Finite Group Databasebeta

← #26107167

Subgroups — #26107167

Order 12032.

628 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243150C211yesmaximal, characteristic
2C2×#15688860C212yesmaximal
3#20241615C211yesmaximal, characteristic
4#20241266C212yesmaximal
5C4×C22⋊Q8×C47C211yesmaximal, characteristic
6C2×C22⋊(C47⋊C8)C2212yes—
7C2×#13093935—21no—
8C2×C47⋊#95—21no—
9C2×C47⋊#95—21no—
10C2×C47⋊#95C2212yes—
11#15688878—21no—
12#15688860C418yescomplemented
13C2×C22⋊Q8×C47C2211yescharacteristic
14C2×C4×Q8×C47C411yescharacteristic, complemented
15C22×C42×C47C2211yescharacteristic
16C47×#391C411yescharacteristic, complemented
17C47×#389C411yescharacteristic, complemented
18C47×#387—21no—
19C47×#385—21no—
20C47×#380C2211yescharacteristic
21C4×C4⋊C4×C47—21no—
22C4×C4⋊C4×C47C411yescharacteristic, complemented
23C4×C22⋊C4×C47—21no—
24C23×Dic47—22no—
25C22×C47⋊C8—22no—
26C22⋊(C47⋊C8)—22no—
27C22⋊(C47⋊C8)C2×C414yes—
28#13093935—42no—
29C47⋊#95—22no—
30C47⋊#95—42no—
31C47⋊#95—42no—
32C47⋊#95C2×C414yes—
33C22×Q8×C47C2×C411yescharacteristic
34C23×C4×C47D411yescharacteristic
35C23×C4×C47D411yescharacteristic
36C23×C4×C47C2311yescharacteristic
37C22⋊Q8×C47—22no—
38C22⋊Q8×C47C2×C414yes—
39C4×Q8×C47—42no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C4⋊C4×C47C2×C411yescharacteristic
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47—21no—
48C2×C42×C47—21no—
49C2×C42×C47—21no—
50C2×C42×C47—42no—
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—
60C24×C47C2×D411yescharacteristic
61C2×Q8×C47—21no—
62C2×Q8×C47—41no—
63C2×Q8×C47—41no—
64C2×Q8×C47C4212yes—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—22no—
69C22×C4×C47—21no—
70C22×C4×C47—22no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—41no—
74C22×C4×C47—41no—
75C22×C4×C47—41no—
76C22×C4×C47—41no—
77C22×C4×C47—41no—
78C22×C4×C47C22⋊C411yescharacteristic
79C22×C4×C47C22⋊C411yescharacteristic
80C22×C4×C47C22⋊C411yescharacteristic
81C22×C4×C47C22⋊C411yescharacteristic
82C22×C4×C47C22×C411yescharacteristic
83C22×C4×C47C22×C412yes—
84C22×C4×C47C2×D411yescharacteristic
85C22×C4×C47#4112yes—
86C22×Dic47—24no—
87C22×Dic47—42no—
88C22×Dic47—42no—
89C22×Dic47—42no—
90C22×Dic47—42no—
91C22×Dic47—42no—
92C22×Dic47—42no—
93C4⋊C4×C47—21no—
94C4⋊C4×C47—24no—
95C4⋊C4×C47—42no—
96C4⋊C4×C47—42no—
97C4⋊C4×C47—41no—
98C4⋊C4×C47—41no—
99C4⋊C4×C47C4212yes—
100C22⋊C4×C47—24no—
101C22⋊C4×C47—41no—
102C22⋊C4×C47—41no—
103C42×C47—22no—
104C42×C47—22no—
105C42×C47—41no—
106C42×C47—41no—
107C42×C47—42no—
108C42×C47—42no—
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×C47C4×D412yes—
118C23×C47AGL(1,ℤ/8)11yescharacteristic
119C23×C47#13711yescharacteristic
120C23×C47#13712yes—
121Q8×C47—42no—
122Q8×C47—42no—
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—22no—
130C2×C4×C47—22no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—42no—
141C2×C4×C47—42no—
142C2×C4×C47—44no—
143C2×C4×C47—42no—
144C2×C4×C47—42no—
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—41no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47#9912yes—
156C2×C4×C47C2×C4211yescharacteristic
157C2×C4×C47#11711yescharacteristic
158C2×C4×C47C4×D412yes—
159C2×Dic47—44no—
160C2×Dic47—44no—
161C2×Dic47—44no—
162C2×Dic47—82no—
163C2×Dic47—82no—
164C2×Dic47—82no—
165C2×Dic47—82no—
166C47⋊C8—44no—
167#8217—471nomaximal, Sylow
168C22×C47—22no—
169C22×C47—22no—
170C22×C47—22no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47—21no—
175C22×C47—22no—
176C22×C47C4×C22⋊C411yescharacteristic
177C22×C47C2×#9911yescharacteristic
178C22×C47#41011yescharacteristic
179C22×C47#41711yescharacteristic
180C22×C47#41911yescharacteristic
181C22×C47#42111yescharacteristic
182C22×C47#42111yescharacteristic
183C22×C47#44012yes—
184C22×C47#44112yes—
185C4×C47—22no—
186C4×C47—21no—
187C4×C47—21no—
188C4×C47—22no—
189C4×C47—42no—
190C4×C47—42no—
191C4×C47—41no—
192C4×C47—41no—
193C4×C47—41no—
194C4×C47—81no—
195C4×C47—81no—
196Dic47—84no—
197C4×C22⋊Q8D4711yescharacteristic
198#1749—471no—
199C2×#328—472no—
200#1434—471no—
201#1294—472no—
202C2×C47—22no—
203C2×C47—22no—
204C2×C47#151111yescharacteristic
205C2×C47#151211yescharacteristic
206C2×C47#151311yescharacteristic
207C2×C47#151511yescharacteristic
208C2×C47C4×#9911yescharacteristic
209C2×C47#175911yescharacteristic
210C2×C47#176311yescharacteristic
211C2×C22⋊Q8C2×D4711yescharacteristic
212C2×C4×Q8Dic4711yescharacteristic, complemented
213C22×C42C2×D4711yescharacteristic
214C2×C22⋊C8—472no—
215C2×#97—941no—
216#391Dic4711yescharacteristic, complemented
217#389Dic4711yescharacteristic, complemented
218#387—21no—
219#385—21no—
220#380C2×D4711yescharacteristic
221C4×C4⋊C4—21no—
222C4×C4⋊C4Dic4711yescharacteristic, complemented
223C4×C22⋊C4—21no—
224C2×#95—472no—
225C2×#95—941no—
226C2×#95—941no—
227#336—941no—
228#328—478no—
229C47#821711yescharacteristic, complemented, Sylow
230C22×Q8C2×Dic4711yescharacteristic
231C23×C4—942no—
232C23×C4C22⋊D4711yescharacteristic
233C23×C4C22⋊D4711yescharacteristic
234C23×C4C22×D4711yescharacteristic
235C22×C8—942no—
236C22⋊Q8—22no—
237C22⋊Q8C4×D4714yes—
238C4×Q8—42no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4C2×Dic4711yescharacteristic
243C2×C4⋊C4C2×Dic4711yescharacteristic
244C2×C4⋊C4C2×Dic4711yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—42no—
250C2×C42C2×Dic4711yescharacteristic
251C2×C42C2×Dic4711yescharacteristic
252C2×C42C22⋊D4711yescharacteristic
253C2×C42C22⋊D4711yescharacteristic
254C22⋊C8—474no—
255C22⋊C8—942no—
256#97—1882no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—474no—
263#95—942no—
264#95—1882no—
265#95—1882no—
266C24C2×C22⋊D4711yescharacteristic
267C2×Q8—21no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8C4×Dic4712yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—944no—
285C22×C4—1882no—
286C22×C4—1882no—
287C22×C4—1882no—
288C22×C4—1882no—
289C22×C4—1882no—
290C22×C4—1882no—
291C22×C4C22⋊Dic4711yescharacteristic
292C22×C4C22⋊Dic4711yescharacteristic
293C22×C4C22⋊Dic4711yescharacteristic
294C22×C4C22⋊Dic4711yescharacteristic
295C22×C4C2×C4×D4712yes—
296C22×C4#1062535012yes—
297C22×C4C22×Dic4711yescharacteristic
298C22×C4C2×C22⋊D4711yescharacteristic
299C2×C8—944no—
300C2×C8—1882no—
301C2×C8—1882no—
302C4⋊C4—21no—
303C4⋊C4—24no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4C4×Dic4712yes—
309C22⋊C4—24no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C42—22no—
313C42—22no—
314C42—41no—
315C42—41no—
316C42—42no—
317C42—42no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23C2×C22⋊Dic4711yescharacteristic
323C23C4×C22⋊D4712yes—
324C23#1309404911yescharacteristic
325C23#1309405411yescharacteristic
326C23#1309405512yes—
327Q8—42no—
328Q8—42no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—3762no—
362C2×C4—3762no—
363C2×C4—3762no—
364C2×C4—3762no—
365C2×C4—1884no—
366C2×C4—1884no—
367C2×C4—1884no—
368C2×C4#1309393912yes—
369C2×C4C2×C4×Dic4711yescharacteristic
370C2×C4#1309398311yescharacteristic
371C2×C4C4×C22⋊D4712yes—
372C8—1884no—
373C22—22no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22C4×C22⋊Dic4711yescharacteristic
382C22C2×#1309393911yescharacteristic
383C22#1568909711yescharacteristic
384C22#1568912411yescharacteristic
385C22#1568913211yescharacteristic
386C22#1568914011yescharacteristic
387C22#1568914011yescharacteristic
388C22#1568922112yes—
389C22#1568922812yes—
390C4—22no—
391C4—22no—
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—42no—
398C4—42no—
399C4—81no—
400C4—81no—
401C4—3764no—
402C2—22no—
403C2—22no—
404C2#2024192111yescharacteristic
405C2#2024192611yescharacteristic
406C2#2024193111yescharacteristic
407C2#2024193911yescharacteristic
408C2C4×#1309393911yescharacteristic
409C2#2024319811yescharacteristic
410C2#2024321811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC2×C4×C47C2×C42
FrattiniC23×C4C22×D47
FittingC4×C22⋊Q8×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenterC4×C22⋊Q8D47
hypocenterC47#8217
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
12. #15688860C4semidirect
14. C2×C4×Q8×C47C4semidirect
16. C47×#391C4semidirect
17. C47×#389C4semidirect
22. C4×C4⋊C4×C47C4semidirect
212. C2×C4×Q8Dic47semidirect
216. #391Dic47semidirect
217. #389Dic47semidirect
222. C4×C4⋊C4Dic47semidirect
229. C47#8217semidirect

Maximal subgroups

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

subgroupindex
#202431502
C2×#156888602
#202416152
#202412662
C4×C22⋊Q8×C472
#821747