Finite Group Databasebeta

← #26225734

Subgroups — #26225734

Order 12032.

990 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689751C211yesmaximal, characteristic
2#20245989C212yesmaximal
3#20244371C218yesmaximal
4#20243465C211yesmaximal, characteristic
5#20243463C212yesmaximal
6C47×#1820C211yesmaximal, characteristic
7#15689751—24no—
8#15689751—24no—
9C2×#13094024C412yescomplemented
10C2×#13094023C412yescomplemented
11C2×#13094016C412yescomplemented
12C2×#13094015C412yescomplemented
13C2×#13094001C2214yes—
14C2×#13093983C2211yescharacteristic
15C22×C4⋊Dic47C2211yescharacteristic
16#15689008—24no—
17#15689006—24no—
18#15688986C2212yes—
19#15688981C2214yes—
20C22⋊C4×Dic47C2214yes—
21#15688890C2214yes—
22#15688889C2218yes—
23C2×#117×C47C2211yescharacteristic
24C47×#381C2212yes—
25C47×#342C2214yes—
26C22×Dic94C2×C414yes—
27C23×Dic47C2312yes—
28#13094024—24no—
29#13094024—24no—
30#13094023—28no—
31#13094016—24no—
32#13094016—24no—
33#13094015—28no—
34#13094001—28no—
35#13094001—28no—
36#13093983—22no—
37#13093983—22no—
38C2×C4⋊Dic47C2×C411yescharacteristic
39C2×C4⋊Dic47C2×C411yescharacteristic
40C2×C4⋊Dic47C2×C414yes—
41C2×C4⋊Dic47C2×C412yes—
42C2×C4⋊Dic47C2×C412yes—
43C2×C4⋊Dic47D414yes—
44C2×C4⋊Dic47D414yes—
45C2×Dic47⋊C4—24no—
46C2×Dic47⋊C4C2×C414yes—
47C2×Dic47⋊C4C2×C414yes—
48C2×C22⋊Dic47C2314yes—
49C2×C22⋊Dic47C2312yes—
50C2×C4×Dic47—24no—
51C2×C4×Dic47C2×C412yes—
52C47⋊#95—22no—
53C47⋊#95—22no—
54C47⋊#95—24no—
55C47⋊#95—24no—
56C23×C4×C47C2311yescharacteristic
57#117×C47—22no—
58#117×C47—22no—
59C2×C4⋊C4×C47C2×C411yescharacteristic
60C2×C4⋊C4×C47C2×C411yescharacteristic
61C2×C22⋊C4×C47C2314yes—
62C2×C22⋊C4×C47C2312yes—
63C2×C42×C47C2×C412yes—
64#95×C47—22no—
65#95×C47—22no—
66C24×C47C2411yescharacteristic
67C22×C4×C47—24no—
68C22×C4×C47—24no—
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C22×C412yes—
72C22×C4×C47C22×C412yes—
73C22×C4×C47C2×D412yes—
74C22×C4×C47C2×D412yes—
75C22×Dic47—24no—
76C22×Dic47—82no—
77C22×Dic47C22×C412yes—
78C22×Dic47C22×C412yes—
79C22×Dic47C22×C414yes—
80C22×Dic47C2×D414yes—
81C22×Dic47C2×D414yes—
82C2×Dic94—28no—
83C2×Dic94—84no—
84C4⋊C4×C47—22no—
85C4⋊C4×C47—22no—
86C22⋊C4×C47—24no—
87C22⋊C4×C47—28no—
88C42×C47—22no—
89C42×C47—22no—
90C22⋊Dic47—24no—
91C22⋊Dic47—28no—
92C4⋊Dic47—21no—
93C4⋊Dic47—21no—
94C4⋊Dic47—21no—
95C4⋊Dic47—21no—
96C4⋊Dic47—28no—
97C4⋊Dic47C22⋊C418yes—
98C4⋊Dic47C22⋊C414yes—
99C4⋊Dic47C22⋊C414yes—
100Dic47⋊C4—28no—
101Dic47⋊C4—28no—
102Dic47⋊C4—84no—
103C4×Dic47—24no—
104C4×Dic47—84no—
105C23×C47—41no—
106C23×C47—41no—
107C23×C47C23×C411yescharacteristic
108C23×C47C22×D412yes—
109C23×C472−1+414yes—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—24no—
113C2×C4×C47—24no—
114C2×C4×C47—22no—
115C2×C4×C47—22no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—44no—
119C2×C4×C47—44no—
120C2×C4×C47—44no—
121C2×C4×C47C2×C22⋊C412yes—
122C2×C4×C47C2×C22⋊C411yescharacteristic
123C2×C4×C47C2×C22⋊C411yescharacteristic
124C2×Dic47—22no—
125C2×Dic47—22no—
126C2×Dic47—44no—
127C2×Dic47—44no—
128C2×Dic47—44no—
129C2×Dic47—42no—
130C2×Dic47—42no—
131C2×Dic47—44no—
132C2×Dic47—44no—
133C2×Dic47—44no—
134C2×Dic47C2×C22⋊C418yes—
135Dic94—88no—
136#21828—471nomaximal, Sylow
137C22×C47—21no—
138C22×C47—22no—
139C22×C47—21no—
140C22×C47—22no—
141C22×C47—44no—
142C22×C47C22×C22⋊C411yescharacteristic
143C22×C47#51912yes—
144C22×C47#53614yes—
145C4×C47—22no—
146C4×C47—21no—
147C4×C47—21no—
148C4×C47—42no—
149C4×C47—42no—
150C4×C47—84no—
151Dic47—28no—
152Dic47—42no—
153Dic47—42no—
154Dic47—84no—
155C2×#544—471no—
156#2313—472no—
157#1996—478no—
158#1820—472no—
159#1820—471no—
160#1820D4711yescharacteristic
161C2×C47—24no—
162C2×C47—22no—
163C2×C47#231011yescharacteristic
164C2×C47#288311yescharacteristic
165C2×C47#288411yescharacteristic
166#544—944no—
167#544—944no—
168C2×C4⋊Q8—472no—
169C2×C4⋊Q8—472no—
170C2×#125—472no—
171C2×#125—472no—
172C2×C22⋊Q8—474no—
173C2×#117—471no—
174C2×#117C2×D4711yescharacteristic
175C22×C4⋊C4—471no—
176#387—944no—
177#387—944no—
178#381—472no—
179#381C2×D4712yes—
180#380—474no—
181C4×C22⋊C4—474no—
182#342—478no—
183#342—474no—
184#342C2×D4714yes—
185C47#2182811yescharacteristic, complemented, Sylow
186C22×Q8—474no—
187C23×C4—472no—
188C23×C4C22×D4711yescharacteristic
189C4⋊Q8—944no—
190C4⋊Q8—944no—
191C4⋊Q8—948no—
192#125—944no—
193#125—944no—
194#125—948no—
195C22⋊Q8—948no—
196C22⋊Q8—948no—
197#117—22no—
198#117—22no—
199#117—942no—
200#117—942no—
201C2×C4⋊C4—472no—
202C2×C4⋊C4—472no—
203C2×C4⋊C4—474no—
204C2×C4⋊C4—474no—
205C2×C4⋊C4—471no—
206C2×C4⋊C4—471no—
207C2×C4⋊C4—474no—
208C2×C4⋊C4—474no—
209C2×C4⋊C4—474no—
210C2×C4⋊C4—944no—
211C2×C4⋊C4C4×D4711yescharacteristic
212C2×C4⋊C4C4×D4711yescharacteristic
213C2×C22⋊C4—472no—
214C2×C22⋊C4—474no—
215C2×C22⋊C4C22×D4714yes—
216C2×C22⋊C4C22×D4712yes—
217C2×C42—472no—
218C2×C42—944no—
219C2×C42C4×D4712yes—
220#95—22no—
221#95—22no—
222#95—944no—
223#95—944no—
224#95—942no—
225#95—942no—
226C24C23×D4711yescharacteristic
227C2×Q8—948no—
228C2×Q8—3764no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—474no—
232C22×C4—474no—
233C22×C4—474no—
234C22×C4—472no—
235C22×C4—472no—
236C22×C4—944no—
237C22×C4—3762no—
238C22×C4C2×C4×D4711yescharacteristic
239C22×C4C2×C4×D4711yescharacteristic
240C22×C4C2×C4×D4712yes—
241C22×C4C2×C4×D4712yes—
242C22×C4D4×D4712yes—
243C22×C4D4×D4712yes—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—474no—
247C4⋊C4—474no—
248C4⋊C4—478no—
249C4⋊C4—948no—
250C4⋊C4—948no—
251C4⋊C4—948no—
252C4⋊C4—941no—
253C4⋊C4—941no—
254C4⋊C4—941no—
255C4⋊C4—941no—
256C4⋊C4—3764no—
257C22⋊C4—24no—
258C22⋊C4—28no—
259C22⋊C4—944no—
260C22⋊C4—948no—
261C42—22no—
262C42—22no—
263C42—944no—
264C42—3764no—
265C23—41no—
266C23—41no—
267C23C22×C4×D4711yescharacteristic
268C23C2×D4×D4712yes—
269C23#1309407114yes—
270Q8—3768no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—478no—
283C2×C4—942no—
284C2×C4—942no—
285C2×C4—1884no—
286C2×C4—1884no—
287C2×C4—1884no—
288C2×C4—1884no—
289C2×C4—1884no—
290C2×C4—1884no—
291C2×C4—1882no—
292C2×C4—1882no—
293C2×C4C22⋊C4×D4711yescharacteristic
294C2×C4C22⋊C4×D4711yescharacteristic
295C2×C4C22⋊C4×D4712yes—
296C22—21no—
297C22—22no—
298C22—21no—
299C22—22no—
300C22—44no—
301C22C2×C22⋊C4×D4711yescharacteristic
302C22#1568960012yes—
303C22#1568969514yes—
304C4—21no—
305C4—21no—
306C4—22no—
307C4—42no—
308C4—42no—
309C4—84no—
310C4—948no—
311C4—3764no—
312C4—1882no—
313C4—1882no—
314C2—24no—
315C2—22no—
316C2#2024597311yescharacteristic
317C2#2025184711yescharacteristic
318C2#2025185011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC47×#1820C2
socleC22×C47C22×C22⋊C4
prime radical, cosocleC24×C47C24
hypercenter#1820D47
hypocenterC47#21828
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
9. C2×#13094024C4semidirect
10. C2×#13094023C4semidirect
11. C2×#13094016C4semidirect
12. C2×#13094015C4semidirect
185. C47#21828semidirect

Maximal subgroups

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

subgroupindex
C2×#156897512
#202459892
#202443712
#202434652
#202434632
C47×#18202
#2182847