Finite Group Databasebeta

← #26113674

Subgroups — #26113674

Order 12032.

514 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 123 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#20252429C211yesmaximal, characteristic
2#20243380C211yesmaximal, characteristic
3#20242069C212yesmaximal, complemented
4#20241291C212yesmaximal, complemented
5C47×#2563C211yesmaximal, characteristic
6#15689201—24no—
7#15689176C2212yes—
8#15689147C2212yes—
9C2×#13093941—21no—
10#15689087—22no—
11#15688892—21no—
12#15688872—24no—
13C2×C4⋊Q8×C47C2211yescharacteristic
14C2×C4×D4×C47C2211yescharacteristic
15C47×#393C414yes—
16C47×#390—21no—
17C47×#382—21no—
18C47×#380—22no—
19C4×C4⋊C4×C47C2211yescharacteristic
20C2×#10625329—24no—
21C2×#10625329—22no—
22C4⋊(C47⋊C8)—22no—
23C4⋊(C47⋊C8)—22no—
24C4⋊(C47⋊C8)D414yes—
25#13093941—42no—
26C22⋊(C47⋊C8)—24no—
27C22⋊(C47⋊C8)—22no—
28C4×C47⋊C8—22no—
29C22×Q8×C47C2×C412yes—
30C22×D4×C47D411yescharacteristic
31C23×C4×C47D412yes—
32C4⋊Q8×C47—22no—
33C4⋊Q8×C47—22no—
34C4×D4×C47—22no—
35C4×D4×C47—22no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47D411yescharacteristic
40C2×C22⋊C4×C47—24no—
41C2×C22⋊C4×C47C2×C412yes—
42C2×C42×C47—21no—
43C2×C42×C47C2311yescharacteristic
44#95×C47—24no—
45#95×C47C2×C412yes—
46C24×C47C22⋊C412yes—
47C2×Q8×C47—24no—
48C2×Q8×C47—82no—
49C2×D4×C47—22no—
50C2×D4×C47—21no—
51C2×D4×C47—21no—
52C2×D4×C47—22no—
53C22×C4×C47—22no—
54C22×C4×C47—22no—
55C22×C4×C47—22no—
56C22×C4×C47—22no—
57C22×C4×C47—44no—
58C22×C4×C47C22⋊C412yes—
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C4⋊C4×C47—22no—
63C4⋊C4×C47—42no—
64C4⋊C4×C47—41no—
65C4⋊C4×C47—42no—
66C4⋊C4×C47—41no—
67C22⋊C4×C47—24no—
68C22⋊C4×C47—84no—
69C42×C47—41no—
70C42×C47—41no—
71C42×C47C2×D412yes—
72C42×C47#4112yes—
73#10625329—42no—
74#10625329—42no—
75#10625329—84no—
76C2×C47⋊C8—24no—
77C2×C47⋊C8—42no—
78C23×C47—22no—
79C23×C47—22no—
80C23×C47—22no—
81C23×C47—42no—
82C23×C47—42no—
83C23×C47C2×C22⋊C411yescharacteristic
84Q8×C47—84no—
85D4×C47—42no—
86D4×C47—42no—
87C2×C4×C47—22no—
88C2×C4×C47—22no—
89C2×C4×C47—41no—
90C2×C4×C47—44no—
91C2×C4×C47—42no—
92C2×C4×C47—42no—
93C2×C4×C47—44no—
94C2×C4×C47—42no—
95C2×C4×C47—42no—
96C2×C4×C47—42no—
97C2×C4×C47—82no—
98C2×C4×C47C4×D412yes—
99C2×C4×C47#12011yescharacteristic
100C2×C4×C47#12112yes—
101C2×C4×C47#12311yescharacteristic
102C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
103C2×C4×C47#13711yescharacteristic
104C2×C4×C47#13712yes—
105C47⋊C8—44no—
106C47⋊C8—82no—
107#9399—471nomaximal, Sylow
108C22×C47—21no—
109C22×C47—21no—
110C22×C47—44no—
111C22×C47—42no—
112C22×C47—42no—
113C22×C47—42no—
114C22×C47—82no—
115C22×C47#38611yescharacteristic
116C22×C47#41711yescharacteristic
117C22×C47#41911yescharacteristic
118C4×C47—21no—
119C4×C47—22no—
120C4×C47—42no—
121C4×C47—82no—
122C4×C47—82no—
123C4×C47#45512yes—
124C4×C47#46812yes—
125C4×C47#47012yes—
126#2975—471no—
127#2563D4711yescharacteristic
128#1800—471no—
129#1548—472no—
130#1303—472no—
131C2×C47—21no—
132C2×C47—21no—
133C2×C47—82no—
134C2×C47#188511yescharacteristic
135C2×C47#189411yescharacteristic
136C2×C47#191411yescharacteristic
137C2×C4⋊Q8C2×D4711yescharacteristic
138C2×C4×D4C2×D4711yescharacteristic
139#436—944no—
140#431—472no—
141#423—472no—
142C2×#100—941no—
143#407—942no—
144#393Dic4714yes—
145#390—21no—
146#382—21no—
147#380—22no—
148C4×C4⋊C4C2×D4711yescharacteristic
149#343—941no—
150#333—944no—
151C47#939911yescharacteristic, complemented, Sylow
152C22×Q8C2×Dic4712yes—
153C22×D4C22⋊D4711yescharacteristic
154C23×C4C22⋊D4712yes—
155C2×M16—944no—
156C2×M16—942no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4×D4—22no—
160C4×D4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4C22⋊D4711yescharacteristic
165C2×C22⋊C4—24no—
166C2×C22⋊C4C2×Dic4712yes—
167C2×C42—21no—
168C2×C42C22×D4711yescharacteristic
169C4⋊C8—474no—
170C4⋊C8—942no—
171C4⋊C8—942no—
172#100—1882no—
173C22⋊C8—944no—
174C22⋊C8—942no—
175C4×C8—942no—
176#95—24no—
177#95C2×Dic4712yes—
178C24C22⋊Dic4712yes—
179C2×Q8—24no—
180C2×Q8—82no—
181C2×D4—22no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—44no—
190C22×C4C22⋊Dic4712yes—
191C22×C4C22×Dic4711yescharacteristic
192C22×C4C2×C22⋊D4711yescharacteristic
193C22×C4C2×C22⋊D4711yescharacteristic
194M16—3764no—
195M16—1882no—
196M16—1882no—
197C2×C8—944no—
198C2×C8—1882no—
199C4⋊C4—22no—
200C4⋊C4—42no—
201C4⋊C4—41no—
202C4⋊C4—42no—
203C4⋊C4—41no—
204C22⋊C4—24no—
205C22⋊C4—84no—
206C42—41no—
207C42—41no—
208C42D4×D4712yes—
209C42#1062535212yes—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×C22⋊Dic4711yescharacteristic
216Q8—84no—
217D4—42no—
218D4—42no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—82no—
230C2×C4D4×Dic4712yes—
231C2×C4#1309399211yescharacteristic
232C2×C4#1309399612yes—
233C2×C4#1309400511yescharacteristic
234C2×C4#1309404911yescharacteristic
235C2×C4#1309405411yescharacteristic
236C2×C4#1309405512yes—
237C8—3762no—
238C8—1884no—
239C22—21no—
240C22—21no—
241C22—44no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—82no—
246C22#1568900311yescharacteristic
247C22#1568912411yescharacteristic
248C22#1568913211yescharacteristic
249C4—21no—
250C4—22no—
251C4—42no—
252C4—82no—
253C4—82no—
254C4#1568930012yes—
255C4#1568937112yes—
256C4#1568937912yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2#2024377711yescharacteristic
261C2#2024382211yescharacteristic
262C2#2024393611yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2563C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#2563D47
hypocenterC47#9399
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2563

Complements

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

NH ≅ G/Ntype
3. #20242069C2semidirect
4. #20241291C2semidirect
151. C47#9399semidirect

Maximal subgroups

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

subgroupindex
#202524292
#202433802
#202420692
#202412912
C47×#25632
#939947