Finite Group Databasebeta

← #30599294

Subgroups — #30599294

Order 15616.

514 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 108 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#21461676C212yesmaximal
2#21460697C214yesmaximal, complemented
3#21448225C211yesmaximal, characteristic
4C2×#14519844C2214yes—
5C2×#14519834—22no—
6#16840628—22no—
7C2×#14519735C412yes—
8C2×C42×D61C2211yescharacteristic
9#16839631C2212yes—
10#16839628C411yescharacteristic
11#16839621—22no—
12#16839613—22no—
13C61×#382C411yescharacteristic
14C22×#92420—24no—
15#14519844—28no—
16#14519834—44no—
17C22×C4×D61C2×C411yescharacteristic
18C22×C4×D61D412yes—
19#14519735—44no—
20C2×C4⋊Dic61C2×C412yes—
21C2×#11758196—22no—
22C2×#11758196—22no—
23C2×#11758198—22no—
24C2×C4×Dic61—22no—
25C2×C4×Dic61D412yes—
26C2×C4×Dic61C2311yescharacteristic
27C42×D61—24no—
28#14519684—22no—
29C2×C4⋊C4×C61C2×C412yes—
30C2×C42×C61C2×C411yescharacteristic
31#95×C61—22no—
32C23×D61C22⋊C411yescharacteristic
33C22×C4×C61—22no—
34C22×C4×C61C22⋊C412yes—
35C22×C4×C61C22×C411yescharacteristic
36C22×Dic61—22no—
37C22×Dic61C22⋊C411yescharacteristic
38C22×Dic61C2×D412yes—
39C2×C4×D61—22no—
40C2×C4×D61—22no—
41C2×C4×D61—22no—
42C2×C4×D61—24no—
43C2×C4×D61—24no—
44C2×C4×D61—22no—
45C2×C4×D61—22no—
46C2×#92420—44no—
47C2×#92420—44no—
48C2×#92420—44no—
49C4⋊C4×C61—42no—
50C42×C61—22no—
51#11758198—44no—
52C4⋊Dic61—42no—
53#11758196—44no—
54#11758196—42no—
55#11758196—42no—
56C4×Dic61—24no—
57C4×Dic61—44no—
58C4×Dic61#4114yes—
59C23×C61C2×C22⋊C411yescharacteristic
60C22×D61—21no—
61C22×D61—21no—
62C22×D61—22no—
63C22×D61—22no—
64C22×D61#10012yes—
65C2×C4×C61—21no—
66C2×C4×C61—21no—
67C2×C4×C61—22no—
68C2×C4×C61—22no—
69C2×C4×C61—22no—
70C2×C4×C61—42no—
71C2×C4×C61—42no—
72C2×C4×C61—42no—
73C2×C4×C61#11712yes—
74C2×Dic61—21no—
75C2×Dic61—21no—
76C2×Dic61—42no—
77C2×Dic61—42no—
78C2×Dic61—42no—
79C2×Dic61#10112yes—
80C2×Dic61#12314yes—
81C2×Dic61#13514yes—
82C2×Dic61AGL(1,ℤ/8)12yes—
83C2×Dic61#13712yes—
84C4×D61—28no—
85C4×D61—44no—
86C4×D61—44no—
87#92420—84no—
88#8729—611nomaximal, Sylow
89C22×C61#38111yescharacteristic
90C22×C61C2×#10011yescharacteristic
91C22×C61C2×#10111yescharacteristic
92C22×C61#41612yes—
93C22×C61#41712yes—
94C2×D61—24no—
95C2×D61—22no—
96C2×D61—41no—
97C2×D61—41no—
98C2×D61—42no—
99C4×C61—22no—
100C4×C61—42no—
101C4×C61—82no—
102Dic61—22no—
103Dic61—82no—
104Dic61#48214yes—
105Dic61#48514yes—
106#2285—611no—
107#1797—612no—
108#1296—614no—
109C2×C61#154411yescharacteristic
110C2×C61#155112yes—
111C2×C61#156111yescharacteristic
112C2×C61#182111yescharacteristic
113C2×C61#182512yes—
114D61—42no—
115C2×#117—612no—
116C22×C42—611no—
117C2×C4⋊C8—614no—
118C2×C22⋊C8—1222no—
119#384—612no—
120#382—611no—
121#382#692311yescharacteristic
122#380—1222no—
123C4×C4⋊C4—1222no—
124#336—1222no—
125C61#872911yescharacteristic, complemented, Sylow
126C23×C4—612no—
127C23×C4—611no—
128C22×C8—1224no—
129#117—2444no—
130C2×C4⋊C4—612no—
131C2×C4⋊C4—1222no—
132C2×C4⋊C4—1222no—
133C2×C4⋊C4C2×#692312yes—
134C2×C22⋊C4—1222no—
135C2×C42—611no—
136C2×C42—612no—
137C2×C42—1224no—
138C2×C42—1222no—
139C2×C42C2×#692311yescharacteristic
140C4⋊C8—1228no—
141C22⋊C8—2444no—
142#95—22no—
143#95—1222no—
144C24—611no—
145C22×C4—22no—
146C22×C4—612no—
147C22×C4—611no—
148C22×C4—1222no—
149C22×C4—1222no—
150C22×C4—1224no—
151C22×C4—1224no—
152C22×C4—1222no—
153C22×C4—1222no—
154C22×C4—1222no—
155C22×C4—1222no—
156C22×C4C22⋊#692312yes—
157C22×C4C22×#692311yescharacteristic
158C2×C8—2444no—
159C2×C8—2444no—
160C2×C8—2444no—
161C4⋊C4—42no—
162C4⋊C4—2442no—
163C4⋊C4—2444no—
164C4⋊C4—2442no—
165C4⋊C4—2442no—
166C22⋊C4—2444no—
167C42—22no—
168C42—614no—
169C42—1224no—
170C42—2444no—
171C23—612no—
172C23—1221no—
173C23—1221no—
174C23—1222no—
175C23—1222no—
176C23C2×C22⋊#692311yescharacteristic
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—614no—
186C2×C4—614no—
187C2×C4—612no—
188C2×C4—612no—
189C2×C4—612no—
190C2×C4—1228no—
191C2×C4—1221no—
192C2×C4—1221no—
193C2×C4—2444no—
194C2×C4—2444no—
195C2×C4—2442no—
196C2×C4—2442no—
197C2×C4—2442no—
198C2×C4#1451985612yes—
199C8—4884no—
200C22—1224no—
201C22—1222no—
202C22—2441no—
203C22—2441no—
204C22—2442no—
205C22#1684070511yescharacteristic
206C22D61⋊#10011yescharacteristic
207C22D61⋊#10111yescharacteristic
208C22#1684075212yes—
209C22#1684075312yes—
210C4—22no—
211C4—42no—
212C4—82no—
213C4—614no—
214C4—614no—
215C4—1222no—
216C4—4882no—
217C2—2442no—
218C2#2146132511yescharacteristic
219C2#2146133912yes—
220C2#2146135911yescharacteristic
221C2#2146171911yescharacteristic
222C2#2146172412yes—
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#382C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#382#6923
hypocenterC61#8729
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
2. #21460697C2semidirect
125. C61#8729semidirect

Maximal subgroups

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

subgroupindex
#214616762
#214606972
#214482252
#872961