Finite Group Databasebeta

← #26214515

Subgroups — #26214515

Order 12032.

906 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×#15689184C211yesmaximal, characteristic
2#20245886C211yesmaximal, characteristic
3#20243260C212yesmaximal, complemented
4#20243255C212yesmaximal, complemented
5#20243169C212yesmaximal
6#20243164C212yesmaximal
7#20243141C211yesmaximal, characteristic, complemented
8#20243136C211yesmaximal, characteristic, complemented
9#20243074C211yesmaximal, characteristic, complemented
10C4×#13093934C211yesmaximal, characteristic, complemented
11C2×C47×#405C211yesmaximal, characteristic
12#15689184C418yescomplemented
13C2×C4⋊(C47⋊C8)C2212yes—
14C2×C22⋊(C47⋊C8)C2212yes—
15C2×#13093935C2211yescharacteristic
16C2×#13093934C2212yes—
17C2×#13093934C2212yes—
18C2×#13093934C2212yes—
19C2×C4×C47⋊C8C2211yescharacteristic
20C2×C8×Dic47C2212yes—
21#15689016C414yes—
22#15689009C414yes—
23C4×C4⋊Dic47C2212yes—
24C4×C22⋊Dic47C2212yes—
25#15688969C2211yescharacteristic
26C42×Dic47C2211yescharacteristic
27#15688877C2212yes—
28#15688877C2212yes—
29#15688877C2212yes—
30#15688877C2212yes—
31C22×M16×C47C2211yescharacteristic
32C23×C8×C47C2211yescharacteristic
33C2×#117×C47C2211yescharacteristic
34C47×#405C418yescomplemented
35C2×#97×C47C2211yescharacteristic
36C2×#97×C47C2211yescharacteristic
37C2×C4×C8×C47C2211yescharacteristic
38C2×C4×C8×C47C2211yescharacteristic
39C22×C47⋊C8C2312yes—
40C22×C47⋊C8C2312yes—
41C2×C4⋊Dic47C2×C414yes—
42C2×C22⋊Dic47C2×C414yes—
43C2×C4×Dic47C2×C414yes—
44C2×C4×Dic47C2312yes—
45C2×C4×Dic47C2312yes—
46C4⋊(C47⋊C8)C2×C418yes—
47C22⋊(C47⋊C8)C2×C418yes—
48#13093935C2×C414yes—
49#13093934—24no—
50#13093934—24no—
51#13093934—24no—
52C4×C47⋊C8C2×C414yes—
53C8×Dic47—24no—
54C47⋊#95C2×C414yes—
55C47⋊#95C2×C414yes—
56C47⋊#95C2×C414yes—
57C23×C4×C47C2311yescharacteristic
58C2×M16×C47—24no—
59C2×M16×C47C2×C414yes—
60C22×C8×C47—24no—
61C22×C8×C47C2×C414yes—
62C22×C8×C47C2311yescharacteristic
63C22×C8×C47C2311yescharacteristic
64C22×C8×C47C2311yescharacteristic
65C22×C8×C47C2311yescharacteristic
66#117×C47C2×C414yes—
67#117×C47C2×C414yes—
68C2×C4⋊C4×C47C2×C412yes—
69C2×C22⋊C4×C47C2×C412yes—
70C2×C42×C47C2311yescharacteristic
71C2×C42×C47C2311yescharacteristic
72#97×C47C2×C414yes—
73#97×C47C2×C414yes—
74C4×C8×C47C2×C414yes—
75C4×C8×C47C2×C414yes—
76C24×C47C22×C411yescharacteristic
77C22×C4×C47—22no—
78C22×C4×C47—22no—
79C22×C4×C47C22×C412yes—
80C22×C4×C47C22×C412yes—
81C22×C4×C47C22×C411yescharacteristic
82C22×C4×C47C22×C412yes—
83C22×C4×C47C22×C412yes—
84C22×C4×C47C2411yescharacteristic
85C22×Dic47C22×C414yes—
86C22×Dic47C22×C414yes—
87M16×C47—28no—
88C2×C8×C47—22no—
89C2×C8×C47—22no—
90C2×C8×C47—28no—
91C2×C8×C47C22×C412yes—
92C2×C8×C47C22×C412yes—
93C2×C8×C47C22×C412yes—
94C2×C8×C47C22×C412yes—
95C2×C8×C47#4114yes—
96C2×C8×C47#4114yes—
97C4⋊C4×C47C4218yes—
98C22⋊C4×C47C4218yes—
99C42×C47C22×C412yes—
100C42×C47C22×C412yes—
101C42×C47C22×C412yes—
102C42×C47C22×C412yes—
103C22⋊Dic47—44no—
104C4⋊Dic47—44no—
105C4×Dic47—22no—
106C4×Dic47—22no—
107C4×Dic47—22no—
108C4×Dic47—22no—
109C4×Dic47—44no—
110C2×C47⋊C8—22no—
111C2×C47⋊C8—22no—
112C2×C47⋊C8—22no—
113C2×C47⋊C8—22no—
114C2×C47⋊C8C22×C414yes—
115C2×C47⋊C8C22×C414yes—
116C23×C47—22no—
117C23×C47—24no—
118C23×C47C2×C4212yes—
119C23×C47C23×C411yescharacteristic
120C2×C4×C47—22no—
121C2×C4×C47—24no—
122C2×C4×C47—24no—
123C2×C4×C47—22no—
124C2×C4×C47—22no—
125C2×C4×C47—24no—
126C2×C4×C47C2×C4214yes—
127C2×C4×C47C2×C4214yes—
128C2×C4×C47C2×C4212yes—
129C2×C4×C47C23×C411yescharacteristic
130C2×C4×C47C23×C411yescharacteristic
131C2×C4×C47C2×#4112yes—
132C2×C4×C47C2×#4112yes—
133C2×Dic47—24no—
134C2×Dic47—24no—
135C2×Dic47—24no—
136C2×Dic47—24no—
137C2×Dic47—24no—
138C2×Dic47—24no—
139C8×C47—22no—
140C8×C47—22no—
141C8×C47#11714yes—
142C8×C47#11714yes—
143C47⋊C8—24no—
144C47⋊C8—24no—
145#20475—471nomaximal, Sylow
146C22×C47—24no—
147C22×C47—22no—
148C22×C47—44no—
149C22×C47C22×C4211yescharacteristic
150C22×C47C4×#4112yes—
151C22×C47#56814yes—
152C4×C47—24no—
153C4×C47—24no—
154C4×C47—22no—
155C4×C47C2×#11711yescharacteristic
156C4×C47C2×#11711yescharacteristic
157C4×C47C4×#4112yes—
158Dic47—44no—
159Dic47—44no—
160C2×#433—471no—
161C2×#405D4711yescharacteristic
162#2298—471no—
163#1771—472no—
164#1770—472no—
165#1753—472no—
166#1752—472no—
167#1746—471no—
168#1745—471no—
169#1726—471no—
170C4×#97—471no—
171C2×C47—42no—
172C2×C47C4×#11711yescharacteristic
173C2×C47#287512yes—
174C2×C47#296714yes—
175C22×M16C2×D4711yescharacteristic
176C23×C8C2×D4711yescharacteristic
177C2×#117C2×D4711yescharacteristic
178#433—478no—
179C2×C4⋊C8—472no—
180C2×C22⋊C8—472no—
181#405Dic4718yescomplemented
182C2×#97—471no—
183C2×#97—472no—
184C2×#97—472no—
185C2×#97—472no—
186C2×#97C2×D4711yescharacteristic
187C2×#97C2×D4711yescharacteristic
188C2×C4×C8—471no—
189C2×C4×C8—472no—
190C2×C4×C8C2×D4711yescharacteristic
191C2×C4×C8C2×D4711yescharacteristic
192#388—474no—
193#387—474no—
194C4×C4⋊C4—472no—
195C4×C22⋊C4—472no—
196#376—471no—
197C43—471no—
198#336—472no—
199#336—472no—
200#336—472no—
201#336—472no—
202C47#2047511yescharacteristic, complemented, Sylow
203C23×C4C22×D4711yescharacteristic
204C2×M16—24no—
205C2×M16C2×Dic4714yes—
206C22×C8—24no—
207C22×C8—472no—
208C22×C8—472no—
209C22×C8C2×Dic4714yes—
210C22×C8C22×D4711yescharacteristic
211C22×C8C22×D4711yescharacteristic
212C22×C8C22×D4711yescharacteristic
213C22×C8C22×D4711yescharacteristic
214#117C4×D4714yes—
215#117C2×Dic4714yes—
216C2×C4⋊C4—474no—
217C2×C4⋊C4C4×D4712yes—
218C2×C22⋊C4—474no—
219C2×C22⋊C4C4×D4712yes—
220C2×C42—472no—
221C2×C42—472no—
222C2×C42—474no—
223C2×C42C22×D4711yescharacteristic
224C2×C42C22×D4711yescharacteristic
225C4⋊C8—478no—
226C22⋊C8—478no—
227#97—474no—
228#97—944no—
229#97—944no—
230#97—944no—
231#97C2×Dic4714yes—
232#97C2×Dic4714yes—
233C4×C8—474no—
234C4×C8—944no—
235C4×C8C2×Dic4714yes—
236C4×C8C2×Dic4714yes—
237#95—474no—
238#95—474no—
239#95—474no—
240C24C2×C4×D4711yescharacteristic
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—474no—
244C22×C4—474no—
245C22×C4C2×C4×D4711yescharacteristic
246C22×C4C2×C4×D4712yes—
247C22×C4C2×C4×D4712yes—
248C22×C4C2×C4×D4712yes—
249C22×C4C22×Dic4712yes—
250C22×C4C23×D4711yescharacteristic
251M16—28no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—28no—
255C2×C8—474no—
256C2×C8—474no—
257C2×C8—942no—
258C2×C8—942no—
259C2×C8—942no—
260C2×C8—942no—
261C2×C8#1062535014yes—
262C2×C8#1062535014yes—
263C2×C8C22×Dic4712yes—
264C2×C8C22×Dic4712yes—
265C2×C8C22×Dic4712yes—
266C2×C8C22×Dic4712yes—
267C4⋊C4—1884no—
268C4⋊C4C4×Dic4718yes—
269C22⋊C4—1884no—
270C22⋊C4C4×Dic4718yes—
271C42—942no—
272C42—942no—
273C42—942no—
274C42—942no—
275C42—1884no—
276C42C2×C4×D4712yes—
277C42C2×C4×D4712yes—
278C42C22×Dic4712yes—
279C42C22×Dic4712yes—
280C23—22no—
281C23—24no—
282C23C2×C4×Dic4712yes—
283C23C22×C4×D4711yescharacteristic
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—944no—
291C2×C4—944no—
292C2×C4—944no—
293C2×C4—944no—
294C2×C4—944no—
295C2×C4—944no—
296C2×C4C2×C4×Dic4714yes—
297C2×C4C2×C4×Dic4714yes—
298C2×C4C2×C4×Dic4712yes—
299C2×C4C22×C4×D4711yescharacteristic
300C2×C4C23×Dic4711yescharacteristic
301C2×C4C2×#1062535012yes—
302C2×C4C2×#1062535012yes—
303C8—22no—
304C8—22no—
305C8—944no—
306C8—944no—
307C8#1309398314yes—
308C8#1309398314yes—
309C22—24no—
310C22—22no—
311C22—44no—
312C22C22×C4×Dic4711yescharacteristic
313C22C4×#1062535012yes—
314C22#1568988814yes—
315C4—24no—
316C4—24no—
317C4—22no—
318C4—1884no—
319C4—1884no—
320C4C2×#1309398311yescharacteristic
321C4C2×#1309398311yescharacteristic
322C4C4×#1062535012yes—
323C2—42no—
324C2C4×#1309398311yescharacteristic
325C2#2025181412yes—
326C2#2025236914yes—
subgroupG/H
center, FrattiniC22×C4C23×D47
commutatorC22×C47C22×C42
FittingC2×C47×#405C2
socleC23×C47C23×C4
prime radicalC24×C47C22×C4
cosocleC22×C4×C47C24
hypercenterC2×#405D47
hypocenterC47#20475
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
3. #20243260C2semidirect
4. #20243255C2semidirect
7. #20243141C2semidirect
8. #20243136C2semidirect
9. #20243074C2semidirect
10. C4×#13093934C2semidirect
12. #15689184C4semidirect
34. C47×#405C4semidirect
181. #405Dic47semidirect
202. C47#20475semidirect

Maximal subgroups

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

subgroupindex
C2×#156891842
#202458862
#202432602
#202432552
#202431692
#202431642
#202431412
#202431362
#202430742
C4×#130939342
C2×C47×#4052
#2047547