Finite Group Databasebeta

← #30603844

Subgroups — #30603844

Order 15616.

686 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 112 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#21461050C212yesmaximal, complemented
2#21460747C214yesmaximal, complemented
3#21448401C211yesmaximal, characteristic
4#16840778—22no—
5#16840607C2214yescomplemented
6#16840605—22no—
7C2×#14519736C411yescharacteristic
8C2×#14519736C2211yescharacteristic
9#16839665—22no—
10#16839665C412yes—
11#16839665C2212yes—
12#16839618—22no—
13C23×D4×C61C411yescharacteristic
14C4⋊#92420—42no—
15C22⋊#92420—24no—
16C4×#92420—42no—
17#14519736—42no—
18#14519736—42no—
19C2×C4⋊Dic61—21no—
20C2×C4⋊Dic61C2×C411yescharacteristic
21C2×C4⋊Dic61C2311yescharacteristic
22C2×C22⋊Dic61—21no—
23C2×C22⋊Dic61—24no—
24C2×C22⋊Dic61—24no—
25C2×C22⋊Dic61D412yes—
26C2×C4×Dic61C2×C412yes—
27C2×C4×Dic61D412yes—
28C25×C61—21no—
29C22×D4×C61—21no—
30C22×D4×C61—22no—
31C22×D4×C61—42no—
32C22×D4×C61—42no—
33C22×D4×C61C814yescomplemented
34C22×D4×C61C2×C412yes—
35C23×C4×C61C2×C411yescharacteristic
36C24×C61—22no—
37C24×C61—24no—
38C24×C61—42no—
39C24×C61—42no—
40C24×C61—82no—
41C24×C61—82no—
42C24×C61C22⋊C411yescharacteristic
43C2×D4×C61—24no—
44C2×D4×C61—22no—
45C2×D4×C61—48no—
46C2×D4×C61—42no—
47C2×D4×C61—42no—
48C2×D4×C61—48no—
49C2×D4×C61—42no—
50C2×D4×C61—42no—
51C2×D4×C61C2×C814yes—
52C22×C4×C61—22no—
53C22×C4×C61—22no—
54C22×C4×C61C22⋊C411yescharacteristic
55C22×C4×C61C2×C812yes—
56C22×C4×C61M1612yes—
57C22×C4×C61C22×C411yescharacteristic
58C22×Dic61—22no—
59C22×Dic61C22⋊C412yes—
60C22×Dic61C2×D412yes—
61C2×#92420—44no—
62C22⋊Dic61—42no—
63C22⋊Dic61—42no—
64C22⋊Dic61—84no—
65C22⋊Dic61—84no—
66C4⋊Dic61—41no—
67C4⋊Dic61—41no—
68C4⋊Dic61—42no—
69C4×Dic61—22no—
70C4×Dic61—22no—
71C4×Dic61—42no—
72C23×C61—21no—
73C23×C61—21no—
74C23×C61—24no—
75C23×C61—41no—
76C23×C61—41no—
77C23×C61—44no—
78C23×C61—44no—
79C23×C61—48no—
80C23×C61—42no—
81C23×C61—42no—
82C23×C61—88no—
83C23×C61—88no—
84C23×C61—84no—
85C23×C61—84no—
86C23×C61C22⋊C812yes—
87C23×C61C2×C22⋊C411yescharacteristic
88D4×C61—44no—
89D4×C61—44no—
90D4×C61—48no—
91C2×C4×C61—21no—
92C2×C4×C61—21no—
93C2×C4×C61—24no—
94C2×C4×C61—24no—
95C2×C4×C61C22⋊C812yes—
96C2×C4×C61#9912yes—
97C2×C4×C61#10012yes—
98C2×C4×C61C22×C811yescharacteristic
99C2×C4×C61C2×M1611yescharacteristic
100C2×Dic61—22no—
101C2×Dic61—22no—
102C2×Dic61—42no—
103C2×Dic61—42no—
104C2×Dic61—42no—
105C2×Dic61—42no—
106C2×Dic61—42no—
107#92420—84no—
108#11433—611nomaximal, Sylow
109C22×C61—21no—
110C22×C61—21no—
111C22×C61—22no—
112C22×C61—42no—
113C22×C61—42no—
114C22×C61—48no—
115C22×C61—41no—
116C22×C61—41no—
117C22×C61—44no—
118C22×C61—84no—
119C22×C61—816no—
120C22×C61—82no—
121C22×C61—82no—
122C22×C61C2×C22⋊C811yescharacteristic
123C22×C61C2×#9911yescharacteristic
124C22×C61C2×#10011yescharacteristic
125C4×C61—21no—
126C4×C61—21no—
127C4×C61#32314yes—
128Dic61—42no—
129Dic61—42no—
130Dic61—82no—
131#2309—611no—
132#1458—612no—
133#1318—614no—
134C2×C61—21no—
135C2×C61—21no—
136C2×C61—42no—
137C2×C61—84no—
138C2×C61C2×#32311yescharacteristic
139C2×C61#212811yescharacteristic
140C2×C61#213411yescharacteristic
141C23×D4#692311yescharacteristic
142C2×#117—611no—
143C2×#117—611no—
144#431—1222no—
145#390—612no—
146#390—612no—
147#390—1222no—
148#379—1222no—
149#324—614no—
150#323—1222no—
151C61#1143311yescharacteristic, complemented, Sylow
152C25—21no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4#9242014yescomplemented
158C22×D4C2×#692312yes—
159C23×C4C2×#692311yescharacteristic
160#117—2442no—
161#117—2442no—
162C2×C4⋊C4—611no—
163C2×C4⋊C4—611no—
164C2×C4⋊C4—1221no—
165C2×C22⋊C4—612no—
166C2×C22⋊C4—1224no—
167C2×C22⋊C4—1224no—
168C2×C22⋊C4—1221no—
169C2×C42—612no—
170C2×C42—612no—
171C4⋊C8—2442no—
172C22⋊C8—1224no—
173C4×C8—2442no—
174C24—22no—
175C24—24no—
176C24—42no—
177C24—42no—
178C24—82no—
179C24—82no—
180C24C22⋊#692311yescharacteristic
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—48no—
186C2×D4—48no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4C2×#9242014yes—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—612no—
193C22×C4—612no—
194C22×C4—1222no—
195C22×C4C2×#9242012yes—
196C22×C4#1175821712yes—
197C22×C4C22⋊#692311yescharacteristic
198C22×C4C22×#692311yescharacteristic
199C2×C8—2444no—
200C4⋊C4—2441no—
201C4⋊C4—2441no—
202C4⋊C4—2442no—
203C22⋊C4—4884no—
204C22⋊C4—4884no—
205C22⋊C4—2442no—
206C22⋊C4—2442no—
207C42—1222no—
208C42—1222no—
209C42—2442no—
210C23—21no—
211C23—21no—
212C23—24no—
213C23—41no—
214C23—41no—
215C23—44no—
216C23—44no—
217C23—48no—
218C23—42no—
219C23—42no—
220C23—88no—
221C23—88no—
222C23—84no—
223C23—84no—
224C23C22⋊#9242012yes—
225C23C2×C22⋊#692311yescharacteristic
226D4—44no—
227D4—44no—
228D4—48no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—1222no—
234C2×C4—1222no—
235C2×C4—2442no—
236C2×C4—2442no—
237C2×C4—2442no—
238C2×C4—2442no—
239C2×C4—2442no—
240C2×C4C22⋊#9242012yes—
241C2×C4#1451983512yes—
242C2×C4#1451983712yes—
243C2×C4C22×#9242011yescharacteristic
244C2×C4C2×#1175821711yescharacteristic
245C8—4884no—
246C22—21no—
247C22—21no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—48no—
252C22—41no—
253C22—41no—
254C22—44no—
255C22—84no—
256C22—816no—
257C22—82no—
258C22—82no—
259C22C2×C22⋊#9242011yescharacteristic
260C22C2×#1451983511yescharacteristic
261C22C2×#1451983711yescharacteristic
262C4—21no—
263C4—21no—
264C4—4882no—
265C4—2442no—
266C4—2442no—
267C4#1684060514yes—
268C2—21no—
269C2—21no—
270C2—42no—
271C2—84no—
272C2C2×#1684060511yescharacteristic
273C2#2146209811yescharacteristic
274C2#2146211811yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
Fitting, prime radicalC23×D4×C61C4
socleC22×C61C2×#99
cosocleC2×C4⋊Dic61C23
hypercenterC23×D4#6923
hypocenterC61#11433
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #21461050C2semidirect
2. #21460747C2semidirect
5. #16840607C22semidirect
33. C22×D4×C61C8semidirect
151. C61#11433semidirect
157. C22×D4#92420semidirect

Maximal subgroups

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

subgroupindex
#214610502
#214607472
#214484012
#1143361