Finite Group Databasebeta

← #26213687

Subgroups — #26213687

Order 12032.

2270 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093983C212yesmaximal
2#20243108C218yesmaximal, complemented
3#20243105C214yesmaximal, complemented
4C23×M16×C47C211yesmaximal, characteristic
5C2×#13093983—22no—
6C2×#13093983—22no—
7C2×#13093983—28no—
8C2×#13093983C4116yescomplemented
9C2×#13093983C2214yes—
10C2×#13093983C2214yes—
11C22×C4⋊Dic47C2214yes—
12C22×C22⋊Dic47—22no—
13C22×C4×Dic47—22no—
14#15688886C22116yescomplemented
15C24×C4×C47C2211yescharacteristic
16C22×M16×C47—28no—
17C22×M16×C47C418yescomplemented
18C22×M16×C47C2214yes—
19C23×C8×C47C2212yes—
20C23×Dic47—24no—
21#13093983—216no—
22#13093983—216no—
23#13093983—28no—
24#13093983—28no—
25#13093983C2×C4116yes—
26#13093983C2×C4116yescomplemented
27C2×C4⋊Dic47—24no—
28C2×C4⋊Dic47—28no—
29C2×C4⋊Dic47C2×C4116yescomplemented
30C2×C4⋊Dic47C2318yes—
31C2×C22⋊Dic47—216no—
32C2×C22⋊Dic47—24no—
33C2×C22⋊Dic47—24no—
34C2×C4×Dic47—216no—
35C2×C4×Dic47—24no—
36C2×C4×Dic47—24no—
37C25×C47D411yescharacteristic
38C23×C4×C47C2×C414yes—
39C23×C4×C47C2×C418yes—
40C23×C4×C47D412yes—
41C23×C4×C47D411yescharacteristic
42C23×C4×C47D418yes—
43C23×C4×C47Q814yes—
44C23×C4×C47C2312yes—
45C23×C4×C47C2311yescharacteristic
46C2×M16×C47—232no—
47C2×M16×C47—28no—
48C2×M16×C47—28no—
49C2×M16×C47C2×C4116yescomplemented
50C22×C8×C47—28no—
51C22×C8×C47C2×C418yescomplemented
52C22×C8×C47C2314yes—
53C24×C47—44no—
54C24×C47C22⋊C418yes—
55C24×C47C4⋊C414yes—
56C24×C47C2×D411yescharacteristic
57C24×C47C2×D412yes—
58C22×C4×C47C42116yes—
59C22×C4×C47C22⋊C418yes—
60C22×C4×C47C22⋊C418yes—
61C22×C4×C47C22⋊C418yes—
62C22×C4×C47C22⋊C4116yes—
63C22×C4×C47C4⋊C418yes—
64C22×C4×C47C4⋊C414yes—
65C22×C4×C47C4⋊C4132yes—
66C22×C4×C47C22×C412yes—
67C22×C4×C47C22×C414yes—
68C22×C4×C47C22×C414yes—
69C22×C4×C47C22×C414yes—
70C22×C4×C47C22×C414yes—
71C22×C4×C47C2×D412yes—
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D414yes—
74C22×C4×C47C2×D414yes—
75C22×C4×C47C2×D414yes—
76C22×C4×C47C2×Q814yes—
77C22×C4×C47C2×Q812yes—
78C22×C4×C47C2411yescharacteristic
79C22×Dic47—216no—
80C22×Dic47—28no—
81C22×Dic47—416no—
82M16×C47—232no—
83C2×C8×C47—216no—
84C2×C8×C47—28no—
85C2×C8×C47C22×C418yes—
86C22⋊Dic47—216no—
87C22⋊Dic47—216no—
88C4⋊Dic47—216no—
89C4⋊Dic47—28no—
90C4⋊Dic47C22×C4116yes—
91C4×Dic47—216no—
92C4×Dic47—216no—
93C23×C47—28no—
94C23×C47—24no—
95C23×C47—416no—
96C23×C47—48no—
97C23×C47#95116yes—
98C23×C47C2×C22⋊C414yes—
99C23×C47C2×C22⋊C414yes—
100C23×C47C2×C22⋊C414yes—
101C23×C47C2×C4⋊C412yes—
102C23×C47C2×C4⋊C414yes—
103C23×C47C22×D411yescharacteristic
104C2×C4×C47#95132yes—
105C2×C4×C47#95116yes—
106C2×C4×C47C2×C4214yes—
107C2×C4×C47C2×C4218yes—
108C2×C4×C47C2×C22⋊C414yes—
109C2×C4×C47C2×C22⋊C414yes—
110C2×C4×C47C2×C22⋊C414yes—
111C2×C4×C47C2×C22⋊C414yes—
112C2×C4×C47C2×C22⋊C418yes—
113C2×C4×C47C2×C4⋊C418yes—
114C2×C4×C47C2×C4⋊C418yes—
115C2×C4×C47C2×C4⋊C418yes—
116C2×C4×C47C2×C4⋊C414yes—
117C2×C4×C47C2×C4⋊C412yes—
118C2×C4×C47C23×C412yes—
119C2×C4×C47C23×C411yescharacteristic
120C2×C4×C47C22×D412yes—
121C2×C4×C47C22×Q811yescharacteristic
122C2×Dic47—216no—
123C2×Dic47—432no—
124C2×Dic47—416no—
125C8×C47—28no—
126#20347—471nomaximal, Sylow
127C22×C47—28no—
128C22×C47—28no—
129C22×C47—28no—
130C22×C47—28no—
131C22×C47—24no—
132C22×C47—416no—
133C22×C47C2×#9514yes—
134C22×C47C2×#9518yes—
135C22×C47C22×C22⋊C412yes—
136C22×C47C22×C4⋊C411yescharacteristic
137C22×C47#57514yes—
138C4×C47C2×#9514yes—
139C4×C47C2×#9518yes—
140C4×C47C22×C4211yescharacteristic
141C4×C47C22×C22⋊C411yescharacteristic
142C4×C47C22×C4⋊C412yes—
143Dic47—416no—
144C23×M16D4711yescharacteristic
145C22×#117—472no—
146#1736—478no—
147#1735—474no—
148C2×C47—28no—
149C2×C47—24no—
150C2×C47C22×#9511yescharacteristic
151C2×C47#289614yes—
152C2×C47#291112yes—
153C24×C4C2×D4711yescharacteristic
154C22×M16—28no—
155C22×M16Dic4718yescomplemented
156C22×M16C2×D4714yes—
157C23×C8C2×D4712yes—
158C2×#117—4716no—
159C2×#117—474no—
160C2×#117—474no—
161C2×#117—942no—
162C2×#117—942no—
163C2×#117—948no—
164C22×C4⋊C4—474no—
165C22×C22⋊C4—942no—
166C22×C42—942no—
167#340—4716no—
168C47#2034711yescharacteristic, complemented, Sylow
169C25D18811yescharacteristic
170C23×C4—944no—
171C23×C4Dic9414yes—
172C23×C4C4×D4718yes—
173C23×C4D18811yescharacteristic
174C23×C4D18812yes—
175C23×C4C2×Dic4714yes—
176C23×C4C22⋊D4718yes—
177C23×C4C22×D4712yes—
178C23×C4C22×D4711yescharacteristic
179C2×M16—232no—
180C2×M16—28no—
181C2×M16—28no—
182C2×M16C2×Dic47116yescomplemented
183C22×C8—28no—
184C22×C8C2×Dic4718yescomplemented
185C22×C8C22×D4714yes—
186#117—4716no—
187#117—4716no—
188#117—9416no—
189#117—9416no—
190#117—948no—
191#117—948no—
192C2×C4⋊C4—4716no—
193C2×C4⋊C4—478no—
194C2×C4⋊C4—944no—
195C2×C4⋊C4—948no—
196C2×C22⋊C4—9416no—
197C2×C22⋊C4—944no—
198C2×C22⋊C4—944no—
199C2×C42—9416no—
200C2×C42—944no—
201C2×C42—944no—
202C24—44no—
203C24C4⋊Dic4714yes—
204C24(C2×D47)⋊C418yes—
205C24C2×D18811yescharacteristic
206C24C2×D18812yes—
207C22×C4—9416no—
208C22×C4—948no—
209C22×C4—18816no—
210C22×C4C4×Dic47116yes—
211C22×C4Dic47⋊C4132yes—
212C22×C4C4⋊Dic4714yes—
213C22×C4C4⋊Dic4718yes—
214C22×C4(C2×D47)⋊C418yes—
215C22×C4(C2×D47)⋊C418yes—
216C22×C4(C2×D47)⋊C418yes—
217C22×C4C22⋊Dic47116yes—
218C22×C4C2×Dic9414yes—
219C22×C4C2×Dic9412yes—
220C22×C4C2×C4×D4714yes—
221C22×C4C2×C4×D4714yes—
222C22×C4C2×C4×D4714yes—
223C22×C4C2×D18812yes—
224C22×C4C2×D18811yescharacteristic
225C22×C4C22×Dic4712yes—
226C22×C4C22×Dic4714yes—
227C22×C4C2×C22⋊D4714yes—
228C22×C4C2×C22⋊D4714yes—
229C22×C4C2×C22⋊D4714yes—
230C22×C4C23×D4711yescharacteristic
231M16—232no—
232C2×C8—216no—
233C2×C8—28no—
234C2×C8C22×Dic4718yes—
235C4⋊C4—4716no—
236C4⋊C4—9416no—
237C4⋊C4—948no—
238C22⋊C4—9416no—
239C22⋊C4—9416no—
240C42—9416no—
241C42—9416no—
242C23—28no—
243C23—24no—
244C23—48no—
245C23—416no—
246C23C47⋊#95116yes—
247C23C2×(C2×D47)⋊C414yes—
248C23C2×(C2×D47)⋊C414yes—
249C23C2×(C2×D47)⋊C414yes—
250C23C2×C4⋊Dic4712yes—
251C23C2×C4⋊Dic4714yes—
252C23C22×D18811yescharacteristic
253C2×C4—9416no—
254C2×C4—18832no—
255C2×C4—18816no—
256C2×C4C47⋊#95132yes—
257C2×C4C47⋊#95116yes—
258C2×C4C2×C4×Dic4718yes—
259C2×C4C2×C4×Dic4714yes—
260C2×C4C2×(C2×D47)⋊C414yes—
261C2×C4C2×(C2×D47)⋊C414yes—
262C2×C4C2×(C2×D47)⋊C414yes—
263C2×C4C2×C22⋊Dic4714yes—
264C2×C4C2×C22⋊Dic4718yes—
265C2×C4C2×Dic47⋊C418yes—
266C2×C4C2×Dic47⋊C418yes—
267C2×C4C2×Dic47⋊C418yes—
268C2×C4C2×C4⋊Dic4714yes—
269C2×C4C2×C4⋊Dic4712yes—
270C2×C4C22×C4×D4712yes—
271C2×C4C23×Dic4711yescharacteristic
272C2×C4C22×C22⋊D4712yes—
273C2×C4C22×Dic9411yescharacteristic
274C8—28no—
275C22—28no—
276C22—28no—
277C22—28no—
278C22—28no—
279C22—24no—
280C22—416no—
281C22C2×C47⋊#9514yes—
282C22C2×C47⋊#9518yes—
283C22C22×(C2×D47)⋊C412yes—
284C22C22×C4⋊Dic4711yescharacteristic
285C22#1568992314yes—
286C4—18816no—
287C4C2×C47⋊#9518yes—
288C4C2×C47⋊#9514yes—
289C4C22×C4×Dic4711yescharacteristic
290C4C22×C22⋊Dic4711yescharacteristic
291C4C22×Dic47⋊C412yes—
292C2—28no—
293C2—24no—
294C2C22×C47⋊#9511yescharacteristic
295C2#2025190814yes—
296C2#2025197812yes—
subgroupG/H
centerC22×C4C2×D188
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC23×M16×C47C2
socleC23×C47C22×D4
prime radicalC25×C47D4
cosocleC22×C4×C47C24
hypercenterC23×M16D47
hypocenterC47#20347
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
2. #20243108C2semidirect
3. #20243105C2semidirect
8. C2×#13093983C4semidirect
14. #15688886C22semidirect
17. C22×M16×C47C4semidirect
26. #13093983C2×C4semidirect
29. C2×C4⋊Dic47C2×C4semidirect
49. C2×M16×C47C2×C4semidirect
51. C22×C8×C47C2×C4semidirect
155. C22×M16Dic47semidirect
168. C47#20347semidirect
182. C2×M16C2×Dic47semidirect
184. C22×C8C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C22×#130939832
#202431082
#202431052
C23×M16×C472
#2034747