Finite Group Databasebeta

← #26125574

Subgroups — #26125574

Order 12032.

392 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 103 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#20246104C211yesmaximal, characteristic
2#20241712C211yesmaximal, characteristic
3#20241331C212yesmaximal
4#20241321C212yesmaximal
5C47×#1461C211yesmaximal, characteristic
6C2×#13094022C411yescharacteristic
7C2×C4×Dic94C411yescharacteristic
8#15689176—21no—
9#15689030C411yescharacteristic
10#15689028C2211yescharacteristic
11#15689007—21no—
12#15689007—21no—
13#15689004—21no—
14#15689004—21no—
15#15688988C2211yescharacteristic
16#15688969C411yescharacteristic
17#15688849—21no—
18#15688849C2212yes—
19C2×C4×Q8×C47C2211yescharacteristic
20C47×#431—21no—
21C47×#324—21no—
22C47×#324C2212yes—
23C22×Dic94C2×C411yescharacteristic
24#13094022—22no—
25#13094022C814yescomplemented
26C4×Dic94—41no—
27C4×Dic94—41no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×C4×Dic47—21no—
33C2×C4×Dic47C2×C411yescharacteristic
34C2×C4×Dic47C2×C411yescharacteristic
35C4⋊(C47⋊C8)—41no—
36C22⋊(C47⋊C8)—22no—
37C4×C47⋊C8—41no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95—21no—
46C22×Q8×C47C2×C411yescharacteristic
47C4×Q8×C47—41no—
48C4×Q8×C47—41no—
49C2×C4⋊C4×C47D411yescharacteristic
50C2×C4⋊C4×C47D411yescharacteristic
51C2×C4⋊C4×C47C2311yescharacteristic
52C2×C42×C47C2×C411yescharacteristic
53C2×C42×C47D411yescharacteristic
54C2×C42×C47D411yescharacteristic
55C4⋊C8×C47—41no—
56C22⋊C8×C47—22no—
57C4×C8×C47—41no—
58C2×Q8×C47—41no—
59C2×Q8×C47—41no—
60C2×Q8×C47C2×C812yes—
61C2×Q8×C47M1612yes—
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×Dic47—21no—
70C22×Dic47—21no—
71C22×Dic47—21no—
72C22×Dic47—21no—
73C2×Dic94—21no—
74C2×Dic94—41no—
75C2×Dic94—41no—
76C2×Dic94C2×C812yes—
77C2×C8×C47—42no—
78C4⋊C4×C47—41no—
79C4⋊C4×C47—41no—
80C4⋊C4×C47—41no—
81C42×C47—21no—
82C42×C47—21no—
83C42×C47—21no—
84C42×C47—21no—
85C42×C47—21no—
86C42×C47—21no—
87C4⋊Dic47—41no—
88Dic47⋊C4—24no—
89Dic47⋊C4—24no—
90Dic47⋊C4—41no—
91Dic47⋊C4—41no—
92C4×Dic47—21no—
93C4×Dic47—42no—
94C4×Dic47—41no—
95C4×Dic47C2×C812yes—
96C2×C47⋊C8—42no—
97C23×C47C2×C22⋊C411yescharacteristic
98Q8×C47—42no—
99Q8×C47—42no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47C22⋊C812yes—
114C2×C4×C47C22⋊C812yes—
115C2×C4×C47#9911yescharacteristic
116C2×C4×C47#9911yescharacteristic
117C2×C4×C47C22×C811yescharacteristic
118C2×C4×C47C2×M1611yescharacteristic
119C2×Dic47—22no—
120C2×Dic47—22no—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—42no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130Dic94—42no—
131Dic94—42no—
132C8×C47—82no—
133C47⋊C8—82no—
134#11482—471nomaximal, Sylow
135C22×C47—21no—
136C22×C47—21no—
137C22×C47C2×C22⋊C811yescharacteristic
138C22×C47C2×#9911yescharacteristic
139C22×C47#41311yescharacteristic
140C4×C47—21no—
141C4×C47—21no—
142C4×C47—21no—
143C4×C47—21no—
144C4×C47—22no—
145C4×C47—22no—
146C4×C47—41no—
147C4×C47—41no—
148C4×C47—41no—
149Dic47—42no—
150Dic47—42no—
151Dic47—81no—
152Dic47—81no—
153#2327—471no—
154#1461—471no—
155#1461D4711yescharacteristic
156#1324—472no—
157#1319—472no—
158C2×C47—21no—
159C2×C47—21no—
160C2×C47#145911yescharacteristic
161C2×C47#212911yescharacteristic
162C2×C47#213311yescharacteristic
163C2×C4⋊Q8—471no—
164C2×C4×Q8—471no—
165C2×C4×Q8C2×D4711yescharacteristic
166#431—21no—
167#431—941no—
168#391—471no—
169#391—471no—
170#387—941no—
171#387—941no—
172#387—941no—
173#387—941no—
174#382—471no—
175#376—471no—
176#324—21no—
177#324—472no—
178#324—941no—
179#324C2×D4712yes—
180C47#1148211yescharacteristic, complemented, Sylow
181C22×Q8—471no—
182C22×Q8C4×D4711yescharacteristic
183C4⋊Q8—474no—
184C4⋊Q8—942no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—1881no—
188C4×Q8—1881no—
189C2×C4⋊C4—471no—
190C2×C4⋊C4—941no—
191C2×C4⋊C4—941no—
192C2×C4⋊C4—941no—
193C2×C4⋊C4D18811yescharacteristic
194C2×C4⋊C4C22⋊D4711yescharacteristic
195C2×C4⋊C4C22×D4711yescharacteristic
196C2×C42—471no—
197C2×C42—471no—
198C2×C42—941no—
199C2×C42C4×D4711yescharacteristic
200C2×C42D18811yescharacteristic
201C2×C42C22⋊D4711yescharacteristic
202C4⋊C8—41no—
203C4⋊C8—1881no—
204C22⋊C8—22no—
205C22⋊C8—942no—
206C4×C8—41no—
207C4×C8—1881no—
208#95—941no—
209#95—941no—
210#95—941no—
211#95—941no—
212#95—941no—
213#95—941no—
214#95—941no—
215#95—941no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—472no—
219C2×Q8—941no—
220C2×Q8—1881no—
221C2×Q8—1881no—
222C2×Q8C8×D4712yes—
223C2×Q8#1062532412yes—
224C22×C4—941no—
225C22×C4—941no—
226C22×C4—941no—
227C22×C4—941no—
228C22×C4(C2×D47)⋊C411yescharacteristic
229C22×C4(C2×D47)⋊C411yescharacteristic
230C22×C4(C2×D47)⋊C411yescharacteristic
231C22×C4(C2×D47)⋊C411yescharacteristic
232C22×C4C2×C4×D4711yescharacteristic
233C22×C4C2×D18811yescharacteristic
234C22×C4C2×C22⋊D4711yescharacteristic
235C2×C8—42no—
236C2×C8—1882no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—944no—
241C4⋊C4—944no—
242C4⋊C4—1881no—
243C4⋊C4—1881no—
244C4⋊C4—1881no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—472no—
252C42—941no—
253C42—1881no—
254C42—1882no—
255C23C2×(C2×D47)⋊C411yescharacteristic
256Q8—42no—
257Q8—42no—
258Q8—1882no—
259Q8—1882no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—942no—
274C2×C4—942no—
275C2×C4—1881no—
276C2×C4—1881no—
277C2×C4—1882no—
278C2×C4—1881no—
279C2×C4—1881no—
280C2×C4—1881no—
281C2×C4—1881no—
282C2×C4—1881no—
283C2×C4—1881no—
284C2×C4(C2×D47)⋊C812yes—
285C2×C4(C2×D47)⋊C812yes—
286C2×C4#1309393811yescharacteristic
287C2×C4#1309393811yescharacteristic
288C2×C4C2×C8×D4711yescharacteristic
289C2×C4C2×#1062532411yescharacteristic
290C8—82no—
291C8—3762no—
292C22—21no—
293C22—21no—
294C22C2×(C2×D47)⋊C811yescharacteristic
295C22C2×#1309393811yescharacteristic
296C22#1568910511yescharacteristic
297C4—22no—
298C4—22no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—3761no—
307C4—3761no—
308C4—1882no—
309C4—1882no—
310C2—21no—
311C2—21no—
312C2#2024170411yescharacteristic
313C2#2024506211yescharacteristic
314C2#2024507511yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC2×C4×C47C22×C8
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1461C2
socleC22×C47C2×#99
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1461D47
hypocenterC47#11482
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
25. #13094022C8semidirect
180. C47#11482semidirect

Maximal subgroups

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

subgroupindex
#202461042
#202417122
#202413312
#202413212
C47×#14612
#1148247