Finite Group Databasebeta

← #26192383

Subgroups — #26192383

Order 12032.

1630 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1630 conjugacy classes of subgroups — too many to draw. Only its 195 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253762C211yesmaximal, characteristic, complemented
2#20253487C212yesmaximal, complemented
3#20242567C218yesmaximal, complemented
4#20242102C211yesmaximal, characteristic, complemented
5#20241831C212yesmaximal, complemented
6C2×D42×C47C211yesmaximal, characteristic
7#15689680C2211yescharacteristic, complemented
8#15689487—22no—
9#15689469C2214yescomplemented
10#15689458—22no—
11#15689403C2214yescomplemented
12#15689328C2214yescomplemented
13#15689274—22no—
14#15689223—24no—
15#15689180C2212yescomplemented
16#15689128—22no—
17C4×#10625329C2211yescharacteristic, complemented
18#15688867C2214yescomplemented
19D4⋊(C47⋊C8)C2218yescomplemented
20C23×D4×C47—22no—
21D42×C47—22no—
22D42×C47—24no—
23D42×C47C2214yes—
24C2×#127×C47C2211yescharacteristic
25C2×#121×C47—22no—
26C2×#120×C47—22no—
27C2×C4×D4×C47C2212yes—
28C2×#10625335—24no—
29C2×#10625329—21no—
30#13094024C2312yescomplemented
31#13094016—21no—
32#13094003—22no—
33C4×Dic94—22no—
34#13093983—21no—
35#13093957—24no—
36C4⋊(C47⋊C8)C2314yescomplemented
37#13093947—24no—
38#13093945—24no—
39#13093945—28no—
40C22⋊(C47⋊C8)—22no—
41#13093935—22no—
42#13093935—21no—
43C4×C47⋊C8C2312yescomplemented
44C25×C47—22no—
45C22×D4×C47—24no—
46C22×D4×C47—28no—
47C22×D4×C47—28no—
48C22×D4×C47—22no—
49C22×D4×C47—416no—
50C22×D4×C47D412yes—
51C22×D4×C47D412yes—
52C23×C4×C47—22no—
53#127×C47—22no—
54#127×C47D412yes—
55#127×C47C2312yes—
56#121×C47—28no—
57#121×C47—28no—
58#120×C47—28no—
59#120×C47—28no—
60C4×D4×C47—22no—
61C4×D4×C47—22no—
62C4×D4×C47D414yes—
63C4×D4×C47C2314yes—
64C2×C4⋊C4×C47D412yes—
65C2×C22⋊C4×C47—22no—
66C2×C22⋊C4×C47—22no—
67C2×C42×C47C2311yescharacteristic
68C24×C47—22no—
69C24×C47—28no—
70C24×C47—22no—
71C24×C47—416no—
72C24×C47—84no—
73C2×D4×C47—28no—
74C2×D4×C47—216no—
75C2×D4×C47—22no—
76C2×D4×C47—24no—
77C2×D4×C47—22no—
78C2×D4×C47—24no—
79C2×D4×C47—48no—
80C2×D4×C47—44no—
81C2×D4×C47—432no—
82C2×D4×C47—416no—
83C2×D4×C47—416no—
84C2×D4×C47—416no—
85C2×D4×C47C2×D414yes—
86C2×D4×C47C2×D414yes—
87C22×C4×C47—22no—
88C22×C4×C47—24no—
89C22×C4×C47—24no—
90C22×C4×C47—48no—
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×D412yes—
93C2×Dic94—22no—
94C4⋊C4×C47—22no—
95C4⋊C4×C47C2×D414yes—
96C22⋊C4×C47—28no—
97C22⋊C4×C47—24no—
98C22⋊C4×C47—24no—
99C42×C47C2×D411yescharacteristic
100C42×C47C2×D412yes—
101C42×C47C2411yescharacteristic
102C22⋊Dic47—41no—
103#10625335—48no—
104C4⋊Dic47—22no—
105C4⋊Dic47—22no—
106Dic47⋊C4—42no—
107C4×Dic47—41no—
108#10625329—42no—
109C2×C47⋊C8—22no—
110C2×C47⋊C8—24no—
111C23×C47—28no—
112C23×C47—24no—
113C23×C47—24no—
114C23×C47—44no—
115C23×C47—48no—
116C23×C47—432no—
117C23×C47—44no—
118C23×C47—44no—
119C23×C47—88no—
120C23×C47—832no—
121C23×C47#12011yescharacteristic
122D4×C47—28no—
123D4×C47—24no—
124D4×C47—44no—
125D4×C47—48no—
126D4×C47—432no—
127D4×C47—48no—
128D4×C47—82no—
129D4×C47AGL(1,ℤ/8)18yes—
130C2×C4×C47—22no—
131C2×C4×C47—24no—
132C2×C4×C47—24no—
133C2×C4×C47—48no—
134C2×C4×C47—44no—
135C2×C4×C47—48no—
136C2×C4×C47#12011yescharacteristic
137C2×C4×C47#12012yes—
138C2×C4×C47C22×D411yescharacteristic
139C2×C4×C47C22×D412yes—
140C2×Dic47—42no—
141Dic94—44no—
142C47⋊C8—42no—
143C47⋊C8—44no—
144#18529—471nomaximal, Sylow
145C22×C47—22no—
146C22×C47—28no—
147C22×C47—416no—
148C22×C47—44no—
149C22×C47—44no—
150C22×C47—48no—
151C22×C47—44no—
152C22×C47—88no—
153C22×C47—832no—
154C22×C47#45512yes—
155C22×C47C2×#12011yescharacteristic
156C4×C47—22no—
157C4×C47—21no—
158C4×C47—21no—
159C4×C47—44no—
160C4×C47C2×AGL(1,ℤ/8)14yes—
161Dic47—82no—
162C2×D42D4711yescharacteristic
163#3138—471no—
164#3110—472no—
165#1639—478no—
166#1555—471no—
167#1490—472no—
168C2×C47—22no—
169C2×C47—48no—
170C2×C47—84no—
171C2×C47#300312yes—
172C2×C47C2×#45511yescharacteristic
173C23×D4—22no—
174D42—22no—
175D42—24no—
176D42C2×D4714yes—
177#533—471no—
178C2×#127C2×D4711yescharacteristic
179C2×#121—22no—
180C2×#120—22no—
181C2×C4×D4C2×D4712yes—
182#490—942no—
183#486—474no—
184#484—942no—
185#474—474no—
186#461—474no—
187#450—942no—
188#440—944no—
189#432—472no—
190#418—942no—
191C4×M16—471no—
192#331—474no—
193D4⋊C8—478no—
194C47#1852911yescharacteristic, complemented, Sylow
195C25—22no—
196C22×D4—24no—
197C22×D4—28no—
198C22×D4—28no—
199C22×D4—22no—
200C22×D4—416no—
201C22×D4C22⋊D4712yes—
202C22×D4C22⋊D4712yes—
203C23×C4—22no—
204C2×SD16—944no—
205C2×M16—941no—
206C4⋊Q8—472no—
207#127—22no—
208#127C22⋊D4712yes—
209#127C22×D4712yes—
210#125—941no—
211C22⋊Q8—942no—
212#121—28no—
213#121—28no—
214#120—28no—
215#120—28no—
216C4×Q8—942no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4C22⋊D4714yes—
220C4×D4C22×D4714yes—
221#117—941no—
222C2×C4⋊C4C22⋊D4712yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C42C22×D4711yescharacteristic
226#107—944no—
227C4⋊C8—474no—
228#103—944no—
229#102—948no—
230#102—944no—
231C22⋊C8—942no—
232#97—941no—
233#97—942no—
234C4×C8—472no—
235C24—22no—
236C24—28no—
237C24—22no—
238C24—416no—
239C24—84no—
240C2×Q8—942no—
241C2×D4—28no—
242C2×D4—216no—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—22no—
246C2×D4—24no—
247C2×D4—48no—
248C2×D4—44no—
249C2×D4—432no—
250C2×D4—416no—
251C2×D4—416no—
252C2×D4—416no—
253C2×D4C2×C22⋊D4714yes—
254C2×D4C2×C22⋊D4714yes—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—48no—
259C22×C4C2×C22⋊D4712yes—
260C22×C4C2×C22⋊D4711yescharacteristic
261SD16—1888no—
262M16—1882no—
263C2×C8—942no—
264C2×C8—944no—
265C4⋊C4—22no—
266C4⋊C4—942no—
267C4⋊C4—942no—
268C4⋊C4—1882no—
269C4⋊C4C2×C22⋊D4714yes—
270C22⋊C4—28no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—1881no—
274C42—1881no—
275C42C2×C22⋊D4711yescharacteristic
276C42C2×C22⋊D4712yes—
277C42C23×D4711yescharacteristic
278C23—28no—
279C23—24no—
280C23—24no—
281C23—44no—
282C23—48no—
283C23—432no—
284C23—44no—
285C23—44no—
286C23—88no—
287C23—832no—
288C23#1309399311yescharacteristic
289Q8—1884no—
290D4—28no—
291D4—24no—
292D4—44no—
293D4—48no—
294D4—432no—
295D4—48no—
296D4—82no—
297D4#1309405018yes—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—48no—
302C2×C4—44no—
303C2×C4—48no—
304C2×C4—1882no—
305C2×C4#1309399311yescharacteristic
306C2×C4#1309399312yes—
307C2×C4C22×C22⋊D4711yescharacteristic
308C2×C4C22×C22⋊D4712yes—
309C8—1882no—
310C8—1884no—
311C22—22no—
312C22—28no—
313C22—416no—
314C22—44no—
315C22—44no—
316C22—44no—
317C22—48no—
318C22—88no—
319C22—832no—
320C22#1568930112yes—
321C22C2×#1309399311yescharacteristic
322C4—21no—
323C4—21no—
324C4—22no—
325C4—44no—
326C4—3762no—
327C4C2×#1309405014yes—
328C2—22no—
329C2—48no—
330C2—84no—
331C2#2025261512yes—
332C2C2×#1568930111yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
Fitting, prime radicalC2×D42×C47C2
socleC22×C47C2×#120
hypercenterC2×D42D47
hypocenterC47#18529
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D42

Complements

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

NH ≅ G/Ntype
1. #20253762C2semidirect
2. #20253487C2semidirect
3. #20242567C2semidirect
4. #20242102C2semidirect
5. #20241831C2semidirect
7. #15689680C22semidirect
9. #15689469C22semidirect
11. #15689403C22semidirect
12. #15689328C22semidirect
15. #15689180C22semidirect
17. C4×#10625329C22semidirect
18. #15688867C22semidirect
19. D4⋊(C47⋊C8)C22semidirect
30. #13094024C23semidirect
36. C4⋊(C47⋊C8)C23semidirect
43. C4×C47⋊C8C23semidirect
194. C47#18529semidirect

Maximal subgroups

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

subgroupindex
#202537622
#202534872
#202425672
#202421022
#202418312
C2×D42×C472
#1852947