Finite Group Databasebeta

← #30598713

Subgroups — #30598713

Order 15616.

690 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 92 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#21461720C211yesmaximal, characteristic
2#21461716C211yesmaximal, characteristic
3#21461229C214yesmaximal
4C2×#16840373C211yesmaximal, characteristic
5#16840734C2212yes—
6#16840705C2212yes—
7#16840642—22no—
8#16840640—22no—
9#16840616—28no—
10#16840373—28no—
11C2×#14519772—22no—
12C2×#14519769C411yescharacteristic, complemented
13C2×#14519767—22no—
14C2×#14519753C2212yes—
15C2×Q8×Dic61C412yescomplemented
16C2×#14519734C2211yescharacteristic
17C2×C4⋊Q8×C61C411yescharacteristic, complemented
18C2×#11758217—22no—
19C2×C22⋊#6923—22no—
20#14519834—24no—
21(C2×#6923)⋊C4—24no—
22C22×C4×D61C2311yescharacteristic
23#14519772—216no—
24#14519769—24no—
25#14519767—216no—
26#14519753—24no—
27#14519753D418yes—
28Q8×Dic61—24no—
29Q8×Dic61—24no—
30#14519734—24no—
31C2×C4⋊Dic61—22no—
32C2×C4⋊Dic61C2×C412yes—
33C2×#11758196—22no—
34C2×#11758196—22no—
35C2×#11758196D412yes—
36C2×#11758198—22no—
37C2×#11758198D412yes—
38C2×C4×Dic61—22no—
39C2×C4×Dic61C2×C411yescharacteristic
40C22×Q8×C61C2×C412yes—
41C4⋊Q8×C61—24no—
42C2×C4⋊C4×C61—22no—
43C2×C42×C61C2×C411yescharacteristic
44C23×D61C2×D411yescharacteristic
45C22×#6923—42no—
46C2×Q8×C61—22no—
47C2×Q8×C61—44no—
48C2×Q8×C61C22⋊C414yes—
49C22×C4×C61—22no—
50C22×C4×C61C22⋊C412yes—
51C22×C4×C61C22×C411yescharacteristic
52C22×Dic61—22no—
53C22×Dic61C22⋊C412yes—
54C22×Dic61C2×D411yescharacteristic
55C2×C4×D61—42no—
56C2×C4×D61C2×D414yes—
57C22⋊#6923—44no—
58#11758217—44no—
59C2×#92420—42no—
60C4⋊C4×C61—28no—
61C42×C61—22no—
62#11758198—28no—
63#11758198—24no—
64C4⋊Dic61—28no—
65C4⋊Dic61—22no—
66C4⋊Dic61C22⋊C414yes—
67#11758196—28no—
68#11758196—28no—
69#11758196—24no—
70C4×Dic61—28no—
71C4×Dic61—22no—
72C23×C61C2×C22⋊C411yescharacteristic
73C22×D61—22no—
74C22×D61—42no—
75C22×D61#12012yes—
76C2×#6923—82no—
77C2×#6923—84no—
78Q8×C61—44no—
79Q8×C61—44no—
80C2×C4×C61—22no—
81C2×C4×C61—24no—
82C2×C4×C61—22no—
83C2×C4×C61—44no—
84C2×C4×C61—42no—
85C2×C4×C61C2×C22⋊C412yes—
86C2×Dic61—22no—
87C2×Dic61—22no—
88C2×Dic61—24no—
89C2×Dic61—44no—
90C2×Dic61—42no—
91C2×Dic61#12012yes—
92C4×D61—44no—
93#92420—82no—
94#8521—611nomaximal, Sylow
95C22×C61—22no—
96C22×C61#37911yescharacteristic
97C22×C61#41012yes—
98C2×D61—24no—
99C2×D61—22no—
100C2×D61—82no—
101#6923—162no—
102C4×C61—22no—
103C4×C61—42no—
104C4×C61—44no—
105Dic61—22no—
106Dic61—42no—
107Dic61—44no—
108C2×#541—611no—
109#1822—611no—
110#1820—611no—
111#1511—614no—
112C2×C61—22no—
113C2×C61#178211yescharacteristic
114C2×C61#179612yes—
115D61—42no—
116#541—1228no—
117C2×C4⋊Q8#692311yescharacteristic, complemented
118C2×#126—1222no—
119C2×#125—611no—
120C2×#125—1222no—
121C2×C22⋊Q8—612no—
122C2×C4×Q8—612no—
123C2×#117—611no—
124#407—612no—
125#381—612no—
126#343—1222no—
127#342—1222no—
128#328—1228no—
129C61#852111yescharacteristic, complemented, Sylow
130C22×Q8C2×#692312yes—
131C23×C4—611no—
132C2×M16—1222no—
133C4⋊Q8—24no—
134#126—12216no—
135#125—1224no—
136#125—12216no—
137C22⋊Q8—618no—
138C22⋊Q8—1224no—
139C4×Q8—1224no—
140C4×Q8—1224no—
141#117—1224no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—612no—
144C2×C4⋊C4—612no—
145C2×C4⋊C4—1222no—
146C2×C4⋊C4—1222no—
147C2×C4⋊C4—1222no—
148C2×C22⋊C4—612no—
149C2×C22⋊C4—1222no—
150C2×C22⋊C4—1222no—
151C2×C42—611no—
152C2×C42—1222no—
153C2×C42C2×#692311yescharacteristic
154C22⋊C8—1224no—
155#95—1224no—
156C24—611no—
157C2×Q8—22no—
158C2×Q8—44no—
159C2×Q8C22⋊#692314yes—
160C22×C4—22no—
161C22×C4—611no—
162C22×C4—614no—
163C22×C4—612no—
164C22×C4—1222no—
165C22×C4—2442no—
166C22×C4—2442no—
167C22×C4C22⋊#692312yes—
168C22×C4C22×#692311yescharacteristic
169M16—2444no—
170C2×C8—2442no—
171C4⋊C4—28no—
172C4⋊C4—614no—
173C4⋊C4—1224no—
174C4⋊C4—1228no—
175C4⋊C4—1228no—
176C4⋊C4—1228no—
177C4⋊C4—1222no—
178C22⋊C4—1224no—
179C22⋊C4—1228no—
180C22⋊C4—2444no—
181C42—22no—
182C42—1228no—
183C42—1222no—
184C23—612no—
185C23—1222no—
186C23—2442no—
187C23C2×C22⋊#692311yescharacteristic
188Q8—44no—
189Q8—44no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—612no—
196C2×C4—1222no—
197C2×C4—1222no—
198C2×C4—1224no—
199C2×C4—4882no—
200C2×C4—4884no—
201C2×C4—2444no—
202C2×C4—2444no—
203C2×C4—2442no—
204C2×C4C2×C22⋊#692312yes—
205C8—4882no—
206C22—22no—
207C22—1222no—
208C22—1224no—
209C22—4882no—
210C22#1684070311yescharacteristic
211C22#1684074212yes—
212C4—22no—
213C4—44no—
214C4—42no—
215C4—1222no—
216C4—9762no—
217C4—2442no—
218C4—2444no—
219C2—22no—
220C2—2442no—
221C2#2146164911yescharacteristic
222C2#2146167112yes—
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4⋊Q8×C61C4
socleC22×C61#379
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenterC2×C4⋊Q8#6923
hypocenterC61#8521
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
12. C2×#14519769C4semidirect
15. C2×Q8×Dic61C4semidirect
17. C2×C4⋊Q8×C61C4semidirect
117. C2×C4⋊Q8#6923semidirect
129. C61#8521semidirect

Maximal subgroups

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

subgroupindex
#214617202
#214617162
#214612292
C2×#168403732
#852161