Finite Group Databasebeta

← #26131468

Subgroups — #26131468

Order 12032.

512 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 135 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×#15688920C212yesmaximal
2C4×#13093943C212yesmaximal, complemented
3#20241515C212yesmaximal, complemented
4C47×#2326C211yesmaximal, characteristic
5C2×#13093943C2214yes—
6C4×#10625329—22no—
7#15688920C418yescomplemented
8#15688892—22no—
9C2×#124×C47C2211yescharacteristic
10C2×C4×Q8×C47C2211yescharacteristic
11C2×C4×D4×C47C411yescharacteristic
12C47×#391C2211yescharacteristic
13C47×#390C411yescharacteristic
14C47×#388—22no—
15C47×#385—22no—
16C47×#384C411yescharacteristic
17C47×#376C411yescharacteristic
18C2×#10625329—24no—
19#13093943—24no—
20#13093943C2×C418yescomplemented
21#13093935—42no—
22C4×C47⋊C8—42no—
23C22×Q8×C47C2311yescharacteristic
24C22×D4×C47C2×C411yescharacteristic
25C23×C4×C47—21no—
26#124×C47—22no—
27#124×C47C2×C414yes—
28C4×Q8×C47—42no—
29C4×D4×C47—42no—
30C2×C4⋊C4×C47—22no—
31C2×C4⋊C4×C47C2×C411yescharacteristic
32C2×C4⋊C4×C47C2×C411yescharacteristic
33C2×C4⋊C4×C47D412yes—
34C2×C22⋊C4×C47—22no—
35C2×C22⋊C4×C47—21no—
36C2×C22⋊C4×C47—22no—
37C2×C42×C47C2×C411yescharacteristic
38C2×C42×C47C2×C411yescharacteristic
39C2×C42×C47C2×C411yescharacteristic
40C2×C42×C47D412yes—
41#95×C47—21no—
42#95×C47—21no—
43#95×C47—22no—
44C24×C47—21no—
45C2×Q8×C47—41no—
46C2×Q8×C47—41no—
47C2×Q8×C47C22×C412yes—
48C2×Q8×C47#4112yes—
49C2×D4×C47—21no—
50C2×D4×C47—41no—
51C2×D4×C47—41no—
52C2×D4×C47C4212yes—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—42no—
57C22×C4×C47—82no—
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47C22⋊C412yes—
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D412yes—
63C4⋊C4×C47—41no—
64C4⋊C4×C47—42no—
65C4⋊C4×C47—41no—
66C4⋊C4×C47—82no—
67C22⋊C4×C47—28no—
68C22⋊C4×C47—42no—
69C22⋊C4×C47—82no—
70C42×C47—22no—
71C42×C47—22no—
72C42×C47—22no—
73C42×C47—21no—
74C42×C47—41no—
75C42×C47C4212yes—
76#10625329—28no—
77#10625329—44no—
78C2×C47⋊C8—44no—
79C23×C47—22no—
80C23×C47—42no—
81C23×C47—42no—
82C23×C47—41no—
83C23×C47—41no—
84C23×C47C2×C22⋊C411yescharacteristic
85Q8×C47—42no—
86Q8×C47—42no—
87D4×C47—42no—
88D4×C47—42no—
89C2×C4×C47—22no—
90C2×C4×C47—21no—
91C2×C4×C47—22no—
92C2×C4×C47—21no—
93C2×C4×C47—22no—
94C2×C4×C47—21no—
95C2×C4×C47—22no—
96C2×C4×C47—41no—
97C2×C4×C47—41no—
98C2×C4×C47—41no—
99C2×C4×C47—42no—
100C2×C4×C47—41no—
101C2×C4×C47—41no—
102C2×C4×C47—42no—
103C2×C4×C47—41no—
104C2×C4×C47—41no—
105C2×C4×C47—41no—
106C2×C4×C47—84no—
107C2×C4×C47#9912yes—
108C2×C4×C47C2×C4211yescharacteristic
109C2×C4×C47#11711yescharacteristic
110C2×C4×C47C4×D414yes—
111C47⋊C8—48no—
112#12632—471nomaximal, Sylow
113C22×C47—21no—
114C22×C47—21no—
115C22×C47—44no—
116C22×C47—42no—
117C22×C47—82no—
118C22×C47—81no—
119C22×C47—81no—
120C22×C47C4×C22⋊C411yescharacteristic
121C22×C47C2×#9911yescharacteristic
122C22×C47#41011yescharacteristic
123C4×C47—22no—
124C4×C47—24no—
125C4×C47—21no—
126C4×C47—21no—
127C4×C47—42no—
128C4×C47—42no—
129C4×C47—41no—
130C4×C47—81no—
131C4×C47—81no—
132#2326D4711yescharacteristic
133C2×#355—472no—
134C4×#101—472no—
135#1394—472no—
136C2×C47—21no—
137C2×C47—21no—
138C2×C47—82no—
139C2×C47C4×#9911yescharacteristic
140C2×C47#213311yescharacteristic
141C2×C47#213411yescharacteristic
142C2×#124C2×D4711yescharacteristic
143C2×C4×Q8C2×D4711yescharacteristic
144C2×C4×D4Dic4711yescharacteristic
145C2×#101—474no—
146C4×M16—942no—
147#391C2×D4711yescharacteristic
148#390Dic4711yescharacteristic
149#388—22no—
150#385—22no—
151#384Dic4711yescharacteristic
152#376Dic4711yescharacteristic
153#355—478no—
154#343—942no—
155C47#1263211yescharacteristic, complemented, Sylow
156C22×Q8C22×D4711yescharacteristic
157C22×D4C2×Dic4711yescharacteristic
158C23×C4—21no—
159C2×M16—944no—
160#124—22no—
161#124C4×D4714yes—
162C4×Q8—42no—
163C4×D4—42no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C2×Dic4711yescharacteristic
166C2×C4⋊C4C2×Dic4711yescharacteristic
167C2×C4⋊C4C22⋊D4712yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C42C2×Dic4711yescharacteristic
172C2×C42C2×Dic4711yescharacteristic
173C2×C42C2×Dic4711yescharacteristic
174C2×C42C22⋊D4712yes—
175#101—478no—
176#101—944no—
177#97—1882no—
178C4×C8—1882no—
179#95—21no—
180#95—21no—
181#95—22no—
182C24—21no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8C2×C4×D4712yes—
186C2×Q8#1062535012yes—
187C2×D4—21no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4C4×Dic4712yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—42no—
195C22×C4—82no—
196C22×C4C22⋊Dic4711yescharacteristic
197C22×C4C22⋊Dic4712yes—
198C22×C4C22⋊Dic4711yescharacteristic
199C22×C4C22×Dic4711yescharacteristic
200C22×C4C2×C22⋊D4712yes—
201M16—948no—
202M16—1884no—
203C2×C8—1884no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—82no—
208C22⋊C4—28no—
209C22⋊C4—42no—
210C22⋊C4—82no—
211C42—22no—
212C42—22no—
213C42—22no—
214C42—21no—
215C42—41no—
216C42C4×Dic4712yes—
217C23—22no—
218C23—42no—
219C23—42no—
220C23—41no—
221C23—41no—
222C23C2×C22⋊Dic4711yescharacteristic
223Q8—42no—
224Q8—42no—
225D4—42no—
226D4—42no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—84no—
245C2×C4#1309393912yes—
246C2×C4C2×C4×Dic4711yescharacteristic
247C2×C4#1309398311yescharacteristic
248C2×C4C4×C22⋊D4714yes—
249C8—1888no—
250C22—21no—
251C22—21no—
252C22—44no—
253C22—42no—
254C22—82no—
255C22—81no—
256C22—81no—
257C22C4×C22⋊Dic4711yescharacteristic
258C22C2×#1309393911yescharacteristic
259C22#1568909711yescharacteristic
260C4—22no—
261C4—24no—
262C4—21no—
263C4—21no—
264C4—42no—
265C4—41no—
266C4—42no—
267C4—81no—
268C4—81no—
269C2—21no—
270C2—21no—
271C2—82no—
272C2C4×#1309393911yescharacteristic
273C2#2024507711yescharacteristic
274C2#2024508011yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC2×C4×C47C2×C42
FrattiniC22×Q8C22×D47
FittingC47×#2326C2
socleC22×C47C2×#99
prime radicalC22×D4×C47C2×C4
cosocleC22×Q8×C47C23
hypercenter#2326D47
hypocenterC47#12632
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. C4×#13093943C2semidirect
3. #20241515C2semidirect
7. #15688920C4semidirect
20. #13093943C2×C4semidirect
155. C47#12632semidirect

Maximal subgroups

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

subgroupindex
C2×#156889202
C4×#130939432
#202415152
C47×#23262
#1263247