Finite Group Databasebeta

← #26126494

Subgroups — #26126494

Order 12032.

452 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 107 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#20249154C211yesmaximal, characteristic
2#20243127C211yesmaximal, characteristic
3#20241395C214yesmaximal, complemented
4C47×#1458C211yesmaximal, characteristic
5C22×#10625329—21no—
6C2×#13094024C411yescharacteristic
7C2×#13094003—21no—
8#15689053—21no—
9#15689053C2212yes—
10#15689048—21no—
11#15689048C412yes—
12#15689029—21no—
13#15688989C411yescharacteristic
14#15688892—21no—
15#15688892C2212yes—
16C2×#117×C47C2211yescharacteristic
17C47×#431—21no—
18C47×#324C2212yes—
19C47×#323—21no—
20C22×Dic94—21no—
21C2×#10625329—24no—
22C2×#10625329—22no—
23C2×#10625329—41no—
24C2×#10625329—42no—
25C22×C47⋊C8—41no—
26#13094024—41no—
27#13094024—41no—
28#13094003—42no—
29#13094003—42no—
30C2×C4⋊Dic47—21no—
31C2×C4⋊Dic47—21no—
32C2×C4⋊Dic47C2×C412yes—
33C2×Dic47⋊C4—22no—
34C2×C22⋊Dic47—22no—
35C47⋊#95—21no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C47⋊#95C2×C412yes—
39C23×C4×C47D411yescharacteristic
40#117×C47—41no—
41#117×C47—41no—
42C2×C4⋊C4×C47D411yescharacteristic
43C2×C4⋊C4×C47C2311yescharacteristic
44C2×C22⋊C4×C47C2×C411yescharacteristic
45C2×C22⋊C4×C47D411yescharacteristic
46C2×C42×C47C2×C411yescharacteristic
47C2×C42×C47D411yescharacteristic
48C4⋊C8×C47—41no—
49C22⋊C8×C47—22no—
50C4×C8×C47—41no—
51C24×C47C22⋊C411yescharacteristic
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—22no—
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47D812yes—
59C22×C4×C47SD1612yes—
60C22×C4×C47C22×C411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×Dic47—22no—
64C22×Dic47—22no—
65C2×Dic94—41no—
66C2×Dic94—41no—
67C2×Dic94—42no—
68C2×Dic94—42no—
69C2×C8×C47—42no—
70C4⋊C4×C47—41no—
71C4⋊C4×C47—41no—
72C22⋊C4×C47—41no—
73C22⋊C4×C47—41no—
74C42×C47—21no—
75C42×C47—21no—
76C42×C47—21no—
77C42×C47—21no—
78C22⋊Dic47—44no—
79C4⋊Dic47—42no—
80C4⋊Dic47—42no—
81C4⋊Dic47—42no—
82Dic47⋊C4—42no—
83Dic47⋊C4—42no—
84#10625329—24no—
85#10625329—44no—
86#10625329—42no—
87C2×C47⋊C8—42no—
88C2×C47⋊C8—44no—
89C23×C47—21no—
90C23×C47—21no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47#10212yes—
94C23×C47C2×C22⋊C411yescharacteristic
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47#9911yescharacteristic
110C2×C4×C47#9911yescharacteristic
111C2×C4×C47#10212yes—
112C2×C4×C47#10412yes—
113C2×C4×C47C2×D811yescharacteristic
114C2×C4×C47C2×SD1611yescharacteristic
115C2×Dic47—42no—
116C2×Dic47—42no—
117C2×Dic47—42no—
118C2×Dic47—42no—
119C2×Dic47—42no—
120C2×Dic47—42no—
121Dic94—82no—
122Dic94—82no—
123C8×C47—82no—
124C47⋊C8—44no—
125#11700—471nomaximal, Sylow
126C22×C47—21no—
127C22×C47—21no—
128C22×C47—22no—
129C22×C47—42no—
130C22×C47—41no—
131C22×C47—41no—
132C22×C47—81no—
133C22×C47C2×#9911yescharacteristic
134C22×C47C2×#10211yescharacteristic
135C22×C47C2×#10411yescharacteristic
136C4×C47—21no—
137C4×C47—21no—
138C4×C47—41no—
139C4×C47—41no—
140C4×C47—41no—
141C4×C47—41no—
142C4×C47#32712yes—
143C4×C47#32712yes—
144Dic47—82no—
145Dic47—82no—
146#2621—471no—
147#1742—471no—
148#1458D4711yescharacteristic
149#1346—474no—
150C2×C47—21no—
151C2×C47—21no—
152C2×C47—42no—
153C2×C47C2×#32711yescharacteristic
154C2×C47#212311yescharacteristic
155C2×C47#213511yescharacteristic
156C22×M16—941no—
157C2×C4⋊Q8—471no—
158C2×C22⋊Q8—941no—
159C2×#117C2×D4711yescharacteristic
160#431—21no—
161#398—472no—
162#398—941no—
163#397—472no—
164#397—941no—
165#391—941no—
166#382—471no—
167#343—472no—
168#343—941no—
169#324C2×D4712yes—
170#323—21no—
171C47#1170011yescharacteristic, complemented, Sylow
172C22×Q8—941no—
173C23×C4D18811yescharacteristic
174C2×M16—944no—
175C2×M16—942no—
176C2×M16—1881no—
177C2×M16—1882no—
178C22×C8—1881no—
179C4⋊Q8—1881no—
180C4⋊Q8—1881no—
181C22⋊Q8—1882no—
182C22⋊Q8—1882no—
183#117—41no—
184#117—41no—
185C2×C4⋊C4—472no—
186C2×C4⋊C4—941no—
187C2×C4⋊C4—941no—
188C2×C4⋊C4—942no—
189C2×C4⋊C4D18811yescharacteristic
190C2×C4⋊C4C22×D4711yescharacteristic
191C2×C22⋊C4—942no—
192C2×C22⋊C4C4×D4711yescharacteristic
193C2×C22⋊C4C22⋊D4711yescharacteristic
194C2×C42C4×D4711yescharacteristic
195C2×C42C22⋊D4711yescharacteristic
196C4⋊C8—41no—
197C22⋊C8—22no—
198C4×C8—41no—
199#95—472no—
200#95—941no—
201#95—942no—
202#95—942no—
203C24(C2×D47)⋊C411yescharacteristic
204C2×Q8—1881no—
205C2×Q8—1881no—
206C2×Q8—1882no—
207C2×Q8—1882no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—942no—
212C22×C4—942no—
213C22×C4#1062532512yes—
214C22×C4D37612yes—
215C22×C4(C2×D47)⋊C411yescharacteristic
216C22×C4(C2×D47)⋊C411yescharacteristic
217C22×C4(C2×D47)⋊C411yescharacteristic
218C22×C4C2×C4×D4711yescharacteristic
219C22×C4C2×D18811yescharacteristic
220C22×C4C2×C22⋊D4711yescharacteristic
221M16—944no—
222M16—1884no—
223M16—1882no—
224C2×C8—42no—
225C2×C8—1882no—
226C2×C8—1884no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—1882no—
230C4⋊C4—1882no—
231C4⋊C4—1882no—
232C4⋊C4—1882no—
233C4⋊C4—1882no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C22⋊C4—1884no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23#1309394612yes—
246C23C2×(C2×D47)⋊C411yescharacteristic
247Q8—3762no—
248Q8—3762no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—1882no—
264C2×C4—1882no—
265C2×C4—1882no—
266C2×C4—1882no—
267C2×C4—1882no—
268C2×C4—1882no—
269C2×C4#1309393811yescharacteristic
270C2×C4#1309393811yescharacteristic
271C2×C4#1309394612yes—
272C2×C4#1309395012yes—
273C2×C4C2×D37611yescharacteristic
274C2×C4C2×#1062532511yescharacteristic
275C8—82no—
276C8—1884no—
277C22—21no—
278C22—21no—
279C22—22no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22—81no—
284C22C2×#1309393811yescharacteristic
285C22C2×#1309394611yescharacteristic
286C22C2×#1309395011yescharacteristic
287C4—21no—
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—3762no—
294C4—3762no—
295C4#1568885812yes—
296C4#1568885812yes—
297C2—21no—
298C2—21no—
299C2—42no—
300C2C2×#1568885811yescharacteristic
301C2#2024504111yescharacteristic
302C2#2024508211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1458C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1458D47
hypocenterC47#11700
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #20241395C2semidirect
171. C47#11700semidirect

Maximal subgroups

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

subgroupindex
#202491542
#202431272
#202413952
C47×#14582
#1170047