Finite Group Databasebeta

← #26132287

Subgroups — #26132287

Order 12032.

432 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 101 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#20246108C211yesmaximal, characteristic
2C2×#15688922C211yesmaximal, characteristic
3#20243382C212yesmaximal
4#20241514C212yesmaximal
5C2×C47×#356C211yesmaximal, characteristic
6C2×Q8×Dic47C411yescharacteristic
7C2×Q8×Dic47C2211yescharacteristic
8#15689146—22no—
9C2×#13093943C2212yes—
10#15689030C411yescharacteristic
11#15689030C2211yescharacteristic
12#15689009—22no—
13#15689009—22no—
14#15688989C411yescharacteristic
15#15688969C411yescharacteristic
16#15688922—22no—
17#15688891—22no—
18C2×C4⋊Q8×C47C2211yescharacteristic
19C2×#101×C47C2212yes—
20C47×#356—22no—
21C2×#10625329—22no—
22Q8×Dic47—42no—
23Q8×Dic47—42no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47C2×C411yescharacteristic
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×C4⋊Dic47D412yes—
28C2×C4×Dic47—21no—
29C2×C4×Dic47C2×C411yescharacteristic
30C2×C4×Dic47C2×C411yescharacteristic
31C2×C4×Dic47D412yes—
32Dic47⋊C8—44no—
33#13093943—22no—
34#13093943—22no—
35C47⋊#95—22no—
36C47⋊#95—21no—
37C47⋊#95—21no—
38C47⋊#95—22no—
39C47⋊#95—22no—
40C22×Q8×C47C2×C411yescharacteristic
41C22×Q8×C47C2311yescharacteristic
42C2×M16×C47—22no—
43C4⋊Q8×C47—22no—
44C4⋊Q8×C47D412yes—
45C4⋊Q8×C47Q812yes—
46C2×C4⋊C4×C47—22no—
47C2×C42×C47C2×C411yescharacteristic
48#101×C47—22no—
49#101×C47—22no—
50C2×Q8×C47—21no—
51C2×Q8×C47—41no—
52C2×Q8×C47—41no—
53C2×Q8×C47—41no—
54C2×Q8×C47—41no—
55C2×Q8×C47C4⋊C412yes—
56C2×Q8×C47C2×D411yescharacteristic
57C2×Q8×C47C2×Q811yescharacteristic
58C2×Q8×C47#4112yes—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D412yes—
63C22×Dic47—21no—
64C22×Dic47—21no—
65C22×Dic47C22⋊C412yes—
66C22×Dic47C22⋊C411yescharacteristic
67C22×Dic47C22⋊C411yescharacteristic
68M16×C47—42no—
69M16×C47—42no—
70C2×C8×C47—42no—
71C4⋊C4×C47—24no—
72C4⋊C4×C47—24no—
73C42×C47—21no—
74C42×C47C4⋊C412yes—
75C4⋊Dic47—41no—
76C4⋊Dic47—42no—
77C4⋊Dic47—42no—
78C4⋊Dic47—41no—
79C4×Dic47—24no—
80C4×Dic47—22no—
81C4×Dic47—42no—
82C4×Dic47—41no—
83#10625329—42no—
84#10625329—42no—
85C2×C47⋊C8—42no—
86C23×C47C2×C22⋊C411yescharacteristic
87Q8×C47—41no—
88Q8×C47—41no—
89Q8×C47—42no—
90Q8×C47—42no—
91Q8×C47—42no—
92C2×C4×C47—22no—
93C2×C4×C47—21no—
94C2×C4×C47—22no—
95C2×C4×C47—21no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47—41no—
101C2×C4×C47—41no—
102C2×C4×C47C2×C4⋊C411yescharacteristic
103C2×C4×C47#11711yescharacteristic
104C2×C4×C47#12112yes—
105C2×C4×C47C22⋊Q812yes—
106C2×Dic47—21no—
107C2×Dic47—22no—
108C2×Dic47—41no—
109C2×Dic47—42no—
110C2×Dic47—41no—
111C2×Dic47—41no—
112C2×Dic47—41no—
113C2×Dic47—42no—
114C2×Dic47—41no—
115C2×Dic47—41no—
116C2×Dic47#9912yes—
117C8×C47—82no—
118C47⋊C8—82no—
119#12804—471nomaximal, Sylow
120C22×C47—21no—
121C22×C47—21no—
122C22×C47#38011yescharacteristic
123C22×C47C2×#9911yescharacteristic
124C22×C47#41011yescharacteristic
125C4×C47—22no—
126C4×C47—21no—
127C4×C47—21no—
128C4×C47—22no—
129C4×C47—42no—
130C4×C47—42no—
131Dic47—22no—
132Dic47—41no—
133Dic47—42no—
134Dic47—81no—
135Dic47—81no—
136#2327—471no—
137C2×#356—471no—
138C2×#356D4711yescharacteristic
139#1801—472no—
140#1394—472no—
141C2×C47—21no—
142C2×C47—21no—
143C2×C47#179911yescharacteristic
144C2×C47#213311yescharacteristic
145C2×C47#213511yescharacteristic
146C2×C4⋊Q8C2×D4711yescharacteristic
147C2×C4×Q8—471no—
148C2×C4×Q8—471no—
149#423—942no—
150C2×#101—472no—
151C2×#101C2×D4712yes—
152#391—471no—
153#391—471no—
154#387—942no—
155#387—942no—
156#382—471no—
157#376—471no—
158#356—22no—
159#356—942no—
160#343—942no—
161C47#1280411yescharacteristic, complemented, Sylow
162C22×Q8C4×D4711yescharacteristic
163C22×Q8C22×D4711yescharacteristic
164C2×M16—22no—
165C2×M16—942no—
166C4⋊Q8—22no—
167C4⋊Q8Dic9412yes—
168C4⋊Q8C22⋊D4712yes—
169C4×Q8—1882no—
170C4×Q8—1882no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—471no—
173C2×C4⋊C4—472no—
174C2×C4⋊C4—471no—
175C2×C4⋊C4—941no—
176C2×C42—471no—
177C2×C42—472no—
178C2×C42—471no—
179C2×C42—941no—
180C2×C42C4×D4711yescharacteristic
181C4⋊C8—1884no—
182#101—22no—
183#101—22no—
184#101—942no—
185#101—942no—
186#95—941no—
187#95—941no—
188#95—942no—
189#95—942no—
190#95—942no—
191C2×Q8—21no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8Dic47⋊C412yes—
197C2×Q8C2×Dic9411yescharacteristic
198C2×Q8#1062535212yes—
199C2×Q8C2×C22⋊D4711yescharacteristic
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—471no—
203C22×C4—472no—
204C22×C4—471no—
205C22×C4—941no—
206C22×C4—941no—
207C22×C4C2×C4×D4711yescharacteristic
208C22×C4D4×D4712yes—
209M16—42no—
210M16—42no—
211M16—1882no—
212M16—1882no—
213C2×C8—42no—
214C2×C8—1882no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—1881no—
218C4⋊C4—1882no—
219C4⋊C4—1881no—
220C4⋊C4—1882no—
221C42—21no—
222C42—944no—
223C42—942no—
224C42—1881no—
225C42—1882no—
226C42Dic47⋊C412yes—
227C23C22⋊C4×D4711yescharacteristic
228Q8—41no—
229Q8—42no—
230Q8—41no—
231Q8—42no—
232Q8—42no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—472no—
244C2×C4—941no—
245C2×C4—942no—
246C2×C4—1881no—
247C2×C4—1881no—
248C2×C4—1881no—
249C2×C4—1882no—
250C2×C4—1881no—
251C2×C4—1881no—
252C2×C4—1882no—
253C2×C4—1881no—
254C2×C4C2×Dic47⋊C411yescharacteristic
255C2×C4#1309398011yescharacteristic
256C2×C4#1309399612yes—
257C2×C4#1309400112yes—
258C8—82no—
259C8—3762no—
260C22—21no—
261C22—21no—
262C22#1568898111yescharacteristic
263C22#99×D4711yescharacteristic
264C22#1568909511yescharacteristic
265C4—22no—
266C4—21no—
267C4—21no—
268C4—22no—
269C4—42no—
270C4—42no—
271C4—942no—
272C4—3761no—
273C4—3761no—
274C4—1881no—
275C4—1882no—
276C2—21no—
277C2—21no—
278C2#2024337411yescharacteristic
279C2#2024507611yescharacteristic
280C2#2024508111yescharacteristic
subgroupG/H
centerC22#99×D47
commutatorC22×C4×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC2×C47×#356C2
socleC22×C47C2×#99
prime radicalC23×C47C2×C22⋊C4
cosocleC22×Q8×C47C23
hypercenterC2×#356D47
hypocenterC47#12804
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
161. C47#12804semidirect

Maximal subgroups

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

subgroupindex
#202461082
C2×#156889222
#202433822
#202415142
C2×C47×#3562
#1280447