Finite Group Databasebeta

← #26125477

Subgroups — #26125477

Order 12032.

374 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 97 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×#15689173C211yesmaximal, characteristic
2#20245188C212yesmaximal, complemented
3#20241327C212yesmaximal, complemented
4#20241312C211yesmaximal, characteristic
5C47×#1314C211yesmaximal, characteristic
6C22×C4×Dic47C411yescharacteristic
7#15689173—22no—
8#15689173—22no—
9C2×Dic47⋊C8C411yescharacteristic
10C2×Dic47⋊C8C2211yescharacteristic
11C2×C22⋊(C47⋊C8)C411yescharacteristic
12C2×C8×Dic47C411yescharacteristic
13C2×C8×Dic47C2211yescharacteristic
14Dic47⋊C16—22no—
15C22⋊(C47⋊C16)C2212yes—
16#15688898—22no—
17C2×C22⋊C8×C47C2211yescharacteristic
18C47×C22⋊C16C2212yes—
19C23×Dic47C811yescharacteristic
20C23×Dic47C811yescharacteristic
21C22×C47⋊C8C2×C411yescharacteristic
22C22×C47⋊C8C2×C411yescharacteristic
23C2×C4×Dic47—21no—
24C2×C4×Dic47—21no—
25C2×C4×Dic47—21no—
26C2×C4×Dic47—21no—
27C2×C4×Dic47C811yescharacteristic
28C2×C4×Dic47C811yescharacteristic
29C2×C4×Dic47C2×C411yescharacteristic
30C2×C4×Dic47C2×C411yescharacteristic
31C2×C47⋊C16—22no—
32Dic47⋊C8—22no—
33Dic47⋊C8—22no—
34C22⋊(C47⋊C8)—22no—
35C8×Dic47—21no—
36C8×Dic47—21no—
37C8×Dic47D412yes—
38C8×Dic47D412yes—
39C23×C4×C47C2×C411yescharacteristic
40C22×C8×C47C2×C411yescharacteristic
41C22×C8×C47C2311yescharacteristic
42C2×C16×C47—22no—
43C22⋊C8×C47—22no—
44C24×C47C2×C811yescharacteristic
45C22×C4×C47—21no—
46C22×C4×C47—21no—
47C22×C4×C47—41no—
48C22×C4×C47—41no—
49C22×C4×C47C2×C811yescharacteristic
50C22×C4×C47C22×C411yescharacteristic
51C22×Dic47—22no—
52C22×Dic47—21no—
53C22×Dic47—21no—
54C22×Dic47—22no—
55C22×Dic47—21no—
56C22×Dic47—21no—
57C22×Dic47—42no—
58C22×Dic47C2×C811yescharacteristic
59C22×Dic47C2×C811yescharacteristic
60C22×Dic47C2×C811yescharacteristic
61C22×Dic47C2×C811yescharacteristic
62C2×C8×C47—21no—
63C2×C8×C47—41no—
64C2×C8×C47—41no—
65C2×C8×C47C2×D412yes—
66C4×Dic47—21no—
67C4×Dic47—21no—
68C4×Dic47—21no—
69C4×Dic47—21no—
70C4×Dic47—21no—
71C4×Dic47—21no—
72C4×Dic47C22⋊C411yescharacteristic
73C4×Dic47C22⋊C411yescharacteristic
74C4×Dic47M1611yescharacteristic
75C4×Dic47M1611yescharacteristic
76C2×C47⋊C8—21no—
77C2×C47⋊C8—41no—
78C2×C47⋊C8—41no—
79C2×C47⋊C8C22⋊C412yes—
80C16×C47—42no—
81C47⋊C16—42no—
82C23×C47—21no—
83C23×C47—21no—
84C23×C47—41no—
85C23×C47—41no—
86C23×C47M3212yes—
87C23×C47C22×C811yescharacteristic
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—21no—
91C2×C4×C47—42no—
92C2×C4×C47—41no—
93C2×C4×C47—41no—
94C2×C4×C47—41no—
95C2×C4×C47C2×C22⋊C411yescharacteristic
96C2×C4×C47C2×M1611yescharacteristic
97C2×Dic47—24no—
98C2×Dic47—24no—
99C2×Dic47—21no—
100C2×Dic47—21no—
101C2×Dic47—21no—
102C2×Dic47—21no—
103C2×Dic47—22no—
104C2×Dic47—21no—
105C2×Dic47—42no—
106C2×Dic47—42no—
107C2×Dic47—41no—
108C2×Dic47C22⋊C811yescharacteristic
109C2×Dic47C22⋊C811yescharacteristic
110C2×Dic47C22⋊C811yescharacteristic
111C2×Dic47C22⋊C811yescharacteristic
112C2×Dic47M3212yes—
113C8×C47—22no—
114C8×C47—41no—
115C47⋊C8—22no—
116C47⋊C8—41no—
117#11460—471nomaximal, Sylow
118C22×C47—21no—
119C22×C47—21no—
120C22×C47—22no—
121C22×C47—42no—
122C22×C47—42no—
123C22×C47—41no—
124C22×C47—41no—
125C22×C47C2×C22⋊C811yescharacteristic
126C22×C47C2×M3211yescharacteristic
127C22×C47C2×M3211yescharacteristic
128C4×C47—21no—
129C4×C47—41no—
130C4×C47#41011yescharacteristic
131C4×C47#41311yescharacteristic
132Dic47—21no—
133Dic47—21no—
134Dic47—22no—
135Dic47—21no—
136Dic47—41no—
137Dic47#34912yes—
138C2×#431—471no—
139#2163—472no—
140#1322—472no—
141#1314—471no—
142#1314D4711yescharacteristic
143C2×C47—21no—
144C2×C47—21no—
145C2×C47—42no—
146C2×C47#145811yescharacteristic
147C2×C47#211211yescharacteristic
148C2×C47C2×#34911yescharacteristic
149C22×C42—471no—
150#431—942no—
151#431—942no—
152C2×C4⋊C8—471no—
153C2×C4⋊C8—471no—
154C2×C22⋊C8—471no—
155C2×C22⋊C8C2×D4711yescharacteristic
156C2×C4×C8—471no—
157C2×C4×C8—471no—
158C4⋊C16—942no—
159C22⋊C16—472no—
160C22⋊C16C2×D4712yes—
161#346—942no—
162C47#1146011yescharacteristic, complemented, Sylow
163C23×C4—471no—
164C23×C4—471no—
165C23×C4C4×D4711yescharacteristic
166C22×C8—471no—
167C22×C8—471no—
168C22×C8C4×D4711yescharacteristic
169C22×C8C22×D4711yescharacteristic
170C2×C42—471no—
171C2×C42—471no—
172C2×C42—471no—
173C2×C42—471no—
174C2×C42—941no—
175C2×C42—941no—
176C2×C42—941no—
177C2×C42—941no—
178C2×C16—22no—
179C2×C16—942no—
180C4⋊C8—942no—
181C4⋊C8—942no—
182C22⋊C8—22no—
183C22⋊C8—942no—
184C4×C8—472no—
185C4×C8—472no—
186C4×C8—941no—
187C4×C8—941no—
188C24C8×D4711yescharacteristic
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—471no—
194C22×C4—471no—
195C22×C4—471no—
196C22×C4—471no—
197C22×C4—941no—
198C22×C4—941no—
199C22×C4—942no—
200C22×C4—941no—
201C22×C4—941no—
202C22×C4—942no—
203C22×C4—1882no—
204C22×C4C8×D4711yescharacteristic
205C22×C4C2×C4×D4711yescharacteristic
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—472no—
210C2×C8—941no—
211C2×C8—1881no—
212C2×C8—1881no—
213C2×C8D4×D4712yes—
214C42—471no—
215C42—471no—
216C42—471no—
217C42—471no—
218C42—941no—
219C42—941no—
220C42—941no—
221C42—941no—
222C42—941no—
223C42—941no—
224C16—42no—
225C16—1882no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23#1309396312yes—
231C23C2×C8×D4711yescharacteristic
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—471no—
240C2×C4—472no—
241C2×C4—471no—
242C2×C4—471no—
243C2×C4—471no—
244C2×C4—942no—
245C2×C4—944no—
246C2×C4—944no—
247C2×C4—941no—
248C2×C4—941no—
249C2×C4—941no—
250C2×C4—941no—
251C2×C4—941no—
252C2×C4—1882no—
253C2×C4—1882no—
254C2×C4—1881no—
255C2×C4C22⋊C4×D4711yescharacteristic
256C2×C4M16×D4711yescharacteristic
257C8—22no—
258C8—41no—
259C8—942no—
260C8—1881no—
261C22—21no—
262C22—21no—
263C22—22no—
264C22—42no—
265C22—42no—
266C22—41no—
267C22—41no—
268C22C22⋊C8×D4711yescharacteristic
269C22M32×D4711yescharacteristic
270C22C2×#1309396311yescharacteristic
271C4—21no—
272C4—41no—
273C4—472no—
274C4—942no—
275C4—941no—
276C4—941no—
277C4—941no—
278C4—1881no—
279C4#1568909511yescharacteristic
280C4#1568910411yescharacteristic
281C2—21no—
282C2—21no—
283C2—42no—
284C2#2024169911yescharacteristic
285C2#2024499711yescharacteristic
286C2#349×D4711yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D47
commutatorC23×C47C22×C8
FrattiniC22×C8C22×D47
FittingC47×#1314C2
socleC22×C47C2×C22⋊C8
prime radicalC24×C47C2×C8
cosocleC22×C8×C47C23
hypercenter#1314D47
hypocenterC47#11460
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. #20245188C2semidirect
3. #20241327C2semidirect
162. C47#11460semidirect

Maximal subgroups

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

subgroupindex
C2×#156891732
#202451882
#202413272
#202413122
C47×#13142
#1146047