Finite Group Databasebeta

← #20250405

Subgroups — #20250405

Order 6016.

528 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#13094001C212yesmaximal, complemented
2#15689059C212yesmaximal, complemented
3#15689056C212yesmaximal, complemented
4#15689046C212yesmaximal, complemented
5#15689029C211yesmaximal, characteristic, complemented
6#15689021C211yesmaximal, characteristic, complemented
7#15689003C212yesmaximal
8#15688998C212yesmaximal, complemented
9C47×#390C211yesmaximal, characteristic
10C22×Dic94C2211yescharacteristic
11C23×Dic47C2212yes—
12#13094001—28no—
13C2×C4⋊Dic47C2212yes—
14C2×C4⋊Dic47C2211yescharacteristic
15C2×C4⋊Dic47C2211yescharacteristic
16C2×C4⋊Dic47C2211yescharacteristic
17C2×Dic47⋊C4C2212yes—
18C2×Dic47⋊C4C2212yes—
19C2×Dic47⋊C4C2212yes—
20C2×C22⋊Dic47C2212yes—
21C2×C22⋊Dic47C2212yes—
22C2×C22⋊Dic47C2212yes—
23C47⋊#95C2212yes—
24C47⋊#95C2212yes—
25C47⋊#95C2212yes—
26C47⋊#95C2212yes—
27C22×D4×C47C2211yescharacteristic
28C2×C4⋊C4×C47C2211yescharacteristic
29C2×C22⋊C4×C47C2212yes—
30C2×C22⋊C4×C47C2212yes—
31C2×C42×C47C2211yescharacteristic
32C24×C47C2312yes—
33C2×D4×C47—41no—
34C2×D4×C47—41no—
35C2×D4×C47Q814yescomplemented
36C22×C4×C47C2312yes—
37C22×C4×C47C2311yescharacteristic
38C22×C4×C47C2311yescharacteristic
39C22×C4×C47C2311yescharacteristic
40C22×Dic47—24no—
41C22×Dic47—42no—
42C22×Dic47—42no—
43C22×Dic47C2312yes—
44C22×Dic47C2312yes—
45C22×Dic47C2312yes—
46C22×Dic47C2312yes—
47C2×Dic94—41no—
48C2×Dic94—41no—
49C2×Dic94D414yescomplemented
50C4⋊C4×C47—41no—
51C22⋊C4×C47—24no—
52C22⋊C4×C47—24no—
53C42×C47—41no—
54C22⋊Dic47—24no—
55C22⋊Dic47—42no—
56C22⋊Dic47—42no—
57C4⋊Dic47—22no—
58C4⋊Dic47—24no—
59C4⋊Dic47—41no—
60C4⋊Dic47—41no—
61Dic47⋊C4—24no—
62Dic47⋊C4—24no—
63Dic47⋊C4—42no—
64C23×C47—22no—
65C23×C47—22no—
66C23×C47—22no—
67C23×C47—22no—
68C23×C47—22no—
69C23×C47—22no—
70C23×C47C2×Q814yes—
71C23×C47C2411yescharacteristic
72D4×C47—44no—
73C2×C4×C47—21no—
74C2×C4×C47—21no—
75C2×C4×C47—21no—
76C2×C4×C47—21no—
77C2×C4×C47—21no—
78C2×C4×C47—21no—
79C2×C4×C47—22no—
80C2×C4×C47—22no—
81C2×C4×C47—22no—
82C2×C4×C47C2×D412yes—
83C2×C4×C47C2×Q812yes—
84C2×C4×C47#4112yes—
85C2×Dic47—22no—
86C2×Dic47—22no—
87C2×Dic47—22no—
88C2×Dic47—22no—
89C2×Dic47—22no—
90C2×Dic47—22no—
91C2×Dic47—22no—
92C2×Dic47—22no—
93C2×Dic47—22no—
94C2×Dic47—22no—
95C2×Dic47—22no—
96C2×Dic47—48no—
97C2×Dic47C2×D414yes—
98Dic94—44no—
99C22×C47—24no—
100C22×C47—24no—
101C22×C47—24no—
102C22×C47—42no—
103C22×C47—42no—
104C22×C47—42no—
105C22×C47—42no—
106C22×C47C22×D411yescharacteristic
107C22×C47C22×Q811yescharacteristic
108C22×C47C2×#4111yescharacteristic
109C22×C47H(5,𝔽2)11yescharacteristic
110C22×C47H(5,𝔽2)11yescharacteristic
111C22×C47H(5,𝔽2)11yescharacteristic
112C22×C472−1+411yescharacteristic
113C4×C47—22no—
114C4×C47—22no—
115C4×C47—22no—
116C4×C47—42no—
117Dic47—24no—
118Dic47—42no—
119Dic47—42no—
120Dic47—42no—
121#2732—471nomaximal, Sylow
122C2×C47—44no—
123C2×C47#53411yescharacteristic
124C2×C47#53911yescharacteristic
125C2×C47#54311yescharacteristic
126C2×C47D4×Q811yescharacteristic
127C2×C47#55011yescharacteristic
128C2×C47#55411yescharacteristic
129C2×C47#56211yescharacteristic
130C2×C22⋊Q8—472no—
131#399—472no—
132#398—472no—
133#396—472no—
134#391—471no—
135#390D4711yescharacteristic
136#389—471no—
137#386—472no—
138#385—472no—
139C47#273211yescharacteristic, complemented, Sylow
140C22×Q8—471no—
141C22×D4C2×D4711yescharacteristic
142C23×C4—472no—
143C22⋊Q8—948no—
144C2×C4⋊C4—472no—
145C2×C4⋊C4—472no—
146C2×C4⋊C4—472no—
147C2×C4⋊C4—471no—
148C2×C4⋊C4—472no—
149C2×C4⋊C4—471no—
150C2×C4⋊C4—471no—
151C2×C4⋊C4C2×D4711yescharacteristic
152C2×C22⋊C4—472no—
153C2×C22⋊C4—472no—
154C2×C22⋊C4—472no—
155C2×C22⋊C4C2×D4712yes—
156C2×C22⋊C4C2×D4712yes—
157C2×C42C2×D4711yescharacteristic
158#95—472no—
159#95—472no—
160#95—472no—
161#95—472no—
162C24C22×D4712yes—
163C2×Q8—474no—
164C2×Q8—1881no—
165C2×Q8—1881no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4Dic9414yescomplemented
169C22×C4—472no—
170C22×C4—472no—
171C22×C4—472no—
172C22×C4—472no—
173C22×C4—944no—
174C22×C4—1882no—
175C22×C4—1882no—
176C22×C4C22×D4712yes—
177C22×C4C22×D4711yescharacteristic
178C22×C4C22×D4711yescharacteristic
179C22×C4C22×D4711yescharacteristic
180C4⋊C4—41no—
181C4⋊C4—944no—
182C4⋊C4—944no—
183C4⋊C4—944no—
184C4⋊C4—942no—
185C4⋊C4—1882no—
186C4⋊C4—1881no—
187C4⋊C4—1881no—
188C22⋊C4—24no—
189C22⋊C4—24no—
190C22⋊C4—944no—
191C22⋊C4—1882no—
192C22⋊C4—1882no—
193C42—41no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23C2×Dic9414yes—
201C23C23×D4711yescharacteristic
202Q8—1884no—
203D4—44no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—474no—
214C2×C4—942no—
215C2×C4—942no—
216C2×C4—942no—
217C2×C4—942no—
218C2×C4—942no—
219C2×C4—942no—
220C2×C4—942no—
221C2×C4—942no—
222C2×C4—942no—
223C2×C4—942no—
224C2×C4—942no—
225C2×C4—1888no—
226C2×C4C2×Dic9412yes—
227C2×C4D4×D4712yes—
228C2×C4#1062535212yes—
229C22—24no—
230C22—24no—
231C22—24no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22C2×D4×D4711yescharacteristic
237C22C22×Dic9411yescharacteristic
238C22C2×#1062535211yescharacteristic
239C22#1309406911yescharacteristic
240C22#1309406911yescharacteristic
241C22#1309407011yescharacteristic
242C22#1309407111yescharacteristic
243C4—22no—
244C4—22no—
245C4—22no—
246C4—42no—
247C4—944no—
248C4—1882no—
249C4—1882no—
250C4—1882no—
251C2—44no—
252C2#1568968111yescharacteristic
253C2#1568971811yescharacteristic
254C2#1568974911yescharacteristic
255C2D4×Dic9411yescharacteristic
256C2#1568979311yescharacteristic
257C2#1568983011yescharacteristic
258C2#1568986711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC47×#390C2
hypercenter#390D47
hypocenterC47#2732
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#13094001C2semidirect
2. #15689059C2semidirect
3. #15689056C2semidirect
4. #15689046C2semidirect
5. #15689029C2semidirect
6. #15689021C2semidirect
8. #15688998C2semidirect
35. C2×D4×C47Q8semidirect
49. C2×Dic94D4semidirect
139. C47#2732semidirect
168. C2×D4Dic94semidirect