Finite Group Databasebeta

← #26125940

Subgroups — #26125940

Order 12032.

416 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 83 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#20248584C211yesmaximal, characteristic
2#20241706C211yesmaximal, characteristic
3#20241333C212yesmaximal, complemented
4#20241331C212yesmaximal, complemented
5C47×#1465C211yesmaximal, characteristic
6C2×#13094016C411yescharacteristic
7C2×C22⋊(C47⋊C8)—21no—
8#15689046—21no—
9#15689046—21no—
10#15689026C411yescharacteristic
11#15689026C411yescharacteristic
12#15688996—21no—
13#15688996—21no—
14#15688992C2211yescharacteristic
15#15688988C2211yescharacteristic
16C4×C4⋊Dic47C411yescharacteristic
17#15688849C2212yes—
18#15688847—21no—
19C2×C4×D4×C47C2211yescharacteristic
20C47×#407—21no—
21C47×#324C2212yes—
22C47×#323—21no—
23C22×C47⋊C8—41no—
24#13094016—41no—
25#13094016—41no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×C22⋊Dic47—21no—
34C2×C22⋊Dic47—21no—
35C2×C22⋊Dic47—21no—
36C2×C22⋊Dic47—21no—
37C2×C4×Dic47C2×C411yescharacteristic
38C2×C4×Dic47C2×C411yescharacteristic
39C22⋊(C47⋊C8)—22no—
40C22⋊(C47⋊C8)—41no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C22×D4×C47C2×C411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C2×M16×C47—41no—
48C4×D4×C47—41no—
49C4×D4×C47—41no—
50C2×C4⋊C4×C47C2311yescharacteristic
51C2×C22⋊C4×C47D411yescharacteristic
52C2×C22⋊C4×C47D411yescharacteristic
53C2×C42×C47C2×C411yescharacteristic
54C22⋊C8×C47—22no—
55C22⋊C8×C47—41no—
56C24×C47C22⋊C411yescharacteristic
57C24×C47C22⋊C411yescharacteristic
58C2×D4×C47—22no—
59C2×D4×C47—22no—
60C2×D4×C47—41no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—81no—
66C22×C4×C47—81no—
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×Dic47—21no—
73C22×Dic47—21no—
74C22×Dic47—21no—
75C22×Dic47—21no—
76M16×C47—82no—
77C2×C8×C47—42no—
78C4⋊C4×C47—41no—
79C22⋊C4×C47—41no—
80C22⋊C4×C47—41no—
81C42×C47—21no—
82C42×C47—21no—
83C22⋊Dic47—42no—
84C22⋊Dic47—42no—
85C22⋊Dic47—81no—
86C22⋊Dic47—81no—
87C4⋊Dic47—42no—
88C4⋊Dic47—41no—
89C4⋊Dic47—41no—
90Dic47⋊C4—42no—
91Dic47⋊C4—41no—
92Dic47⋊C4—41no—
93Dic47⋊C4—81no—
94Dic47⋊C4—81no—
95C4×Dic47—22no—
96C4×Dic47—22no—
97C2×C47⋊C8—42no—
98C2×C47⋊C8—82no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47—41no—
105C23×C47—41no—
106C23×C47—41no—
107C23×C47—41no—
108C23×C47#10112yes—
109C23×C47C2×C22⋊C411yescharacteristic
110D4×C47—44no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—41no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—81no—
123C2×C4×C47—81no—
124C2×C4×C47#9911yescharacteristic
125C2×C4×C47#9911yescharacteristic
126C2×C4×C47#13112yes—
127C2×Dic47—22no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—41no—
132C2×Dic47—41no—
133C2×Dic47—41no—
134C2×Dic47—41no—
135C2×Dic47—41no—
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C8×C47—82no—
140C47⋊C8—82no—
141#11572—471nomaximal, Sylow
142C22×C47—21no—
143C22×C47—21no—
144C22×C47—22no—
145C22×C47—42no—
146C22×C47—42no—
147C22×C47—41no—
148C22×C47—41no—
149C22×C47—41no—
150C22×C47—41no—
151C22×C47—41no—
152C22×C47—81no—
153C22×C47—81no—
154C22×C47#40811yescharacteristic
155C22×C47C2×#9911yescharacteristic
156C22×C47C2×#10111yescharacteristic
157C4×C47—21no—
158C4×C47—21no—
159C4×C47—22no—
160C4×C47—41no—
161C4×C47—41no—
162C4×C47—41no—
163Dic47—42no—
164Dic47—81no—
165Dic47—81no—
166Dic47—81no—
167#2566—471no—
168#1465D4711yescharacteristic
169#1459—471no—
170#1325—472no—
171#1324—472no—
172C2×C47—21no—
173C2×C47—21no—
174C2×C47—42no—
175C2×C47—81no—
176C2×C47#147211yescharacteristic
177C2×C47#212711yescharacteristic
178C2×C47#213211yescharacteristic
179C2×#125—471no—
180C2×C4×D4C2×D4711yescharacteristic
181#407—21no—
182C2×C22⋊C8—941no—
183#396—941no—
184#396—941no—
185#390—471no—
186#390—471no—
187#385—941no—
188#385—941no—
189#384—471no—
190#382—471no—
191C4×C4⋊C4—471no—
192#324—472no—
193#324C2×D4712yes—
194#323—21no—
195#323—941no—
196C47#1157211yescharacteristic, complemented, Sylow
197C22×D4C4×D4711yescharacteristic
198C23×C4D18811yescharacteristic
199C23×C4C22⋊D4711yescharacteristic
200C2×M16—41no—
201C22×C8—1881no—
202#125—1881no—
203#125—1881no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—471no—
207C2×C4⋊C4—471no—
208C2×C4⋊C4—941no—
209C2×C4⋊C4—941no—
210C2×C4⋊C4—941no—
211C2×C4⋊C4—941no—
212C2×C4⋊C4—941no—
213C2×C4⋊C4C22×D4711yescharacteristic
214C2×C22⋊C4—941no—
215C2×C22⋊C4—941no—
216C2×C22⋊C4—941no—
217C2×C22⋊C4—941no—
218C2×C22⋊C4D18811yescharacteristic
219C2×C22⋊C4C22⋊D4711yescharacteristic
220C2×C42—471no—
221C2×C42—471no—
222C2×C42C4×D4711yescharacteristic
223C22⋊C8—22no—
224C22⋊C8—41no—
225C22⋊C8—942no—
226C22⋊C8—1881no—
227#95—941no—
228#95—941no—
229#95—941no—
230C24(C2×D47)⋊C411yescharacteristic
231C24(C2×D47)⋊C411yescharacteristic
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—41no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—941no—
242C22×C4—941no—
243C22×C4—941no—
244C22×C4—941no—
245C22×C4(C2×D47)⋊C411yescharacteristic
246C22×C4(C2×D47)⋊C411yescharacteristic
247C22×C4C2×C4×D4711yescharacteristic
248C22×C4C2×D18811yescharacteristic
249C22×C4C2×C22⋊D4711yescharacteristic
250M16—82no—
251C2×C8—42no—
252C2×C8—3762no—
253C2×C8—1882no—
254C4⋊C4—41no—
255C4⋊C4—3761no—
256C4⋊C4—3761no—
257C4⋊C4—1881no—
258C4⋊C4—1881no—
259C4⋊C4—1882no—
260C4⋊C4—1882no—
261C4⋊C4—1881no—
262C4⋊C4—1881no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—3761no—
266C22⋊C4—3761no—
267C22⋊C4—1882no—
268C22⋊C4—1882no—
269C42—21no—
270C42—21no—
271C42—942no—
272C42—942no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23#1309394212yes—
283C23C2×(C2×D47)⋊C411yescharacteristic
284D4—44no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—942no—
299C2×C4—1881no—
300C2×C4—1881no—
301C2×C4—1881no—
302C2×C4—1881no—
303C2×C4—1881no—
304C2×C4—1881no—
305C2×C4—1881no—
306C2×C4—1881no—
307C2×C4—1881no—
308C2×C4—1881no—
309C2×C4—1881no—
310C2×C4#1309393811yescharacteristic
311C2×C4#1309393811yescharacteristic
312C2×C4#1309403212yes—
313C8—82no—
314C8—3762no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—81no—
326C22—81no—
327C22#1568909111yescharacteristic
328C22C2×#1309393811yescharacteristic
329C22C2×#1309394211yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—3761no—
337C4—3761no—
338C4—3761no—
339C4—1882no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—81no—
344C2#2024175311yescharacteristic
345C2#2024505511yescharacteristic
346C2#2024507111yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1465C2
socleC22×C47C2×#99
prime radicalC22×D4×C47C2×C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1465D47
hypocenterC47#11572
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #20241333C2semidirect
4. #20241331C2semidirect
196. C47#11572semidirect

Maximal subgroups

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

subgroupindex
#202485842
#202417062
#202413332
#202413312
C47×#14652
#1157247