Finite Group Databasebeta

← #26106399

Subgroups — #26106399

Order 12032.

788 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252017C211yesmaximal, characteristic
2C4×#13093980C212yesmaximal
3#20241274C212yesmaximal
4#20241273C211yesmaximal, characteristic
5C47×#1297C211yesmaximal, characteristic
6C2×#13093980—22no—
7C22×C4×Dic47C2211yescharacteristic
8#15689177—22no—
9#15689173—22no—
10#15689146C412yescomplemented
11#15689087C2211yescharacteristic
12M16×Dic47C412yescomplemented
13C4×Dic47⋊C4—22no—
14#15688974C414yescomplemented
15C22⋊C4×Dic47—22no—
16C4×C22⋊Dic47C2212yes—
17#15688968C414yescomplemented
18C42×Dic47—22no—
19C47×#407C2211yescharacteristic
20C4×C22⋊C4×C47C2212yes—
21C23×Dic47D412yes—
22C2×#10625329C2×C412yes—
23#13093980—216no—
24C2×Dic47⋊C4—24no—
25C2×C22⋊Dic47—22no—
26C2×C4×Dic47—24no—
27C2×C4×Dic47—22no—
28C2×C4×Dic47—24no—
29C2×C4×Dic47C2×C411yescharacteristic
30C2×C4×Dic47C2×C411yescharacteristic
31C2×C4×Dic47C2×C412yes—
32C2×C4×Dic47C2×C412yes—
33C2×C4×Dic47D412yes—
34C2×C4×Dic47D412yes—
35C2×C4×Dic47Q812yes—
36Dic47⋊C8—24no—
37C22⋊(C47⋊C8)—22no—
38#13093934—22no—
39C8×Dic47—22no—
40C47⋊#95—24no—
41C47⋊#95C2×C414yes—
42C23×C4×C47C2311yescharacteristic
43C2×M16×C47C2×C412yes—
44C2×C22⋊C4×C47—22no—
45C2×C42×C47—22no—
46C22⋊C8×C47—22no—
47#95×C47C2×C414yes—
48C24×C47C2×D411yescharacteristic
49C22×C4×C47—24no—
50C22×C4×C47—42no—
51C22×C4×C47C22×C411yescharacteristic
52C22×C4×C47C22×C412yes—
53C22×C4×C47C2×D412yes—
54C22×C4×C47C2×Q811yescharacteristic
55C22×Dic47—28no—
56C22×Dic47—24no—
57C22×Dic47C22⋊C412yes—
58C22×Dic47C22⋊C412yes—
59C22×Dic47C22⋊C412yes—
60C22×Dic47C22⋊C412yes—
61C22×Dic47C4⋊C412yes—
62C22×Dic47C4⋊C412yes—
63M16×C47—42no—
64C2×C8×C47—22no—
65C22⋊C4×C47—28no—
66C42×C47—44no—
67C22⋊Dic47—28no—
68Dic47⋊C4—216no—
69C4×Dic47—216no—
70C4×Dic47—44no—
71C4×Dic47—48no—
72C4×Dic47C4211yescharacteristic
73C4×Dic47C4212yes—
74C4×Dic47C4211yescharacteristic
75C4×Dic47C22⋊C411yescharacteristic
76C4×Dic47C22⋊C411yescharacteristic
77C4×Dic47C22⋊C412yes—
78C4×Dic47C4⋊C412yes—
79C4×Dic47C4⋊C412yes—
80C4×Dic47C4⋊C412yes—
81C4×Dic47C4⋊C412yes—
82#10625329—42no—
83C2×C47⋊C8—22no—
84C23×C47—24no—
85C23×C47—41no—
86C23×C47C2×C22⋊C412yes—
87C23×C47C2×C4⋊C411yescharacteristic
88C2×C4×C47—22no—
89C2×C4×C47—24no—
90C2×C4×C47—28no—
91C2×C4×C47—42no—
92C2×C4×C47—41no—
93C2×C4×C47—44no—
94C2×C4×C47—44no—
95C2×C4×C47C2×C4211yescharacteristic
96C2×C4×C47C2×C22⋊C411yescharacteristic
97C2×C4×C47C2×C4⋊C412yes—
98C2×Dic47—24no—
99C2×Dic47—24no—
100C2×Dic47—24no—
101C2×Dic47—24no—
102C2×Dic47—24no—
103C2×Dic47—28no—
104C2×Dic47—44no—
105C2×Dic47—44no—
106C2×Dic47#9512yes—
107C2×Dic47#9512yes—
108C2×Dic47#9512yes—
109C2×Dic47#9512yes—
110C2×Dic47#10418yes—
111C8×C47—42no—
112C47⋊C8—42no—
113#8048—471nomaximal, Sylow
114C22×C47—22no—
115C22×C47—22no—
116C22×C47—22no—
117C22×C47—21no—
118C22×C47—44no—
119C22×C47C2×#9511yescharacteristic
120C22×C47C2×#10414yes—
121C4×C47—21no—
122C4×C47—22no—
123C4×C47—48no—
124C4×C47#41012yes—
125Dic47—22no—
126Dic47—22no—
127Dic47—48no—
128Dic47#33914yes—
129Dic47#33914yes—
130#2920—471no—
131C4×#117—472no—
132#1297—471no—
133#1297—472no—
134#1297D4711yescharacteristic
135C2×C47—24no—
136C2×C47—41no—
137C2×C47C2×#33912yes—
138C2×C47#174011yescharacteristic
139C2×#117—942no—
140C22×C42—471no—
141#432—942no—
142#431—942no—
143#423—472no—
144#407—471no—
145#407C2×D4711yescharacteristic
146C4×M16—472no—
147C4×C4⋊C4—474no—
148C4×C4⋊C4—942no—
149C4×C22⋊C4—472no—
150C4×C22⋊C4—942no—
151C4×C22⋊C4C2×D4712yes—
152#376—474no—
153C43—942no—
154C47#804811yescharacteristic, complemented, Sylow
155C23×C4—472no—
156C23×C4C22×D4711yescharacteristic
157C2×M16—472no—
158C2×M16C4×D4712yes—
159#117—9416no—
160C2×C4⋊C4—944no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—942no—
163C2×C42—22no—
164C2×C42—472no—
165C2×C42—472no—
166C2×C42—471no—
167C2×C42—471no—
168C2×C42—472no—
169C2×C42—472no—
170C2×C42—472no—
171C2×C42—944no—
172C2×C42—944no—
173C2×C42—942no—
174C4⋊C8—944no—
175C22⋊C8—22no—
176C22⋊C8—942no—
177#97—942no—
178C4×C8—942no—
179#95—474no—
180#95—944no—
181#95C4×D4714yes—
182C24D4×D4711yescharacteristic
183C22×C4—24no—
184C22×C4—42no—
185C22×C4—472no—
186C22×C4—472no—
187C22×C4—472no—
188C22×C4—472no—
189C22×C4—472no—
190C22×C4—472no—
191C22×C4—948no—
192C22×C4—944no—
193C22×C4C2×C4×D4711yescharacteristic
194C22×C4C2×C4×D4712yes—
195C22×C4D4×D4712yes—
196C22×C4Q8×D4711yescharacteristic
197M16—42no—
198M16—1882no—
199C2×C8—22no—
200C2×C8—942no—
201C4⋊C4—9416no—
202C22⋊C4—28no—
203C22⋊C4—948no—
204C42—44no—
205C42—471no—
206C42—472no—
207C42—471no—
208C42—471no—
209C42—471no—
210C42—472no—
211C42—472no—
212C42—472no—
213C42—472no—
214C42—472no—
215C42—9416no—
216C42—1884no—
217C42—1888no—
218C23—24no—
219C23—41no—
220C23C22⋊C4×D4712yes—
221C23C4⋊C4×D4711yescharacteristic
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—28no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—472no—
230C2×C4—472no—
231C2×C4—478no—
232C2×C4—472no—
233C2×C4—472no—
234C2×C4—944no—
235C2×C4—944no—
236C2×C4—944no—
237C2×C4—944no—
238C2×C4—944no—
239C2×C4—948no—
240C2×C4—1884no—
241C2×C4—1884no—
242C2×C4C42×D4711yescharacteristic
243C2×C4C22⋊C4×D4711yescharacteristic
244C2×C4C4⋊C4×D4712yes—
245C8—42no—
246C8—1882no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—21no—
251C22—44no—
252C22#95×D4711yescharacteristic
253C22#104×D4714yes—
254C4—21no—
255C4—22no—
256C4—48no—
257C4—474no—
258C4—474no—
259C4—942no—
260C4—942no—
261C4—1888no—
262C4#1568909512yes—
263C2—24no—
264C2—41no—
265C2#339×D4712yes—
266C2#2024311911yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D47
commutatorC2×C4×C47C2×C42
FrattiniC23×C4C22×D47
FittingC47×#1297C2
socleC22×C47C2×#95
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1297D47
hypocenterC47#8048
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
10. #15689146C4semidirect
12. M16×Dic47C4semidirect
14. #15688974C4semidirect
17. #15688968C4semidirect
154. C47#8048semidirect

Maximal subgroups

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

subgroupindex
#202520172
C4×#130939802
#202412742
#202412732
C47×#12972
#804847