Finite Group Databasebeta

← #26111839

Subgroups — #26111839

Order 12032.

632 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689176C211yesmaximal, characteristic
2C2×#15688860C212yesmaximal
3#20241615C211yesmaximal, characteristic
4#20241266C212yesmaximal
5C47×#2855C211yesmaximal, characteristic
6#15689176—24no—
7C2×C4⋊(C47⋊C8)—21no—
8C2×C22⋊(C47⋊C8)C2212yes—
9C2×C4×C47⋊C8—21no—
10C2×C47⋊#95—21no—
11C2×C47⋊#95—21no—
12C2×C47⋊#95C2212yes—
13#15688860—24no—
14C2×C22⋊Q8×C47C2211yescharacteristic
15C22×C42×C47C2211yescharacteristic
16C47×#391C411yescharacteristic, complemented
17C47×#391C411yescharacteristic, complemented
18C47×#389C411yescharacteristic, complemented
19C47×#389C411yescharacteristic, complemented
20C47×#387—21no—
21C47×#387—21no—
22C47×#385—21no—
23C47×#381—21no—
24C47×#380C2211yescharacteristic
25C23×Dic47—22no—
26C22×C47⋊C8—22no—
27C4⋊(C47⋊C8)—24no—
28C22⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)D414yes—
30C4×C47⋊C8—24no—
31C47⋊#95—24no—
32C47⋊#95—22no—
33C47⋊#95—22no—
34C47⋊#95—42no—
35C22×Q8×C47C2×C411yescharacteristic
36C23×C4×C47D411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C23×C4×C47C2311yescharacteristic
39C22⋊Q8×C47—22no—
40C22⋊Q8×C47—22no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47C2×C411yescharacteristic
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C4⋊C4×C47C2×C411yescharacteristic
47C2×C22⋊C4×C47—21no—
48C2×C22⋊C4×C47—21no—
49C2×C42×C47—22no—
50C2×C42×C47—22no—
51C2×C42×C47C2×C411yescharacteristic
52C2×C42×C47C2×C411yescharacteristic
53C2×C42×C47D411yescharacteristic
54C2×C42×C47D411yescharacteristic
55#95×C47—21no—
56#95×C47—21no—
57#95×C47—21no—
58#95×C47—21no—
59#95×C47—21no—
60#95×C47—21no—
61#95×C47—21no—
62C24×C47C2×D411yescharacteristic
63C2×Q8×C47—22no—
64C2×Q8×C47—41no—
65C2×Q8×C47—41no—
66C22×C4×C47—22no—
67C22×C4×C47—22no—
68C22×C4×C47—22no—
69C22×C4×C47—22no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47—41no—
75C22×C4×C47—41no—
76C22×C4×C47—41no—
77C22×C4×C47C22⋊C411yescharacteristic
78C22×C4×C47C22⋊C411yescharacteristic
79C22×C4×C47C22⋊C411yescharacteristic
80C22×C4×C47C22⋊C411yescharacteristic
81C22×C4×C47SD1612yes—
82C22×C4×C47Q1612yes—
83C22×C4×C47C22×C411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×C4×C47C2×D412yes—
86C22×C4×C47#4112yes—
87C22×Dic47—24no—
88C22×Dic47—42no—
89C22×Dic47—42no—
90C22×Dic47—42no—
91C22×Dic47—42no—
92C22×Dic47—42no—
93C22×Dic47—42no—
94C4⋊C4×C47—22no—
95C4⋊C4×C47—42no—
96C4⋊C4×C47—41no—
97C4⋊C4×C47—41no—
98C4⋊C4×C47—42no—
99C4⋊C4×C47—41no—
100C4⋊C4×C47—41no—
101C22⋊C4×C47—42no—
102C22⋊C4×C47—42no—
103C42×C47—22no—
104C42×C47—22no—
105C42×C47—21no—
106C42×C47—21no—
107C42×C47—21no—
108C42×C47—21no—
109C2×C47⋊C8—24no—
110C2×C47⋊C8—42no—
111C2×C47⋊C8—42no—
112C23×C47—21no—
113C23×C47—21no—
114C23×C47—21no—
115C23×C47—21no—
116C23×C47C2×C22⋊C411yescharacteristic
117C23×C47#12112yes—
118C23×C47C2×SD1611yescharacteristic
119C23×C47C2×Q1611yescharacteristic
120C23×C47#13512yes—
121Q8×C47—82no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—24no—
133C2×C4×C47—24no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—42no—
138C2×C4×C47—42no—
139C2×C4×C47—44no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47#10312yes—
153C2×C4×C47#10312yes—
154C2×C4×C47#10412yes—
155C2×C4×C47C4×D412yes—
156C2×C4×C47#12011yescharacteristic
157C2×C4×C47#12311yescharacteristic
158C2×Dic47—44no—
159C2×Dic47—44no—
160C2×Dic47—44no—
161C2×Dic47—82no—
162C2×Dic47—82no—
163C2×Dic47—82no—
164C2×Dic47—82no—
165C47⋊C8—44no—
166#9112—471nomaximal, Sylow
167C22×C47—22no—
168C22×C47—22no—
169C22×C47—22no—
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47—22no—
175C22×C47#38611yescharacteristic
176C22×C47#41011yescharacteristic
177C22×C47#41011yescharacteristic
178C22×C47C2×#10311yescharacteristic
179C22×C47#41611yescharacteristic
180C22×C47C2×#10411yescharacteristic
181C22×C47#42111yescharacteristic
182C22×C47#46512yes—
183C22×C47#46712yes—
184C4×C47—22no—
185C4×C47—22no—
186C4×C47—22no—
187C4×C47—21no—
188C4×C47—22no—
189C4×C47—21no—
190C4×C47—41no—
191C4×C47—81no—
192C4×C47—81no—
193C4×C47—81no—
194C4×C47—81no—
195Dic47—84no—
196C2×#431—471no—
197#2855D4711yescharacteristic
198C2×#328—472no—
199#1434—471no—
200#1294—472no—
201C2×C47—22no—
202C2×C47—22no—
203C2×C47#150011yescharacteristic
204C2×C47#150311yescharacteristic
205C2×C47#151811yescharacteristic
206C2×C47#152111yescharacteristic
207C2×C47#187111yescharacteristic
208C2×C47#189211yescharacteristic
209C2×C47#189911yescharacteristic
210C2×C22⋊Q8C2×D4711yescharacteristic
211C22×C42C2×D4711yescharacteristic
212#431—944no—
213C2×C4⋊C8—941no—
214C2×C22⋊C8—472no—
215C2×C4×C8—941no—
216#391Dic4711yescharacteristic, complemented
217#391Dic4711yescharacteristic, complemented
218#389Dic4711yescharacteristic, complemented
219#389Dic4711yescharacteristic, complemented
220#387—21no—
221#387—21no—
222#385—21no—
223#381—21no—
224#380C2×D4711yescharacteristic
225C2×#95—472no—
226C2×#95—941no—
227C2×#95—941no—
228#328—944no—
229C47#911211yescharacteristic, complemented, Sylow
230C22×Q8C2×Dic4711yescharacteristic
231C23×C4—942no—
232C23×C4C22⋊D4711yescharacteristic
233C23×C4C22⋊D4711yescharacteristic
234C23×C4C22×D4711yescharacteristic
235C22×C8—942no—
236C22⋊Q8—22no—
237C22⋊Q8—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4C2×Dic4711yescharacteristic
242C2×C4⋊C4C2×Dic4711yescharacteristic
243C2×C4⋊C4C2×Dic4711yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42C2×Dic4711yescharacteristic
249C2×C42C2×Dic4711yescharacteristic
250C2×C42C22⋊D4711yescharacteristic
251C2×C42C22⋊D4711yescharacteristic
252C4⋊C8—944no—
253C22⋊C8—474no—
254C22⋊C8—942no—
255C4×C8—944no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—944no—
264#95—942no—
265#95—942no—
266#95—1882no—
267C24C2×C22⋊D4711yescharacteristic
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—41no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—944no—
283C22×C4—1882no—
284C22×C4—1882no—
285C22×C4—1882no—
286C22×C4—1882no—
287C22×C4—1882no—
288C22×C4—1882no—
289C22×C4#1062533612yes—
290C22×C4#1062533712yes—
291C22×C4C22⋊Dic4711yescharacteristic
292C22×C4C22⋊Dic4711yescharacteristic
293C22×C4C22⋊Dic4711yescharacteristic
294C22×C4C22⋊Dic4711yescharacteristic
295C22×C4D4×D4712yes—
296C22×C4#1062535212yes—
297C22×C4C22×Dic4711yescharacteristic
298C22×C4C2×C22⋊D4711yescharacteristic
299C2×C8—944no—
300C2×C8—1882no—
301C2×C8—1882no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C42—22no—
312C42—22no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C2×C22⋊Dic4711yescharacteristic
322C23#1309399612yes—
323C23C2×#1062533611yescharacteristic
324C23C2×#1062533711yescharacteristic
325C23#1309404412yes—
326Q8—82no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—3762no—
358C2×C4—3762no—
359C2×C4—3762no—
360C2×C4—3762no—
361C2×C4—1884no—
362C2×C4—1884no—
363C2×C4—1884no—
364C2×C4#1309394812yes—
365C2×C4#1309394812yes—
366C2×C4#1309395112yes—
367C2×C4D4×Dic4712yes—
368C2×C4#1309399211yescharacteristic
369C2×C4#1309400511yescharacteristic
370C8—1884no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22#1568900311yescharacteristic
380C22#1568909711yescharacteristic
381C22#1568909711yescharacteristic
382C22C2×#1309394811yescharacteristic
383C22#1568911911yescharacteristic
384C22C2×#1309395111yescharacteristic
385C22#1568914011yescharacteristic
386C22#1568935412yes—
387C22#1568936212yes—
388C4—22no—
389C4—22no—
390C4—21no—
391C4—22no—
392C4—21no—
393C4—22no—
394C4—41no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—3764no—
400C2—22no—
401C2—22no—
402C2#2024187411yescharacteristic
403C2#2024188711yescharacteristic
404C2#2024195211yescharacteristic
405C2#2024196511yescharacteristic
406C2#2024369111yescharacteristic
407C2#2024381211yescharacteristic
408C2#2024384711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2855C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#2855D47
hypocenterC47#9112
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
16. C47×#391C4semidirect
17. C47×#391C4semidirect
18. C47×#389C4semidirect
19. C47×#389C4semidirect
216. #391Dic47semidirect
217. #391Dic47semidirect
218. #389Dic47semidirect
219. #389Dic47semidirect
229. C47#9112semidirect

Maximal subgroups

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

subgroupindex
C2×#156891762
C2×#156888602
#202416152
#202412662
C47×#28552
#911247