Finite Group Databasebeta

← #30596909

Subgroups — #30596909

Order 15616.

726 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 176 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×#16840785C212yesmaximal
2#21460697C214yesmaximal, complemented
3#384×D61C211yesmaximal, characteristic
4#16840785—24no—
5#16840785C418yescomplemented
6C2×#14519844C2214yes—
7C2×#14519834—22no—
8C2×#14519832—22no—
9C2×C4⋊C4×D61—22no—
10C2×C4⋊C4×D61C412yes—
11C2×C42×D61C2211yescharacteristic
12#16839632C411yescharacteristic
13#16839631—22no—
14#16839631C2212yes—
15C61×#384C411yescharacteristic
16C22×#92420—24no—
17#14519844—24no—
18#14519844C2×C418yescomplemented
19#14519834—28no—
20#14519832—28no—
21C22×C4×D61—22no—
22C22×C4×D61C2×C411yescharacteristic
23C22×C4×D61D412yes—
24C2×C4⋊Dic61—22no—
25C2×C4⋊Dic61C2×C412yes—
26C2×#11758196—22no—
27C2×#11758196—24no—
28C4⋊C4×D61—44no—
29C4⋊C4×D61—48no—
30C2×C4×Dic61D412yes—
31C2×C4×Dic61C2311yescharacteristic
32C42×D61C2×C414yes—
33C42×D61D412yes—
34C42×D61Q812yes—
35C2×C4⋊C4×C61—22no—
36C2×C4⋊C4×C61C2×C412yes—
37C2×C42×C61C2×C411yescharacteristic
38C23×D61C22⋊C411yescharacteristic
39C22×C4×C61—22no—
40C22×C4×C61C22⋊C412yes—
41C22×C4×C61C22×C411yescharacteristic
42C22×Dic61—22no—
43C22×Dic61C22⋊C411yescharacteristic
44C22×Dic61C2×D412yes—
45C2×C4×D61—22no—
46C2×C4×D61—22no—
47C2×C4×D61—24no—
48C2×C4×D61—24no—
49C2×C4×D61—48no—
50C2×C4×D61—44no—
51C2×C4×D61C4212yes—
52C2×C4×D61C22⋊C414yes—
53C2×C4×D61C4⋊C412yes—
54C2×C4×D61C4⋊C414yes—
55C2×#92420—28no—
56C2×#92420—44no—
57C2×#92420—44no—
58C4⋊C4×C61—44no—
59C4⋊C4×C61—42no—
60C42×C61C4212yes—
61C42×C61C4⋊C412yes—
62C4⋊Dic61—44no—
63C4⋊Dic61—42no—
64#11758196—44no—
65#11758196—48no—
66C4×Dic61C22⋊C414yes—
67C4×Dic61C4⋊C414yes—
68C4×Dic61C22×C412yes—
69C4×Dic61C2×D411yescharacteristic
70C4×Dic61C2×Q811yescharacteristic
71C23×C61C2×C22⋊C411yescharacteristic
72C22×D61—22no—
73C22×D61—22no—
74C22×D61#9512yes—
75C22×D61#10012yes—
76C22×D61#10112yes—
77C2×C4×C61—21no—
78C2×C4×C61—21no—
79C2×C4×C61—22no—
80C2×C4×C61—22no—
81C2×C4×C61—42no—
82C2×C4×C61—44no—
83C2×C4×C61#9514yes—
84C2×C4×C61C2×C4211yescharacteristic
85C2×C4×C61C2×C4⋊C411yescharacteristic
86C2×Dic61—21no—
87C2×Dic61—21no—
88C2×Dic61—42no—
89C2×Dic61—44no—
90C2×Dic61#9512yes—
91C2×Dic61C2×C22⋊C412yes—
92C2×Dic61C2×C4⋊C412yes—
93C2×Dic61AGL(1,ℤ/8)14yes—
94C2×Dic61#13714yes—
95C4×D61—24no—
96C4×D61—24no—
97C4×D61—28no—
98C4×D61—28no—
99C4×D61—88no—
100#92420—48no—
101#8040—611nomaximal, Sylow
102C22×C61C2×#9511yescharacteristic
103C22×C61C2×#10011yescharacteristic
104C22×C61C2×#10111yescharacteristic
105C22×C61#41812yes—
106C22×C61#41912yes—
107C2×D61—22no—
108C2×D61—24no—
109C2×D61—22no—
110C2×D61—24no—
111C2×D61#34314yes—
112C4×C61—24no—
113C4×C61—21no—
114C4×C61—21no—
115C4×C61—82no—
116Dic61—22no—
117Dic61—82no—
118Dic61#41714yes—
119Dic61#42814yes—
120C2×#432—612no—
121C2×#384—611no—
122#1296—614no—
123C2×C61#155511yescharacteristic
124C2×C61#155612yes—
125C2×C61#155811yescharacteristic
126C2×C61#173612yes—
127C2×C61C2×#34311yescharacteristic
128D61—24no—
129C22×C4⋊C4—612no—
130C22×C4⋊C4—1222no—
131C22×C42—611no—
132#432—618no—
133#432—1224no—
134C2×C4⋊C8—614no—
135C2×C22⋊C8—1222no—
136C2×#97—1222no—
137#384—612no—
138#384—611no—
139#384—1222no—
140#384#692311yescharacteristic
141C61#804011yescharacteristic, complemented, Sylow
142C23×C4—612no—
143C23×C4—611no—
144C23×C4—1222no—
145C22×C8—1224no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—612no—
148C2×C4⋊C4—1222no—
149C2×C4⋊C4—1224no—
150C2×C4⋊C4—1222no—
151C2×C4⋊C4—2444no—
152C2×C4⋊C4—2448no—
153C2×C4⋊C4C2×#692312yes—
154C2×C42—612no—
155C2×C42—611no—
156C2×C42—614no—
157C2×C42—612no—
158C2×C42—612no—
159C2×C42C2×#692311yescharacteristic
160C4⋊C8—618no—
161C4⋊C8—1224no—
162C22⋊C8—1228no—
163#97—1228no—
164C24—611no—
165C22×C4—22no—
166C22×C4—612no—
167C22×C4—611no—
168C22×C4—614no—
169C22×C4—614no—
170C22×C4—612no—
171C22×C4—612no—
172C22×C4—1224no—
173C22×C4—1224no—
174C22×C4—1222no—
175C22×C4—1222no—
176C22×C4—1222no—
177C22×C4—2444no—
178C22×C4—2448no—
179C22×C4C22⋊#692312yes—
180C22×C4C22×#692311yescharacteristic
181C2×C8—1228no—
182C2×C8—2444no—
183C2×C8—2444no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—2442no—
187C4⋊C4—2444no—
188C4⋊C4—2448no—
189C4⋊C4—2444no—
190C42—614no—
191C42—614no—
192C42—612no—
193C42—611no—
194C42—611no—
195C42C4×#692312yes—
196C42C4⋊#692312yes—
197C23—612no—
198C23—612no—
199C23—612no—
200C23—1222no—
201C23—1222no—
202C23C2×C22⋊#692311yescharacteristic
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—614no—
210C2×C4—614no—
211C2×C4—612no—
212C2×C4—612no—
213C2×C4—612no—
214C2×C4—1228no—
215C2×C4—1228no—
216C2×C4—1224no—
217C2×C4—1224no—
218C2×C4—1221no—
219C2×C4—1221no—
220C2×C4—4888no—
221C2×C4—2442no—
222C2×C4—2444no—
223C2×C4(C2×#6923)⋊C414yes—
224C2×C4C2×C4×#692311yescharacteristic
225C2×C4C2×C4⋊#692311yescharacteristic
226C8—2448no—
227C22—614no—
228C22—1224no—
229C22—1222no—
230C22—1224no—
231C22—1222no—
232C22C2×(C2×#6923)⋊C411yescharacteristic
233C22D61⋊#10011yescharacteristic
234C22D61⋊#10111yescharacteristic
235C22#1684075412yes—
236C22#1684075512yes—
237C4—24no—
238C4—21no—
239C4—21no—
240C4—82no—
241C4—614no—
242C4—614no—
243C4—1222no—
244C4—4882no—
245C2—1224no—
246C2#2146134711yescharacteristic
247C2#2146134912yes—
248C2#2146135311yescharacteristic
249C2#2146150612yes—
250C2#2146152211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#384C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#384#6923
hypocenterC61#8040
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
2. #21460697C2semidirect
5. #16840785C4semidirect
18. #14519844C2×C4semidirect
141. C61#8040semidirect

Maximal subgroups

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

subgroupindex
C2×#168407852
#214606972
#384×D612
#804061