Finite Group Databasebeta

← #26125691

Subgroups — #26125691

Order 12032.

418 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#20246008C211yesmaximal, characteristic
2#20241722C211yesmaximal, characteristic
3#20241333C214yesmaximal, complemented
4C47×#1464C211yesmaximal, characteristic
5C2×#13093983C411yescharacteristic
6C22×C4⋊Dic47C411yescharacteristic
7#15689180—21no—
8#15689021—21no—
9#15689021—21no—
10#15688995—21no—
11#15688995—21no—
12#15688992C2212yes—
13#15688988C412yes—
14#15688849C2212yes—
15#15688847—21no—
16C2×#117×C47C2211yescharacteristic
17C47×#432—21no—
18C47×#324C2212yes—
19C47×#323—21no—
20C23×Dic47—21no—
21#13093983—24no—
22C2×C4⋊Dic47—21no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—41no—
25C2×C4⋊Dic47—41no—
26C2×C4⋊Dic47C2×C412yes—
27C2×Dic47⋊C4—22no—
28C2×Dic47⋊C4—22no—
29C2×C22⋊Dic47—21no—
30C2×C4×Dic47C2×C412yes—
31C4⋊(C47⋊C8)—41no—
32C22⋊(C47⋊C8)—22no—
33#13093935—41no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C23×C4×C47C2×C411yescharacteristic
37#117×C47—41no—
38#117×C47—41no—
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C4⋊C4×C47C2311yescharacteristic
41C2×C22⋊C4×C47D411yescharacteristic
42C2×C22⋊C4×C47D411yescharacteristic
43C2×C42×C47D411yescharacteristic
44C2×C42×C47D411yescharacteristic
45C4⋊C8×C47—41no—
46C22⋊C8×C47—22no—
47#97×C47—41no—
48C24×C47C22⋊C411yescharacteristic
49C22×C4×C47—22no—
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47—22no—
53C22×C4×C47C22⋊C411yescharacteristic
54C22×C4×C47C22⋊C411yescharacteristic
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C22×C411yescharacteristic
57C22×C4×C47C2×D411yescharacteristic
58C22×C4×C47C2×D411yescharacteristic
59C22×Dic47—22no—
60C22×Dic47—22no—
61C22×Dic47—42no—
62C22×Dic47—82no—
63C2×C8×C47—42no—
64C4⋊C4×C47—41no—
65C4⋊C4×C47—41no—
66C22⋊C4×C47—41no—
67C22⋊C4×C47—41no—
68C42×C47—21no—
69C42×C47—21no—
70C42×C47—21no—
71C42×C47—21no—
72C22⋊Dic47—24no—
73C4⋊Dic47—24no—
74C4⋊Dic47—41no—
75C4⋊Dic47—41no—
76C4⋊Dic47—42no—
77Dic47⋊C4—82no—
78Dic47⋊C4—82no—
79C4×Dic47—22no—
80C4×Dic47M1614yes—
81C2×C47⋊C8—42no—
82C23×C47—21no—
83C23×C47—21no—
84C23×C47—41no—
85C23×C47—41no—
86C23×C47#10112yes—
87C23×C47C2×C22⋊C411yescharacteristic
88C2×C4×C47—24no—
89C2×C4×C47—24no—
90C2×C4×C47—21no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—21no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47—41no—
101C2×C4×C47—41no—
102C2×C4×C47#9911yescharacteristic
103C2×C4×C47#9911yescharacteristic
104C2×C4×C47C2×M1612yes—
105C2×Dic47—24no—
106C2×Dic47—42no—
107C2×Dic47—42no—
108C2×Dic47—42no—
109C2×Dic47—42no—
110C2×Dic47—42no—
111C2×Dic47—84no—
112C8×C47—82no—
113C47⋊C8—82no—
114#11516—471nomaximal, Sylow
115C22×C47—21no—
116C22×C47—21no—
117C22×C47—22no—
118C22×C47—42no—
119C22×C47—42no—
120C22×C47—41no—
121C22×C47—41no—
122C22×C47#40711yescharacteristic
123C22×C47C2×#9911yescharacteristic
124C22×C47C2×#10111yescharacteristic
125C4×C47—21no—
126C4×C47—21no—
127C4×C47—22no—
128C4×C47—41no—
129C4×C47—41no—
130C4×C47—41no—
131C4×C47—41no—
132Dic47—44no—
133Dic47—82no—
134#2317—471no—
135#1464—471no—
136#1464D4711yescharacteristic
137#1325—474no—
138C2×C47—21no—
139C2×C47—21no—
140C2×C47—42no—
141C2×C47#146311yescharacteristic
142C2×C47#212811yescharacteristic
143C2×C47#213511yescharacteristic
144C2×#117—471no—
145C2×#117C2×D4711yescharacteristic
146C22×C4⋊C4—471no—
147#432—21no—
148#432—941no—
149#389—941no—
150#389—941no—
151#385—941no—
152#385—941no—
153#384—472no—
154#382—472no—
155#324—472no—
156#324C2×D4712yes—
157#323—21no—
158#323—941no—
159C47#1151611yescharacteristic, complemented, Sylow
160C23×C4—941no—
161C23×C4C4×D4711yescharacteristic
162#117—41no—
163#117—41no—
164#117—944no—
165C2×C4⋊C4—472no—
166C2×C4⋊C4—941no—
167C2×C4⋊C4—942no—
168C2×C4⋊C4—942no—
169C2×C4⋊C4—941no—
170C2×C4⋊C4—1881no—
171C2×C4⋊C4—1881no—
172C2×C4⋊C4C4×D4711yescharacteristic
173C2×C4⋊C4C22×D4711yescharacteristic
174C2×C22⋊C4—941no—
175C2×C22⋊C4D18811yescharacteristic
176C2×C22⋊C4C22⋊D4711yescharacteristic
177C2×C42—472no—
178C2×C42D18811yescharacteristic
179C2×C42C22⋊D4711yescharacteristic
180C4⋊C8—41no—
181C4⋊C8—1881no—
182C22⋊C8—22no—
183C22⋊C8—942no—
184#97—41no—
185#97—1881no—
186#95—942no—
187#95—942no—
188C24(C2×D47)⋊C411yescharacteristic
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—942no—
194C22×C4—942no—
195C22×C4—3762no—
196C22×C4—1882no—
197C22×C4(C2×D47)⋊C411yescharacteristic
198C22×C4(C2×D47)⋊C411yescharacteristic
199C22×C4(C2×D47)⋊C411yescharacteristic
200C22×C4C2×C4×D4711yescharacteristic
201C22×C4C2×D18811yescharacteristic
202C22×C4C2×C22⋊D4711yescharacteristic
203C2×C8—42no—
204C2×C8—1882no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—944no—
208C4⋊C4—3762no—
209C4⋊C4—3762no—
210C4⋊C4—1882no—
211C4⋊C4—1881no—
212C4⋊C4—1881no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—944no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—474no—
221C42—942no—
222C23—21no—
223C23—21no—
224C23—41no—
225C23—41no—
226C23#1309394212yes—
227C23C2×(C2×D47)⋊C411yescharacteristic
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—944no—
243C2×C4—3764no—
244C2×C4—1882no—
245C2×C4—1882no—
246C2×C4—1882no—
247C2×C4—1882no—
248C2×C4—1882no—
249C2×C4#1309393811yescharacteristic
250C2×C4#1309393811yescharacteristic
251C2×C4M16×D4712yes—
252C8—82no—
253C8—3762no—
254C22—21no—
255C22—21no—
256C22—22no—
257C22—42no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22#1568908611yescharacteristic
262C22C2×#1309393811yescharacteristic
263C22C2×#1309394211yescharacteristic
264C4—21no—
265C4—21no—
266C4—22no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—3762no—
272C4—1884no—
273C2—21no—
274C2—21no—
275C2—42no—
276C2#2024171711yescharacteristic
277C2#2024505911yescharacteristic
278C2#2024508211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1464C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1464D47
hypocenterC47#11516
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #20241333C2semidirect
159. C47#11516semidirect

Maximal subgroups

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

subgroupindex
#202460082
#202417222
#202413332
C47×#14642
#1151647