Finite Group Databasebeta

← #26126682

Subgroups — #26126682

Order 12032.

356 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20245187C211yesmaximal, characteristic, complemented
2C4×#13093956C211yesmaximal, characteristic, complemented
3#20241400C212yesmaximal, complemented
4#20241397C212yesmaximal
5C47×#3168C211yesmaximal, characteristic
6C2×#13093956—21no—
7C4×C4⋊Dic47—21no—
8C4⋊(C47⋊C16)—21no—
9C22⋊(C47⋊C16)C2212yes—
10C4×C47⋊C16—21no—
11#15688886C2212yes—
12C47×#529—21no—
13C47×#499C411yescharacteristic, complemented
14C47×#495C411yescharacteristic, complemented
15C47×#488—21no—
16C47×#467C2211yescharacteristic
17C47×#466C2211yescharacteristic
18C47×#465—21no—
19C4×Q16×C47C411yescharacteristic, complemented
20C4×SD16×C47—21no—
21C4×D8×C47C411yescharacteristic, complemented
22C2×C4×C8×C47C2211yescharacteristic
23C2×C4⋊Dic47—22no—
24C2×C4×Dic47—41no—
25C2×C47⋊C16—22no—
26#13093956—24no—
27C47⋊#95—41no—
28C2×Q16×C47C2×C411yescharacteristic
29C2×SD16×C47—21no—
30C2×D8×C47C2×C411yescharacteristic
31C22×C8×C47D411yescharacteristic
32C22×C8×C47C2311yescharacteristic
33#126×C47—41no—
34#125×C47—21no—
35#124×C47—21no—
36#123×C47—41no—
37C22⋊Q8×C47—21no—
38#121×C47—21no—
39C4×Q8×C47—21no—
40C4×D4×C47—21no—
41C4×D4×C47—41no—
42#117×C47—41no—
43C2×C42×C47D411yescharacteristic
44#107×C47C2×C411yescharacteristic
45#107×C47C2×C411yescharacteristic
46#106×C47—21no—
47#103×C47—21no—
48#103×C47—21no—
49#102×C47—21no—
50#102×C47—21no—
51C4×C8×C47C2×C411yescharacteristic
52C4×C8×C47C2×C411yescharacteristic
53C4×C8×C47D411yescharacteristic
54C4×C8×C47D411yescharacteristic
55C2×Q8×C47—21no—
56C2×D4×C47—21no—
57C2×D4×C47—41no—
58C22×C4×C47—41no—
59C22×C4×C47D811yescharacteristic
60C22×C4×C47D811yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×Dic47—42no—
63Q16×C47—41no—
64SD16×C47—42no—
65D8×C47—41no—
66C2×C8×C47—22no—
67C2×C8×C47—41no—
68C2×C8×C47C22⋊C411yescharacteristic
69C2×C8×C47C22⋊C411yescharacteristic
70C2×C8×C47C22×C411yescharacteristic
71C2×C8×C47C2×D411yescharacteristic
72C4⋊C4×C47—21no—
73C4⋊C4×C47—21no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—41no—
76C4⋊C4×C47—41no—
77C4⋊C4×C47—41no—
78C22⋊C4×C47—41no—
79C22⋊C4×C47—41no—
80C22⋊C4×C47—41no—
81C22⋊C4×C47—41no—
82C22⋊C4×C47—41no—
83C42×C47—41no—
84C42×C47C22⋊C411yescharacteristic
85C42×C47C22⋊C411yescharacteristic
86C42×C47SD1611yescharacteristic
87C42×C47SD1611yescharacteristic
88C4⋊Dic47—24no—
89C4⋊Dic47—42no—
90C4×Dic47—82no—
91C47⋊C16—42no—
92C23×C47—41no—
93C23×C47C2×D811yescharacteristic
94Q8×C47—41no—
95Q8×C47—41no—
96D4×C47—41no—
97D4×C47—41no—
98D4×C47—82no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—41no—
104C2×C4×C47—41no—
105C2×C4×C47—41no—
106C2×C4×C47—82no—
107C2×C4×C47#10211yescharacteristic
108C2×C4×C47#10211yescharacteristic
109C2×C4×C47#10211yescharacteristic
110C2×C4×C47#10211yescharacteristic
111C2×C4×C47C2×C22⋊C411yescharacteristic
112C2×C4×C47C2×SD1611yescharacteristic
113C2×Dic47—44no—
114C2×Dic47—82no—
115C2×Dic47—82no—
116C8×C47—21no—
117C8×C47—21no—
118C8×C47—21no—
119C8×C47#10412yes—
120#11741—471nomaximal, Sylow
121C22×C47—21no—
122C22×C47—21no—
123C22×C47—81no—
124C22×C47—81no—
125C22×C47—81no—
126C22×C47C2×#10211yescharacteristic
127C22×C47#50812yes—
128C4×C47—21no—
129C4×C47—21no—
130C4×C47—21no—
131C4×C47—21no—
132C4×C47—81no—
133C4×C47—81no—
134C4×C47—81no—
135C4×C47#36012yes—
136C4×C47#41011yescharacteristic
137C4×C47C2×#10411yescharacteristic
138Dic47—84no—
139#3168D4711yescharacteristic
140#2162—471no—
141C4×#106—471no—
142#1348—472no—
143#1347—472no—
144C2×C47—22no—
145C2×C47—161no—
146C2×C47#150211yescharacteristic
147C2×C47#213811yescharacteristic
148C2×C47C2×#36011yescharacteristic
149#529—21no—
150#499Dic4711yescharacteristic, complemented
151#495Dic4711yescharacteristic, complemented
152#488—21no—
153#467C2×D4711yescharacteristic
154#466C2×D4711yescharacteristic
155#465—21no—
156C4×Q16Dic4711yescharacteristic, complemented
157C4×SD16—21no—
158C4×D8Dic4711yescharacteristic, complemented
159C2×#106—941no—
160C2×C4×C8C2×D4711yescharacteristic
161C4×C4⋊C4—941no—
162C4⋊C16—941no—
163C22⋊C16—472no—
164C4×C16—941no—
165#340—472no—
166C47#1174111yescharacteristic, complemented, Sylow
167C2×Q16C2×Dic4711yescharacteristic
168C2×SD16—21no—
169C2×D8C2×Dic4711yescharacteristic
170C22×C8C22⋊D4711yescharacteristic
171C22×C8C22×D4711yescharacteristic
172#126—41no—
173#125—21no—
174#124—21no—
175#123—41no—
176C22⋊Q8—21no—
177#121—21no—
178C4×Q8—21no—
179C4×D4—21no—
180C4×D4—41no—
181#117—41no—
182C2×C4⋊C4—942no—
183C2×C42—1881no—
184C2×C42C22⋊D4711yescharacteristic
185C2×C16—942no—
186#107C2×Dic4711yescharacteristic
187#107C2×Dic4711yescharacteristic
188#106—21no—
189#106—944no—
190#103—21no—
191#103—21no—
192#102—21no—
193#102—21no—
194C4×C8C2×Dic4711yescharacteristic
195C4×C8C2×Dic4711yescharacteristic
196C4×C8C22⋊D4711yescharacteristic
197C4×C8C22⋊D4711yescharacteristic
198#95—1881no—
199C2×Q8—21no—
200C2×D4—21no—
201C2×D4—41no—
202C22×C4—41no—
203C22×C4—1882no—
204C22×C4#1062533411yescharacteristic
205C22×C4#1062533411yescharacteristic
206C22×C4C2×C22⋊D4711yescharacteristic
207Q16—41no—
208SD16—42no—
209D8—41no—
210C2×C8—22no—
211C2×C8—41no—
212C2×C8C22⋊Dic4711yescharacteristic
213C2×C8C22⋊Dic4711yescharacteristic
214C2×C8C22×Dic4711yescharacteristic
215C2×C8C2×C22⋊D4711yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—944no—
223C4⋊C4—1882no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C42—41no—
230C42—3762no—
231C42#1062533511yescharacteristic
232C42#1062533511yescharacteristic
233C42C22⋊Dic4711yescharacteristic
234C42C22⋊Dic4711yescharacteristic
235C16—1882no—
236C23—41no—
237C23C2×#1062533411yescharacteristic
238Q8—41no—
239Q8—41no—
240D4—41no—
241D4—41no—
242D4—82no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—82no—
251C2×C4—3762no—
252C2×C4—3762no—
253C2×C4—1884no—
254C2×C4#1309394511yescharacteristic
255C2×C4#1309394511yescharacteristic
256C2×C4#1309394511yescharacteristic
257C2×C4#1309394511yescharacteristic
258C2×C4C2×C22⋊Dic4711yescharacteristic
259C2×C4C2×#1062533511yescharacteristic
260C8—21no—
261C8—21no—
262C8—21no—
263C8#1309395112yes—
264C22—21no—
265C22—21no—
266C22—81no—
267C22—81no—
268C22—81no—
269C22C2×#1309394511yescharacteristic
270C22#1568955312yes—
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—81no—
276C4—81no—
277C4—81no—
278C4—3764no—
279C4#1568893312yes—
280C4#1568909711yescharacteristic
281C4C2×#1309395111yescharacteristic
282C2—22no—
283C2—161no—
284C2#2024188311yescharacteristic
285C2#2024509511yescharacteristic
286C2C2×#1568893311yescharacteristic
subgroupG/H
centerC2×C4#13093945
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#3168C2
socleC22×C47C2×#102
prime radicalC47×#466C22
cosocleC22×C8×C47C23
hypercenter#3168D47
hypocenterC47#11741
LCSG ▷ C2×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. #20245187C2semidirect
2. C4×#13093956C2semidirect
3. #20241400C2semidirect
13. C47×#499C4semidirect
14. C47×#495C4semidirect
19. C4×Q16×C47C4semidirect
21. C4×D8×C47C4semidirect
150. #499Dic47semidirect
151. #495Dic47semidirect
156. C4×Q16Dic47semidirect
158. C4×D8Dic47semidirect
166. C47#11741semidirect

Maximal subgroups

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

subgroupindex
#202451872
C4×#130939562
#202414002
#202413972
C47×#31682
#1174147