Finite Group Databasebeta

← #30599348

Subgroups — #30599348

Order 15616.

786 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 76 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#21461645C212yesmaximal
2#21460718C214yesmaximal, complemented
3#21448400C211yesmaximal, characteristic
4C2×#14519835—24no—
5#16840703C2214yes—
6C23×C22⋊D61C2211yescharacteristic
7C2×#14519733C412yescomplemented
8#16839662—24no—
9#16839618C411yescharacteristic, complemented
10#16839617C2212yes—
11C61×#379C411yescharacteristic, complemented
12C2×C22⋊#6923—24no—
13C2×C22⋊#6923—28no—
14#14519835—48no—
15C24×D61C2311yescharacteristic
16C22×C22⋊D61—24no—
17C22×C22⋊D61—44no—
18C22×C22⋊D61D414yes—
19C23×Dic61C2×C411yescharacteristic
20#14519733—44no—
21C2×#11758196—22no—
22C2×C22⋊Dic61—22no—
23C2×C22⋊Dic61C2×C412yes—
24C2×#11758198—24no—
25C2×C4×Dic61—22no—
26C25×C61C2×C411yescharacteristic
27C2×C22⋊C4×C61—22no—
28C2×C22⋊C4×C61C2×C412yes—
29C24×C61—22no—
30C24×C61—42no—
31C24×C61—82no—
32C24×C61C22⋊C412yes—
33C24×C61C22×C411yescharacteristic
34C23×D61—42no—
35C23×D61—82no—
36C23×D61C2×D412yes—
37C23×D61#4114yes—
38C22×#6923—44no—
39C22×C4×C61—22no—
40C2×C22⋊D61—28no—
41C2×C22⋊D61—28no—
42C2×C22⋊D61—44no—
43C2×C22⋊D61—416no—
44C22×Dic61—22no—
45C22×Dic61—24no—
46C22×Dic61—22no—
47C22×Dic61C22⋊C412yes—
48C22⋊#6923—48no—
49C22⋊#6923—88no—
50C22⋊C4×C61—42no—
51C22⋊C4×C61—82no—
52C22⋊Dic61—42no—
53C22⋊Dic61—82no—
54#11758198—84no—
55#11758196—44no—
56C4×Dic61—44no—
57C23×C61—22no—
58C23×C61—22no—
59C23×C61—24no—
60C23×C61—42no—
61C23×C61—44no—
62C23×C61—42no—
63C23×C61—42no—
64C23×C61—84no—
65C23×C61—88no—
66C23×C61C2×C22⋊C411yescharacteristic
67C23×C61#11712yes—
68C22×D61—24no—
69C22×D61—44no—
70C22×D61—48no—
71C22×D61—44no—
72C22×D61—88no—
73C22×D61#12314yes—
74C2×#6923—84no—
75C2×#6923—88no—
76C2×C4×C61—44no—
77C2×C4×C61—42no—
78C22⋊D61—416no—
79C2×Dic61—28no—
80C2×Dic61—22no—
81C2×Dic61—22no—
82C2×Dic61—44no—
83C2×Dic61—42no—
84C2×Dic61#9914yes—
85#8748—611nomaximal, Sylow
86C22×C61—22no—
87C22×C61—22no—
88C22×C61—24no—
89C22×C61—44no—
90C22×C61—42no—
91C22×C61—42no—
92C22×C61—42no—
93C22×C61—84no—
94C22×C61—88no—
95C22×C61#38111yescharacteristic
96C22×C61C2×#9912yes—
97C2×D61—24no—
98C2×D61—28no—
99C2×D61—84no—
100C2×D61—88no—
101#6923—164no—
102C4×C61—82no—
103Dic61—24no—
104Dic61—82no—
105#2309—611no—
106#1781—612no—
107#1304—614no—
108C2×C61—22no—
109C2×C61—42no—
110C2×C61—82no—
111C2×C61#181911yescharacteristic
112C2×C61#182812yes—
113D61—44no—
114C23×D4—611no—
115C2×#117—612no—
116C2×#99—1224no—
117#390—1224no—
118#379—611no—
119#379—612no—
120#379—614no—
121#379#692311yescharacteristic, complemented
122C61#874811yescharacteristic, complemented, Sylow
123C25—611no—
124C25C2×#692311yescharacteristic
125C22×D4—614no—
126C22×D4—1224no—
127C22×D4—2444no—
128C23×C4—611no—
129#117—2444no—
130C2×C4⋊C4—1222no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—612no—
133C2×C22⋊C4—1224no—
134C2×C22⋊C4—1222no—
135C2×C22⋊C4—1228no—
136C2×C22⋊C4—1224no—
137C2×C22⋊C4C2×#692312yes—
138C2×C42—1222no—
139#99—2448no—
140C24—22no—
141C24—42no—
142C24—82no—
143C24—612no—
144C24—614no—
145C24—4882no—
146C24—2442no—
147C24C22⋊#692312yes—
148C24C22×#692311yescharacteristic
149C2×D4—1228no—
150C2×D4—1228no—
151C2×D4—2444no—
152C2×D4—24416no—
153C22×C4—22no—
154C22×C4—612no—
155C22×C4—1224no—
156C22×C4—1222no—
157C22×C4—1222no—
158C22×C4—2444no—
159C4⋊C4—2444no—
160C22⋊C4—42no—
161C22⋊C4—82no—
162C22⋊C4—4884no—
163C22⋊C4—4888no—
164C22⋊C4—4882no—
165C22⋊C4—2448no—
166C22⋊C4—2442no—
167C42—2444no—
168C23—22no—
169C23—22no—
170C23—24no—
171C23—42no—
172C23—44no—
173C23—42no—
174C23—42no—
175C23—84no—
176C23—88no—
177C23—614no—
178C23—1224no—
179C23—4888no—
180C23—2444no—
181C23—2448no—
182C23—2444no—
183C23C2×C22⋊#692311yescharacteristic
184C23#1451985612yes—
185D4—24416no—
186C2×C4—44no—
187C2×C4—42no—
188C2×C4—614no—
189C2×C4—1228no—
190C2×C4—1222no—
191C2×C4—1222no—
192C2×C4—4888no—
193C2×C4—4884no—
194C2×C4—2444no—
195C2×C4—2442no—
196C22—22no—
197C22—22no—
198C22—24no—
199C22—44no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—84no—
204C22—88no—
205C22—1224no—
206C22—1228no—
207C22—4884no—
208C22—4888no—
209C22#1684070511yescharacteristic
210C22D61⋊#9912yes—
211C4—82no—
212C4—1224no—
213C4—9764no—
214C4—4882no—
215C2—22no—
216C2—42no—
217C2—82no—
218C2—2444no—
219C2#2146171511yescharacteristic
220C2#2146173012yes—
subgroupG/H
centerC22#16840705
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC61×#379C4
socleC22×C61#381
prime radicalC23×C22⋊D61C22
cosocleC24×D61C23
hypercenter#379#6923
hypocenterC61#8748
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ #379

Complements

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

NH ≅ G/Ntype
2. #21460718C2semidirect
7. C2×#14519733C4semidirect
9. #16839618C4semidirect
11. C61×#379C4semidirect
121. #379#6923semidirect
122. C61#8748semidirect

Maximal subgroups

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

subgroupindex
#214616452
#214607182
#214484002
#874861