Finite Group Databasebeta

← #26125918

Subgroups — #26125918

Order 12032.

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

normal subgroups only

This group has 462 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#20248387C211yesmaximal, characteristic
2#20241712C211yesmaximal, characteristic, complemented
3#20241319C214yesmaximal, complemented
4C47×#1467C211yesmaximal, characteristic, complemented
5C2×#13094021C411yescharacteristic
6C2×#13094013C412yes—
7#15689176—21no—
8#15689040—21no—
9#15689040—21no—
10#15689025—21no—
11#15689025—21no—
12#15689024C2212yes—
13C4×C4⋊Dic47C411yescharacteristic
14#15688849—21no—
15#15688849C2212yes—
16C2×C4×Q8×C47C2211yescharacteristic
17C47×#432—21no—
18C47×#324—21no—
19C47×#324C2212yes—
20C22×D188—21no—
21C22×D188—21no—
22C22×D188C2×C412yes—
23#13094021—41no—
24#13094021—41no—
25#13094013—44no—
26C2×C4⋊Dic47—21no—
27C2×(C2×D47)⋊C4—22no—
28C2×(C2×D47)⋊C4—22no—
29C2×(C2×D47)⋊C4—22no—
30C2×(C2×D47)⋊C4—22no—
31C2×C4×Dic47C2×C412yes—
32C4⋊(C47⋊C8)—41no—
33C22⋊(C47⋊C8)—22no—
34C4×C47⋊C8—41no—
35C47⋊#95—21no—
36C22×Q8×C47C2×C411yescharacteristic
37C4×Q8×C47—41no—
38C4×Q8×C47—41no—
39C2×C4⋊C4×C47D411yescharacteristic
40C2×C4⋊C4×C47D411yescharacteristic
41C2×C4⋊C4×C47C2311yescharacteristic
42C2×C42×C47C2×C411yescharacteristic
43C2×C42×C47D411yescharacteristic
44C2×C42×C47D411yescharacteristic
45C4⋊C8×C47—41no—
46C22⋊C8×C47—22no—
47#97×C47—41no—
48C23×D47—22no—
49C2×Q8×C47—22no—
50C2×Q8×C47—22no—
51C2×Q8×C47—41no—
52C22×C4×C47C22⋊C411yescharacteristic
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—
60C2×D188—24no—
61C2×D188—42no—
62C2×D188—42no—
63C2×D188—42no—
64C2×D188—42no—
65C2×D188—42no—
66C2×D188—41no—
67C2×D188—41no—
68C2×D188—82no—
69C2×C8×C47—42no—
70C4⋊C4×C47—41no—
71C4⋊C4×C47—41no—
72C4⋊C4×C47—41no—
73C42×C47—21no—
74C42×C47—21no—
75C42×C47—21no—
76C42×C47—21no—
77C42×C47—21no—
78C42×C47—21no—
79(C2×D47)⋊C4—44no—
80(C2×D47)⋊C4—44no—
81(C2×D47)⋊C4—82no—
82(C2×D47)⋊C4—82no—
83C4⋊Dic47—42no—
84C4×Dic47—42no—
85C2×C47⋊C8—42no—
86C23×C47C2×C22⋊C411yescharacteristic
87C22×D47—42no—
88C22×D47—42no—
89C22×D47—42no—
90C22×D47—42no—
91C22×D47—42no—
92C22×D47—42no—
93C22×D47—42no—
94Q8×C47—44no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—22no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47#9911yescharacteristic
110C2×C4×C47#9911yescharacteristic
111C2×C4×C47#10012yes—
112C2×C4×C47#13112yes—
113C2×Dic47—42no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116D188—84no—
117D188—82no—
118D188—84no—
119D188—82no—
120C8×C47—82no—
121C47⋊C8—82no—
122#11567—471nomaximal, Sylow
123C22×C47—21no—
124C22×C47—21no—
125C22×C47#40811yescharacteristic
126C22×C47C2×#9911yescharacteristic
127C22×C47C2×#10011yescharacteristic
128C2×D47—82no—
129C2×D47—82no—
130C2×D47—82no—
131C2×D47—82no—
132C2×D47—82no—
133C2×D47—82no—
134C2×D47—82no—
135C4×C47—21no—
136C4×C47—22no—
137C4×C47—22no—
138C4×C47—41no—
139C4×C47—41no—
140C4×C47—41no—
141C4×C47—41no—
142C4×C47#35512yes—
143Dic47—82no—
144#2548—471no—
145#1467D4711yescharacteristic, complemented
146#1461—471no—
147#1318—474no—
148C2×C47—21no—
149C2×C47—21no—
150C2×C47#147111yescharacteristic
151C2×C47#212811yescharacteristic
152C2×C47C2×#35511yescharacteristic
153D47—162no—
154C2×#127—471no—
155C2×#124—472no—
156C2×C4×Q8C2×D4711yescharacteristic
157#432—21no—
158#431—941no—
159#394—941no—
160#394—941no—
161#390—472no—
162#390—941no—
163#390—941no—
164C4×C4⋊C4—471no—
165#324—21no—
166#324—472no—
167#324—941no—
168#324C2×D4712yes—
169C47#1156711yescharacteristic, complemented, Sylow
170C22×Q8C4×D4711yescharacteristic
171C22×D4—472no—
172C22×D4—941no—
173C22×D4—941no—
174#127—1881no—
175#127—1881no—
176#124—1884no—
177C4×Q8—41no—
178C4×Q8—41no—
179C2×C4⋊C4—941no—
180C2×C4⋊C4D18811yescharacteristic
181C2×C4⋊C4C22⋊D4711yescharacteristic
182C2×C4⋊C4C22×D4711yescharacteristic
183C2×C22⋊C4—942no—
184C2×C22⋊C4—942no—
185C2×C22⋊C4—942no—
186C2×C22⋊C4—942no—
187C2×C42—472no—
188C2×C42C4×D4711yescharacteristic
189C2×C42D18811yescharacteristic
190C2×C42C22⋊D4711yescharacteristic
191C4⋊C8—41no—
192C4⋊C8—1881no—
193C22⋊C8—22no—
194C22⋊C8—942no—
195#97—41no—
196C4×C8—1881no—
197#95—941no—
198C24—942no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—41no—
202C2×D4—944no—
203C2×D4—3762no—
204C2×D4—1882no—
205C2×D4—1882no—
206C2×D4—1882no—
207C2×D4—1882no—
208C2×D4—1882no—
209C2×D4—1881no—
210C2×D4—1881no—
211C22×C4—942no—
212C22×C4(C2×D47)⋊C411yescharacteristic
213C22×C4(C2×D47)⋊C411yescharacteristic
214C22×C4(C2×D47)⋊C411yescharacteristic
215C22×C4(C2×D47)⋊C411yescharacteristic
216C22×C4C2×C4×D4711yescharacteristic
217C22×C4C2×D18811yescharacteristic
218C22×C4C2×C22⋊D4711yescharacteristic
219C2×C8—42no—
220C2×C8—1882no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—1882no—
225C22⋊C4—3762no—
226C22⋊C4—3762no—
227C22⋊C4—1884no—
228C22⋊C4—1884no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—1882no—
236C23—1882no—
237C23—1882no—
238C23—1882no—
239C23—1882no—
240C23—1882no—
241C23—1882no—
242C23—1882no—
243C23C2×(C2×D47)⋊C411yescharacteristic
244Q8—44no—
245D4—3764no—
246D4—3762no—
247D4—3764no—
248D4—3762no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—1882no—
264C2×C4—1882no—
265C2×C4—1882no—
266C2×C4#1309393811yescharacteristic
267C2×C4#1309393811yescharacteristic
268C2×C4#1309394012yes—
269C2×C4#1309403212yes—
270C8—82no—
271C8—3762no—
272C22—21no—
273C22—21no—
274C22—3762no—
275C22—3762no—
276C22—3762no—
277C22—3762no—
278C22—3762no—
279C22—3762no—
280C22—3762no—
281C22#1568909111yescharacteristic
282C22C2×#1309393811yescharacteristic
283C22C2×#1309394011yescharacteristic
284C4—21no—
285C4—22no—
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—3762no—
292C4#1568891912yes—
293C2—21no—
294C2—21no—
295C2—7522no—
296C2#2024174911yescharacteristic
297C2#2024505911yescharacteristic
298C2C2×#1568891911yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1467C2
socleC22×C47C2×#99
prime radicalC2×#13094021C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1467D47
hypocenterC47#11567
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. #20241712C2semidirect
3. #20241319C2semidirect
4. C47×#1467C2semidirect
145. #1467D47semidirect
169. C47#11567semidirect

Maximal subgroups

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

subgroupindex
#202483872
#202417122
#202413192
C47×#14672
#1156747