Finite Group Databasebeta

← #26216021

Subgroups — #26216021

Order 12032.

1602 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1602 conjugacy classes of subgroups — too many to draw. Only its 275 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
1C2×#15689597C211yesmaximal, characteristic
2#20245965C212yesmaximal
3#20243465C211yesmaximal, characteristic, complemented
4#20243463C212yesmaximal, complemented
5#20243461C214yesmaximal, complemented
6#342×D47C214yesmaximal, complemented
7C47×#1820C211yesmaximal, characteristic, complemented
8C23×D188C2211yescharacteristic
9#15689597—28no—
10C2×C4×D188C414yescomplemented
11C2×#13093986C414yescomplemented
12C2×#13093983C2211yescharacteristic
13C2×C22⋊C4×D47C2214yes—
14C2×C22⋊C4×D47C2212yes—
15C2×C22⋊C4×D47C2212yes—
16#15689001—28no—
17#15688986C2212yes—
18#15688978C2214yes—
19#15688890C2214yes—
20#15688889C2218yes—
21C2×#117×C47C2211yescharacteristic
22C47×#381C2212yes—
23C47×#342C2214yes—
24C24×D47C2311yescharacteristic
25C24×D47C2311yescharacteristic
26C22×D188—28no—
27C22×D188C2×C412yes—
28C22×D188C2×C412yes—
29C22×D188D418yes—
30C22×C4×D47—24no—
31C22×C4×D47C2×C414yes—
32C22×C4×D47C2×C414yescomplemented
33C4×D188—216no—
34#13093986—216no—
35#13093983—24no—
36C2×C4⋊Dic47C2×C412yescomplemented
37C2×C22⋊Dic47C2314yes—
38C2×C22⋊Dic47C2312yes—
39C2×(C2×D47)⋊C4—24no—
40C2×(C2×D47)⋊C4—28no—
41C2×(C2×D47)⋊C4C2×C414yes—
42C2×(C2×D47)⋊C4C2×C414yescomplemented
43C22⋊C4×D47—28no—
44C22⋊C4×D47—28no—
45C22⋊C4×D47—216no—
46C2×C4×Dic47C2×C412yescomplemented
47C47⋊#95—24no—
48C23×C4×C47C2311yescharacteristic
49#117×C47—24no—
50C2×C4⋊C4×C47C2×C412yescomplemented
51C2×C22⋊C4×C47C2314yes—
52C2×C22⋊C4×C47C2312yes—
53C2×C42×C47C2×C412yescomplemented
54#95×C47—24no—
55C24×C47C2411yescharacteristic
56C23×D47—24no—
57C23×D47—82no—
58C23×D47—82no—
59C23×D47C22×C412yes—
60C23×D47C22×C412yes—
61C23×D47C2×D414yes—
62C23×D47C2×D414yes—
63C23×D47#4118yes—
64C22×C4×C47—24no—
65C22×C4×C47—24no—
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C22×C411yescharacteristic
68C22×C4×C47C22×C414yes—
69C22×C4×C47C2×D414yes—
70C22×Dic47—24no—
71C22×Dic47C22×C414yes—
72C2×D188—28no—
73C2×D188—28no—
74C2×D188—216no—
75C2×D188—216no—
76C2×D188C22⋊C418yes—
77C2×D188C22⋊C418yes—
78C2×C4×D47—28no—
79C2×C4×D47—28no—
80C2×C4×D47—48no—
81C2×C4×D47—88no—
82C2×C4×D47—84no—
83C2×C4×D47—84no—
84C4⋊C4×C47—24no—
85C22⋊C4×C47—24no—
86C22⋊C4×C47—28no—
87C42×C47—24no—
88C22⋊Dic47—24no—
89C22⋊Dic47—28no—
90(C2×D47)⋊C4—28no—
91(C2×D47)⋊C4—28no—
92(C2×D47)⋊C4—48no—
93(C2×D47)⋊C4—88no—
94C4⋊Dic47—24no—
95C4×Dic47—24no—
96C23×C47—42no—
97C23×C47C23×C411yescharacteristic
98C23×C47C22×D412yes—
99C23×C47C2×#4114yes—
100C22×D47—24no—
101C22×D47—28no—
102C22×D47—44no—
103C22×D47—44no—
104C22×D47—48no—
105C22×D47—48no—
106C22×D47—88no—
107C22×D47—88no—
108C22×D47C2×C22⋊C414yes—
109C22×D47C2×C22⋊C414yes—
110C22×D47#11718yes—
111C22×D47#12318yes—
112C22×D47#12318yes—
113C2×C4×C47—22no—
114C2×C4×C47—22no—
115C2×C4×C47—28no—
116C2×C4×C47—24no—
117C2×C4×C47—44no—
118C2×C4×C47—48no—
119C2×C4×C47—44no—
120C2×C4×C47C2×C22⋊C412yes—
121C2×C4×C47C2×C22⋊C412yes—
122C2×Dic47—24no—
123C2×Dic47—44no—
124C2×Dic47—48no—
125C2×Dic47—44no—
126D188—216no—
127D188—216no—
128C4×D47—88no—
129C4×D47—88no—
130C4×D47—816no—
131#20689—471nomaximal, Sylow
132C22×C47—22no—
133C22×C47—24no—
134C22×C47—44no—
135C22×C47C22×C22⋊C411yescharacteristic
136C22×C47C2×#11712yes—
137C22×C47C2×#12314yes—
138C2×D47—28no—
139C2×D47—28no—
140C2×D47—24no—
141C2×D47—48no—
142C2×D47—48no—
143C2×D47—48no—
144C2×D47—48no—
145C2×D47—88no—
146C2×D47#38118yes—
147C4×C47—22no—
148C4×C47—22no—
149C4×C47—44no—
150C4×C47—84no—
151Dic47—44no—
152Dic47—84no—
153C2×#518—471no—
154#2308—472no—
155#1820—472no—
156#1820—471no—
157#1820D4711yescharacteristic, complemented
158#1819—474no—
159C2×#342—474no—
160C2×C47—24no—
161C2×C47—22no—
162C2×C47C2×#38111yescharacteristic
163C2×C47#288312yes—
164D47—28no—
165D47—44no—
166C23×D4—471no—
167#518—948no—
168C2×C4×D4—474no—
169C2×C4×D4—474no—
170C2×#117—471no—
171C2×#117C2×D4711yescharacteristic
172C22×C22⋊C4—472no—
173C22×C22⋊C4—472no—
174C22×C22⋊C4—474no—
175#386—948no—
176#381—472no—
177#381C2×D4712yes—
178#379—474no—
179#342—478no—
180#342—474no—
181#342C2×D4714yes—
182C47#2068911yescharacteristic, complemented, Sylow
183C25—471no—
184C25—471no—
185C22×D4—472no—
186C22×D4—472no—
187C22×D4—478no—
188C22×D4—948no—
189C23×C4—474no—
190C23×C4—474no—
191C23×C4—944no—
192C23×C4C22×D4711yescharacteristic
193C4×D4—9416no—
194C4×D4—9416no—
195#117—24no—
196#117—944no—
197C2×C4⋊C4—472no—
198C2×C4⋊C4C4×D4712yescomplemented
199C2×C22⋊C4—472no—
200C2×C22⋊C4—474no—
201C2×C22⋊C4—474no—
202C2×C22⋊C4—474no—
203C2×C22⋊C4—948no—
204C2×C22⋊C4—948no—
205C2×C22⋊C4—9416no—
206C2×C22⋊C4—948no—
207C2×C22⋊C4—944no—
208C2×C22⋊C4C22×D4714yes—
209C2×C22⋊C4C22×D4712yes—
210C2×C42—472no—
211C2×C42C4×D4712yescomplemented
212#95—24no—
213#95—944no—
214C24—478no—
215C24—472no—
216C24—474no—
217C24—472no—
218C24—474no—
219C24—944no—
220C24—3762no—
221C24—3762no—
222C24C23×D4711yescharacteristic
223C2×D4—478no—
224C2×D4—478no—
225C2×D4—948no—
226C2×D4—948no—
227C2×D4—9416no—
228C2×D4—9416no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—474no—
232C22×C4—948no—
233C22×C4—948no—
234C22×C4—944no—
235C22×C4—3768no—
236C22×C4—3764no—
237C22×C4—3764no—
238C22×C4—1888no—
239C22×C4C2×C4×D4711yescharacteristic
240C22×C4C2×C4×D4711yescharacteristic
241C22×C4C2×C4×D4714yes—
242C22×C4D4×D4714yes—
243C4⋊C4—24no—
244C4⋊C4—944no—
245C22⋊C4—24no—
246C22⋊C4—28no—
247C22⋊C4—948no—
248C22⋊C4—948no—
249C22⋊C4—944no—
250C22⋊C4—948no—
251C22⋊C4—3768no—
252C22⋊C4—1888no—
253C42—24no—
254C42—944no—
255C23—42no—
256C23—474no—
257C23—478no—
258C23—478no—
259C23—478no—
260C23—474no—
261C23—948no—
262C23—944no—
263C23—3768no—
264C23—3768no—
265C23—1888no—
266C23—1884no—
267C23—1884no—
268C23—1888no—
269C23C22×C4×D4711yescharacteristic
270C23C2×D4×D4712yes—
271C23#41×D4714yes—
272D4—9416no—
273D4—9416no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—28no—
277C2×C4—24no—
278C2×C4—44no—
279C2×C4—48no—
280C2×C4—44no—
281C2×C4—944no—
282C2×C4—3768no—
283C2×C4—3768no—
284C2×C4—37616no—
285C2×C4—1888no—
286C2×C4—1884no—
287C2×C4—1884no—
288C2×C4C22⋊C4×D4712yes—
289C2×C4C22⋊C4×D4712yes—
290C22—22no—
291C22—24no—
292C22—44no—
293C22—478no—
294C22—948no—
295C22—944no—
296C22—948no—
297C22—3768no—
298C22—1888no—
299C22—1888no—
300C22—1888no—
301C22—1888no—
302C22C2×C22⋊C4×D4711yescharacteristic
303C22#117×D4712yes—
304C22#123×D4714yes—
305C4—22no—
306C4—22no—
307C4—44no—
308C4—84no—
309C4—3764no—
310C4—1884no—
311C2—24no—
312C2—22no—
313C2—948no—
314C2—1884no—
315C2#381×D4711yescharacteristic
316C2#2025184712yes—
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC47×#1820C2
socleC22×C47C22×C22⋊C4
prime radicalC23×D188C22
cosocleC24×C47C24
hypercenter#1820D47
hypocenterC47#20689
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
3. #20243465C2semidirect
4. #20243463C2semidirect
5. #20243461C2semidirect
6. #342×D47C2semidirect
7. C47×#1820C2semidirect
10. C2×C4×D188C4semidirect
11. C2×#13093986C4semidirect
32. C22×C4×D47C2×C4semidirect
36. C2×C4⋊Dic47C2×C4semidirect
42. C2×(C2×D47)⋊C4C2×C4semidirect
46. C2×C4×Dic47C2×C4semidirect
50. C2×C4⋊C4×C47C2×C4semidirect
53. C2×C42×C47C2×C4semidirect
157. #1820D47semidirect
182. C47#20689semidirect
198. C2×C4⋊C4C4×D47semidirect
211. C2×C42C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156895972
#202459652
#202434652
#202434632
#202434612
#342×D472
C47×#18202
#2068947