Finite Group Databasebeta

← #26125928

Subgroups — #26125928

Order 12032.

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

normal subgroups only

This group has 480 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#20248341C211yesmaximal, characteristic
2#20241726C211yesmaximal, characteristic
3#20241333C214yesmaximal, complemented
4C47×#1459C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic
6C2×#13094003—21no—
7C2×#13094003—21no—
8#15689087—21no—
9#15689059—21no—
10#15689059—21no—
11#15689026C412yes—
12#15688992C2212yes—
13C4×C4⋊Dic47C411yescharacteristic
14#15688849C2212yes—
15#15688847—21no—
16C2×C4×D4×C47C2211yescharacteristic
17C2×C22⋊C8×C47—21no—
18C47×#324C2212yes—
19C47×#323—21no—
20C22×Dic94—21no—
21C2×#10625329—41no—
22#13094024—41no—
23#13094024—41no—
24#13094003—44no—
25#13094003—44no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47C2×C412yes—
29C2×Dic47⋊C4—22no—
30C2×Dic47⋊C4—22no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—22no—
33C2×C4×Dic47C2×C412yes—
34C22⋊(C47⋊C8)—22no—
35C22⋊(C47⋊C8)—41no—
36C47⋊#95—21no—
37C22×D4×C47C2×C411yescharacteristic
38C23×C4×C47D411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C22×C8×C47—41no—
41C4×D4×C47—41no—
42C4×D4×C47—41no—
43C2×C4⋊C4×C47C2311yescharacteristic
44C2×C22⋊C4×C47D411yescharacteristic
45C2×C22⋊C4×C47D411yescharacteristic
46C2×C42×C47C2×C411yescharacteristic
47C22⋊C8×C47—22no—
48C22⋊C8×C47—41no—
49C24×C47C22⋊C411yescharacteristic
50C24×C47C22⋊C411yescharacteristic
51C2×D4×C47—22no—
52C2×D4×C47—22no—
53C2×D4×C47—41no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—42no—
59C22×C4×C47—42no—
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×Dic47—22no—
66C22×Dic47—22no—
67C2×Dic94—42no—
68C2×Dic94—42no—
69C2×Dic94—41no—
70C2×Dic94—41no—
71C2×C8×C47—42no—
72C2×C8×C47—82no—
73C4⋊C4×C47—41no—
74C22⋊C4×C47—41no—
75C22⋊C4×C47—41no—
76C42×C47—21no—
77C42×C47—21no—
78C22⋊Dic47—44no—
79C22⋊Dic47—44no—
80C4⋊Dic47—24no—
81C4⋊Dic47—42no—
82C4⋊Dic47—42no—
83Dic47⋊C4—44no—
84Dic47⋊C4—44no—
85C4×Dic47—24no—
86#10625329—82no—
87C2×C47⋊C8—42no—
88C23×C47—21no—
89C23×C47—21no—
90C23×C47—21no—
91C23×C47—21no—
92C23×C47—21no—
93C23×C47—41no—
94C23×C47—41no—
95C23×C47—41no—
96C23×C47—41no—
97C23×C47#10112yes—
98C23×C47C2×C22⋊C411yescharacteristic
99D4×C47—44no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—22no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—42no—
107C2×C4×C47—42no—
108C2×C4×C47—41no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—81no—
112C2×C4×C47—81no—
113C2×C4×C47#9911yescharacteristic
114C2×C4×C47#9911yescharacteristic
115C2×C4×C47#13112yes—
116C2×Dic47—24no—
117C2×Dic47—42no—
118C2×Dic47—42no—
119C2×Dic47—42no—
120C2×Dic47—42no—
121C2×Dic47—42no—
122Dic94—82no—
123Dic94—82no—
124C8×C47—82no—
125C47⋊C8—82no—
126#11570—471nomaximal, Sylow
127C22×C47—21no—
128C22×C47—21no—
129C22×C47—22no—
130C22×C47—42no—
131C22×C47—42no—
132C22×C47—41no—
133C22×C47—41no—
134C22×C47—41no—
135C22×C47—41no—
136C22×C47—41no—
137C22×C47—81no—
138C22×C47—81no—
139C22×C47#40811yescharacteristic
140C22×C47C2×#9911yescharacteristic
141C22×C47C2×#10111yescharacteristic
142C4×C47—21no—
143C4×C47—22no—
144C4×C47—41no—
145C4×C47—41no—
146C4×C47—41no—
147C4×C47#35312yes—
148Dic47—44no—
149Dic47—82no—
150#2542—471no—
151#1465—471no—
152#1459D4711yescharacteristic
153#1325—474no—
154C2×C47—21no—
155C2×C47—21no—
156C2×C47—42no—
157C2×C47—81no—
158C2×C47#147211yescharacteristic
159C2×C47C2×#35311yescharacteristic
160C2×C47#213511yescharacteristic
161C2×C4⋊Q8—471no—
162C2×C22⋊Q8—941no—
163C2×C22⋊Q8—941no—
164C2×C4×D4C2×D4711yescharacteristic
165#407—941no—
166C2×C22⋊C8—21no—
167#399—941no—
168#399—941no—
169#390—472no—
170#384—472no—
171C4×C4⋊C4—471no—
172#324—472no—
173#324C2×D4712yes—
174#323—21no—
175#323—941no—
176C47#1157011yescharacteristic, complemented, Sylow
177C22×Q8—941no—
178C22×D4C4×D4711yescharacteristic
179C23×C4D18811yescharacteristic
180C23×C4C22⋊D4711yescharacteristic
181C2×M16—1881no—
182C22×C8—41no—
183C4⋊Q8—1881no—
184C4⋊Q8—1881no—
185C22⋊Q8—1884no—
186C22⋊Q8—1884no—
187C4×D4—41no—
188C4×D4—41no—
189C2×C4⋊C4—472no—
190C2×C4⋊C4—941no—
191C2×C4⋊C4—941no—
192C2×C4⋊C4—942no—
193C2×C4⋊C4—942no—
194C2×C4⋊C4C22×D4711yescharacteristic
195C2×C22⋊C4—942no—
196C2×C22⋊C4—942no—
197C2×C22⋊C4D18811yescharacteristic
198C2×C22⋊C4C22⋊D4711yescharacteristic
199C2×C42—472no—
200C2×C42C4×D4711yescharacteristic
201C22⋊C8—22no—
202C22⋊C8—41no—
203C22⋊C8—942no—
204C22⋊C8—1881no—
205#95—941no—
206C24(C2×D47)⋊C411yescharacteristic
207C24(C2×D47)⋊C411yescharacteristic
208C2×Q8—1882no—
209C2×Q8—1882no—
210C2×Q8—1881no—
211C2×Q8—1881no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—41no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—942no—
222C22×C4—942no—
223C22×C4(C2×D47)⋊C411yescharacteristic
224C22×C4(C2×D47)⋊C411yescharacteristic
225C22×C4C2×C4×D4711yescharacteristic
226C22×C4C2×D18811yescharacteristic
227C22×C4C2×C22⋊D4711yescharacteristic
228M16—3762no—
229C2×C8—42no—
230C2×C8—82no—
231C2×C8—1882no—
232C4⋊C4—41no—
233C4⋊C4—944no—
234C4⋊C4—1882no—
235C4⋊C4—1882no—
236C4⋊C4—1884no—
237C4⋊C4—1884no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—1884no—
241C22⋊C4—1884no—
242C42—21no—
243C42—21no—
244C42—944no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23#1309394212yes—
255C23C2×(C2×D47)⋊C411yescharacteristic
256Q8—3762no—
257Q8—3762no—
258D4—44no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—944no—
273C2×C4—1882no—
274C2×C4—1882no—
275C2×C4—1882no—
276C2×C4—1882no—
277C2×C4—1882no—
278C2×C4#1309393811yescharacteristic
279C2×C4#1309393811yescharacteristic
280C2×C4#1309403012yes—
281C8—82no—
282C8—3762no—
283C22—21no—
284C22—21no—
285C22—22no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22—81no—
295C22#1568908911yescharacteristic
296C22C2×#1309393811yescharacteristic
297C22C2×#1309394211yescharacteristic
298C4—21no—
299C4—22no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—3762no—
304C4—1884no—
305C4#1568891512yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2#2024175111yescharacteristic
311C2C2×#1568891511yescharacteristic
312C2#2024508211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1459C2
socleC22×C47C2×#99
prime radicalC22×D4×C47C2×C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1459D47
hypocenterC47#11570
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #20241333C2semidirect
176. C47#11570semidirect

Maximal subgroups

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

subgroupindex
#202483412
#202417262
#202413332
C47×#14592
#1157047