Finite Group Databasebeta

← #26107336

Subgroups — #26107336

Order 12032.

400 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243139C211yesmaximal, characteristic, complemented
2#20241976C211yesmaximal, characteristic
3#20241874C211yesmaximal, characteristic
4C8×#10625329C211yesmaximal, characteristic, complemented
5#20241286C211yesmaximal, characteristic, complemented
6#20241280C211yesmaximal, characteristic, complemented
7C47×#2966C211yesmaximal, characteristic
8#15689176C2211yescharacteristic
9C4×#10625329C2211yescharacteristic
10C2×#13093935C2211yescharacteristic
11#15688969C2211yescharacteristic
12#15688877—21no—
13#15688877—21no—
14#15688865C412yescomplemented
15#15688863C412yescomplemented
16#15688860C412yes—
17#15688857C412yes—
18#15688845—21no—
19C8×C47⋊C8—21no—
20C47×#529C2211yescharacteristic
21C47×#446C411yescharacteristic, complemented
22C8×Q8×C47C411yescharacteristic, complemented
23C47×#436C411yescharacteristic, complemented
24C47×#435—21no—
25C8×D4×C47—21no—
26C8×D4×C47C411yescharacteristic, complemented
27C47×#433—21no—
28C47×#432C2211yescharacteristic
29C47×#405—21no—
30C2×C4×C8×C47C2211yescharacteristic
31C2×#10625329—21no—
32C22×C47⋊C8—21no—
33C2×C4×Dic47—21no—
34C4⋊(C47⋊C8)—21no—
35C22⋊(C47⋊C8)C2×C412yes—
36#13093935—21no—
37#13093935C2×C412yes—
38#13093935C2×C412yes—
39C4×C47⋊C8—21no—
40C4×C47⋊C8—21no—
41C47⋊#95—21no—
42C47⋊#95C2×C412yes—
43C2×M16×C47—21no—
44C22×C8×C47—21no—
45C22×C8×C47D411yescharacteristic
46C22×C8×C47D411yescharacteristic
47#126×C47—21no—
48#125×C47C2×C411yescharacteristic
49#124×C47C2×C411yescharacteristic
50#123×C47—21no—
51C22⋊Q8×C47C2×C411yescharacteristic
52#121×C47C2×C411yescharacteristic
53C4×Q8×C47C2×C411yescharacteristic
54C4×D4×C47—21no—
55C4×D4×C47C2×C411yescharacteristic
56#117×C47—21no—
57C2×C42×C47C2311yescharacteristic
58C4⋊C8×C47—21no—
59C4⋊C8×C47—21no—
60C4⋊C8×C47C2×C411yescharacteristic
61C4⋊C8×C47C2×C411yescharacteristic
62C22⋊C8×C47—21no—
63C22⋊C8×C47—21no—
64C22⋊C8×C47—21no—
65#97×C47—21no—
66#97×C47—21no—
67C4×C8×C47—21no—
68C4×C8×C47C2×C411yescharacteristic
69C4×C8×C47C2×C411yescharacteristic
70C4×C8×C47D411yescharacteristic
71C4×C8×C47D411yescharacteristic
72C2×Q8×C47C4211yescharacteristic
73C2×D4×C47—21no—
74C2×D4×C47C4211yescharacteristic
75C22×C4×C47—21no—
76C22×C4×C47C22×C411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47#4111yescharacteristic
79C22×Dic47—22no—
80M16×C47—42no—
81C2×C8×C47—21no—
82C2×C8×C47—21no—
83C2×C8×C47—41no—
84C2×C8×C47—41no—
85C2×C8×C47—42no—
86C2×C8×C47C22⋊C411yescharacteristic
87C2×C8×C47C22⋊C411yescharacteristic
88C2×C8×C47C22⋊C411yescharacteristic
89C2×C8×C47C22⋊C411yescharacteristic
90C4⋊C4×C47—21no—
91C4⋊C4×C47—21no—
92C4⋊C4×C47—21no—
93C4⋊C4×C47—21no—
94C4⋊C4×C47C4211yescharacteristic
95C4⋊C4×C47C4211yescharacteristic
96C22⋊C4×C47—21no—
97C22⋊C4×C47—21no—
98C22⋊C4×C47—21no—
99C22⋊C4×C47—21no—
100C22⋊C4×C47—21no—
101C42×C47—21no—
102C42×C47C22×C411yescharacteristic
103C42×C47C22×C411yescharacteristic
104C42×C47C2×D411yescharacteristic
105C42×C47#4111yescharacteristic
106C4×Dic47—41no—
107C4×Dic47—41no—
108#10625329—24no—
109C2×C47⋊C8—22no—
110C2×C47⋊C8—22no—
111C2×C47⋊C8—22no—
112C2×C47⋊C8—41no—
113C2×C47⋊C8—41no—
114C23×C47—21no—
115C23×C47C4×D411yescharacteristic
116Q8×C47—41no—
117D4×C47—41no—
118D4×C47—42no—
119C2×C4×C47—22no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47C2×C4211yescharacteristic
129C2×C4×C47C2×C22⋊C411yescharacteristic
130C2×C4×C47#11711yescharacteristic
131C2×C4×C47C4×D411yescharacteristic
132C2×C4×C47C4×D411yescharacteristic
133C2×C4×C47C4×D411yescharacteristic
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—42no—
137C8×C47—21no—
138C8×C47—21no—
139C8×C47—21no—
140C8×C47—21no—
141C8×C47—41no—
142C8×C47—41no—
143C47⋊C8—24no—
144C47⋊C8—42no—
145C47⋊C8—42no—
146#8250—471nomaximal, Sylow
147C22×C47—21no—
148C22×C47—21no—
149C22×C47—41no—
150C22×C47—41no—
151C22×C47—41no—
152C22×C47C4×C22⋊C411yescharacteristic
153C22×C47#44312yes—
154C4×C47—21no—
155C4×C47—21no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47—21no—
159C4×C47—41no—
160C4×C47—41no—
161C4×C47—41no—
162C4×C47#42111yescharacteristic
163C4×C47#42111yescharacteristic
164Dic47—82no—
165#2966D4711yescharacteristic
166#1746—471no—
167#1524—471no—
168#1500—471no—
169C8×M16—471no—
170#1302—471no—
171#1300—471no—
172C2×C47—22no—
173C2×C47—81no—
174C2×C47#175711yescharacteristic
175C2×C47#175911yescharacteristic
176C2×C47#176411yescharacteristic
177#529C2×D4711yescharacteristic
178#446Dic4711yescharacteristic, complemented
179C8×Q8Dic4711yescharacteristic, complemented
180#436Dic4711yescharacteristic, complemented
181#435—21no—
182C8×D4—21no—
183C8×D4Dic4711yescharacteristic, complemented
184#433—21no—
185#432C2×D4711yescharacteristic
186#431—471no—
187#405—21no—
188C4×M16—471no—
189C2×#97—471no—
190C2×C4×C8C2×D4711yescharacteristic
191#376—471no—
192#336—941no—
193#336—941no—
194#330—472no—
195#329—472no—
196#328—472no—
197#327—472no—
198#322—941no—
199C82—941no—
200C47#825011yescharacteristic, complemented, Sylow
201C2×M16—21no—
202C2×M16—941no—
203C22×C8—21no—
204C22×C8—941no—
205C22×C8C22⋊D4711yescharacteristic
206C22×C8C22⋊D4711yescharacteristic
207#126—21no—
208#125C4×D4711yescharacteristic
209#124C4×D4711yescharacteristic
210#123—21no—
211C22⋊Q8C4×D4711yescharacteristic
212#121C4×D4711yescharacteristic
213C4×Q8C2×Dic4711yescharacteristic
214C4×D4—21no—
215C4×D4C2×Dic4711yescharacteristic
216#117—21no—
217C2×C42—941no—
218C2×C42C22×D4711yescharacteristic
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—941no—
222C4⋊C8C2×Dic4711yescharacteristic
223C4⋊C8C2×Dic4711yescharacteristic
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—472no—
228#97—21no—
229#97—21no—
230#97—472no—
231#97—472no—
232#97—941no—
233C4×C8—21no—
234C4×C8—941no—
235C4×C8—941no—
236C4×C8C2×Dic4711yescharacteristic
237C4×C8C2×Dic4711yescharacteristic
238C4×C8C22⋊D4711yescharacteristic
239C4×C8C22⋊D4711yescharacteristic
240#95—472no—
241#95—941no—
242C2×Q8C4×Dic4711yescharacteristic
243C2×D4—21no—
244C2×D4C4×Dic4711yescharacteristic
245C22×C4—21no—
246C22×C4—942no—
247C22×C4C2×C4×D4711yescharacteristic
248C22×C4#1062535011yescharacteristic
249C22×C4C2×C22⋊D4711yescharacteristic
250M16—42no—
251M16—944no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—42no—
257C2×C8—942no—
258C2×C8—942no—
259C2×C8—942no—
260C2×C8—1881no—
261C2×C8—1881no—
262C2×C8C22⋊Dic4711yescharacteristic
263C2×C8C22⋊Dic4711yescharacteristic
264C2×C8C22⋊Dic4711yescharacteristic
265C2×C8C22⋊Dic4711yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4C4×Dic4711yescharacteristic
271C4⋊C4C4×Dic4711yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C42—21no—
278C42—1881no—
279C42—1881no—
280C42C2×C4×D4711yescharacteristic
281C42#1062535011yescharacteristic
282C42C22×Dic4711yescharacteristic
283C42C2×C22⋊D4711yescharacteristic
284C23—21no—
285C23C4×C22⋊D4711yescharacteristic
286Q8—41no—
287D4—41no—
288D4—42no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—1882no—
299C2×C4—1882no—
300C2×C4—1882no—
301C2×C4C2×C4×Dic4711yescharacteristic
302C2×C4C2×C22⋊Dic4711yescharacteristic
303C2×C4#1309398311yescharacteristic
304C2×C4C4×C22⋊D4711yescharacteristic
305C2×C4C4×C22⋊D4711yescharacteristic
306C2×C4C4×C22⋊D4711yescharacteristic
307C8—21no—
308C8—21no—
309C8—21no—
310C8—21no—
311C8—41no—
312C8—41no—
313C8—944no—
314C8—1882no—
315C8—1882no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22C4×C22⋊Dic4711yescharacteristic
322C22#1568923812yes—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—3762no—
332C4#1568914011yescharacteristic
333C4#1568914011yescharacteristic
334C2—22no—
335C2—81no—
336C2#2024318811yescharacteristic
337C2#2024319811yescharacteristic
338C2#2024322311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊Dic47
commutatorC2×C4×C47C2×C42
FrattiniC2×C42C22×D47
FittingC47×#2966C2
socleC22×C47C4×C22⋊C4
prime radical#121×C47C2×C4
cosocleC2×C42×C47C23
hypercenter#2966D47
hypocenterC47#8250
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #2966

Complements

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

NH ≅ G/Ntype
1. #20243139C2semidirect
4. C8×#10625329C2semidirect
5. #20241286C2semidirect
6. #20241280C2semidirect
14. #15688865C4semidirect
15. #15688863C4semidirect
21. C47×#446C4semidirect
22. C8×Q8×C47C4semidirect
23. C47×#436C4semidirect
26. C8×D4×C47C4semidirect
178. #446Dic47semidirect
179. C8×Q8Dic47semidirect
180. #436Dic47semidirect
183. C8×D4Dic47semidirect
200. C47#8250semidirect

Maximal subgroups

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

subgroupindex
#202431392
#202419762
#202418742
C8×#106253292
#202412862
#202412802
C47×#29662
#825047