Finite Group Databasebeta

← #26135351

Subgroups — #26135351

Order 12032.

418 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689524C211yesmaximal, characteristic, complemented
2C2×#15688925C211yesmaximal, characteristic
3C2×#15688923C211yesmaximal, characteristic
4#20244252C211yesmaximal, characteristic
5#20241479C211yesmaximal, characteristic, complemented
6#20241465C211yesmaximal, characteristic, complemented
7C47×#1380C211yesmaximal, characteristic, complemented
8C22×D376C2211yescharacteristic
9C2×#13094015—21no—
10C22×C47⋊C16C2211yescharacteristic
11#15689524—22no—
12#15689524—22no—
13C2×#13093958C2211yescharacteristic
14C2×#13093955—21no—
15C2×#13093944—21no—
16C2×C8×Dic47C2211yescharacteristic
17#15689024—21no—
18#15688925—22no—
19#15688923—22no—
20C22×C16×C47C2211yescharacteristic
21C2×#107×C47C2211yescharacteristic
22C2×#107×C47C2211yescharacteristic
23C22×D188—21no—
24C2×D376—21no—
25C2×D376—21no—
26C2×D376—21no—
27C2×D376—21no—
28C2×D376—21no—
29C2×D376Q812yes—
30C22×C47⋊C8D411yescharacteristic
31#13094015—24no—
32#13094015—42no—
33C2×C4⋊Dic47—21no—
34C2×C4⋊Dic47—41no—
35C2×Dic47⋊C4—41no—
36C2×(C2×D47)⋊C4—41no—
37C2×(C2×D47)⋊C4—41no—
38C2×C4×Dic47D411yescharacteristic
39C2×C47⋊C16—21no—
40C2×C47⋊C16—21no—
41C2×C47⋊C16—21no—
42#13093958—21no—
43#13093958D412yes—
44#13093955—24no—
45#13093944—42no—
46C8×Dic47—21no—
47C8×Dic47—21no—
48C22×C8×C47C2311yescharacteristic
49C2×C4⋊C4×C47—21no—
50C2×C4⋊C4×C47—21no—
51C2×C16×C47—21no—
52C2×C16×C47—21no—
53C2×C16×C47—21no—
54#107×C47—21no—
55#107×C47—21no—
56#107×C47—21no—
57#107×C47D412yes—
58C23×D47—41no—
59C22×C4×C47—41no—
60C22×C4×C47—41no—
61C22×C4×C47C2×D411yescharacteristic
62C22×Dic47—41no—
63C22×Dic47D811yescharacteristic
64C22×Dic47D811yescharacteristic
65C2×D188—22no—
66C2×D188—41no—
67C2×D188—41no—
68C2×D188—41no—
69C2×D188—41no—
70C2×D188—41no—
71C2×C8×C47C2×D411yescharacteristic
72C2×C8×C47C2×D411yescharacteristic
73C2×C8×C47C2×Q811yescharacteristic
74C2×C8×C47#4111yescharacteristic
75C2×C8×C47#4111yescharacteristic
76C2×C8×C47#4111yescharacteristic
77C4⋊C4×C47—22no—
78C4⋊C4×C47—41no—
79C4⋊C4×C47—41no—
80C4⋊C4×C47—41no—
81(C2×D47)⋊C4—82no—
82(C2×D47)⋊C4—82no—
83C4⋊Dic47—22no—
84C4⋊Dic47—44no—
85C4⋊Dic47—41no—
86Dic47⋊C4—44no—
87C4×Dic47—21no—
88C4×Dic47—21no—
89C2×C47⋊C8—21no—
90C2×C47⋊C8—21no—
91C2×C47⋊C8SD1612yes—
92D376—22no—
93D376—22no—
94D376—41no—
95D376—41no—
96C16×C47—41no—
97C47⋊C16—41no—
98C23×C47C2×D811yescharacteristic
99C22×D47—42no—
100C22×D47—81no—
101C22×D47—81no—
102C22×D47—81no—
103C22×D47—81no—
104C22×D47—81no—
105C22×D47—81no—
106C2×C4×C47—42no—
107C2×C4×C47—81no—
108C2×C4×C47—81no—
109C2×C4×C47—81no—
110C2×C4×C47—81no—
111C2×C4×C47—81no—
112C2×C4×C47#12011yescharacteristic
113C2×C4×C47C22⋊Q811yescharacteristic
114C2×C4×C47#12411yescharacteristic
115C2×C4×C47C2×SD1611yescharacteristic
116C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
117C2×C4×C47#13711yescharacteristic
118C2×Dic47—21no—
119C2×Dic47—21no—
120C2×Dic47—21no—
121C2×Dic47—21no—
122C2×Dic47—21no—
123C2×Dic47—21no—
124C2×Dic47—42no—
125C2×Dic47—81no—
126C2×Dic47—81no—
127D188—42no—
128D188—42no—
129D188—81no—
130D188—81no—
131C8×C47C22⋊Q811yescharacteristic
132C8×C47C22⋊Q811yescharacteristic
133C8×C47#12411yescharacteristic
134C8×C47#12411yescharacteristic
135C47⋊C8—22no—
136#13365—471nomaximal, Sylow
137C22×C47#44711yescharacteristic
138C22×C47#47411yescharacteristic
139C22×C47#48611yescharacteristic
140C22×C47#50811yescharacteristic
141C22×C47#50811yescharacteristic
142C22×C47#50911yescharacteristic
143C22×C47#50911yescharacteristic
144C2×D47—82no—
145C2×D47—82no—
146C2×D47—82no—
147C2×D47—161no—
148C2×D47—161no—
149C2×D47—161no—
150C2×D47—161no—
151C4×C47—82no—
152C4×C47—161no—
153C4×C47#39311yescharacteristic
154C4×C47#44811yescharacteristic
155C4×C47#47611yescharacteristic
156C4×C47#48911yescharacteristic
157Dic47—41no—
158Dic47—41no—
159Dic47—82no—
160C2×#499—471no—
161C2×#357—471no—
162C2×#357—471no—
163#1970—471no—
164#1380—471no—
165#1380D4711yescharacteristic, complemented
166#1374—471no—
167C2×C47#202311yescharacteristic
168C2×C47#218611yescharacteristic
169C2×C47#218611yescharacteristic
170C2×C47#222811yescharacteristic
171C2×C47#222811yescharacteristic
172C2×C47#224211yescharacteristic
173C2×C47#224411yescharacteristic
174D47—162no—
175C22×D8—471no—
176C2×#125—941no—
177C22×C16—471no—
178C22×C16C2×D4711yescharacteristic
179#499—942no—
180#499—942no—
181C2×#107—471no—
182C2×#107C2×D4711yescharacteristic
183C2×#107C2×D4711yescharacteristic
184C2×#106—941no—
185C2×#102—941no—
186C2×C4×C8—471no—
187#390—941no—
188#357—942no—
189#357—942no—
190C47#1336511yescharacteristic, complemented, Sylow
191C22×D4—941no—
192C2×D8—472no—
193C2×D8—941no—
194C2×D8—941no—
195C2×D8—941no—
196C2×D8—941no—
197C2×D8—941no—
198C22×C8—471no—
199C22×C8C22×D4711yescharacteristic
200#125—944no—
201#125—1882no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—941no—
205C2×C4⋊C4—1881no—
206C2×C4⋊C4—1881no—
207C2×C22⋊C4—1881no—
208C2×C22⋊C4—1881no—
209C2×C42—471no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—941no—
214C2×C16—941no—
215C2×C16—941no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—472no—
220#107—941no—
221#107D18812yes—
222#106—944no—
223#102—1882no—
224C4×C8—941no—
225C4×C8—941no—
226C24—1881no—
227C2×D4—942no—
228C2×D4—1881no—
229C2×D4—1881no—
230C2×D4—1881no—
231C2×D4—1881no—
232C2×D4—1881no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—471no—
236C22×C4—471no—
237C22×C4—1881no—
238C22×C4D4×D4711yescharacteristic
239D8—942no—
240D8—942no—
241D8—1881no—
242D8—1881no—
243C2×C8—472no—
244C2×C8—941no—
245C2×C8—941no—
246C2×C8C2×D18811yescharacteristic
247C2×C8#1062535011yescharacteristic
248C2×C8#1062535011yescharacteristic
249C2×C8D4×D4711yescharacteristic
250C2×C8#1062535211yescharacteristic
251C2×C8Q8×D4711yescharacteristic
252C4⋊C4—22no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—942no—
257C4⋊C4—1884no—
258C4⋊C4—1884no—
259C4⋊C4—1881no—
260C22⋊C4—3762no—
261C22⋊C4—3762no—
262C42—941no—
263C42—941no—
264C16—41no—
265C16—1881no—
266C23—3761no—
267C23—3761no—
268C23—3761no—
269C23—3761no—
270C23—3761no—
271C23—3761no—
272C23—1882no—
273C23D8×D4711yescharacteristic
274D4—3761no—
275D4—3761no—
276D4—1882no—
277D4—1882no—
278C2×C4—42no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—941no—
285C2×C4—941no—
286C2×C4—941no—
287C2×C4—941no—
288C2×C4—941no—
289C2×C4—941no—
290C2×C4—3761no—
291C2×C4—3761no—
292C2×C4—1882no—
293C2×C4#1309399411yescharacteristic
294C2×C4#1309400211yescharacteristic
295C2×C4#1309401011yescharacteristic
296C2×C4SD16×D4711yescharacteristic
297C2×C4#1309405111yescharacteristic
298C2×C4#1309405611yescharacteristic
299C8—942no—
300C8#1309400211yescharacteristic
301C8#1309400411yescharacteristic
302C8#1309401011yescharacteristic
303C8#1309401211yescharacteristic
304C22—3762no—
305C22—3762no—
306C22—3762no—
307C22—7521no—
308C22—7521no—
309C22—7521no—
310C22—7521no—
311C22#1568925811yescharacteristic
312C22#1568940211yescharacteristic
313C22#1568946811yescharacteristic
314C22#1568955411yescharacteristic
315C22#1568955611yescharacteristic
316C22#1568955711yescharacteristic
317C22#1568956011yescharacteristic
318C4—82no—
319C4—161no—
320C4—3762no—
321C4—1881no—
322C4—1881no—
323C4#1568903611yescharacteristic
324C4#1568926311yescharacteristic
325C4#1568941211yescharacteristic
326C4#1568948411yescharacteristic
327C2—7522no—
328C2#2024452311yescharacteristic
329C2#2024531211yescharacteristic
330C2#2024531411yescharacteristic
331C2#2024554411yescharacteristic
332C2#2024554811yescharacteristic
333C2#2024562111yescharacteristic
334C2#2024562811yescharacteristic
subgroupG/H
centerC23D8×D47
commutator, cosocleC22×C8×C47C23
FrattiniC22×C8C22×D47
FittingC47×#1380C2
socleC23×C47C2×D8
prime radicalC22×D376C22
hypercenter#1380D47
hypocenterC47#13365
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
1. C2×#15689524C2semidirect
5. #20241479C2semidirect
6. #20241465C2semidirect
7. C47×#1380C2semidirect
165. #1380D47semidirect
190. C47#13365semidirect

Maximal subgroups

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

subgroupindex
C2×#156895242
C2×#156889252
C2×#156889232
#202442522
#202414792
#202414652
C47×#13802
#1336547