Finite Group Databasebeta

← #26125568

Subgroups — #26125568

Order 12032.

440 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 119 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#20241712C212yesmaximal, complemented
2#20241334C212yesmaximal, complemented
3#20241320C212yesmaximal
4C47×#2326C211yesmaximal, characteristic
5#15689176—22no—
6#15688849—22no—
7#15688849C2214yes—
8C2×#124×C47C411yescharacteristic
9C2×C4×Q8×C47C2211yescharacteristic
10C2×C4×D4×C47C411yescharacteristic
11C47×#391C411yescharacteristic
12C47×#390C2211yescharacteristic
13C47×#388—22no—
14C47×#385—22no—
15C47×#384C2211yescharacteristic
16C47×#376C411yescharacteristic
17C4⋊(C47⋊C8)—42no—
18C22⋊(C47⋊C8)—24no—
19C4×C47⋊C8—42no—
20C22×Q8×C47C2×C411yescharacteristic
21C22×D4×C47C2×C411yescharacteristic
22C23×C4×C47—21no—
23#124×C47—22no—
24#124×C47C814yescomplemented
25C4×Q8×C47—42no—
26C4×D4×C47—42no—
27C2×C4⋊C4×C47—22no—
28C2×C4⋊C4×C47C2×C411yescharacteristic
29C2×C4⋊C4×C47D412yes—
30C2×C4⋊C4×C47C2311yescharacteristic
31C2×C22⋊C4×C47—22no—
32C2×C22⋊C4×C47—21no—
33C2×C22⋊C4×C47—22no—
34C2×C42×C47C2×C411yescharacteristic
35C2×C42×C47C2×C411yescharacteristic
36C2×C42×C47C2×C411yescharacteristic
37C2×C42×C47D412yes—
38#95×C47—21no—
39#95×C47—21no—
40#95×C47—22no—
41C24×C47—21no—
42C2×Q8×C47—41no—
43C2×Q8×C47—41no—
44C2×Q8×C47C2×C812yes—
45C2×Q8×C47M1612yes—
46C2×D4×C47—21no—
47C2×D4×C47—41no—
48C2×D4×C47—41no—
49C2×D4×C47C2×C812yes—
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47—21no—
53C22×C4×C47—42no—
54C22×C4×C47—82no—
55C22×C4×C47C22⋊C412yes—
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47C22×C411yescharacteristic
59C22×C4×C47C2×D412yes—
60C4⋊C4×C47—42no—
61C4⋊C4×C47—41no—
62C4⋊C4×C47—41no—
63C4⋊C4×C47—82no—
64C22⋊C4×C47—28no—
65C22⋊C4×C47—42no—
66C22⋊C4×C47—82no—
67C42×C47—22no—
68C42×C47—22no—
69C42×C47—22no—
70C42×C47—21no—
71C42×C47—41no—
72C42×C47C2×C812yes—
73C2×C47⋊C8—44no—
74C23×C47—22no—
75C23×C47—42no—
76C23×C47—42no—
77C23×C47—41no—
78C23×C47—41no—
79C23×C47C2×C22⋊C411yescharacteristic
80Q8×C47—42no—
81Q8×C47—42no—
82D4×C47—42no—
83D4×C47—42no—
84C2×C4×C47—21no—
85C2×C4×C47—22no—
86C2×C4×C47—21no—
87C2×C4×C47—22no—
88C2×C4×C47—22no—
89C2×C4×C47—22no—
90C2×C4×C47—21no—
91C2×C4×C47—41no—
92C2×C4×C47—41no—
93C2×C4×C47—41no—
94C2×C4×C47—42no—
95C2×C4×C47—41no—
96C2×C4×C47—41no—
97C2×C4×C47—42no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47—41no—
101C2×C4×C47—84no—
102C2×C4×C47C22⋊C814yes—
103C2×C4×C47#9912yes—
104C2×C4×C47C22×C811yescharacteristic
105C2×C4×C47C2×M1611yescharacteristic
106C47⋊C8—84no—
107#11480—471nomaximal, Sylow
108C22×C47—21no—
109C22×C47—21no—
110C22×C47—44no—
111C22×C47—42no—
112C22×C47—82no—
113C22×C47—81no—
114C22×C47—81no—
115C22×C47C2×C22⋊C811yescharacteristic
116C22×C47C2×#9911yescharacteristic
117C22×C47#41311yescharacteristic
118C4×C47—22no—
119C4×C47—24no—
120C4×C47—21no—
121C4×C47—21no—
122C4×C47—42no—
123C4×C47—42no—
124C4×C47—41no—
125C4×C47—81no—
126C4×C47—81no—
127#2326D4711yescharacteristic
128#1461—472no—
129#1325—472no—
130#1318—472no—
131C2×C47—21no—
132C2×C47—21no—
133C2×C47—82no—
134C2×C47#145911yescharacteristic
135C2×C47#212811yescharacteristic
136C2×C47#213311yescharacteristic
137C2×#124Dic4711yescharacteristic
138C2×C4×Q8C2×D4711yescharacteristic
139C2×C4×D4Dic4711yescharacteristic
140#431—942no—
141#391Dic4711yescharacteristic
142#390C2×D4711yescharacteristic
143#388—22no—
144#385—22no—
145#384C2×D4711yescharacteristic
146#376Dic4711yescharacteristic
147#324—474no—
148#324—942no—
149C47#1148011yescharacteristic, complemented, Sylow
150C22×Q8C2×Dic4711yescharacteristic
151C22×D4C2×Dic4711yescharacteristic
152C23×C4—21no—
153#124—22no—
154#124C47⋊C814yescomplemented
155C4×Q8—42no—
156C4×D4—42no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C2×Dic4711yescharacteristic
159C2×C4⋊C4C22⋊D4712yes—
160C2×C4⋊C4C22×D4711yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—22no—
164C2×C42C2×Dic4711yescharacteristic
165C2×C42C2×Dic4711yescharacteristic
166C2×C42C2×Dic4711yescharacteristic
167C2×C42C22⋊D4712yes—
168C4⋊C8—1882no—
169C22⋊C8—944no—
170C4×C8—1882no—
171#95—21no—
172#95—22no—
173#95—21no—
174C24—21no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8C2×C47⋊C812yes—
178C2×Q8#1062532912yes—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4C2×C47⋊C812yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4C22⋊Dic4712yes—
189C22×C4C22⋊Dic4711yescharacteristic
190C22×C4C22⋊Dic4711yescharacteristic
191C22×C4C22×Dic4711yescharacteristic
192C22×C4C2×C22⋊D4712yes—
193C2×C8—1884no—
194C4⋊C4—42no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—82no—
198C22⋊C4—28no—
199C22⋊C4—42no—
200C22⋊C4—82no—
201C42—22no—
202C42—22no—
203C42—22no—
204C42—21no—
205C42—41no—
206C42C2×C47⋊C812yes—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23C2×C22⋊Dic4711yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—42no—
216D4—42no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—84no—
235C2×C4C22⋊(C47⋊C8)14yes—
236C2×C4#1309393912yes—
237C2×C4C22×C47⋊C811yescharacteristic
238C2×C4C2×#1062532911yescharacteristic
239C8—3764no—
240C22—21no—
241C22—21no—
242C22—44no—
243C22—42no—
244C22—82no—
245C22—81no—
246C22—81no—
247C22C2×C22⋊(C47⋊C8)11yescharacteristic
248C22C2×#1309393911yescharacteristic
249C22#1568910711yescharacteristic
250C4—24no—
251C4—21no—
252C4—21no—
253C4—22no—
254C4—42no—
255C4—42no—
256C4—41no—
257C4—81no—
258C4—81no—
259C2—21no—
260C2—21no—
261C2—82no—
262C2#2024170611yescharacteristic
263C2#2024506011yescharacteristic
264C2#2024507711yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC2×C4×C47C22×C8
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2326C2
socleC22×C47C2×#99
prime radicalC22×D4×C47C2×C4
cosocleC2×C4⋊C4×C47C23
hypercenter#2326D47
hypocenterC47#11480
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2326

Complements

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

NH ≅ G/Ntype
1. #20241712C2semidirect
2. #20241334C2semidirect
24. #124×C47C8semidirect
149. C47#11480semidirect
154. #124C47⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202417122
#202413342
#202413202
C47×#23262
#1148047