Finite Group Databasebeta

← #26136072

Subgroups — #26136072

Order 12032.

378 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688925C211yesmaximal, characteristic
2C2×#15688923C211yesmaximal, characteristic
3#20244252C211yesmaximal, characteristic
4#20243546C211yesmaximal, characteristic, complemented
5#20241479C211yesmaximal, characteristic, complemented
6#20241465C211yesmaximal, characteristic, complemented
7C47×#1380C211yesmaximal, characteristic, complemented
8C22×D376C2211yescharacteristic
9C22×C47⋊C16C2211yescharacteristic
10C2×#13093958C2211yescharacteristic
11C2×#13093957—21no—
12C2×#13093944—21no—
13C2×C8×Dic47C2211yescharacteristic
14#15689024—21no—
15#15688992—21no—
16#15688925—22no—
17#15688923—22no—
18C22×C16×C47C2211yescharacteristic
19C2×#107×C47C2211yescharacteristic
20C2×#107×C47C2211yescharacteristic
21C22×D188—21no—
22C2×D376—21no—
23C2×D376—21no—
24C2×D376—21no—
25C2×D376—21no—
26C2×D376—21no—
27C2×D376D412yes—
28C22×C47⋊C8D411yescharacteristic
29C2×C4⋊Dic47—21no—
30C2×Dic47⋊C4—41no—
31C2×Dic47⋊C4—41no—
32C2×(C2×D47)⋊C4—41no—
33C2×(C2×D47)⋊C4—41no—
34C2×C4×Dic47D411yescharacteristic
35C2×C47⋊C16—21no—
36C2×C47⋊C16—21no—
37C2×C47⋊C16—21no—
38#13093958—21no—
39#13093958—21no—
40#13093957—42no—
41#13093944—42no—
42C8×Dic47—21no—
43C8×Dic47—21no—
44C22×C8×C47C2311yescharacteristic
45C2×C4⋊C4×C47—21no—
46C2×C4⋊C4×C47—21no—
47C2×C16×C47—21no—
48C2×C16×C47—21no—
49C2×C16×C47—21no—
50#107×C47—21no—
51#107×C47—21no—
52#107×C47—21no—
53#107×C47—21no—
54C23×D47—41no—
55C22×C4×C47—41no—
56C22×C4×C47—41no—
57C22×C4×C47C2×D411yescharacteristic
58C22×Dic47—41no—
59C22×Dic47D811yescharacteristic
60C22×Dic47D811yescharacteristic
61C2×D188—22no—
62C2×D188—41no—
63C2×D188—41no—
64C2×D188—41no—
65C2×D188—41no—
66C2×D188—41no—
67C2×C8×C47C2×D411yescharacteristic
68C2×C8×C47#4111yescharacteristic
69C2×C8×C47#4111yescharacteristic
70C2×C8×C47#4111yescharacteristic
71C2×C8×C47#4111yescharacteristic
72C2×C8×C47#4111yescharacteristic
73C4⋊C4×C47—41no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—41no—
76C4⋊C4×C47—41no—
77(C2×D47)⋊C4—82no—
78(C2×D47)⋊C4—82no—
79C4⋊Dic47—41no—
80C4⋊Dic47—41no—
81Dic47⋊C4—82no—
82Dic47⋊C4—82no—
83C4×Dic47—21no—
84C4×Dic47—21no—
85C2×C47⋊C8—21no—
86C2×C47⋊C8—21no—
87C2×C47⋊C8SD1612yes—
88D376—22no—
89D376—22no—
90D376—41no—
91D376—41no—
92C16×C47—41no—
93C47⋊C16—41no—
94C23×C47C2×D811yescharacteristic
95C22×D47—42no—
96C22×D47—81no—
97C22×D47—81no—
98C22×D47—81no—
99C22×D47—81no—
100C22×D47—81no—
101C22×D47—81no—
102C2×C4×C47—81no—
103C2×C4×C47—81no—
104C2×C4×C47—81no—
105C2×C4×C47—81no—
106C2×C4×C47—81no—
107C2×C4×C47—81no—
108C2×C4×C47#12111yescharacteristic
109C2×C4×C47#12311yescharacteristic
110C2×C4×C47#12411yescharacteristic
111C2×C4×C47C2×SD1611yescharacteristic
112C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
113C2×C4×C47#13711yescharacteristic
114C2×Dic47—21no—
115C2×Dic47—21no—
116C2×Dic47—21no—
117C2×Dic47—21no—
118C2×Dic47—81no—
119C2×Dic47—81no—
120C2×Dic47—81no—
121C2×Dic47D1612yes—
122C2×Dic47SD3212yes—
123D188—42no—
124D188—42no—
125D188—81no—
126D188—81no—
127C8×C47#12311yescharacteristic
128C8×C47#12311yescharacteristic
129C8×C47#12611yescharacteristic
130C8×C47#12611yescharacteristic
131C47⋊C8—22no—
132#13480—471nomaximal, Sylow
133C22×C47#45911yescharacteristic
134C22×C47#48011yescharacteristic
135C22×C47#48611yescharacteristic
136C22×C47C2×D1611yescharacteristic
137C22×C47C2×SD3211yescharacteristic
138C22×C47#50911yescharacteristic
139C22×C47#51011yescharacteristic
140C2×D47—82no—
141C2×D47—82no—
142C2×D47—82no—
143C2×D47—161no—
144C2×D47—161no—
145C2×D47—161no—
146C2×D47—161no—
147C4×C47—161no—
148C4×C47—161no—
149C4×C47#39711yescharacteristic
150C4×C47#46111yescharacteristic
151C4×C47#48111yescharacteristic
152C4×C47#48911yescharacteristic
153Dic47—22no—
154Dic47—22no—
155Dic47—161no—
156C2×#357—471no—
157C2×#357—471no—
158#1970—471no—
159#1845—471no—
160#1380—471no—
161#1380D4711yescharacteristic, complemented
162#1374—471no—
163C2×C47#207111yescharacteristic
164C2×C47#220611yescharacteristic
165C2×C47#220811yescharacteristic
166C2×C47#223211yescharacteristic
167C2×C47#223311yescharacteristic
168C2×C47#224011yescharacteristic
169C2×C47#224311yescharacteristic
170D47—162no—
171C22×D8—471no—
172C22×C16—471no—
173C22×C16C2×D4711yescharacteristic
174C2×#107—471no—
175C2×#107—941no—
176C2×#107C2×D4711yescharacteristic
177C2×#107C2×D4711yescharacteristic
178C2×#102—941no—
179C2×C4×C8—471no—
180#390—941no—
181#384—941no—
182#357—942no—
183#357—942no—
184C47#1348011yescharacteristic, complemented, Sylow
185C22×D4—941no—
186C2×D8—472no—
187C2×D8—941no—
188C2×D8—941no—
189C2×D8—941no—
190C2×D8—941no—
191C2×D8—941no—
192C22×C8—471no—
193C22×C8C22×D4711yescharacteristic
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—941no—
197C2×C4⋊C4—1881no—
198C2×C4⋊C4—1881no—
199C2×C22⋊C4—1881no—
200C2×C22⋊C4—1881no—
201C2×C42—471no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—941no—
206C2×C16—941no—
207C2×C16—941no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—941no—
213#107—941no—
214#107—1882no—
215#102—1882no—
216C4×C8—941no—
217C4×C8—941no—
218C24—1881no—
219C2×D4—942no—
220C2×D4—1881no—
221C2×D4—1881no—
222C2×D4—1881no—
223C2×D4—1881no—
224C2×D4—1881no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—471no—
228C22×C4—471no—
229C22×C4—1881no—
230C22×C4D4×D4711yescharacteristic
231D8—942no—
232D8—942no—
233D8—1881no—
234D8—1881no—
235C2×C8—472no—
236C2×C8—941no—
237C2×C8—941no—
238C2×C8#1062535011yescharacteristic
239C2×C8#1062535011yescharacteristic
240C2×C8#1062535011yescharacteristic
241C2×C8D4×D4711yescharacteristic
242C2×C8#1062535211yescharacteristic
243C2×C8#1062535411yescharacteristic
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—3762no—
249C4⋊C4—3762no—
250C4⋊C4—1881no—
251C4⋊C4—1881no—
252C22⋊C4—3762no—
253C22⋊C4—3762no—
254C42—941no—
255C42—941no—
256C16—41no—
257C16—1881no—
258C23—3761no—
259C23—3761no—
260C23—3761no—
261C23—3761no—
262C23—3761no—
263C23—3761no—
264C23—1882no—
265C23D8×D4711yescharacteristic
266D4—3761no—
267D4—3761no—
268D4—1882no—
269D4—1882no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—472no—
277C2×C4—472no—
278C2×C4—941no—
279C2×C4—941no—
280C2×C4—941no—
281C2×C4—941no—
282C2×C4—3761no—
283C2×C4—3761no—
284C2×C4—3761no—
285C2×C4#1309399711yescharacteristic
286C2×C4#1309400911yescharacteristic
287C2×C4#1309401011yescharacteristic
288C2×C4SD16×D4711yescharacteristic
289C2×C4#1309405111yescharacteristic
290C2×C4#1309405611yescharacteristic
291C8—942no—
292C8#1309400611yescharacteristic
293C8#1309400911yescharacteristic
294C8#1309401811yescharacteristic
295C8#1309401911yescharacteristic
296C22—3762no—
297C22—3762no—
298C22—3762no—
299C22—7521no—
300C22—7521no—
301C22—7521no—
302C22—7521no—
303C22#1568931711yescharacteristic
304C22#1568943811yescharacteristic
305C22#1568946811yescharacteristic
306C22D16×D4711yescharacteristic
307C22SD32×D4711yescharacteristic
308C22#1568956011yescharacteristic
309C22#1568956511yescharacteristic
310C4—161no—
311C4—161no—
312C4—942no—
313C4—942no—
314C4—7521no—
315C4#1568905211yescharacteristic
316C4#1568932711yescharacteristic
317C4#1568944311yescharacteristic
318C4#1568948411yescharacteristic
319C2—7522no—
320C2#2024479511yescharacteristic
321C2#2024542211yescharacteristic
322C2#2024543211yescharacteristic
323C2#2024557011yescharacteristic
324C2#2024557511yescharacteristic
325C2#2024561011yescharacteristic
326C2#2024562611yescharacteristic
subgroupG/H
centerC23D8×D47
commutator, cosocleC22×C8×C47C23
FrattiniC22×C8C22×D47
FittingC47×#1380C2
socleC23×C47C2×D8
prime radicalC22×D376C22
hypercenter#1380D47
hypocenterC47#13480
LCSG ▷ C22×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
4. #20243546C2semidirect
5. #20241479C2semidirect
6. #20241465C2semidirect
7. C47×#1380C2semidirect
161. #1380D47semidirect
184. C47#13480semidirect

Maximal subgroups

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

subgroupindex
C2×#156889252
C2×#156889232
#202442522
#202435462
#202414792
#202414652
C47×#13802
#1348047