Finite Group Databasebeta

← #26111660

Subgroups — #26111660

Order 12032.

964 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
2#20243298C211yesmaximal, characteristic
3C2×#15688849C212yesmaximal, complemented
4#20241231C212yesmaximal, complemented
5C2×C47×#385C211yesmaximal, characteristic
6C23×C47⋊C8—21no—
7C2×C22⋊(C47⋊C8)—24no—
8C2×C22⋊(C47⋊C8)—21no—
9C2×C22⋊(C47⋊C8)—21no—
10C2×C22⋊(C47⋊C8)—21no—
11C2×C22⋊(C47⋊C8)C2212yes—
12C2×C22⋊(C47⋊C8)C2212yes—
13#15688849—24no—
14C24×C4×C47C2211yescharacteristic
15C22×C4⋊C4×C47C411yescharacteristic
16C22×C4⋊C4×C47C2211yescharacteristic
17C22×C22⋊C4×C47C411yescharacteristic
18C22×C22⋊C4×C47C411yescharacteristic
19C47×#385—21no—
20C47×#385—21no—
21C47×#385—21no—
22C47×#385—21no—
23C2×#95×C47C411yescharacteristic
24C2×#95×C47C2211yescharacteristic
25C22×C47⋊C8—24no—
26C22×C47⋊C8—22no—
27C22×C47⋊C8—22no—
28C22×C47⋊C8—42no—
29C22×C47⋊C8—42no—
30C22⋊(C47⋊C8)—24no—
31C22⋊(C47⋊C8)—22no—
32C22⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)—22no—
34C22⋊(C47⋊C8)—42no—
35C22⋊(C47⋊C8)—42no—
36C22⋊(C47⋊C8)D414yes—
37C25×C47C2×C411yescharacteristic
38C23×C4×C47—22no—
39C23×C4×C47—22no—
40C23×C4×C47—42no—
41C23×C4×C47—42no—
42C23×C4×C47C2×C411yescharacteristic
43C23×C4×C47C2×C411yescharacteristic
44C23×C4×C47C2×C411yescharacteristic
45C23×C4×C47C2×C411yescharacteristic
46C23×C4×C47C2×C411yescharacteristic
47C23×C4×C47D411yescharacteristic
48C23×C4×C47D411yescharacteristic
49C23×C4×C47D411yescharacteristic
50C23×C4×C47D411yescharacteristic
51C23×C4×C47C2311yescharacteristic
52C2×C4⋊C4×C47—21no—
53C2×C4⋊C4×C47—21no—
54C2×C4⋊C4×C47—22no—
55C2×C4⋊C4×C47—21no—
56C2×C4⋊C4×C47—21no—
57C2×C4⋊C4×C47—22no—
58C2×C4⋊C4×C47—42no—
59C2×C22⋊C4×C47—21no—
60C2×C22⋊C4×C47—21no—
61C2×C22⋊C4×C47—22no—
62C2×C22⋊C4×C47—21no—
63C2×C22⋊C4×C47—21no—
64C2×C22⋊C4×C47—42no—
65C2×C22⋊C4×C47C814yescomplemented
66#95×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×C47C2×C812yes—
79C24×C47M1612yes—
80C24×C47C22×C411yescharacteristic
81C22×C4×C47—22no—
82C22×C4×C47—22no—
83C22×C4×C47—21no—
84C22×C4×C47—21no—
85C22×C4×C47—22no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—22no—
89C22×C4×C47—22no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—22no—
93C22×C4×C47—22no—
94C22×C4×C47—22no—
95C22×C4×C47—21no—
96C22×C4×C47—22no—
97C22×C4×C47—22no—
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—41no—
104C22×C4×C47—41no—
105C22×C4×C47—41no—
106C22×C4×C47—41no—
107C22×C4×C47—41no—
108C22×C4×C47—41no—
109C22×C4×C47—41no—
110C22×C4×C47—48no—
111C22×C4×C47—48no—
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×C47C22⋊C411yescharacteristic
119C22×C4×C47C22⋊C411yescharacteristic
120C22×C4×C47C2×C812yescomplemented
121C22×C4×C47C2×C812yescomplemented
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2×D412yes—
125C22×C4×C47#4112yes—
126C4⋊C4×C47—42no—
127C4⋊C4×C47—42no—
128C4⋊C4×C47—41no—
129C4⋊C4×C47—42no—
130C4⋊C4×C47—41no—
131C22⋊C4×C47—24no—
132C22⋊C4×C47—24no—
133C22⋊C4×C47—42no—
134C22⋊C4×C47—42no—
135C2×C47⋊C8—24no—
136C2×C47⋊C8—48no—
137C2×C47⋊C8—42no—
138C2×C47⋊C8—42no—
139C2×C47⋊C8—42no—
140C2×C47⋊C8—42no—
141C2×C47⋊C8—42no—
142C23×C47—21no—
143C23×C47—21no—
144C23×C47—21no—
145C23×C47—21no—
146C23×C47—21no—
147C23×C47—21no—
148C23×C47—21no—
149C23×C47—21no—
150C23×C47—24no—
151C23×C47—24no—
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—42no—
161C23×C47—42no—
162C23×C47—44no—
163C23×C47—44no—
164C23×C47—44no—
165C23×C47—44no—
166C23×C47C22⋊C812yes—
167C23×C47C22⋊C812yes—
168C23×C47#9912yes—
169C23×C47#10112yes—
170C23×C47C2×C22⋊C411yescharacteristic
171C23×C47C4×D412yes—
172C23×C47C22×C811yescharacteristic
173C23×C47C2×M1611yescharacteristic
174C23×C47#13112yes—
175C2×C4×C47—24no—
176C2×C4×C47—24no—
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—22no—
185C2×C4×C47—22no—
186C2×C4×C47—21no—
187C2×C4×C47—21no—
188C2×C4×C47—24no—
189C2×C4×C47—42no—
190C2×C4×C47—42no—
191C2×C4×C47—44no—
192C2×C4×C47—44no—
193C2×C4×C47—44no—
194C2×C4×C47—42no—
195C2×C4×C47—42no—
196C2×C4×C47—42no—
197C2×C4×C47—41no—
198C2×C4×C47—41no—
199C2×C4×C47—41no—
200C2×C4×C47—41no—
201C2×C4×C47—42no—
202C2×C4×C47—42no—
203C2×C4×C47—41no—
204C2×C4×C47—41no—
205C2×C4×C47—41no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—82no—
210C2×C4×C47—82no—
211C2×C4×C47#9912yes—
212C2×C4×C47#12011yescharacteristic
213C2×C4×C47#12112yes—
214C2×C4×C47#12311yescharacteristic
215C47⋊C8—44no—
216C47⋊C8—82no—
217#9081—471nomaximal, Sylow
218C22×C47—22no—
219C22×C47—22no—
220C22×C47—22no—
221C22×C47—22no—
222C22×C47—22no—
223C22×C47—21no—
224C22×C47—21no—
225C22×C47—21no—
226C22×C47—21no—
227C22×C47—22no—
228C22×C47—22no—
229C22×C47—22no—
230C22×C47—22no—
231C22×C47—22no—
232C22×C47—22no—
233C22×C47—22no—
234C22×C47—22no—
235C22×C47—48no—
236C22×C47—48no—
237C22×C47—41no—
238C22×C47—41no—
239C22×C47—41no—
240C22×C47—41no—
241C22×C47#32414yescomplemented
242C22×C47#38611yescharacteristic
243C22×C47C2×C22⋊C811yescharacteristic
244C22×C47#40811yescharacteristic
245C22×C47C2×#9911yescharacteristic
246C22×C47C2×#9911yescharacteristic
247C22×C47C2×#10111yescharacteristic
248C22×C47#41311yescharacteristic
249C22×C47C8×D412yes—
250C22×C47#43512yes—
251C4×C47—21no—
252C4×C47—22no—
253C4×C47—22no—
254C4×C47—21no—
255C4×C47—42no—
256C4×C47—41no—
257C4×C47—42no—
258C4×C47—81no—
259C4×C47—81no—
260C22×C22⋊C8—471no—
261C2×#385D4711yescharacteristic
262#1779—471no—
263C2×#324—472no—
264#1280—472no—
265C2×C47—24no—
266C2×C47—22no—
267C2×C47—22no—
268C2×C47—42no—
269C2×C47C2×#32411yescharacteristic
270C2×C47#145911yescharacteristic
271C2×C47#146811yescharacteristic
272C2×C47#147211yescharacteristic
273C2×C47#186911yescharacteristic
274C2×C47#189811yescharacteristic
275C2×C47#190111yescharacteristic
276C24×C4C2×D4711yescharacteristic
277C23×C8—941no—
278C22×C4⋊C4Dic4711yescharacteristic
279C22×C4⋊C4C2×D4711yescharacteristic
280C22×C22⋊C4Dic4711yescharacteristic
281C22×C22⋊C4Dic4711yescharacteristic
282C2×C22⋊C8—472no—
283C2×C22⋊C8—472no—
284C2×C22⋊C8—944no—
285C2×C22⋊C8—941no—
286C2×C22⋊C8—941no—
287C2×C22⋊C8—941no—
288#385—21no—
289#385—21no—
290#385—21no—
291#385—21no—
292C2×#95Dic4711yescharacteristic
293C2×#95C2×D4711yescharacteristic
294#324—944no—
295C47#908111yescharacteristic, complemented, Sylow
296C25C2×Dic4711yescharacteristic
297C23×C4—22no—
298C23×C4—22no—
299C23×C4—42no—
300C23×C4—42no—
301C23×C4C2×Dic4711yescharacteristic
302C23×C4C2×Dic4711yescharacteristic
303C23×C4C2×Dic4711yescharacteristic
304C23×C4C2×Dic4711yescharacteristic
305C23×C4C2×Dic4711yescharacteristic
306C23×C4C22⋊D4711yescharacteristic
307C23×C4C22⋊D4711yescharacteristic
308C23×C4C22⋊D4711yescharacteristic
309C23×C4C22⋊D4711yescharacteristic
310C23×C4C22×D4711yescharacteristic
311C22×C8—944no—
312C22×C8—942no—
313C22×C8—942no—
314C22×C8—1882no—
315C22×C8—1882no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—42no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—42no—
329C2×C22⋊C4C47⋊C814yescomplemented
330C22⋊C8—474no—
331C22⋊C8—944no—
332C22⋊C8—942no—
333C22⋊C8—942no—
334C22⋊C8—942no—
335C22⋊C8—1882no—
336C22⋊C8—1882no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—41no—
346C24—41no—
347C24C2×C47⋊C812yes—
348C24#1062532912yes—
349C24C22⋊Dic4711yescharacteristic
350C24C22⋊Dic4711yescharacteristic
351C24C22×Dic4711yescharacteristic
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4C2×C47⋊C812yescomplemented
390C22×C4C2×C47⋊C812yescomplemented
391C22×C4C22⋊Dic4711yescharacteristic
392C22×C4C22⋊Dic4711yescharacteristic
393C22×C4D4×D4712yes—
394C22×C4#1062535212yes—
395C22×C4C2×C22⋊D4711yescharacteristic
396C22×C4C2×C22⋊D4711yescharacteristic
397C2×C8—944no—
398C2×C8—1882no—
399C2×C8—1882no—
400C2×C8—1882no—
401C2×C8—1888no—
402C2×C8—1882no—
403C2×C8—1882no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4—41no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—24no—
427C23—24no—
428C23—21no—
429C23—24no—
430C23—24no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23C22⋊(C47⋊C8)12yes—
438C23C22⋊(C47⋊C8)12yes—
439C23#1309393912yes—
440C23#1309394312yes—
441C23C2×C22⋊Dic4711yescharacteristic
442C23D4×Dic4712yes—
443C23C22×C47⋊C811yescharacteristic
444C23C2×#1062532911yescharacteristic
445C23#1309403112yes—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—24no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4#1309393912yes—
483C2×C4#1309399211yescharacteristic
484C2×C4#1309399612yes—
485C2×C4#1309400511yescharacteristic
486C8—3762no—
487C8—1884no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—48no—
506C22—48no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22#1568884914yescomplemented
512C22#1568900311yescharacteristic
513C22C2×C22⋊(C47⋊C8)11yescharacteristic
514C22#1568909011yescharacteristic
515C22C2×#1309393911yescharacteristic
516C22C2×#1309393911yescharacteristic
517C22C2×#1309394311yescharacteristic
518C22#1568910711yescharacteristic
519C22D4×C47⋊C812yes—
520C22#1568919512yes—
521C4—22no—
522C4—21no—
523C4—22no—
524C4—21no—
525C4—42no—
526C4—41no—
527C4—42no—
528C4—81no—
529C4—81no—
530C2—24no—
531C2—22no—
532C2—22no—
533C2—42no—
534C2C2×#1568884911yescharacteristic
535C2#2024170611yescharacteristic
536C2#2024173611yescharacteristic
537C2#2024175211yescharacteristic
538C2#2024368111yescharacteristic
539C2#2024384211yescharacteristic
540C2#2024385711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC23×C47C22×C8
FrattiniC23×C4C22×D47
FittingC2×C47×#385C2
socleC23×C47C2×C22⋊C4
prime radicalC25×C47C2×C4
cosocleC23×C4×C47C23
hypercenterC2×#385D47
hypocenterC47#9081
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
3. C2×#15688849C2semidirect
4. #20241231C2semidirect
65. C2×C22⋊C4×C47C8semidirect
120. C22×C4×C47C2×C8semidirect
121. C22×C4×C47C2×C8semidirect
241. C22×C47#324semidirect
295. C47#9081semidirect
329. C2×C22⋊C4C47⋊C8semidirect
389. C22×C4C2×C47⋊C8semidirect
390. C22×C4C2×C47⋊C8semidirect
511. C22#15688849semidirect

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
#202432982
C2×#156888492
#202412312
C2×C47×#3852
#908147