Finite Group Databasebeta

← #26125849

Subgroups — #26125849

Order 12032.

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

normal subgroups only

This group has 366 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#20241722C211yesmaximal, characteristic
2#20241702C211yesmaximal, characteristic
3#20241334C212yesmaximal, complemented
4#20241322C212yesmaximal, complemented
5C47×#2499C211yesmaximal, characteristic
6#15689180—21no—
7#15689176—21no—
8#15688849C2212yes—
9#15688849C2212yes—
10#15688847—21no—
11#15688847—21no—
12C2×C22⋊Q8×C47C411yescharacteristic
13C2×#117×C47C2211yescharacteristic
14C47×#397—21no—
15C47×#397—21no—
16C47×#391C411yescharacteristic
17C47×#391C2211yescharacteristic
18C47×#387—21no—
19C47×#387—21no—
20C47×#384C2211yescharacteristic
21C47×#382C411yescharacteristic
22C4×C22⋊C4×C47C411yescharacteristic
23C4⋊(C47⋊C8)—41no—
24C4⋊(C47⋊C8)—41no—
25C22⋊(C47⋊C8)—22no—
26C22⋊(C47⋊C8)—22no—
27#13093935—41no—
28C4×C47⋊C8—41no—
29C22×Q8×C47C2×C411yescharacteristic
30C23×C4×C47C2×C411yescharacteristic
31C22⋊Q8×C47—41no—
32C22⋊Q8×C47—41no—
33#117×C47—41no—
34#117×C47—41no—
35C2×C4⋊C4×C47—21no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47C2×C411yescharacteristic
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C4⋊C4×C47C2311yescharacteristic
41C2×C22⋊C4×C47—21no—
42C2×C22⋊C4×C47—21no—
43C2×C22⋊C4×C47D411yescharacteristic
44C2×C22⋊C4×C47D411yescharacteristic
45C2×C42×C47C2×C411yescharacteristic
46C2×C42×C47C2×C411yescharacteristic
47C2×C42×C47D411yescharacteristic
48C2×C42×C47D411yescharacteristic
49#95×C47—21no—
50#95×C47—21no—
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47—21no—
56C24×C47C22⋊C411yescharacteristic
57C2×Q8×C47—22no—
58C2×Q8×C47—41no—
59C2×Q8×C47—41no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—22no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—41no—
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22×C411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C4⋊C4×C47—41no—
75C4⋊C4×C47—41no—
76C4⋊C4×C47—42no—
77C4⋊C4×C47—42no—
78C4⋊C4×C47—41no—
79C4⋊C4×C47—41no—
80C4⋊C4×C47—41no—
81C22⋊C4×C47—41no—
82C22⋊C4×C47—41no—
83C22⋊C4×C47—42no—
84C22⋊C4×C47—42no—
85C42×C47—21no—
86C42×C47—21no—
87C42×C47—21no—
88C42×C47—21no—
89C42×C47—22no—
90C42×C47—41no—
91C2×C47⋊C8—42no—
92C2×C47⋊C8—42no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—21no—
96C23×C47—41no—
97C23×C47—41no—
98C23×C47C2×C22⋊C411yescharacteristic
99Q8×C47—82no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—22no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—42no—
110C2×C4×C47—42no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47#9911yescharacteristic
127C2×C4×C47#9911yescharacteristic
128C2×C4×C47#13112yes—
129C47⋊C8—82no—
130C47⋊C8—82no—
131#11552—471nomaximal, Sylow
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—41no—
135C22×C47—41no—
136C22×C47—41no—
137C22×C47—81no—
138C22×C47—81no—
139C22×C47#40811yescharacteristic
140C22×C47C2×#9911yescharacteristic
141C22×C47#41311yescharacteristic
142C4×C47—21no—
143C4×C47—21no—
144C4×C47—22no—
145C4×C47—42no—
146C4×C47—41no—
147C4×C47—41no—
148C4×C47—41no—
149C4×C47—41no—
150C4×C47—81no—
151C4×C47—81no—
152C4×C47—81no—
153#2499D4711yescharacteristic
154#1464—471no—
155#1458—471no—
156#1325—472no—
157#1319—472no—
158C2×C47—21no—
159C2×C47—21no—
160C2×C47—81no—
161C2×C47#146811yescharacteristic
162C2×C47#212611yescharacteristic
163C2×C47#213511yescharacteristic
164C2×C22⋊Q8Dic4711yescharacteristic
165C2×#117C2×D4711yescharacteristic
166#432—941no—
167#431—941no—
168#397—21no—
169#397—21no—
170#391Dic4711yescharacteristic
171#391C2×D4711yescharacteristic
172#387—21no—
173#387—21no—
174#384C2×D4711yescharacteristic
175#382Dic4711yescharacteristic
176C4×C22⋊C4Dic4711yescharacteristic
177#324—472no—
178#324—472no—
179#323—941no—
180#323—941no—
181C47#1155211yescharacteristic, complemented, Sylow
182C22×Q8C2×Dic4711yescharacteristic
183C23×C4C2×Dic4711yescharacteristic
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186#117—41no—
187#117—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C2×Dic4711yescharacteristic
192C2×C4⋊C4C2×Dic4711yescharacteristic
193C2×C4⋊C4C22×D4711yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4C22⋊D4711yescharacteristic
197C2×C22⋊C4C22⋊D4711yescharacteristic
198C2×C42C2×Dic4711yescharacteristic
199C2×C42C2×Dic4711yescharacteristic
200C2×C42C22⋊D4711yescharacteristic
201C2×C42C22⋊D4711yescharacteristic
202C4⋊C8—1881no—
203C4⋊C8—1881no—
204C22⋊C8—942no—
205C22⋊C8—942no—
206#97—1881no—
207C4×C8—1881no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215C24C22⋊Dic4711yescharacteristic
216C2×Q8—22no—
217C2×Q8—41no—
218C2×Q8—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4C22⋊Dic4711yescharacteristic
228C22×C4C22⋊Dic4711yescharacteristic
229C22×C4C22⋊Dic4711yescharacteristic
230C22×C4C22×Dic4711yescharacteristic
231C22×C4C2×C22⋊D4711yescharacteristic
232C22×C4C2×C22⋊D4711yescharacteristic
233C2×C8—1882no—
234C2×C8—1882no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—22no—
251C42—41no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23C2×C22⋊Dic4711yescharacteristic
258Q8—82no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4#1309393911yescharacteristic
286C2×C4#1309393911yescharacteristic
287C2×C4#1309403112yes—
288C8—3762no—
289C8—3762no—
290C22—21no—
291C22—21no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#1568909011yescharacteristic
298C22C2×#1309393911yescharacteristic
299C22#1568910711yescharacteristic
300C4—21no—
301C4—21no—
302C4—22no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—81no—
309C4—81no—
310C4—81no—
311C2—21no—
312C2—21no—
313C2—81no—
314C2#2024173611yescharacteristic
315C2#2024505311yescharacteristic
316C2#2024508311yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2499C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#2499D47
hypocenterC47#11552
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2499

Complements

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

NH ≅ G/Ntype
3. #20241334C2semidirect
4. #20241322C2semidirect
181. C47#11552semidirect

Maximal subgroups

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

subgroupindex
#202417222
#202417022
#202413342
#202413222
C47×#24992
#1155247