Finite Group Databasebeta

← #26104258

Subgroups — #26104258

Order 12032.

516 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 143 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#20245957C211yesmaximal, characteristic
2#20243236C212yesmaximal
3#20241214C211yesmaximal, characteristic
4#20241213C212yesmaximal
5C47×#1272C211yesmaximal, characteristic
6C2×#13094022C411yescharacteristic
7C2×Q8×Dic47C411yescharacteristic
8C2×#13093989C411yescharacteristic
9C2×Dic47⋊C8—22no—
10C2×C8×Dic47—22no—
11#15689029C2211yescharacteristic
12#15689020—22no—
13#15688992C411yescharacteristic
14#15688974—22no—
15C42×Dic47C2211yescharacteristic
16#15688878C2212yes—
17C47×#391C2211yescharacteristic
18C47×#336C2212yes—
19C22×Dic94C2×C411yescharacteristic
20C22×C47⋊C8—22no—
21#13094022—22no—
22#13094022C814yescomplemented
23Q8×Dic47—42no—
24#13093989—42no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—22no—
28C2×Dic47⋊C4—22no—
29C2×Dic47⋊C4—21no—
30C2×C4×Dic47—21no—
31C2×C4×Dic47—21no—
32C2×C4×Dic47C2×C411yescharacteristic
33C2×C4×Dic47C2×C411yescharacteristic
34C2×C4×Dic47D412yes—
35C2×C4×Dic47D412yes—
36Dic47⋊C8—28no—
37C8×Dic47—44no—
38C47⋊#95—22no—
39C22×Q8×C47C2×C411yescharacteristic
40C22×C8×C47—22no—
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C42×C47C2311yescharacteristic
43#95×C47—22no—
44C2×Q8×C47—21no—
45C2×Q8×C47—41no—
46C2×Q8×C47—41no—
47C2×Q8×C47C2×C812yes—
48C22×C4×C47—21no—
49C22×C4×C47—21no—
50C22×C4×C47C22×C411yescharacteristic
51C22×C4×C47C2×D412yes—
52C22×Dic47—21no—
53C22×Dic47—21no—
54C22×Dic47C22⋊C411yescharacteristic
55C22×Dic47C22⋊C412yes—
56C22×Dic47C22⋊C411yescharacteristic
57C2×Dic94—21no—
58C2×Dic94—41no—
59C2×Dic94—41no—
60C2×Dic94C2×C812yes—
61C2×C8×C47—24no—
62C2×C8×C47—42no—
63C2×C8×C47—42no—
64C4⋊C4×C47—41no—
65C42×C47—41no—
66C4⋊Dic47—42no—
67C4⋊Dic47—41no—
68Dic47⋊C4—28no—
69Dic47⋊C4—42no—
70Dic47⋊C4—44no—
71C4×Dic47—22no—
72C4×Dic47—24no—
73C4×Dic47—42no—
74C4×Dic47—42no—
75C4×Dic47C2×C812yes—
76C4×Dic47M1612yes—
77C4×Dic47SD1614yes—
78C4×Dic47Q1614yes—
79C2×C47⋊C8—24no—
80C2×C47⋊C8—42no—
81C2×C47⋊C8—42no—
82C23×C47C2×C22⋊C411yescharacteristic
83Q8×C47—42no—
84Q8×C47—42no—
85C2×C4×C47—22no—
86C2×C4×C47—21no—
87C2×C4×C47—21no—
88C2×C4×C47—22no—
89C2×C4×C47—22no—
90C2×C4×C47—41no—
91C2×C4×C47—41no—
92C2×C4×C47—41no—
93C2×C4×C47—42no—
94C2×C4×C47C22×C811yescharacteristic
95C2×C4×C47C2×M1611yescharacteristic
96C2×C4×C47C2×SD1612yes—
97C2×C4×C47C2×Q1612yes—
98C2×Dic47—22no—
99C2×Dic47—21no—
100C2×Dic47—22no—
101C2×Dic47—41no—
102C2×Dic47—41no—
103C2×Dic47—41no—
104C2×Dic47—42no—
105C2×Dic47C22⋊C814yes—
106C2×Dic47#10012yes—
107C2×Dic47#10314yes—
108C2×Dic47#10314yes—
109C2×Dic47#10414yes—
110Dic94—42no—
111Dic94—42no—
112C8×C47—44no—
113C47⋊C8—44no—
114#7571—471nomaximal, Sylow
115C22×C47C2×C22⋊C811yescharacteristic
116C22×C47#41011yescharacteristic
117C22×C47C2×#10011yescharacteristic
118C22×C47C2×#10312yes—
119C22×C47C2×#10412yes—
120C4×C47—21no—
121C4×C47—21no—
122C4×C47—22no—
123C4×C47—22no—
124C4×C47—42no—
125C4×C47—81no—
126Dic47—42no—
127Dic47—41no—
128Dic47—81no—
129Dic47Q8⋊C818yes—
130Dic47#33314yes—
131C4×C4⋊Q8—471no—
132C4×C4⋊C8—472no—
133#1272—471no—
134#1272—472no—
135#1272D4711yescharacteristic
136C2×C47#146011yescharacteristic
137C2×C47C2×Q8⋊C812yes—
138C2×C47#150312yes—
139C2×C47#152511yescharacteristic
140C2×C47C2×#33311yescharacteristic
141C2×C4⋊Q8—471no—
142C2×C4×Q8—471no—
143C2×C4×Q8—471no—
144C2×C4⋊C8—942no—
145C2×C4×C8—942no—
146#391—471no—
147#391C2×D4711yescharacteristic
148#389—942no—
149#384—471no—
150C4×C4⋊C4—942no—
151C43—471no—
152#336—472no—
153#336C2×D4712yes—
154C47#757111yescharacteristic, complemented, Sylow
155C22×Q8—471no—
156C22×Q8C4×D4711yescharacteristic
157C22×C8—22no—
158C22×C8—942no—
159C4⋊Q8—474no—
160C4⋊Q8—942no—
161C4×Q8—1882no—
162C4×Q8—1882no—
163C2×C4⋊C4—471no—
164C2×C4⋊C4—942no—
165C2×C4⋊C4—941no—
166C2×C4⋊C4—941no—
167C2×C4⋊C4—942no—
168C2×C4⋊C4C4×D4711yescharacteristic
169C2×C42—471no—
170C2×C42—472no—
171C2×C42—471no—
172C2×C42—472no—
173C2×C42—941no—
174C2×C42—941no—
175C2×C42C22×D4711yescharacteristic
176C4⋊C8—948no—
177C4×C8—1884no—
178#95—22no—
179#95—942no—
180C2×Q8—21no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—472no—
184C2×Q8—941no—
185C2×Q8—1881no—
186C2×Q8—1881no—
187C2×Q8C8×D4712yes—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—471no—
191C22×C4—472no—
192C22×C4—471no—
193C22×C4—941no—
194C22×C4—941no—
195C22×C4C2×C4×D4711yescharacteristic
196C22×C4D4×D4712yes—
197C2×C8—24no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—944no—
201C2×C8—1882no—
202C2×C8—1882no—
203C4⋊C4—41no—
204C4⋊C4—948no—
205C4⋊C4—1881no—
206C4⋊C4—1884no—
207C4⋊C4—1882no—
208C4⋊C4—1882no—
209C42—41no—
210C42—472no—
211C42—472no—
212C42—474no—
213C42—474no—
214C42—944no—
215C42—942no—
216C42—1882no—
217C42—1882no—
218C23C22⋊C4×D4711yescharacteristic
219Q8—42no—
220Q8—42no—
221Q8—1882no—
222Q8—1882no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—474no—
233C2×C4—472no—
234C2×C4—474no—
235C2×C4—474no—
236C2×C4—474no—
237C2×C4—942no—
238C2×C4—941no—
239C2×C4—942no—
240C2×C4—1881no—
241C2×C4—1882no—
242C2×C4—1881no—
243C2×C4—1881no—
244C2×C4C2×C8×D4711yescharacteristic
245C2×C4M16×D4711yescharacteristic
246C2×C4SD16×D4712yes—
247C2×C4Q16×D4712yes—
248C8—44no—
249C8—1884no—
250C22C22⋊C8×D4711yescharacteristic
251C22#1568909511yescharacteristic
252C22#100×D4711yescharacteristic
253C22#103×D4712yes—
254C22#104×D4712yes—
255C4—21no—
256C4—21no—
257C4—22no—
258C4—22no—
259C4—42no—
260C4—81no—
261C4—478no—
262C4—474no—
263C4—3761no—
264C4—1882no—
265C4—1881no—
266C2#2024170711yescharacteristic
267C2Q8⋊C8×D4712yes—
268C2#2024188512yes—
269C2#2024197811yescharacteristic
270C2#333×D4711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#1272C2
socle, prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1272D47
hypocenterC47#7571
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
22. #13094022C8semidirect
154. C47#7571semidirect

Maximal subgroups

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

subgroupindex
#202459572
#202432362
#202412142
#202412132
C47×#12722
#757147