Finite Group Databasebeta

← #30598194

Subgroups — #30598194

Order 15616.

930 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 164 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#21461121C214yesmaximal, complemented
2#21461050C212yesmaximal
3C2×#16840240C211yesmaximal, characteristic
4#16840778C2214yes—
5Q8⋊#92420—28no—
6#16840607—22no—
7#16840605—22no—
8#16840240—28no—
9C2×Q8×Dic61—22no—
10C2×Q8×Dic61C412yes—
11C2×Q8×Dic61C2212yes—
12C2×#14519736—22no—
13C2×#14519736C411yescharacteristic
14C2×#14519736C2211yescharacteristic
15C23×Q8×C61C411yescharacteristic
16C4⋊#92420—24no—
17C22⋊#92420—24no—
18C4×#92420—24no—
19Q8×Dic61—216no—
20Q8×Dic61—24no—
21Q8×Dic61—24no—
22Q8×Dic61—24no—
23Q8×Dic61D418yes—
24#14519736—216no—
25#14519736—24no—
26#14519736—24no—
27C2×C4⋊Dic61—22no—
28C2×C4⋊Dic61—22no—
29C2×C4⋊Dic61C2×C412yes—
30C2×C4⋊Dic61D412yes—
31C2×C22⋊Dic61—21no—
32C2×C22⋊Dic61D412yes—
33C2×C4×Dic61—21no—
34C2×C4×Dic61—22no—
35C2×C4×Dic61—22no—
36C2×C4×Dic61C2×C411yescharacteristic
37C2×C4×Dic61C2311yescharacteristic
38C22×Q8×C61—24no—
39C22×Q8×C61—24no—
40C22×Q8×C61—21no—
41C22×Q8×C61—22no—
42C22×Q8×C61C814yescomplemented
43C22×Q8×C61C2×C412yes—
44C23×C4×C61—21no—
45C23×C4×C61C2×C411yescharacteristic
46C24×C61C22⋊C411yescharacteristic
47C2×Q8×C61—216no—
48C2×Q8×C61—24no—
49C2×Q8×C61—24no—
50C2×Q8×C61—216no—
51C2×Q8×C61—24no—
52C2×Q8×C61—22no—
53C2×Q8×C61—22no—
54C2×Q8×C61—22no—
55C2×Q8×C61C22⋊C414yes—
56C2×Q8×C61C2×C814yes—
57C2×Q8×C61M1614yes—
58C22×C4×C61—24no—
59C22×C4×C61—24no—
60C22×C4×C61—24no—
61C22×C4×C61—22no—
62C22×C4×C61—22no—
63C22×C4×C61—22no—
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C2×C812yes—
66C22×C4×C61M1612yes—
67C22×C4×C61C22×C411yescharacteristic
68C22×Dic61—22no—
69C22×Dic61C22⋊C412yes—
70C22×Dic61C2×D412yes—
71C2×#92420—44no—
72C22⋊Dic61—24no—
73C22⋊Dic61—24no—
74C4⋊Dic61—28no—
75C4⋊Dic61—22no—
76C4⋊Dic61—28no—
77C4⋊Dic61—24no—
78C4⋊Dic61C22⋊C414yes—
79C4×Dic61—28no—
80C4×Dic61—24no—
81C4×Dic61—22no—
82C4×Dic61—28no—
83C4×Dic61C2×D414yes—
84C23×C61—21no—
85C23×C61—21no—
86C23×C61—42no—
87C23×C61C22⋊C812yes—
88C23×C61C2×C22⋊C411yescharacteristic
89Q8×C61—216no—
90Q8×C61—24no—
91Q8×C61—24no—
92Q8×C61—24no—
93Q8×C61C22⋊C818yes—
94C2×C4×C61—216no—
95C2×C4×C61—24no—
96C2×C4×C61—21no—
97C2×C4×C61—24no—
98C2×C4×C61—24no—
99C2×C4×C61—21no—
100C2×C4×C61—24no—
101C2×C4×C61—24no—
102C2×C4×C61C22⋊C812yes—
103C2×C4×C61C2×C22⋊C412yes—
104C2×C4×C61C22×C811yescharacteristic
105C2×C4×C61C2×M1612yes—
106C2×C4×C61C2×M1611yescharacteristic
107C2×Dic61—22no—
108C2×Dic61—24no—
109C2×Dic61—42no—
110C2×Dic61—44no—
111C2×Dic61—42no—
112C2×Dic61#12014yes—
113#92420—84no—
114#8323—611nomaximal, Sylow
115C22×C61—22no—
116C22×C61—22no—
117C22×C61—42no—
118C22×C61—42no—
119C22×C61—42no—
120C22×C61#37911yescharacteristic
121C22×C61C2×C22⋊C811yescharacteristic
122C22×C61#40711yescharacteristic
123C4×C61—28no—
124C4×C61—22no—
125C4×C61C2×C22⋊C812yes—
126C4×C61#40712yes—
127Dic61—24no—
128Dic61—42no—
129Dic61—44no—
130C2×#519—611no—
131#1479—614no—
132#1458—612no—
133C2×C61—21no—
134C2×C61—21no—
135C2×C61—42no—
136C2×C61#177911yescharacteristic
137C2×C61#179612yes—
138C23×Q8#692311yescharacteristic
139#519—1228no—
140C2×C4×Q8—612no—
141C2×C4×Q8—612no—
142C2×C4×Q8—1222no—
143C2×#117—611no—
144C2×#117—611no—
145C2×#117—1222no—
146#431—614no—
147Q8⋊C8—1228no—
148#324—1222no—
149#323—1222no—
150C61#832311yescharacteristic, complemented, Sylow
151C22×Q8—24no—
152C22×Q8—24no—
153C22×Q8—22no—
154C22×Q8—21no—
155C22×Q8#9242014yescomplemented
156C22×Q8C2×#692312yes—
157C23×C4—21no—
158C23×C4C2×#692311yescharacteristic
159C4×Q8—618no—
160C4×Q8—1224no—
161C4×Q8—1224no—
162C4×Q8—12216no—
163C4×Q8—1224no—
164#117—1224no—
165#117—1224no—
166#117—12216no—
167C2×C4⋊C4—612no—
168C2×C4⋊C4—612no—
169C2×C4⋊C4—1222no—
170C2×C4⋊C4—1222no—
171C2×C22⋊C4—612no—
172C2×C22⋊C4—1221no—
173C2×C42—611no—
174C2×C42—611no—
175C2×C42—1222no—
176C2×C42—1222no—
177C2×C42—1221no—
178C4⋊C8—1224no—
179C22⋊C8—1224no—
180C4×C8—1224no—
181C24C22⋊#692311yescharacteristic
182C2×Q8—24no—
183C2×Q8—216no—
184C2×Q8—24no—
185C2×Q8—216no—
186C2×Q8—24no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8C2×#9242014yes—
191C2×Q8#1175821714yes—
192C2×Q8C22⋊#692314yes—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—612no—
200C22×C4—612no—
201C22×C4—1222no—
202C22×C4C2×#9242012yes—
203C22×C4#1175821712yes—
204C22×C4C22⋊#692311yescharacteristic
205C22×C4C22×#692311yescharacteristic
206C2×C8—2444no—
207C4⋊C4—614no—
208C4⋊C4—1224no—
209C4⋊C4—1228no—
210C4⋊C4—1228no—
211C4⋊C4—1222no—
212C22⋊C4—1224no—
213C22⋊C4—1224no—
214C42—614no—
215C42—1228no—
216C42—1228no—
217C42—1224no—
218C42—1222no—
219C23—21no—
220C23—21no—
221C23—42no—
222C23C22⋊#9242012yes—
223C23C2×C22⋊#692311yescharacteristic
224Q8—216no—
225Q8—24no—
226Q8—24no—
227Q8—24no—
228Q8C22⋊#9242018yes—
229C2×C4—216no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—21no—
233C2×C4—24no—
234C2×C4—21no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—614no—
238C2×C4—1224no—
239C2×C4—1222no—
240C2×C4—2442no—
241C2×C4—2444no—
242C2×C4—2442no—
243C2×C4C22⋊#9242012yes—
244C2×C4C2×C22⋊#692312yes—
245C2×C4C22×#9242011yescharacteristic
246C2×C4C2×#1175821712yes—
247C2×C4C2×#1175821711yescharacteristic
248C8—4884no—
249C22—22no—
250C22—22no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22#1684070311yescharacteristic
255C22C2×C22⋊#9242011yescharacteristic
256C22#1684073211yescharacteristic
257C4—28no—
258C4—22no—
259C4—1224no—
260C4—2442no—
261C4—2444no—
262C4C2×C22⋊#9242012yes—
263C4#1684073212yes—
264C2—21no—
265C2—21no—
266C2—42no—
267C2#2146164111yescharacteristic
268C2#2146167112yes—
subgroupG/H
centerC22#16840703
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC23×Q8×C61C4
socleC22×C61#379
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC23×Q8#6923
hypocenterC61#8323
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. #21461121C2semidirect
42. C22×Q8×C61C8semidirect
150. C61#8323semidirect
155. C22×Q8#92420semidirect

Maximal subgroups

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

subgroupindex
#214611212
#214610502
C2×#168402402
#832361