Finite Group Databasebeta

← #26109607

Subgroups — #26109607

Order 12032.

898 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 898 conjugacy classes of subgroups — too many to draw. Only its 179 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#20243298C212yesmaximal
2#20241231C214yesmaximal, complemented
3C2×C4×C22⋊C4×C47C211yesmaximal, characteristic
4C2×C22⋊(C47⋊C8)—24no—
5C2×C22⋊(C47⋊C8)C2214yes—
6C24×C4×C47C2211yescharacteristic
7C22×C22⋊C4×C47C411yescharacteristic
8C22×C22⋊C4×C47C411yescharacteristic
9C22×C42×C47C412yes—
10C4×C22⋊C4×C47—24no—
11C2×#95×C47C2212yes—
12C22×C47⋊C8—24no—
13C22⋊(C47⋊C8)—28no—
14C22⋊(C47⋊C8)—48no—
15C25×C47C2×C411yescharacteristic
16C23×C4×C47—24no—
17C23×C4×C47—44no—
18C23×C4×C47C2×C411yescharacteristic
19C23×C4×C47C2×C412yes—
20C23×C4×C47C2×C412yes—
21C23×C4×C47D414yes—
22C23×C4×C47C2311yescharacteristic
23C2×C22⋊C4×C47—22no—
24C2×C22⋊C4×C47—22no—
25C2×C22⋊C4×C47—22no—
26C2×C22⋊C4×C47—24no—
27C2×C22⋊C4×C47C814yescomplemented
28C2×C42×C47—22no—
29C2×C42×C47—22no—
30C2×C42×C47—44no—
31#95×C47—24no—
32C24×C47—22no—
33C24×C47—22no—
34C24×C47—42no—
35C24×C47—42no—
36C24×C47C22⋊C412yes—
37C24×C47C2×C812yes—
38C24×C47M1612yes—
39C24×C47C22×C411yescharacteristic
40C22×C4×C47—22no—
41C22×C4×C47—24no—
42C22×C4×C47—22no—
43C22×C4×C47—22no—
44C22×C4×C47—22no—
45C22×C4×C47—22no—
46C22×C4×C47—28no—
47C22×C4×C47—28no—
48C22×C4×C47—44no—
49C22×C4×C47—42no—
50C22×C4×C47—416no—
51C22×C4×C47—42no—
52C22×C4×C47—42no—
53C22×C4×C47—42no—
54C22×C4×C47—42no—
55C22×C4×C47C22⋊C412yes—
56C22×C4×C47C2×C814yes—
57C22×C4×C47M1614yes—
58C22×C4×C47C2×D412yes—
59C22×C4×C47#4114yes—
60C22⋊C4×C47—28no—
61C22⋊C4×C47—28no—
62C42×C47—44no—
63C42×C47—44no—
64C2×C47⋊C8—48no—
65C2×C47⋊C8—44no—
66C23×C47—22no—
67C23×C47—22no—
68C23×C47—22no—
69C23×C47—22no—
70C23×C47—24no—
71C23×C47—24no—
72C23×C47—24no—
73C23×C47—24no—
74C23×C47—44no—
75C23×C47—44no—
76C23×C47—44no—
77C23×C47—44no—
78C23×C47—44no—
79C23×C47—44no—
80C23×C47C22⋊C814yes—
81C23×C47C2×C22⋊C411yescharacteristic
82C23×C47#11712yes—
83C23×C47C22×C811yescharacteristic
84C23×C47C2×M1611yescharacteristic
85C23×C47C2×M1612yes—
86C2×C4×C47—28no—
87C2×C4×C47—22no—
88C2×C4×C47—24no—
89C2×C4×C47—22no—
90C2×C4×C47—24no—
91C2×C4×C47—24no—
92C2×C4×C47—44no—
93C2×C4×C47—416no—
94C2×C4×C47—44no—
95C2×C4×C47—44no—
96C2×C4×C47—42no—
97C2×C4×C47—42no—
98C2×C4×C47—42no—
99C2×C4×C47—42no—
100C2×C4×C47—44no—
101C2×C4×C47—44no—
102C2×C4×C47#9914yes—
103C2×C4×C47#12314yes—
104C47⋊C8—84no—
105#8700—471nomaximal, Sylow
106C22×C47—22no—
107C22×C47—22no—
108C22×C47—22no—
109C22×C47—22no—
110C22×C47—22no—
111C22×C47—22no—
112C22×C47—24no—
113C22×C47—28no—
114C22×C47—48no—
115C22×C47—48no—
116C22×C47—42no—
117C22×C47—42no—
118C22×C47—44no—
119C22×C47#38111yescharacteristic
120C22×C47C2×C22⋊C811yescharacteristic
121C22×C47#40711yescharacteristic
122C22×C47C2×#9912yes—
123C22×C47#41312yes—
124C22×C47#43112yes—
125C22×C47#43212yes—
126C4×C47—24no—
127C4×C47—24no—
128C4×C47—44no—
129C4×C47—44no—
130C2×C4×C22⋊C4D4711yescharacteristic
131#1779—472no—
132#1280—474no—
133C2×C47—22no—
134C2×C47—22no—
135C2×C47—44no—
136C2×C47#145912yes—
137C2×C47#146512yes—
138C2×C47#181711yescharacteristic
139C2×C47#181911yescharacteristic
140C2×C47#182211yescharacteristic
141C24×C4C2×D4711yescharacteristic
142C22×C22⋊C4Dic4711yescharacteristic
143C22×C22⋊C4Dic4711yescharacteristic
144C22×C42Dic4712yes—
145C2×C22⋊C8—474no—
146C2×C22⋊C8—944no—
147C4×C22⋊C4—24no—
148C2×#95C2×D4712yes—
149C47#870011yescharacteristic, complemented, Sylow
150C25C2×Dic4711yescharacteristic
151C23×C4—24no—
152C23×C4—44no—
153C23×C4C2×Dic4711yescharacteristic
154C23×C4C2×Dic4712yes—
155C23×C4C2×Dic4712yes—
156C23×C4C22⋊D4714yes—
157C23×C4C22×D4711yescharacteristic
158C22×C8—944no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4C47⋊C814yescomplemented
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—44no—
167C22⋊C8—948no—
168C22⋊C8—1888no—
169#95—24no—
170C24—22no—
171C24—22no—
172C24—42no—
173C24—42no—
174C24C2×C47⋊C812yes—
175C24#1062532912yes—
176C24C22⋊Dic4712yes—
177C24C22×Dic4711yescharacteristic
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—28no—
185C22×C4—28no—
186C22×C4—44no—
187C22×C4—42no—
188C22×C4—416no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4C2×C47⋊C814yes—
194C22×C4#1062532914yes—
195C22×C4C22⋊Dic4712yes—
196C22×C4#1062535014yes—
197C22×C4C2×C22⋊D4712yes—
198C2×C8—1888no—
199C2×C8—1884no—
200C22⋊C4—28no—
201C22⋊C4—28no—
202C42—44no—
203C42—44no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—24no—
209C23—24no—
210C23—24no—
211C23—24no—
212C23—44no—
213C23—44no—
214C23—44no—
215C23—44no—
216C23—44no—
217C23—44no—
218C23C22⋊(C47⋊C8)14yes—
219C23C2×C22⋊Dic4711yescharacteristic
220C23#1309398312yes—
221C23C22×C47⋊C811yescharacteristic
222C23C2×#1062532911yescharacteristic
223C23C2×#1062532912yes—
224C2×C4—28no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—44no—
231C2×C4—416no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4#1309393914yes—
241C2×C4#1309400714yes—
242C8—3764no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—24no—
250C22—28no—
251C22—48no—
252C22—48no—
253C22—42no—
254C22—42no—
255C22—44no—
256C22#1568898611yescharacteristic
257C22C2×C22⋊(C47⋊C8)11yescharacteristic
258C22#1568908711yescharacteristic
259C22C2×#1309393912yes—
260C22#1568910712yes—
261C22#1568917612yes—
262C22#1568918012yes—
263C4—24no—
264C4—24no—
265C4—44no—
266C4—44no—
267C2—22no—
268C2—22no—
269C2—44no—
270C2#2024170612yes—
271C2#2024172612yes—
272C2#2024345611yescharacteristic
273C2#2024346211yescharacteristic
274C2#2024347111yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC23×C47C22×C8
FrattiniC23×C4C22×D47
FittingC2×C4×C22⋊C4×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC25×C47C2×C4
cosocleC23×C4×C47C23
hypercenterC2×C4×C22⋊C4D47
hypocenterC47#8700
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #20241231C2semidirect
27. C2×C22⋊C4×C47C8semidirect
149. C47#8700semidirect
163. C2×C22⋊C4C47⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202432982
#202412312
C2×C4×C22⋊C4×C472
#870047