Finite Group Databasebeta

← #26198821

Subgroups — #26198821

Order 12032.

662 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253902C211yesmaximal, characteristic, complemented
2#20253397C212yesmaximal
3C2×#15689202C211yesmaximal, characteristic
4#20242912C212yesmaximal, complemented
5#20242902C212yesmaximal, complemented
6#20242888C214yesmaximal, complemented
7#20242043C212yesmaximal, complemented
8C47×#1564C211yesmaximal, characteristic, complemented
9C22×#10625324—21no—
10#15689680C2212yes—
11C2×C4×D188C2211yescharacteristic
12#15689528C2211yescharacteristic, complemented
13#15689528C2211yescharacteristic, complemented
14#15689525—21no—
15#15689525—21no—
16#15689521C2212yescomplemented
17#15689454—22no—
18#15689414C2214yes—
19#15689329C2214yes—
20#15689276—22no—
21C4×#10625325—24no—
22#15689202—21no—
23#15689202—21no—
24#15689202C2212yes—
25#15689202C2212yes—
26#15689161—21no—
27#15689161—21no—
28C2×C4⋊(C47⋊C8)C2211yescharacteristic
29C2×(C2×D47)⋊C8—21no—
30#15688871C2214yescomplemented
31#15688870C2214yescomplemented
32C47×#431C2212yes—
33C2×C4×C8×C47C2211yescharacteristic
34C47×#335C2212yescomplemented
35C47×#334C2212yescomplemented
36C22×D188D411yescharacteristic
37C22×C4×D47—21no—
38C2×#10625325—24no—
39C2×#10625324—22no—
40C2×#10625324—22no—
41C2×#10625324—44no—
42C22×C47⋊C8—21no—
43#13094024C2312yes—
44#13094024C2312yes—
45#13094016—22no—
46#13094003—22no—
47#13094003—22no—
48C4×Dic94—22no—
49C4×Dic94—22no—
50C4×D188—21no—
51C4×D188—21no—
52C4×D188D412yes—
53C4×D188C2312yes—
54#13093983—22no—
55C2×C4⋊Dic47D411yescharacteristic
56C2×(C2×D47)⋊C4—21no—
57#13093958—21no—
58#13093958—21no—
59#13093958—21no—
60#13093958—21no—
61#13093956—22no—
62#13093956—22no—
63#13093956—24no—
64C4⋊(C47⋊C8)—21no—
65C4⋊(C47⋊C8)C2312yes—
66#13093949—24no—
67#13093946—24no—
68(C2×D47)⋊C8—22no—
69(C2×D47)⋊C8—22no—
70C22×C8×C47D411yescharacteristic
71C22×C8×C47D411yescharacteristic
72C2×C42×C47C2311yescharacteristic
73C4⋊C8×C47C2314yes—
74C22⋊C8×C47—22no—
75C4×C8×C47—22no—
76C4×C8×C47D411yescharacteristic
77C4×C8×C47D411yescharacteristic
78C4×C8×C47C2311yescharacteristic
79C4×C8×C47C2311yescharacteristic
80C23×D47—21no—
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47#4111yescharacteristic
84C22×Dic47—21no—
85C2×D188—21no—
86C2×D188—42no—
87C2×D188C2×D412yes—
88C2×C4×D47—22no—
89C2×C4×D47—22no—
90C2×C4×D47—42no—
91C2×C4×D47—42no—
92C2×Dic94—22no—
93C2×Dic94—22no—
94C2×C8×C47—22no—
95C2×C8×C47—24no—
96C2×C8×C47SD1614yescomplemented
97C2×C8×C47C2×D411yescharacteristic
98C2×C8×C47C2×D411yescharacteristic
99C2×C8×C47C2×D411yescharacteristic
100C2×C8×C47C2×D411yescharacteristic
101C42×C47C2×D411yescharacteristic
102C42×C47C2×D411yescharacteristic
103C42×C47#4111yescharacteristic
104C42×C47C2411yescharacteristic
105C22⋊Dic47—42no—
106(C2×D47)⋊C4—22no—
107(C2×D47)⋊C4—22no—
108C4⋊Dic47—21no—
109C4⋊Dic47—22no—
110C4⋊Dic47—22no—
111C4⋊Dic47—22no—
112C4⋊Dic47—22no—
113C4⋊Dic47C2×D412yes—
114Dic47⋊C4—42no—
115Dic47⋊C4—42no—
116C4×Dic47—42no—
117C2×C47⋊C8—22no—
118C2×C47⋊C8—42no—
119#10625325—48no—
120#10625324—44no—
121#10625324—44no—
122C23×C47#12111yescharacteristic
123C22×D47—22no—
124C22×D47—42no—
125C22×D47—42no—
126C22×D47—42no—
127C2×C4×C47—22no—
128C2×C4×C47—41no—
129C2×C4×C47#12111yescharacteristic
130C2×C4×C47#12111yescharacteristic
131C2×C4×C47#12111yescharacteristic
132C2×C4×C47C2×SD1612yes—
133C2×C4×C47#13712yes—
134C2×C4×C47C22×D411yescharacteristic
135C2×C4×C47C22×D411yescharacteristic
136C2×C4×C47C2×#4111yescharacteristic
137C2×Dic47—22no—
138C2×Dic47—42no—
139C2×Dic47—42no—
140C2×Dic47—42no—
141D188—24no—
142D188—42no—
143C4×D47—42no—
144C4×D47—42no—
145C4×D47—44no—
146Dic94—44no—
147Dic94—82no—
148C8×C47—21no—
149C8×C47—21no—
150C8×C47—44no—
151C8×C47C2×SD1612yes—
152C8×C47C2×SD1612yes—
153C47⋊C8—42no—
154#19137—471nomaximal, Sylow
155C22×C47—21no—
156C22×C47—21no—
157C22×C47#46112yes—
158C22×C47C2×#12111yescharacteristic
159C2×D47—42no—
160C2×D47—42no—
161C2×D47—42no—
162C2×D47—84no—
163C4×C47—21no—
164C4×C47—21no—
165C4×C47—21no—
166C4×C47#46112yes—
167C4×C47C22×SD1611yescharacteristic
168C4×C47C2×#13711yescharacteristic
169C4×C47#57512yes—
170Dic47—42no—
171Dic47—82no—
172Dic47—82no—
173#3160—471no—
174#3101—472no—
175C2×#436—471no—
176#1695—472no—
177#1693—472no—
178#1691—474no—
179#1564D4711yescharacteristic, complemented
180#1542—472no—
181C2×C47—22no—
182C2×C47C2×#46111yescharacteristic
183C2×C47#305411yescharacteristic
184C2×C47#306711yescharacteristic
185D47—82no—
186C22×M16—941no—
187#533—472no—
188C2×C4×D4—471no—
189#500—471no—
190#500—471no—
191#499—941no—
192#499—941no—
193#498—472no—
194#483—942no—
195#476—474no—
196#461—474no—
197#450—942no—
198C4×SD16—944no—
199#436—472no—
200#436—472no—
201#436—941no—
202#436—941no—
203#431C2×D4712yes—
204#427—941no—
205#427—941no—
206C2×C4⋊C8—471no—
207C2×C22⋊C8—941no—
208C2×C4×C8C2×D4711yescharacteristic
209#335C2×D4712yescomplemented
210#334C2×D4712yescomplemented
211#333—474no—
212#332—474no—
213C47#1913711yescharacteristic, complemented, Sylow
214C22×D4—471no—
215C23×C4—941no—
216C2×SD16—944no—
217C2×M16—942no—
218C2×M16—942no—
219C2×M16—1884no—
220C22×C8—941no—
221C22×C8D18811yescharacteristic
222C22×C8D18811yescharacteristic
223C4⋊Q8—472no—
224C4⋊Q8—472no—
225#125—942no—
226C22⋊Q8—942no—
227C22⋊Q8—942no—
228C4×Q8—942no—
229C4×Q8—942no—
230C4×D4—472no—
231C4×D4—472no—
232C4×D4—941no—
233C4×D4—941no—
234#117—942no—
235C2×C4⋊C4—471no—
236C2×C22⋊C4—941no—
237C2×C42C22×D4711yescharacteristic
238#107—941no—
239#107—941no—
240#107—941no—
241#107—941no—
242#106—944no—
243#106—942no—
244#106—942no—
245C4⋊C8—472no—
246C4⋊C8—941no—
247C4⋊C8C22×D4714yes—
248#103—944no—
249#102—944no—
250C22⋊C8—22no—
251C22⋊C8—942no—
252C22⋊C8—942no—
253C4×C8—22no—
254C4×C8D18811yescharacteristic
255C4×C8D18811yescharacteristic
256C4×C8C22×D4711yescharacteristic
257C4×C8C22×D4711yescharacteristic
258C24—941no—
259C2×Q8—942no—
260C2×Q8—942no—
261C2×D4—472no—
262C2×D4—941no—
263C2×D4—1882no—
264C22×C4—941no—
265C22×C4—942no—
266C22×C4—942no—
267C22×C4—1882no—
268C22×C4—1882no—
269C22×C4C2×D18811yescharacteristic
270C22×C4D4×D4711yescharacteristic
271C22×C4#1062535411yescharacteristic
272SD16—1888no—
273M16—1884no—
274M16—1884no—
275C2×C8—22no—
276C2×C8—24no—
277C2×C8—942no—
278C2×C8—1882no—
279C2×C8#1062532514yescomplemented
280C2×C8C2×D18811yescharacteristic
281C2×C8C2×D18811yescharacteristic
282C2×C8C2×D18811yescharacteristic
283C2×C8C2×D18811yescharacteristic
284C4⋊C4—472no—
285C4⋊C4—941no—
286C4⋊C4—942no—
287C4⋊C4—942no—
288C4⋊C4—942no—
289C4⋊C4—942no—
290C4⋊C4—1882no—
291C4⋊C4—1882no—
292C22⋊C4—942no—
293C22⋊C4—942no—
294C22⋊C4—1882no—
295C42—1882no—
296C42C2×D18811yescharacteristic
297C42D4×D4711yescharacteristic
298C42#1062535411yescharacteristic
299C42C23×D4711yescharacteristic
300C23—942no—
301C23—1882no—
302C23—1882no—
303C23—1882no—
304C23#1309399911yescharacteristic
305Q8—3762no—
306Q8—1884no—
307D4—944no—
308D4—1882no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—942no—
312C2×C4—1882no—
313C2×C4—1882no—
314C2×C4—1882no—
315C2×C4—1884no—
316C2×C4—1882no—
317C2×C4—1882no—
318C2×C4#1309399911yescharacteristic
319C2×C4#1309399911yescharacteristic
320C2×C4#1309399911yescharacteristic
321C2×C4C2×#1062532512yes—
322C2×C4#1309405712yes—
323C2×C4C2×D4×D4711yescharacteristic
324C2×C4C22×D18811yescharacteristic
325C2×C4C2×#1062535411yescharacteristic
326C8—21no—
327C8—21no—
328C8—44no—
329C8—1882no—
330C8C2×#1062532512yes—
331C8C2×#1062532512yes—
332C22—21no—
333C22—21no—
334C22—3764no—
335C22—1882no—
336C22—1882no—
337C22—1882no—
338C22#1568932912yes—
339C22C2×#1309399911yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—3762no—
344C4—3762no—
345C4—1882no—
346C4#1568932912yes—
347C4C22×#1062532511yescharacteristic
348C4C2×#1309405711yescharacteristic
349C4#1568992012yes—
350C2—22no—
351C2—3762no—
352C2C2×#1568932911yescharacteristic
353C2#2025299811yescharacteristic
354C2#2025310911yescharacteristic
subgroupG/H
centerC2×C4#13093999
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#1564C2
socleC22×C47C2×#121
prime radicalC22×D188D4
hypercenter#1564D47
hypocenterC47#19137
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1564

Complements

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

NH ≅ G/Ntype
1. #20253902C2semidirect
4. #20242912C2semidirect
5. #20242902C2semidirect
6. #20242888C2semidirect
7. #20242043C2semidirect
8. C47×#1564C2semidirect
12. #15689528C22semidirect
13. #15689528C22semidirect
16. #15689521C22semidirect
30. #15688871C22semidirect
31. #15688870C22semidirect
34. C47×#335C22semidirect
35. C47×#334C22semidirect
96. C2×C8×C47SD16semidirect
179. #1564D47semidirect
209. #335C2×D47semidirect
210. #334C2×D47semidirect
213. C47#19137semidirect
279. C2×C8#10625325semidirect

Maximal subgroups

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

subgroupindex
#202539022
#202533972
C2×#156892022
#202429122
#202429022
#202428882
#202420432
C47×#15642
#1913747