Finite Group Databasebeta

← #26125411

Subgroups — #26125411

Order 12032.

450 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C22×Dic47⋊C8C211yesmaximal, characteristic
2#20244999C211yesmaximal, characteristic
3#20241327C214yesmaximal, complemented
4C47×#2112C211yesmaximal, characteristic
5C23×C47⋊C8C411yescharacteristic
6C22×C4×Dic47C411yescharacteristic
7C2×#13093964—21no—
8C2×Dic47⋊C8—21no—
9C2×Dic47⋊C8—21no—
10C2×Dic47⋊C8—21no—
11C2×Dic47⋊C8—21no—
12C2×Dic47⋊C8C412yes—
13C2×Dic47⋊C8C2212yes—
14C22⋊(C47⋊C16)—21no—
15C22⋊(C47⋊C16)C2212yes—
16C23×C8×C47C2211yescharacteristic
17C2×M32×C47—21no—
18C47×C22⋊C16—21no—
19C47×C22⋊C16C2212yes—
20C23×Dic47C811yescharacteristic
21C23×Dic47C811yescharacteristic
22C22×C47⋊C8—21no—
23C22×C47⋊C8—21no—
24C22×C47⋊C8—42no—
25C22×C47⋊C8C2×C412yes—
26C2×C4×Dic47—21no—
27C2×C4×Dic47—21no—
28C2×C4×Dic47—21no—
29C2×C4×Dic47—21no—
30C2×C4×Dic47C812yes—
31C2×C4×Dic47C2×C412yes—
32#13093964—42no—
33C2×C47⋊C16—22no—
34Dic47⋊C8—22no—
35Dic47⋊C8—22no—
36Dic47⋊C8—22no—
37Dic47⋊C8—22no—
38C23×C4×C47C2×C411yescharacteristic
39C22×C8×C47—42no—
40C22×C8×C47C2×C411yescharacteristic
41C22×C8×C47D411yescharacteristic
42C22×C8×C47D411yescharacteristic
43C22×C8×C47D411yescharacteristic
44C22×C8×C47D411yescharacteristic
45C22×C8×C47C2311yescharacteristic
46M32×C47—42no—
47C2×C16×C47—22no—
48C24×C47C2×C811yescharacteristic
49C22×C4×C47—22no—
50C22×C4×C47—22no—
51C22×C4×C47C22⋊C411yescharacteristic
52C22×C4×C47C22⋊C411yescharacteristic
53C22×C4×C47C2×C811yescharacteristic
54C22×C4×C47M1611yescharacteristic
55C22×C4×C47M1611yescharacteristic
56C22×C4×C47C22×C411yescharacteristic
57C22×Dic47—21no—
58C22×Dic47—21no—
59C22×Dic47—24no—
60C22×Dic47—21no—
61C22×Dic47—21no—
62C22×Dic47—42no—
63C22×Dic47C2×C812yes—
64C22×Dic47C2×C812yes—
65C2×C8×C47—22no—
66C2×C8×C47—22no—
67C2×C8×C47—41no—
68C2×C8×C47—41no—
69C2×C8×C47—41no—
70C2×C8×C47—41no—
71C2×C8×C47C22⋊C411yescharacteristic
72C2×C8×C47C22⋊C411yescharacteristic
73C2×C8×C47C2×D411yescharacteristic
74C2×C8×C47C2×D411yescharacteristic
75C4×Dic47—22no—
76C4×Dic47—22no—
77C4×Dic47—22no—
78C4×Dic47—22no—
79C2×C47⋊C8—22no—
80C2×C47⋊C8—42no—
81C2×C47⋊C8—42no—
82C2×C47⋊C8—42no—
83C16×C47—42no—
84C47⋊C16—42no—
85C23×C47—22no—
86C23×C47—22no—
87C23×C47—22no—
88C23×C47C22⋊C811yescharacteristic
89C23×C47C22⋊C811yescharacteristic
90C23×C47C22×C811yescharacteristic
91C2×C4×C47—22no—
92C2×C4×C47—22no—
93C2×C4×C47—22no—
94C2×C4×C47—21no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47C22⋊C811yescharacteristic
99C2×C4×C47C22⋊C811yescharacteristic
100C2×C4×C47#9912yes—
101C2×C4×C47#10112yes—
102C2×C4×C47C2×C22⋊C411yescharacteristic
103C2×C4×C47C2×M1611yescharacteristic
104C2×Dic47—24no—
105C2×Dic47—24no—
106C2×Dic47—22no—
107C2×Dic47—22no—
108C2×Dic47—22no—
109C2×Dic47—42no—
110C2×Dic47—42no—
111C2×Dic47—42no—
112C2×Dic47M3214yescomplemented
113C8×C47—21no—
114C8×C47—21no—
115C8×C47—21no—
116C8×C47—21no—
117C47⋊C8—42no—
118#11440—471nomaximal, Sylow
119C22×C47—28no—
120C22×C47—21no—
121C22×C47—21no—
122C22×C47—21no—
123C22×C47—21no—
124C22×C47—21no—
125C22×C47—21no—
126C22×C47#32412yes—
127C22×C47#34912yes—
128C22×C47C2×C22⋊C811yescharacteristic
129C22×C47C2×M3212yes—
130C4×C47—21no—
131C4×C47—21no—
132C4×C47#32412yes—
133C4×C47C2×#9911yescharacteristic
134C4×C47C2×#10111yescharacteristic
135Dic47—24no—
136Dic47—42no—
137C22×C4⋊C8—471no—
138#2112—471no—
139#2112D4711yescharacteristic
140#1322—474no—
141C2×C47—22no—
142C2×C47—22no—
143C2×C47—22no—
144C2×C47C2×#32411yescharacteristic
145C2×C47#211211yescharacteristic
146C2×C47C2×#34911yescharacteristic
147C23×C8—471no—
148C23×C8C2×D4711yescharacteristic
149C22×C42—471no—
150C2×M32—21no—
151C2×M32—941no—
152C2×C4⋊C8—472no—
153C2×C4⋊C8—472no—
154C2×C4⋊C8—941no—
155C2×C4⋊C8—941no—
156C2×C4⋊C8—941no—
157C2×C4⋊C8—941no—
158C22⋊C16—21no—
159C22⋊C16—472no—
160C22⋊C16—941no—
161C22⋊C16C2×D4712yes—
162C47#1144011yescharacteristic, complemented, Sylow
163C23×C4—471no—
164C23×C4—471no—
165C23×C4C4×D4711yescharacteristic
166C22×C8—42no—
167C22×C8—472no—
168C22×C8—941no—
169C22×C8—941no—
170C22×C8—1882no—
171C22×C8C4×D4711yescharacteristic
172C22×C8D18811yescharacteristic
173C22×C8D18811yescharacteristic
174C22×C8C22⋊D4711yescharacteristic
175C22×C8C22⋊D4711yescharacteristic
176C22×C8C22×D4711yescharacteristic
177C2×C42—472no—
178C2×C42—472no—
179C2×C42—941no—
180C2×C42—941no—
181C2×C42—941no—
182C2×C42—941no—
183M32—42no—
184M32—1882no—
185C2×C16—22no—
186C2×C16—942no—
187C4⋊C8—942no—
188C4⋊C8—942no—
189C4⋊C8—942no—
190C4⋊C8—942no—
191C24C8×D4711yescharacteristic
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—472no—
195C22×C4—472no—
196C22×C4—944no—
197C22×C4—941no—
198C22×C4—941no—
199C22×C4—941no—
200C22×C4—941no—
201C22×C4—1882no—
202C22×C4C8×D4711yescharacteristic
203C22×C4#1062532411yescharacteristic
204C22×C4#1062532411yescharacteristic
205C22×C4(C2×D47)⋊C411yescharacteristic
206C22×C4(C2×D47)⋊C411yescharacteristic
207C22×C4C2×C4×D4711yescharacteristic
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—942no—
215C2×C8—1882no—
216C2×C8—1882no—
217C2×C8—1882no—
218C2×C8(C2×D47)⋊C411yescharacteristic
219C2×C8(C2×D47)⋊C411yescharacteristic
220C2×C8C2×D18811yescharacteristic
221C2×C8C2×C22⋊D4711yescharacteristic
222C42—942no—
223C42—942no—
224C42—942no—
225C42—942no—
226C16—42no—
227C16—1882no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23(C2×D47)⋊C811yescharacteristic
232C23(C2×D47)⋊C811yescharacteristic
233C23C2×C8×D4711yescharacteristic
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—474no—
242C2×C4—944no—
243C2×C4—944no—
244C2×C4—942no—
245C2×C4—942no—
246C2×C4—942no—
247C2×C4—1882no—
248C2×C4—1882no—
249C2×C4—1882no—
250C2×C4(C2×D47)⋊C811yescharacteristic
251C2×C4(C2×D47)⋊C811yescharacteristic
252C2×C4#1309393812yes—
253C2×C4#1309394212yes—
254C2×C4C2×(C2×D47)⋊C411yescharacteristic
255C2×C4C2×#1062532411yescharacteristic
256C8—21no—
257C8—21no—
258C8—21no—
259C8—21no—
260C8—1882no—
261C22—28no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22#1568884812yes—
269C22#1568890412yes—
270C22C2×(C2×D47)⋊C811yescharacteristic
271C22M32×D4712yes—
272C4—21no—
273C4—21no—
274C4—944no—
275C4—1882no—
276C4#1568884812yes—
277C4C2×#1309393811yescharacteristic
278C4C2×#1309394211yescharacteristic
279C2—22no—
280C2—22no—
281C2—22no—
282C2C2×#1568884811yescharacteristic
283C2#2024499811yescharacteristic
284C2C2×#1568890411yescharacteristic
subgroupG/H
centerC2×C4C2×(C2×D47)⋊C4
commutatorC23×C47C22×C8
FrattiniC22×C8C22×D47
FittingC47×#2112C2
socleC22×C47C2×C22⋊C8
prime radicalC24×C47C2×C8
cosocleC22×C8×C47C23
hypercenter#2112D47
hypocenterC47#11440
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
3. #20241327C2semidirect
112. C2×Dic47M32semidirect
162. C47#11440semidirect

Maximal subgroups

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

subgroupindex
C22×Dic47⋊C82
#202449992
#202413272
C47×#21122
#1144047