Finite Group Databasebeta

← #26131459

Subgroups — #26131459

Order 12032.

500 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 115 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#20245957C211yesmaximal, characteristic
2C2×#15688921C212yesmaximal
3#101×Dic47C212yesmaximal
4#20241515C211yesmaximal, characteristic
5C47×#1394C211yesmaximal, characteristic
6C2×#13094022C2211yescharacteristic
7C2×Q8×Dic47C2211yescharacteristic
8C2×#13093989C411yescharacteristic
9C2×#13093943C2212yes—
10M16×Dic47—22no—
11#15689029C411yescharacteristic
12#15689020—22no—
13#15688992C411yescharacteristic
14#15688974—22no—
15C42×Dic47C411yescharacteristic
16#15688921C418yescomplemented
17#15688891—22no—
18C2×#101×C47C2212yes—
19C47×#391C2211yescharacteristic
20C22×Dic94C2×C411yescharacteristic
21C2×#10625329—22no—
22#13094022—21no—
23#13094022—21no—
24#13094022C2×C414yes—
25Q8×Dic47—41no—
26Q8×Dic47—41no—
27#13093989—41no—
28#13093989—41no—
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×C4⋊Dic47D412yes—
31C2×Dic47⋊C4—22no—
32C2×Dic47⋊C4—22no—
33C2×Dic47⋊C4—21no—
34C2×C4×Dic47—21no—
35C2×C4×Dic47—21no—
36C2×C4×Dic47—21no—
37C2×C4×Dic47C2×C411yescharacteristic
38C2×C4×Dic47C2×C411yescharacteristic
39C2×C4×Dic47D412yes—
40#13093943—22no—
41#13093943C2×C414yes—
42#13093934—42no—
43C8×Dic47—42no—
44C47⋊#95—22no—
45C22×Q8×C47C2311yescharacteristic
46C2×M16×C47—22no—
47C2×C4⋊C4×C47C2×C411yescharacteristic
48C2×C42×C47C2×C411yescharacteristic
49#101×C47—22no—
50#101×C47C2×C414yes—
51#95×C47—22no—
52C2×Q8×C47—41no—
53C2×Q8×C47—41no—
54C2×Q8×C47C22×C412yes—
55C2×Q8×C47#4111yescharacteristic
56C2×Q8×C47#4111yescharacteristic
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C2×D412yes—
61C22×Dic47—21no—
62C22×Dic47—21no—
63C22×Dic47C22⋊C411yescharacteristic
64C22×Dic47C22⋊C412yes—
65C22×Dic47C22⋊C411yescharacteristic
66C2×Dic94—21no—
67C2×Dic94—41no—
68C2×Dic94—41no—
69C2×Dic94C4212yes—
70M16×C47—24no—
71M16×C47—42no—
72C2×C8×C47—42no—
73C4⋊C4×C47—41no—
74C42×C47—41no—
75C4⋊Dic47—41no—
76C4⋊Dic47—42no—
77Dic47⋊C4—24no—
78Dic47⋊C4—24no—
79Dic47⋊C4—44no—
80Dic47⋊C4—42no—
81C4×Dic47—24no—
82C4×Dic47—21no—
83C4×Dic47—22no—
84C4×Dic47—22no—
85C4×Dic47—21no—
86C4×Dic47—21no—
87C4×Dic47—42no—
88C4×Dic47—41no—
89C4×Dic47—41no—
90C4×Dic47C4212yes—
91#10625329—24no—
92#10625329—42no—
93C2×C47⋊C8—42no—
94C23×C47C2×C22⋊C411yescharacteristic
95Q8×C47—42no—
96Q8×C47—41no—
97Q8×C47—41no—
98C2×C4×C47—22no—
99C2×C4×C47—21no—
100C2×C4×C47—22no—
101C2×C4×C47—21no—
102C2×C4×C47—22no—
103C2×C4×C47—41no—
104C2×C4×C47—41no—
105C2×C4×C47—42no—
106C2×C4×C47—41no—
107C2×C4×C47C2×C4211yescharacteristic
108C2×C4×C47#11711yescharacteristic
109C2×C4×C47C4×D412yes—
110C2×C4×C47C4×D412yes—
111C2×Dic47—22no—
112C2×Dic47—22no—
113C2×Dic47—22no—
114C2×Dic47—22no—
115C2×Dic47—22no—
116C2×Dic47—21no—
117C2×Dic47—22no—
118C2×Dic47—41no—
119C2×Dic47—41no—
120C2×Dic47—41no—
121C2×Dic47—42no—
122C2×Dic47#9911yescharacteristic
123C2×Dic47#9911yescharacteristic
124Dic94—42no—
125Dic94—42no—
126C8×C47—44no—
127C47⋊C8—44no—
128C4×#356—471nomaximal, Sylow
129C22×C47—21no—
130C22×C47—21no—
131C22×C47C4×C22⋊C411yescharacteristic
132C22×C47C2×#9911yescharacteristic
133C22×C47#41011yescharacteristic
134C4×C47—22no—
135C4×C47—22no—
136C4×C47—21no—
137C4×C47—21no—
138C4×C47—42no—
139C4×C47—81no—
140Dic47—24no—
141Dic47—21no—
142Dic47—41no—
143Dic47—42no—
144Dic47—42no—
145Dic47#35612yescomplemented
146C4×C4⋊Q8—471no—
147C2×#356—472no—
148C4×#101—472no—
149#1394—471no—
150#1394D4711yescharacteristic
151C2×C47—21no—
152C2×C47—21no—
153C2×C47C4×#9911yescharacteristic
154C2×C47C2×#35611yescharacteristic
155C2×C47#213211yescharacteristic
156C2×C4⋊Q8—471no—
157C2×C4×Q8—471no—
158C2×C4×Q8—471no—
159C2×#101—472no—
160C2×#101C2×D4712yes—
161C4×M16—942no—
162#391—471no—
163#391C2×D4711yescharacteristic
164#389—942no—
165#384—471no—
166C4×C4⋊C4—942no—
167C43—471no—
168#356—478no—
169#343—942no—
170C47C4×#35611yescharacteristic, complemented, Sylow
171C22×Q8—471no—
172C22×Q8C22×D4711yescharacteristic
173C2×M16—22no—
174C2×M16—942no—
175C4⋊Q8—474no—
176C4⋊Q8—941no—
177C4⋊Q8—941no—
178C4×Q8—1881no—
179C4×Q8—1881no—
180C4×Q8—1881no—
181C4×Q8—1881no—
182C2×C4⋊C4—471no—
183C2×C4⋊C4—472no—
184C2×C4⋊C4—942no—
185C2×C4⋊C4—941no—
186C2×C4⋊C4—942no—
187C2×C4⋊C4C4×D4711yescharacteristic
188C2×C42—471no—
189C2×C42—471no—
190C2×C42—472no—
191C2×C42—941no—
192C2×C42—941no—
193C2×C42—941no—
194C2×C42C4×D4711yescharacteristic
195#101—22no—
196#101—474no—
197#101—942no—
198#101C4×D4714yes—
199#97—1882no—
200C4×C8—1882no—
201#95—22no—
202#95—942no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—472no—
206C2×Q8—941no—
207C2×Q8—1881no—
208C2×Q8—1881no—
209C2×Q8C2×C4×D4712yes—
210C2×Q8#1062535211yescharacteristic
211C2×Q8#1062535411yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—471no—
215C22×C4—472no—
216C22×C4—471no—
217C22×C4—941no—
218C22×C4—941no—
219C22×C4C2×C4×D4711yescharacteristic
220C22×C4D4×D4712yes—
221M16—24no—
222M16—42no—
223M16—944no—
224M16—1882no—
225C2×C8—42no—
226C2×C8—1882no—
227C4⋊C4—41no—
228C4⋊C4—944no—
229C4⋊C4—944no—
230C4⋊C4—1881no—
231C4⋊C4—1882no—
232C4⋊C4—1884no—
233C4⋊C4—1882no—
234C42—41no—
235C42—472no—
236C42—942no—
237C42—942no—
238C42—941no—
239C42—941no—
240C42—944no—
241C42—941no—
242C42—1881no—
243C42—1881no—
244C42—1882no—
245C23C22⋊C4×D4711yescharacteristic
246Q8—42no—
247Q8—41no—
248Q8—41no—
249Q8—1882no—
250Q8—1882no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—471no—
261C2×C4—471no—
262C2×C4—942no—
263C2×C4—942no—
264C2×C4—942no—
265C2×C4—942no—
266C2×C4—942no—
267C2×C4—941no—
268C2×C4—942no—
269C2×C4—1881no—
270C2×C4—1881no—
271C2×C4—1881no—
272C2×C4—1882no—
273C2×C4C42×D4711yescharacteristic
274C2×C4#1309398211yescharacteristic
275C2×C4(C22⋊D47)⋊C412yes—
276C2×C4#1309398612yes—
277C8—44no—
278C8—1884no—
279C22—21no—
280C22—21no—
281C22#1568897011yescharacteristic
282C22#99×D4711yescharacteristic
283C22#1568909511yescharacteristic
284C4—22no—
285C4—22no—
286C4—21no—
287C4—21no—
288C4—42no—
289C4—81no—
290C4—472no—
291C4—944no—
292C4—941no—
293C4—1881no—
294C4—1882no—
295C4—1882no—
296C2—21no—
297C2—21no—
298C2#2024317511yescharacteristic
299C2#356×D4711yescharacteristic
300C2#2024507011yescharacteristic
subgroupG/H
centerC22#99×D47
commutatorC2×C4×C47C2×C42
FrattiniC22×Q8C22×D47
FittingC47×#1394C2
socleC22×C47C2×#99
prime radicalC23×C47C2×C22⋊C4
cosocleC22×Q8×C47C23
hypercenter#1394D47
hypocenterC47C4×#356
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
16. #15688921C4semidirect
145. Dic47#356semidirect
170. C47C4×#356semidirect

Maximal subgroups

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

subgroupindex
#202459572
C2×#156889212
#101×Dic472
#202415152
C47×#13942
C4×#35647