Finite Group Databasebeta

← #30601835

Subgroups — #30601835

Order 15616.

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

normal subgroups only

This group has 422 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#21461994C212yesmaximal
2C4×#14519845C211yesmaximal, characteristic
3#21461203C212yesmaximal
4#21461029C211yesmaximal, characteristic
5#21456406C211yesmaximal, characteristic
6#16840781C2211yescharacteristic
7C2×#14519845—21no—
8C4×#11758213C2211yescharacteristic
9C4×C4⋊#6923C2212yes—
10#16840636—21no—
11#16840632—24no—
12#16840616—24no—
13#92420⋊C8—21no—
14#16840603—21no—
15#16840317C2212yes—
16#16840142C411yescharacteristic, complemented
17#16840114C412yescomplemented
18#16840004—21no—
19#16840004—21no—
20#16839856—21no—
21#16839856—21no—
22C4×C8×D61C2211yescharacteristic
23C61×#500C411yescharacteristic, complemented
24C2×#11758213D412yes—
25C2×C4⋊#6923—22no—
26C2×C4×#6923—22no—
27#14519845—24no—
28C4⋊#92420—21no—
29#14519834D412yes—
30#14519832—21no—
31C4×#92420—21no—
32C4×#92420—21no—
33(C2×#6923)⋊C4—22no—
34C2×Dic244—21no—
35C2×Dic244—21no—
36C2×C8×D61D411yescharacteristic
37C2×C8×D61D411yescharacteristic
38#14519777C2×C412yes—
39#14519768—22no—
40#14519757—22no—
41#14519753—22no—
42Q8×Dic61—22no—
43#14519742—22no—
44#14519735—22no—
45C42×D61C2311yescharacteristic
46#14519700—22no—
47#14519700—22no—
48C4×#92418C2×C411yescharacteristic
49C8×Dic61D411yescharacteristic
50C8×Dic61D411yescharacteristic
51C4⋊Q8×C61C2×C412yes—
52#107×C61—21no—
53#107×C61—21no—
54C4×C8×C61C2×C411yescharacteristic
55C22×#6923—42no—
56C2×Q8×C61—22no—
57C2×C4×D61C2×D411yescharacteristic
58C2×C4×D61C2×D411yescharacteristic
59C2×C4×D61C2×D411yescharacteristic
60C2×Dic122—22no—
61C2×#92420—22no—
62C2×#92420—22no—
63C4⋊#6923—24no—
64C4⋊#6923—42no—
65C4×#6923—44no—
66#11758213—24no—
67C2×C8×C61C22⋊C411yescharacteristic
68C2×C8×C61C22⋊C411yescharacteristic
69C4⋊C4×C61—22no—
70C4⋊C4×C61—22no—
71C42×C61C22×C411yescharacteristic
72#11758198—42no—
73C4⋊Dic61—22no—
74C4⋊Dic61—22no—
75C4⋊Dic61—42no—
76#11758196—42no—
77C4×Dic61—42no—
78C4×Dic61C2×D411yescharacteristic
79C4×Dic61#4111yescharacteristic
80C4×Dic61#4111yescharacteristic
81C2×#92418C22⋊C411yescharacteristic
82C2×#92418C22⋊C411yescharacteristic
83Dic244—42no—
84Dic244—42no—
85C8×D61—22no—
86C8×D61—41no—
87C22×D61#12711yescharacteristic
88C2×#6923—44no—
89C2×#6923—82no—
90C2×#6923—82no—
91Q8×C61—82no—
92C2×C4×C61—42no—
93C2×C4×C61C2×C22⋊C411yescharacteristic
94C2×C4×C61C4×D411yescharacteristic
95C2×C4×C61C4×D411yescharacteristic
96C2×Dic61—42no—
97C2×Dic61#12111yescharacteristic
98C2×Dic61#12111yescharacteristic
99C2×Dic61#12411yescharacteristic
100C4×D61—21no—
101C4×D61—21no—
102C4×D61—41no—
103C4×D61#13714yes—
104Dic122—42no—
105Dic122—42no—
106#92420—42no—
107#92420—42no—
108C8×C61—21no—
109C8×C61#10412yes—
110#92418—21no—
111#92418—21no—
112#9720—611nomaximal, Sylow
113C22×C61#39011yescharacteristic
114C2×D61—21no—
115C2×D61—21no—
116C2×D61#49712yes—
117#6923—84no—
118C4×C61—21no—
119C4×C61—82no—
120C4×C61C2×#10411yescharacteristic
121C4×C61#42111yescharacteristic
122C4×C61#44112yes—
123Dic61—21no—
124Dic61—21no—
125Dic61—82no—
126Dic61#47012yes—
127#3170—611no—
128#1986—612no—
129C4×#106—611no—
130#1503—612no—
131#1451—611no—
132C2×C61#197111yescharacteristic
133C2×C61#197611yescharacteristic
134C2×C61#198011yescharacteristic
135D61—22no—
136#533—612no—
137#500#692311yescharacteristic, complemented
138#494—611no—
139#487—612no—
140#467—1221no—
141#467—1221no—
142C4×Q16—1221no—
143C4×Q16—1221no—
144#431—611no—
145C2×#106—1221no—
146C4×M16—611no—
147C2×C4×C8—611no—
148C4×C4⋊C4—612no—
149#340—1221no—
150#339—1224no—
151#328—1224no—
152#322—1221no—
153#322—1221no—
154C61#972011yescharacteristic, complemented, Sylow
155C2×Q16—1221no—
156C2×Q16—1221no—
157C2×M16—612no—
158C22×C8—611no—
159C22×C8—611no—
160C4⋊Q8—612no—
161C4⋊Q8C2×#692312yes—
162#125—1222no—
163C22⋊Q8—1222no—
164C22⋊Q8—1222no—
165C4×Q8—1222no—
166C4×Q8—1222no—
167#117—1222no—
168C2×C4⋊C4—1222no—
169C2×C42—611no—
170C2×C42—1222no—
171#107—21no—
172#107—21no—
173#106—1224no—
174C4⋊C8—1221no—
175#103—1222no—
176#103—1222no—
177C22⋊C8—612no—
178#97—1221no—
179C4×C8—611no—
180C4×C8—611no—
181C4×C8—611no—
182C4×C8—1221no—
183C4×C8—1221no—
184C4×C8C2×#692311yescharacteristic
185#95—1222no—
186C2×Q8—22no—
187C2×Q8—1222no—
188C22×C4—611no—
189C22×C4—611no—
190C22×C4—611no—
191C22×C4—2442no—
192Q16—2442no—
193Q16—2442no—
194M16—1224no—
195C2×C8—611no—
196C2×C8—611no—
197C2×C8—1222no—
198C2×C8—1222no—
199C2×C8—1222no—
200C2×C8—2441no—
201C2×C8C22⋊#692311yescharacteristic
202C2×C8C22⋊#692311yescharacteristic
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—1224no—
206C4⋊C4—1222no—
207C4⋊C4—1222no—
208C4⋊C4—2442no—
209C4⋊C4—2442no—
210C4⋊C4—2442no—
211C22⋊C4—2442no—
212C42—611no—
213C42—611no—
214C42—611no—
215C42—2442no—
216C42—2444no—
217C42C22×#692311yescharacteristic
218C23—611no—
219Q8—82no—
220Q8—2442no—
221Q8—2442no—
222C2×C4—42no—
223C2×C4—611no—
224C2×C4—614no—
225C2×C4—611no—
226C2×C4—611no—
227C2×C4—1221no—
228C2×C4—1221no—
229C2×C4—4882no—
230C2×C4—4882no—
231C2×C4—2441no—
232C2×C4—2444no—
233C2×C4—2442no—
234C2×C4C2×C22⋊#692311yescharacteristic
235C2×C4D4×#692311yescharacteristic
236C2×C4D4×#692311yescharacteristic
237C8—21no—
238C8—1221no—
239C8—1221no—
240C8—2442no—
241C8—2442no—
242C8#1451984212yes—
243C22—612no—
244C22—1221no—
245C22—1221no—
246C22#1684071411yescharacteristic
247C4—21no—
248C4—82no—
249C4—612no—
250C4—1221no—
251C4—1221no—
252C4—4884no—
253C4—4882no—
254C4C2×#1451984211yescharacteristic
255C4#1684076111yescharacteristic
256C4#1684080212yes—
257C2—1222no—
258C2#2146196711yescharacteristic
259C2#2146197611yescharacteristic
260C2#2146198211yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#500C4
socleC22×C61#390
prime radicalC22×D61#127
cosocleC42×D61C23
hypercenter#500#6923
hypocenterC61#9720
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
16. #16840142C4semidirect
17. #16840114C4semidirect
23. C61×#500C4semidirect
137. #500#6923semidirect
154. C61#9720semidirect

Maximal subgroups

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

subgroupindex
#214619942
C4×#145198452
#214612032
#214610292
#214564062
#972061