yescharacteristic
18C47×#385—21no—
19C47×#385—21no—
20C47×#385—21no—
21C47×#380C411yescharacteristic, complemented
22C47×#380C411yescharacteristic, complemented
23C47×#380C2211yescharacteristic
24C47×#379—21no—
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)D414yes—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C47⋊#95—42no—
39C25×C47D411yescharacteristic
40C22×Q8×C47C2×C411yescharacteristic
41C23×C4×C47—22no—
42C23×C4×C47—22no—
43C23×C4×C47—42no—
44C23×C4×C47—42no—
45C23×C4×C47C2×C411yescharacteristic
46C23×C4×C47C2×C411yescharacteristic
47C23×C4×C47D411yescharacteristic
48C23×C4×C47D411yescharacteristic
49C23×C4×C47D411yescharacteristic
50C23×C4×C47C2311yescharacteristic
51C22⋊Q8×C47—22no—
52C22⋊Q8×C47—22no—
53C22⋊Q8×C47—42no—
54C22⋊Q8×C47—42no—
55C2×C4⋊C4×C47—21no—
56C2×C4⋊C4×C47—21no—
57C2×C4⋊C4×C47—21no—
58C2×C4⋊C4×C47C2×C411yescharacteristic
59C2×C4⋊C4×C47C2×C411yescharacteristic
60C2×C4⋊C4×C47C2×C411yescharacteristic
61C2×C22⋊C4×C47—21no—
62C2×C22⋊C4×C47—21no—
63C2×C22⋊C4×C47—21no—
64C2×C22⋊C4×C47—21no—
65C2×C22⋊C4×C47—21no—
66C2×C22⋊C4×C47—21no—
67#95×C47—21no—
68#95×C47—21no—
69#95×C47—21no—
70C24×C47—22no—
71C24×C47—22no—
72C24×C47—22no—
73C24×C47—22no—
74C24×C47—41no—
75C24×C47—41no—
76C24×C47C22⋊C411yescharacteristic
77C24×C47C22⋊C411yescharacteristic
78C24×C47SD1612yes—
79C24×C47Q1612yes—
80C24×C47C2×D411yescharacteristic
81C2×Q8×C47—22no—
82C2×Q8×C47—41no—
83C2×Q8×C47—41no—
84C22×C4×C47—22no—
85C22×C4×C47—22no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—22no—
89C22×C4×C47—22no—
90C22×C4×C47—22no—
91C22×C4×C47—22no—
92C22×C4×C47—22no—
93C22×C4×C47—22no—
94C22×C4×C47—21no—
95C22×C4×C47—21no—
96C22×C4×C47—21no—
97C22×C4×C47—21no—
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—41no—
101C22×C4×C47—41no—
102C22×C4×C47—41no—
103C22×C4×C47—41no—
104C22×C4×C47—41no—
105C22×C4×C47—41no—
106C22×C4×C47—41no—
107C22×C4×C47—48no—
108C22×C4×C47—48no—
109C22×C4×C47C22⋊C411yescharacteristic
110C22×C4×C47C22⋊C411yescharacteristic
111C22×C4×C47C22×C411yescharacteristic
112C22×C4×C47C2×D411yescharacteristic
113C22×C4×C47C2×D412yes—
114C22×C4×C47#4112yes—
115C22×Dic47—24no—
116C22×Dic47—42no—
117C22×Dic47—42no—
118C22×Dic47—42no—
119C22×Dic47—42no—
120C22×Dic47—42no—
121C22×Dic47—42no—
122C4⋊C4×C47—22no—
123C4⋊C4×C47—42no—
124C4⋊C4×C47—41no—
125C4⋊C4×C47—41no—
126C4⋊C4×C47—42no—
127C4⋊C4×C47—41no—
128C4⋊C4×C47—41no—
129C22⋊C4×C47—42no—
130C22⋊C4×C47—42no—
131C22⋊C4×C47—42no—
132C22⋊C4×C47—42no—
133C22⋊C4×C47—42no—
134C22⋊C4×C47—42no—
135C22⋊Dic47—48no—
136C2×C47⋊C8—24no—
137C2×C47⋊C8—48no—
138C2×C47⋊C8—42no—
139C2×C47⋊C8—42no—
140C23×C47—21no—
141C23×C47—21no—
142C23×C47—21no—
143C23×C47—21no—
144C23×C47—21no—
145C23×C47—21no—
146C23×C47—21no—
147C23×C47—21no—
148C23×C47—24no—
149C23×C47—24no—
150C23×C47—21no—
151C23×C47—21no—
152C23×C47—21no—
153C23×C47—21no—
154C23×C47—21no—
155C23×C47—21no—
156C23×C47—24no—
157C23×C47—24no—
158C23×C47—42no—
159C23×C47—42no—
160C23×C47—44no—
161C23×C47—44no—
162C23×C47—44no—
163C23×C47—44no—
164C23×C47#9912yes—
165C23×C47#10312yes—
166C23×C47#10312yes—
167C23×C47#10412yes—
168C23×C47C2×C22⋊C411yescharacteristic
169C23×C47#12112yes—
170C23×C47C2×SD1611yescharacteristic
171C23×C47C2×Q1611yescharacteristic
172C23×C47#13512yes—
173Q8×C47—82no—
174C2×C4×C47—24no—
175C2×C4×C47—24no—
176C2×C4×C47—21no—
177C2×C4×C47—21no—
178C2×C4×C47—21no—
179C2×C4×C47—21no—
180C2×C4×C47—21no—
181C2×C4×C47—21no—
182C2×C4×C47—21no—
183C2×C4×C47—21no—
184C2×C4×C47—21no—
185C2×C4×C47—42no—
186C2×C4×C47—42no—
187C2×C4×C47—44no—
188C2×C4×C47—44no—
189C2×C4×C47—44no—
190C2×C4×C47—42no—
191C2×C4×C47—42no—
192C2×C4×C47—41no—
193C2×C4×C47—41no—
194C2×C4×C47—41no—
195C2×C4×C47—41no—
196C2×C4×C47—41no—
197C2×C4×C47—41no—
198C2×C4×C47—41no—
199C2×C4×C47—41no—
200C2×C4×C47—41no—
201C2×C4×C47—41no—
202C2×C4×C47—41no—
203C2×C4×C47—41no—
204C2×C4×C47#9912yes—
205C2×C4×C47C4×D412yes—
206C2×C4×C47#12011yescharacteristic
207C2×C4×C47#12311yescharacteristic
208C2×Dic47—44no—
209C2×Dic47—44no—
210C2×Dic47—44no—
211C2×Dic47—82no—
212C2×Dic47—82no—
213C2×Dic47—82no—
214C2×Dic47—82no—
215C47⋊C8—44no—
216#9111—471nomaximal, Sylow
217C22×C47—22no—
218C22×C47—22no—
219C22×C47—22no—
220C22×C47—22no—
221C22×C47—22no—
222C22×C47—21no—
223C22×C47—21no—
224C22×C47—22no—
225C22×C47—22no—
226C22×C47—22no—
227C22×C47—22no—
228C22×C47—21no—
229C22×C47—22no—
230C22×C47—21no—
231C22×C47—22no—
232C22×C47—22no—
233C22×C47—22no—
234C22×C47—48no—
235C22×C47—48no—
236C22×C47—41no—
237C22×C47—41no—
238C22×C47—41no—
239C22×C47—41no—
240C22×C47#32814yescomplemented
241C22×C47#38611yescharacteristic
242C22×C47C2×#9911yescharacteristic
243C22×C47C2×#9911yescharacteristic
244C22×C47C2×#10311yescharacteristic
245C22×C47#41611yescharacteristic
246C22×C47C2×#10411yescharacteristic
247C22×C47#42111yescharacteristic
248C22×C47#46512yes—
249C22×C47#46712yes—
250C4×C47—21no—
251C4×C47—22no—
252C4×C47—21no—
253C4×C47—22no—
254C4×C47—41no—
255C4×C47—81no—
256C4×C47—81no—
257C4×C47—81no—
258C4×C47—81no—
259Dic47—84no—
260C22×C22⋊C8—471no—
261#2848D4711yescharacteristic
262C2×#328—472no—
263C22⋊#95—471no—
264#1294—472no—
265C2×C47—22no—
266C2×C47—24no—
267C2×C47—22no—
268C2×C47—42no—
269C2×C47C2×#32811yescharacteristic
270C2×C47#150111yescharacteristic
271C2×C47#151611yescharacteristic
272C2×C47#152011yescharacteristic
273C2×C47#187111yescharacteristic
274C2×C47#189211yescharacteristic
275C2×C47#189811yescharacteristic
276C24×C4C2×D4711yescharacteristic
277C23×C8—941no—
278C2×C22⋊Q8Dic4711yescharacteristic, complemented
279C2×C22⋊Q8Dic4711yescharacteristic, complemented
280C2×C22⋊Q8C2×D4711yescharacteristic
281C22×C22⋊C4—941no—
282C2×C22⋊C8—472no—
283C2×C22⋊C8—944no—
284C2×C22⋊C8—941no—
285#385—21no—
286#385—21no—
287#385—21no—
288#380Dic4711yescharacteristic, complemented
289#380Dic4711yescharacteristic, complemented
290#380C2×D4711yescharacteristic
291#379—21no—
292C2×#95—472no—
293C2×#95—941no—
294#328—944no—
295C47#911111yescharacteristic, complemented, Sylow
296C25C22⋊D4711yescharacteristic
297C22×Q8C2×Dic4711yescharacteristic
298C23×C4—22no—
299C23×C4—22no—
300C23×C4—42no—
301C23×C4—42no—
302C23×C4—942no—
303C23×C4C2×Dic4711yescharacteristic
304C23×C4C2×Dic4711yescharacteristic
305C23×C4C22⋊D4711yescharacteristic
306C23×C4C22⋊D4711yescharacteristic
307C23×C4C22⋊D4711yescharacteristic
308C23×C4C22×D4711yescharacteristic
309C22×C8—944no—
310C22×C8—942no—
311C22×C8—1882no—
312C22×C8—1882no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C22⋊Q8—42no—
316C22⋊Q8—42no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4C2×Dic4711yescharacteristic
321C2×C4⋊C4C2×Dic4711yescharacteristic
322C2×C4⋊C4C2×Dic4711yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—944no—
330C2×C22⋊C4—1882no—
331C2×C22⋊C4—1882no—
332C22⋊C8—474no—
333C22⋊C8—944no—
334C22⋊C8—942no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—942no—
339#95—942no—
340#95—1882no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—41no—
346C24—41no—
347C24#1062533612yes—
348C24#1062533712yes—
349C24C22⋊Dic4711yescharacteristic
350C24C22⋊Dic4711yescharacteristic
351C24C2×C22⋊D4711yescharacteristic
352C2×Q8—22no—
353C2×Q8—41no—
354C2×Q8—41no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—944no—
381C22×C4—1882no—
382C22×C4—1882no—
383C22×C4—1882no—
384C22×C4—1882no—
385C22×C4—1882no—
386C22×C4—1882no—
387C22×C4C22⋊Dic4711yescharacteristic
388C22×C4C22⋊Dic4711yescharacteristic
389C22×C4D4×D4712yes—
390C22×C4#1062535212yes—
391C22×C4C22×Dic4711yescharacteristic
392C22×C4C2×C22⋊D4711yescharacteristic
393C2×C8—944no—
394C2×C8—1888no—
395C2×C8—1882no—
396C2×C8—1882no—
397C4⋊C4—22no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—1888no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—42no—
430C23—42no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23#1309393912yes—
436C23#1309394812yes—
437C23#1309394812yes—
438C23#1309395112yes—
439C23C2×C22⋊Dic4711yescharacteristic
440C23#1309399612yes—
441C23C2×#1062533611yescharacteristic
442C23C2×#1062533711yescharacteristic
443C23#1309404412yes—
444Q8—82no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—3762no—
476C2×C4—3762no—
477C2×C4—3762no—
478C2×C4—3762no—
479C2×C4—1884no—
480C2×C4—1884no—
481C2×C4—1884no—
482C2×C4#1309393912yes—
483C2×C4D4×Dic4712yes—
484C2×C4#1309399211yescharacteristic
485C2×C4#1309400511yescharacteristic
486C8—1884no—
487C22—22no—
488C22—22no—
489C22—21no—
490C22—22no—
491C22—21no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—21no—
496C22—21no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—48no—
505C22—48no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22#1568886014yescomplemented
511C22#1568900311yescharacteristic
512C22C2×#1309393911yescharacteristic
513C22C2×#1309393911yescharacteristic
514C22C2×#1309394811yescharacteristic
515C22#1568911911yescharacteristic
516C22C2×#1309395111yescharacteristic
517C22#1568914011yescharacteristic
518C22#1568935412yes—
519C22#1568936212yes—
520C4—21no—
521C4—22no—
522C4—21no—
523C4—22no—
524C4—41no—
525C4—81no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4—3764no—
530C2—22no—
531C2—22no—
532C2—24no—
533C2—42no—
534C2C2×#1568886011yescharacteristic
535C2#2024187911yescharacteristic
536C2#2024194411yescharacteristic
537C2#2024196011yescharacteristic
538C2#2024369111yescharacteristic
539C2#2024381211yescharacteristic
540C2#2024384211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2848C2
socleC23×C47C2×C22⋊C4
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenter#2848D47
hypocenterC47#9111
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
2. C2×#15688860C2semidirect
4. #20241266C2semidirect
15. C2×C22⋊Q8×C47C4semidirect
16. C2×C22⋊Q8×C47C4semidirect
21. C47×#380C4semidirect
22. C47×#380C4semidirect
240. C22×C47#328semidirect
278. C2×C22⋊Q8Dic47semidirect
279. C2×C22⋊Q8Dic47semidirect
288. #380Dic47semidirect
289. #380Dic47semidirect
295. C47#9111semidirect
510. C22#15688860semidirect

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×#156888602
#202416112
#202412662
C47×#28482
#911147