Finite Group Databasebeta

← #30597782

Subgroups — #30597782

Order 15616.

528 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 116 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#21461536C212yesmaximal, complemented
2C2×#16840617C212yesmaximal
3#21460695C212yesmaximal, complemented
4#21448366C211yesmaximal, characteristic
5C2×#14519832C2214yes—
6#16840628—22no—
7#16840628—22no—
8#16840617C418yescomplemented
9C2×#14519764C2211yescharacteristic
10C2×#14519742C411yescharacteristic
11C2×D4×Dic61C411yescharacteristic
12#16839668C411yescharacteristic
13#16839650—22no—
14#16839627C2211yescharacteristic
15C22⋊C4×Dic61—22no—
16C42×Dic61C2211yescharacteristic
17C61×#390C411yescharacteristic
18C22×#92420—24no—
19#14519832—24no—
20#14519832C2×C418yescomplemented
21C22×Dic122C2×C411yescharacteristic
22C23×Dic61—21no—
23#14519764—22no—
24#14519764C2×C414yes—
25#14519742—42no—
26D4×Dic61—42no—
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×#11758196—21no—
29C2×C22⋊Dic61—22no—
30C2×C22⋊Dic61—21no—
31C2×C4×Dic61—21no—
32C2×C4×Dic61C2×C411yescharacteristic
33C2×C4×Dic61D412yes—
34C2×C4×Dic61D412yes—
35C2×C4×Dic61C2311yescharacteristic
36#14519684—22no—
37#14519684—22no—
38C22×D4×C61C2×C411yescharacteristic
39C2×C4⋊C4×C61C2×C411yescharacteristic
40C2×C22⋊C4×C61—22no—
41C2×C42×C61C2×C411yescharacteristic
42C24×C61—21no—
43C2×D4×C61—21no—
44C2×D4×C61—41no—
45C2×D4×C61—41no—
46C2×D4×C61C4212yes—
47C22×C4×C61—21no—
48C22×C4×C61C22⋊C412yes—
49C22×C4×C61C22×C411yescharacteristic
50C22×Dic61—21no—
51C22×Dic61—21no—
52C22×Dic61—44no—
53C22×Dic61—42no—
54C22×Dic61C22⋊C411yescharacteristic
55C22×Dic61C22⋊C411yescharacteristic
56C22×Dic61C2×D412yes—
57C2×Dic122—21no—
58C2×Dic122—41no—
59C2×Dic122—41no—
60C2×Dic122C4212yes—
61C2×#92420—28no—
62C2×#92420—44no—
63C2×#92420—44no—
64C4⋊C4×C61—41no—
65C22⋊C4×C61—44no—
66C42×C61—41no—
67C22⋊Dic61—28no—
68C22⋊Dic61—42no—
69C4⋊Dic61—41no—
70#11758196—42no—
71C4×Dic61—22no—
72C4×Dic61—42no—
73C4×Dic61—42no—
74C4×Dic61—42no—
75C4×Dic61C22×C412yes—
76C4×Dic61#4112yes—
77C23×C61—22no—
78C23×C61—41no—
79C23×C61—41no—
80C23×C61—42no—
81C23×C61—42no—
82C23×C61C2×C22⋊C411yescharacteristic
83D4×C61—42no—
84D4×C61—42no—
85C2×C4×C61—21no—
86C2×C4×C61—21no—
87C2×C4×C61—22no—
88C2×C4×C61—22no—
89C2×C4×C61—22no—
90C2×C4×C61—41no—
91C2×C4×C61—42no—
92C2×C4×C61C2×C4211yescharacteristic
93C2×C4×C61#11711yescharacteristic
94C2×Dic61—22no—
95C2×Dic61—22no—
96C2×Dic61—22no—
97C2×Dic61—22no—
98C2×Dic61—21no—
99C2×Dic61—22no—
100C2×Dic61—41no—
101C2×Dic61—41no—
102C2×Dic61—41no—
103C2×Dic61—42no—
104C2×Dic61—84no—
105C2×Dic61#10012yes—
106C2×Dic61C4×D414yes—
107Dic122—42no—
108Dic122—42no—
109#92420—48no—
110#8224—611nomaximal, Sylow
111C22×C61—42no—
112C22×C61—44no—
113C22×C61—81no—
114C22×C61—81no—
115C22×C61—82no—
116C22×C61C4×C22⋊C411yescharacteristic
117C22×C61C2×#10011yescharacteristic
118C22×C61#41311yescharacteristic
119C22×C61#42112yes—
120C22×C61#42112yes—
121C4×C61—21no—
122C4×C61—21no—
123C4×C61—42no—
124C4×C61—81no—
125Dic61—24no—
126Dic61—22no—
127Dic61—41no—
128Dic61—42no—
129Dic61—81no—
130C4×#124—611no—
131#1745—612no—
132C2×#329—612no—
133#1295—612no—
134C2×C61—82no—
135C2×C61#153311yescharacteristic
136C2×C61#153512yes—
137C2×C61#153611yescharacteristic
138C2×C61C4×#10011yescharacteristic
139C2×C61#175912yes—
140C2×#124—611no—
141C2×C4×Q8—611no—
142C2×C4×D4—611no—
143C2×#97—614no—
144#391—611no—
145#390#692311yescharacteristic
146#388—1222no—
147#382—611no—
148C4×C22⋊C4—1222no—
149C43—611no—
150#336—1222no—
151#336—1222no—
152#329—618no—
153C61#822411yescharacteristic, complemented, Sylow
154C22×Q8—611no—
155C22×D4C2×#692311yescharacteristic
156C23×C4—1221no—
157C22×C8—1224no—
158#124—614no—
159#124—1222no—
160C4×Q8—2442no—
161C4×D4—2442no—
162C2×C4⋊C4—611no—
163C2×C4⋊C4—1221no—
164C2×C4⋊C4C2×#692311yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4—1222no—
167C2×C22⋊C4—1221no—
168C2×C42—611no—
169C2×C42—611no—
170C2×C42—612no—
171C2×C42—612no—
172C2×C42—1221no—
173C2×C42C2×#692311yescharacteristic
174#97—618no—
175#97—1224no—
176#95—1222no—
177#95—1222no—
178C24—21no—
179C2×Q8—612no—
180C2×Q8—1221no—
181C2×Q8—2441no—
182C2×Q8—2441no—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4C4×#692312yes—
187C22×C4—21no—
188C22×C4—612no—
189C22×C4—611no—
190C22×C4—611no—
191C22×C4—1221no—
192C22×C4—1221no—
193C22×C4—2442no—
194C22×C4—2444no—
195C22×C4C22⋊#692312yes—
196C22×C4C22×#692311yescharacteristic
197C2×C8—1228no—
198C2×C8—2444no—
199C2×C8—2444no—
200C4⋊C4—41no—
201C4⋊C4—2441no—
202C4⋊C4—2442no—
203C22⋊C4—44no—
204C22⋊C4—1228no—
205C22⋊C4—2442no—
206C42—41no—
207C42—612no—
208C42—612no—
209C42—1222no—
210C42—2442no—
211C42—2442no—
212C42—2442no—
213C23—22no—
214C23—41no—
215C23—41no—
216C23—42no—
217C23—42no—
218C23C2×C22⋊#692311yescharacteristic
219Q8—2442no—
220Q8—2442no—
221D4—42no—
222D4—42no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—612no—
231C2×C4—614no—
232C2×C4—1222no—
233C2×C4—1222no—
234C2×C4—1222no—
235C2×C4—1221no—
236C2×C4—1222no—
237C2×C4—1222no—
238C2×C4—4884no—
239C2×C4—2441no—
240C2×C4—2442no—
241C2×C4—2441no—
242C2×C4—2441no—
243C2×C4C2×C4×#692311yescharacteristic
244C2×C4#1451985611yescharacteristic
245C8—2448no—
246C22—42no—
247C22—44no—
248C22—81no—
249C22—82no—
250C22—81no—
251C22C4×C22⋊#692311yescharacteristic
252C22D61⋊#10011yescharacteristic
253C22#1684074911yescharacteristic
254C22#1684075912yes—
255C22#1684076012yes—
256C4—21no—
257C4—21no—
258C4—42no—
259C4—81no—
260C4—1222no—
261C4—1224no—
262C4—4881no—
263C4—2441no—
264C4—2442no—
265C2—82no—
266C2#2146129911yescharacteristic
267C2#2146130412yes—
268C2#2146130811yescharacteristic
269C2#2146156911yescharacteristic
270C2#2146158212yes—
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC22×D4×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#8224
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #21461536C2semidirect
3. #21460695C2semidirect
8. #16840617C4semidirect
20. #14519832C2×C4semidirect
153. C61#8224semidirect

Maximal subgroups

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

subgroupindex
#214615362
C2×#168406172
#214606952
#214483662
#822461