Finite Group Databasebeta

← #26108901

Subgroups — #26108901

Order 12032.

1198 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1198 conjugacy classes of subgroups — too many to draw. Only its 147 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#20243309C212yesmaximal
2#20241718C214yesmaximal, complemented
3C22×#123×C47C211yesmaximal, characteristic
4C2×#13093941—24no—
5#15689087C2214yes—
6#15688849—28no—
7C23×D4×C47C2211yescharacteristic
8C24×C4×C47C411yescharacteristic
9C2×#123×C47—24no—
10C2×#123×C47—28no—
11C22×C4⋊C4×C47C2212yes—
12C22×C22⋊C4×C47C411yescharacteristic
13C22×C22⋊C4×C47C412yes—
14C2×#10625329—24no—
15#13093941—48no—
16C22⋊(C47⋊C8)—28no—
17C25×C47C2×C411yescharacteristic
18C25×C47C2×C411yescharacteristic
19C22×D4×C47—24no—
20C22×D4×C47—44no—
21C22×D4×C47D414yes—
22C23×C4×C47—22no—
23C23×C4×C47—24no—
24C23×C4×C47—44no—
25C23×C4×C47C2×C412yes—
26C23×C4×C47C2×C412yes—
27C23×C4×C47C2311yescharacteristic
28#123×C47—232no—
29C2×C4⋊C4×C47—24no—
30C2×C4⋊C4×C47—24no—
31C2×C4⋊C4×C47D418yes—
32C2×C22⋊C4×C47—22no—
33C2×C22⋊C4×C47—24no—
34C2×C22⋊C4×C47—24no—
35C2×C22⋊C4×C47—24no—
36C2×C22⋊C4×C47—24no—
37C24×C47—24no—
38C24×C47—22no—
39C24×C47—22no—
40C24×C47—42no—
41C24×C47—44no—
42C24×C47—44no—
43C24×C47C22⋊C412yes—
44C24×C47C22⋊C412yes—
45C24×C47C22×C411yescharacteristic
46C2×D4×C47—28no—
47C2×D4×C47—28no—
48C2×D4×C47—44no—
49C2×D4×C47—416no—
50C22×C4×C47—28no—
51C22×C4×C47—22no—
52C22×C4×C47—22no—
53C22×C4×C47—22no—
54C22×C4×C47—22no—
55C22×C4×C47—44no—
56C22×C4×C47—42no—
57C22×C4×C47—416no—
58C22×C4×C47—44no—
59C22×C4×C47—44no—
60C22×C4×C47—44no—
61C22×C4×C47C22⋊C414yes—
62C22×C4×C47C22⋊C414yes—
63C22×C4×C47C2×D412yes—
64C22×C4×C47C2×D414yes—
65C4⋊C4×C47—216no—
66C22⋊C4×C47—28no—
67C22⋊C4×C47—216no—
68#10625329—48no—
69C2×C47⋊C8—44no—
70C23×C47—22no—
71C23×C47—24no—
72C23×C47—28no—
73C23×C47—22no—
74C23×C47—22no—
75C23×C47—22no—
76C23×C47—24no—
77C23×C47—44no—
78C23×C47—416no—
79C23×C47—44no—
80C23×C47—44no—
81C23×C47—44no—
82C23×C47—44no—
83C23×C47—44no—
84C23×C47—44no—
85C23×C47—88no—
86C23×C47#9914yes—
87C23×C47C2×C22⋊C411yescharacteristic
88C23×C47C2×C22⋊C412yes—
89D4×C47—416no—
90C2×C4×C47—24no—
91C2×C4×C47—24no—
92C2×C4×C47—24no—
93C2×C4×C47—28no—
94C2×C4×C47—44no—
95C2×C4×C47—44no—
96C2×C4×C47—44no—
97C2×C4×C47—44no—
98C2×C4×C47—416no—
99C2×C4×C47—44no—
100C2×C4×C47—44no—
101C2×C4×C47#12014yes—
102C47⋊C8—84no—
103#8548—471nomaximal, Sylow
104C22×C47—28no—
105C22×C47—22no—
106C22×C47—22no—
107C22×C47—22no—
108C22×C47—24no—
109C22×C47—44no—
110C22×C47—44no—
111C22×C47—416no—
112C22×C47—44no—
113C22×C47—42no—
114C22×C47—44no—
115C22×C47—84no—
116C22×C47—88no—
117C22×C47#37911yescharacteristic
118C22×C47C2×#9912yes—
119C22×C47#41314yes—
120C4×C47—24no—
121C4×C47—44no—
122C4×C47—44no—
123C22×#123D4711yescharacteristic
124#1783—472no—
125#1463—474no—
126C2×C47—22no—
127C2×C47—24no—
128C2×C47—44no—
129C2×C47—82no—
130C2×C47#178111yescharacteristic
131C2×C47#178512yes—
132C23×D4C2×D4711yescharacteristic
133C24×C4Dic4711yescharacteristic
134C2×#123—24no—
135C2×#123—28no—
136C22×C4⋊C4C2×D4712yes—
137C22×C22⋊C4Dic4711yescharacteristic
138C22×C22⋊C4Dic4712yes—
139C2×#100—944no—
140#407—474no—
141#324—948no—
142C47#854811yescharacteristic, complemented, Sylow
143C25C2×Dic4711yescharacteristic
144C25C2×Dic4711yescharacteristic
145C22×D4—24no—
146C22×D4—44no—
147C22×D4C22⋊D4714yes—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—44no—
151C23×C4C2×Dic4712yes—
152C23×C4C2×Dic4712yes—
153C23×C4C22×D4711yescharacteristic
154C2×M16—944no—
155#123—232no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4C22⋊D4718yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164#100—1888no—
165C22⋊C8—948no—
166C24—24no—
167C24—22no—
168C24—22no—
169C24—42no—
170C24—44no—
171C24—44no—
172C24C22⋊Dic4712yes—
173C24C22⋊Dic4712yes—
174C24C22×Dic4711yescharacteristic
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—44no—
178C2×D4—416no—
179C22×C4—28no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4—416no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4C22⋊Dic4714yes—
191C22×C4C22⋊Dic4714yes—
192C22×C4C2×C22⋊D4712yes—
193C22×C4C2×C22⋊D4714yes—
194M16—1888no—
195C2×C8—1884no—
196C4⋊C4—216no—
197C22⋊C4—28no—
198C22⋊C4—216no—
199C23—22no—
200C23—24no—
201C23—28no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—24no—
206C23—44no—
207C23—416no—
208C23—44no—
209C23—44no—
210C23—44no—
211C23—44no—
212C23—44no—
213C23—44no—
214C23—88no—
215C23#1309393914yes—
216C23C2×C22⋊Dic4711yescharacteristic
217C23C2×C22⋊Dic4712yes—
218D4—416no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—28no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—416no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4#1309399314yes—
231C8—3764no—
232C22—28no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—24no—
237C22—44no—
238C22—44no—
239C22—416no—
240C22—44no—
241C22—42no—
242C22—44no—
243C22—84no—
244C22—88no—
245C22#1568897911yescharacteristic
246C22C2×#1309393912yes—
247C22#1568910714yes—
248C4—24no—
249C4—44no—
250C4—44no—
251C2—22no—
252C2—24no—
253C2—44no—
254C2—82no—
255C2#2024330311yescharacteristic
256C2#2024331612yes—
subgroupG/H
centerC22#15688979
commutatorC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC22×#123×C47C2
socleC22×C47#379
prime radicalC23×D4×C47C22
cosocleC23×C4×C47C23
hypercenterC22×#123D47
hypocenterC47#8548
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
2. #20241718C2semidirect
142. C47#8548semidirect

Maximal subgroups

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

subgroupindex
#202433092
#202417182
C22×#123×C472
#854847