Finite Group Databasebeta

← #26108910

Subgroups — #26108910

Order 12032.

1026 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C22×#13094022C211yesmaximal, characteristic
2#20243312C211yesmaximal, characteristic
3#20241717C214yesmaximal
4C47×#1784C211yesmaximal, characteristic
5C23×Dic94C411yescharacteristic
6C2×#13094022—28no—
7C2×#13094022—22no—
8C2×#13094022—22no—
9C22×Dic47⋊C4C412yes—
10C22×Dic47⋊C4C2212yes—
11C22×C4×Dic47C411yescharacteristic
12C2×#13093943—22no—
13#15689087C2212yes—
14#15688848—28no—
15C23×Q8×C47C2211yescharacteristic
16C2×#101×C47—22no—
17C47×#407C2212yes—
18C22×Dic94—22no—
19C22×Dic94—24no—
20C22×Dic94—44no—
21C23×Dic47C2×C412yes—
22C23×Dic47C2×C412yes—
23C2×#10625329—22no—
24#13094022—216no—
25#13094022—216no—
26C2×Dic47⋊C4—24no—
27C2×Dic47⋊C4—22no—
28C2×Dic47⋊C4—22no—
29C2×Dic47⋊C4—24no—
30C2×Dic47⋊C4—24no—
31C2×Dic47⋊C4—24no—
32C2×Dic47⋊C4D418yes—
33C2×C4×Dic47—22no—
34C2×C4×Dic47—24no—
35#13093943—44no—
36C22⋊(C47⋊C8)—24no—
37C22×Q8×C47—24no—
38C22×Q8×C47—44no—
39C22×Q8×C47D412yes—
40C22×Q8×C47D412yes—
41C23×C4×C47C2×C411yescharacteristic
42C23×C4×C47C2×C411yescharacteristic
43C23×C4×C47C2311yescharacteristic
44C2×M16×C47—22no—
45#101×C47—44no—
46C22⋊C8×C47—24no—
47C24×C47C22×C411yescharacteristic
48C2×Q8×C47—24no—
49C2×Q8×C47—24no—
50C2×Q8×C47—24no—
51C2×Q8×C47—24no—
52C2×Q8×C47—42no—
53C2×Q8×C47—42no—
54C2×Q8×C47—416no—
55C22×C4×C47—22no—
56C22×C4×C47—24no—
57C22×C4×C47—22no—
58C22×C4×C47—42no—
59C22×C4×C47—42no—
60C22×C4×C47C22⋊C412yes—
61C22×C4×C47C22⋊C412yes—
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D414yes—
65C22×Dic47—22no—
66C22×Dic47—22no—
67C22×Dic47—28no—
68C22×Dic47—22no—
69C22×Dic47—22no—
70C22×Dic47—44no—
71C22×Dic47C22⋊C414yes—
72C22×Dic47C22⋊C414yes—
73C2×Dic94—28no—
74C2×Dic94—416no—
75C2×Dic94—44no—
76C2×Dic94—44no—
77M16×C47—44no—
78C2×C8×C47—42no—
79Dic47⋊C4—216no—
80Dic47⋊C4—28no—
81Dic47⋊C4—28no—
82C4×Dic47—28no—
83#10625329—44no—
84C2×C47⋊C8—42no—
85C23×C47—22no—
86C23×C47—24no—
87C23×C47C2×C22⋊C412yes—
88C23×C47C2×C22⋊C411yescharacteristic
89Q8×C47—48no—
90Q8×C47—48no—
91C2×C4×C47—28no—
92C2×C4×C47—22no—
93C2×C4×C47—22no—
94C2×C4×C47—22no—
95C2×C4×C47—22no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—44no—
100C2×C4×C47—44no—
101C2×C4×C47#9914yes—
102C2×C4×C47#12012yes—
103C2×C4×C47#12012yes—
104C2×Dic47—28no—
105C2×Dic47—28no—
106C2×Dic47—24no—
107C2×Dic47—24no—
108C2×Dic47—24no—
109C2×Dic47—44no—
110C2×Dic47—44no—
111C2×Dic47—44no—
112C2×Dic47#10118yescomplemented
113Dic94—416no—
114C8×C47—82no—
115C47⋊C8—82no—
116#8551—471nomaximal, Sylow
117C22×C47—28no—
118C22×C47—22no—
119C22×C47—22no—
120C22×C47—22no—
121C22×C47#37911yescharacteristic
122C22×C47C2×#9912yes—
123C22×C47C2×#10114yes—
124C4×C47—22no—
125C4×C47—22no—
126C4×C47—24no—
127C4×C47—42no—
128Dic47—28no—
129Dic47—44no—
130C22×C4⋊Q8—471no—
131#1784—471no—
132#1784D4711yescharacteristic
133#1463—474no—
134C2×C47—22no—
135C2×C47—24no—
136C2×C47#178111yescharacteristic
137C2×C47#178412yes—
138C23×Q8—471no—
139C23×Q8C2×D4711yescharacteristic
140C2×C4⋊Q8—948no—
141C2×C4⋊Q8—942no—
142C2×C4⋊Q8—942no—
143C22×C4⋊C4—472no—
144C22×C4⋊C4—472no—
145C22×C42—471no—
146C2×#101—22no—
147C2×#101—942no—
148#407—472no—
149#407C2×D4712yes—
150#324—948no—
151C47#855111yescharacteristic, complemented, Sylow
152C22×Q8—24no—
153C22×Q8—44no—
154C22×Q8—944no—
155C22×Q8—942no—
156C22×Q8—1884no—
157C22×Q8D18812yes—
158C22×Q8C22⋊D4712yes—
159C23×C4—472no—
160C23×C4—472no—
161C23×C4C4×D4711yescharacteristic
162C23×C4C4×D4711yescharacteristic
163C23×C4C22×D4711yescharacteristic
164C2×M16—22no—
165C2×M16—942no—
166C4⋊Q8—9416no—
167C4⋊Q8—9416no—
168C2×C4⋊C4—478no—
169C2×C4⋊C4—944no—
170C2×C4⋊C4—944no—
171C2×C4⋊C4—944no—
172C2×C4⋊C4—942no—
173C2×C4⋊C4—944no—
174C2×C4⋊C4—942no—
175C2×C42—944no—
176C2×C42—942no—
177#101—44no—
178#101—1884no—
179C22⋊C8—24no—
180C22⋊C8—944no—
181C24C2×C4×D4711yescharacteristic
182C2×Q8—24no—
183C2×Q8—24no—
184C2×Q8—24no—
185C2×Q8—24no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—416no—
189C2×Q8—948no—
190C2×Q8—18816no—
191C2×Q8—1884no—
192C2×Q8—1884no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—474no—
199C22×C4—474no—
200C22×C4—948no—
201C22×C4—942no—
202C22×C4—942no—
203C22×C4—942no—
204C22×C4—942no—
205C22×C4—1884no—
206C22×C4(C2×D47)⋊C412yes—
207C22×C4(C2×D47)⋊C412yes—
208C22×C4C2×D18811yescharacteristic
209C22×C4D4×D4714yes—
210C22×C4C2×C22⋊D4711yescharacteristic
211M16—44no—
212M16—1884no—
213C2×C8—42no—
214C2×C8—1882no—
215C4⋊C4—9416no—
216C4⋊C4—948no—
217C4⋊C4—948no—
218C42—948no—
219C23—22no—
220C23—24no—
221C23C22⋊C4×D4712yes—
222C23C2×(C2×D47)⋊C411yescharacteristic
223Q8—48no—
224Q8—48no—
225Q8—18816no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—478no—
237C2×C4—948no—
238C2×C4—948no—
239C2×C4—944no—
240C2×C4—944no—
241C2×C4—944no—
242C2×C4—1884no—
243C2×C4—1884no—
244C2×C4—1884no—
245C2×C4#1309393814yes—
246C2×C4#1309399212yes—
247C2×C4#1309399412yes—
248C8—82no—
249C8—3762no—
250C22—28no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22#1568897811yescharacteristic
255C22C2×#1309393812yes—
256C22#101×D4714yes—
257C4—22no—
258C4—22no—
259C4—24no—
260C4—42no—
261C4—948no—
262C4—1884no—
263C2—22no—
264C2—24no—
265C2#2024330211yescharacteristic
266C2#2024331012yes—
subgroupG/H
centerC22#15688978
commutator, prime radicalC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1784C2
socleC22×C47#379
cosocleC23×C4×C47C23
hypercenter#1784D47
hypocenterC47#8551
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
112. C2×Dic47#101semidirect
151. C47#8551semidirect

Maximal subgroups

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

subgroupindex
C22×#130940222
#202433122
#202417172
C47×#17842
#855147