Finite Group Databasebeta

← #26107334

Subgroups — #26107334

Order 12032.

436 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252364C211yesmaximal, characteristic
2#20243076C211yesmaximal, characteristic, complemented
3#20241874C211yesmaximal, characteristic
4#20241675C211yesmaximal, characteristic, complemented
5#20241287C211yesmaximal, characteristic, complemented
6#20241282C211yesmaximal, characteristic, complemented
7C47×#1524C211yesmaximal, characteristic
8#15689254C411yescharacteristic, complemented
9Q8×C47⋊C8C411yescharacteristic, complemented
10#15689201C411yescharacteristic, complemented
11#15689195—21no—
12D4×C47⋊C8—21no—
13D4×C47⋊C8C411yescharacteristic, complemented
14#15689184—21no—
15#15689180C2211yescharacteristic
16#15689176C2211yescharacteristic
17#15689079—21no—
18C2×#13093934—21no—
19C2×C4×C47⋊C8C2211yescharacteristic
20C2×C8×Dic47—21no—
21#15688969C2211yescharacteristic
22#15688860C412yes—
23#15688857C412yes—
24#15688845—21no—
25#15688845—21no—
26C47×#529C2211yescharacteristic
27C4×M16×C47C2211yescharacteristic
28C2×#97×C47C2211yescharacteristic
29C47×#330C412yescomplemented
30C47×#329C412yescomplemented
31C2×#10625329—21no—
32C22×C47⋊C8—21no—
33C22×C47⋊C8D411yescharacteristic
34C22×C47⋊C8D411yescharacteristic
35C2×C4×Dic47—21no—
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)C2×C411yescharacteristic
40C4⋊(C47⋊C8)C2×C411yescharacteristic
41C22⋊(C47⋊C8)—21no—
42C22⋊(C47⋊C8)—21no—
43C22⋊(C47⋊C8)—21no—
44C22⋊(C47⋊C8)C2×C412yes—
45#13093935—21no—
46#13093935—21no—
47#13093934—24no—
48C4×C47⋊C8—21no—
49C4×C47⋊C8—21no—
50C4×C47⋊C8C2×C411yescharacteristic
51C4×C47⋊C8C2×C411yescharacteristic
52C4×C47⋊C8D411yescharacteristic
53C4×C47⋊C8D411yescharacteristic
54C8×Dic47—24no—
55C47⋊#95—21no—
56C47⋊#95C2×C412yes—
57C2×M16×C47—21no—
58C22×C8×C47—21no—
59#126×C47—21no—
60#125×C47C2×C411yescharacteristic
61#124×C47C2×C411yescharacteristic
62#123×C47—21no—
63C22⋊Q8×C47C2×C411yescharacteristic
64#121×C47C2×C411yescharacteristic
65C4×Q8×C47C2×C411yescharacteristic
66C4×D4×C47—21no—
67C4×D4×C47C2×C411yescharacteristic
68#117×C47—21no—
69C2×C42×C47C2311yescharacteristic
70#97×C47—21no—
71#97×C47C2×C412yes—
72#97×C47C2×C412yes—
73C4×C8×C47—21no—
74C2×Q8×C47C4211yescharacteristic
75C2×D4×C47—21no—
76C2×D4×C47C4211yescharacteristic
77C22×C4×C47—21no—
78C22×C4×C47C22×C411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47#4111yescharacteristic
81C22×Dic47—22no—
82M16×C47—42no—
83C2×C8×C47—22no—
84C2×C8×C47—22no—
85C2×C8×C47—22no—
86C2×C8×C47—22no—
87C4⋊C4×C47—21no—
88C4⋊C4×C47—21no—
89C4⋊C4×C47—21no—
90C4⋊C4×C47—21no—
91C4⋊C4×C47C4211yescharacteristic
92C4⋊C4×C47C4211yescharacteristic
93C22⋊C4×C47—21no—
94C22⋊C4×C47—21no—
95C22⋊C4×C47—21no—
96C22⋊C4×C47—21no—
97C22⋊C4×C47—21no—
98C42×C47—21no—
99C42×C47C22×C411yescharacteristic
100C42×C47C22×C411yescharacteristic
101C42×C47C2×D411yescharacteristic
102C42×C47#4111yescharacteristic
103C4×Dic47—22no—
104C4×Dic47—22no—
105#10625329—42no—
106C2×C47⋊C8—22no—
107C2×C47⋊C8—22no—
108C2×C47⋊C8—21no—
109C2×C47⋊C8—21no—
110C2×C47⋊C8—22no—
111C2×C47⋊C8—42no—
112C2×C47⋊C8C22⋊C411yescharacteristic
113C2×C47⋊C8C22⋊C411yescharacteristic
114C2×C47⋊C8C22⋊C411yescharacteristic
115C2×C47⋊C8C22⋊C411yescharacteristic
116C23×C47—21no—
117C23×C47C4×D411yescharacteristic
118Q8×C47—41no—
119D4×C47—41no—
120D4×C47—42no—
121C2×C4×C47—22no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47C2×C4211yescharacteristic
131C2×C4×C47C2×C22⋊C411yescharacteristic
132C2×C4×C47#11711yescharacteristic
133C2×C4×C47C4×D411yescharacteristic
134C2×C4×C47C4×D411yescharacteristic
135C2×C4×C47C4×D411yescharacteristic
136C2×Dic47—24no—
137C2×Dic47—42no—
138C2×Dic47—42no—
139C8×C47—24no—
140C8×C47—42no—
141C47⋊C8—21no—
142C47⋊C8—21no—
143C47⋊C8—42no—
144C47⋊C8—41no—
145C47⋊C8—41no—
146C47⋊C8#10412yes—
147C47⋊C8#10412yes—
148#8249—471nomaximal, Sylow
149C22×C47—21no—
150C22×C47—21no—
151C22×C47—41no—
152C22×C47—41no—
153C22×C47—41no—
154C22×C47C4×C22⋊C411yescharacteristic
155C22×C47#44412yes—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47—21no—
159C4×C47—21no—
160C4×C47—21no—
161C4×C47—41no—
162C4×C47—41no—
163C4×C47—41no—
164C4×C47C2×#10411yescharacteristic
165C4×C47C2×#10411yescharacteristic
166Dic47—44no—
167#2966—471no—
168#1726—471no—
169#1524D4711yescharacteristic
170#1500—471no—
171#1451—471no—
172#1302—471no—
173#1300—471no—
174C2×C47—22no—
175C2×C47—81no—
176C2×C47#175711yescharacteristic
177C2×C47C4×#10411yescharacteristic
178C2×C47#176511yescharacteristic
179#529C2×D4711yescharacteristic
180#446—471no—
181C8×Q8—471no—
182#436—471no—
183#435—941no—
184C8×D4—471no—
185C8×D4—941no—
186#433—941no—
187#432—471no—
188#431—471no—
189#405—941no—
190C4×M16C2×D4711yescharacteristic
191C2×#97—941no—
192C2×#97C2×D4711yescharacteristic
193C2×C4×C8—471no—
194C2×C4×C8—941no—
195#376—471no—
196#330Dic4712yescomplemented
197#329Dic4712yescomplemented
198#328—472no—
199#327—472no—
200#322—941no—
201#322—941no—
202C47#824911yescharacteristic, complemented, Sylow
203C2×M16—21no—
204C2×M16—941no—
205C22×C8—21no—
206C22×C8—471no—
207C22×C8—471no—
208C22×C8—941no—
209#126—21no—
210#125C2×Dic4711yescharacteristic
211#124C2×Dic4711yescharacteristic
212#123—21no—
213C22⋊Q8C4×D4711yescharacteristic
214#121C4×D4711yescharacteristic
215C4×Q8C4×D4711yescharacteristic
216C4×D4—21no—
217C4×D4C4×D4711yescharacteristic
218#117—21no—
219C2×C42—941no—
220C2×C42C22×D4711yescharacteristic
221C4⋊C8—471no—
222C4⋊C8—471no—
223C4⋊C8—941no—
224C4⋊C8—941no—
225C4⋊C8—941no—
226C22⋊C8—472no—
227C22⋊C8—941no—
228C22⋊C8—941no—
229C22⋊C8—941no—
230#97—21no—
231#97—941no—
232#97—944no—
233#97—941no—
234#97C2×Dic4712yes—
235#97C2×Dic4712yes—
236C4×C8—21no—
237C4×C8—471no—
238C4×C8—471no—
239C4×C8—471no—
240C4×C8—471no—
241C4×C8—941no—
242C4×C8—944no—
243C4×C8—941no—
244#95—472no—
245#95—941no—
246C2×Q8C4×Dic4711yescharacteristic
247C2×D4—21no—
248C2×D4C4×Dic4711yescharacteristic
249C22×C4—21no—
250C22×C4—942no—
251C22×C4C2×C4×D4711yescharacteristic
252C22×C4D4×D4711yescharacteristic
253C22×C4#1062535211yescharacteristic
254M16—42no—
255M16—1882no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—471no—
261C2×C8—471no—
262C2×C8—471no—
263C2×C8—471no—
264C2×C8—942no—
265C2×C8—942no—
266C2×C8—941no—
267C2×C8—941no—
268C2×C8—942no—
269C2×C8—1882no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4C4×Dic4711yescharacteristic
275C4⋊C4C4×Dic4711yescharacteristic
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C42—21no—
282C42—942no—
283C42—942no—
284C42C2×C4×D4711yescharacteristic
285C42D4×D4711yescharacteristic
286C42#1062535211yescharacteristic
287C42C22×Dic4711yescharacteristic
288C23—21no—
289C23(C22⋊D47)⋊C411yescharacteristic
290Q8—41no—
291D4—41no—
292D4—42no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—944no—
303C2×C4—1882no—
304C2×C4—1882no—
305C2×C4C2×C4×Dic4711yescharacteristic
306C2×C4C22⋊C4×D4711yescharacteristic
307C2×C4#1309398011yescharacteristic
308C2×C4(C22⋊D47)⋊C411yescharacteristic
309C2×C4D4×Dic4711yescharacteristic
310C2×C4D4×Dic4711yescharacteristic
311C8—24no—
312C8—42no—
313C8—472no—
314C8—472no—
315C8—941no—
316C8—941no—
317C8—1882no—
318C8—1881no—
319C8—1881no—
320C22—21no—
321C22—21no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22C22⋊C4×Dic4711yescharacteristic
326C22#1568924212yes—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—1884no—
336C4#104×D4711yescharacteristic
337C4#104×D4711yescharacteristic
338C2—22no—
339C2—81no—
340C2#2024319111yescharacteristic
341C2#104×Dic4711yescharacteristic
342C2#2024323311yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D47
commutatorC2×C4×C47C2×C42
FrattiniC2×C42C22×D47
FittingC47×#1524C2
socleC22×C47C4×C22⋊C4
prime radical#121×C47C2×C4
cosocleC2×C42×C47C23
hypercenter#1524D47
hypocenterC47#8249
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1524

Complements

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

NH ≅ G/Ntype
2. #20243076C2semidirect
4. #20241675C2semidirect
5. #20241287C2semidirect
6. #20241282C2semidirect
8. #15689254C4semidirect
9. Q8×C47⋊C8C4semidirect
10. #15689201C4semidirect
13. D4×C47⋊C8C4semidirect
29. C47×#330C4semidirect
30. C47×#329C4semidirect
196. #330Dic47semidirect
197. #329Dic47semidirect
202. C47#8249semidirect

Maximal subgroups

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

subgroupindex
#202523642
#202430762
#202418742
#202416752
#202412872
#202412822
C47×#15242
#824947