Finite Group Databasebeta

← #30601829

Subgroups — #30601829

Order 15616.

430 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 96 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×#14519846C211yesmaximal, characteristic
3#21461203C212yesmaximal
4C8×#11758213C211yesmaximal, characteristic
5#21456406C211yesmaximal, characteristic
6#16840781C2211yescharacteristic
7C2×#14519846—21no—
8C4×#11758213C2211yescharacteristic
9C4×C4⋊#6923C2212yes—
10#16840636—21no—
11#16840632—24no—
12#16840616—24no—
13#92420⋊C8—21no—
14C8×#92420—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#14519846—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—22no—
67#11758213Q1614yescomplemented
68C2×C8×C61C22⋊C411yescharacteristic
69C2×C8×C61C22⋊C411yescharacteristic
70C4⋊C4×C61—22no—
71C4⋊C4×C61—22no—
72C42×C61C22×C411yescharacteristic
73#11758198—42no—
74C4⋊Dic61—22no—
75C4⋊Dic61—22no—
76C4⋊Dic61—42no—
77#11758196—42no—
78C4×Dic61—42no—
79C4×Dic61C2×D411yescharacteristic
80C4×Dic61#4111yescharacteristic
81C4×Dic61#4111yescharacteristic
82C2×#92418C22⋊C411yescharacteristic
83C2×#92418C22⋊C411yescharacteristic
84Dic244—42no—
85Dic244—42no—
86C8×D61—22no—
87C8×D61—41no—
88C22×D61#12711yescharacteristic
89C2×#6923—44no—
90C2×#6923—82no—
91C2×#6923—82no—
92Q8×C61—82no—
93C2×C4×C61—42no—
94C2×C4×C61C2×C22⋊C411yescharacteristic
95C2×C4×C61C4×D411yescharacteristic
96C2×C4×C61C4×D411yescharacteristic
97C2×Dic61—42no—
98C2×Dic61#12111yescharacteristic
99C2×Dic61#12111yescharacteristic
100C2×Dic61#12411yescharacteristic
101C4×D61—21no—
102C4×D61—21no—
103C4×D61—41no—
104C4×D61C2×Q1612yes—
105C4×D61#13512yes—
106Dic122—42no—
107Dic122—42no—
108#92420—24no—
109#92420—42no—
110C8×C61—21no—
111C8×C61#10412yes—
112#92418—21no—
113#92418—21no—
114#9718—611nomaximal, Sylow
115C22×C61#39011yescharacteristic
116C2×D61—21no—
117C2×D61—21no—
118C2×D61#49411yescharacteristic
119C2×D61#49511yescharacteristic
120#6923—84no—
121C4×C61—21no—
122C4×C61—82no—
123C4×C61C2×#10411yescharacteristic
124C4×C61#42111yescharacteristic
125C4×C61C4×Q1612yes—
126Dic61—21no—
127Dic61—21no—
128Dic61—82no—
129Dic61#46712yes—
130#3170—611no—
131#1986—612no—
132C4×#107—611no—
133#1503—612no—
134C8×M16—611no—
135C2×C61#196911yescharacteristic
136C2×C61#197611yescharacteristic
137C2×C61#198011yescharacteristic
138D61—21no—
139D61—21no—
140#533—612no—
141#500#692311yescharacteristic, complemented
142#494—611no—
143#487—612no—
144#467—1221no—
145#467—1221no—
146C4×Q16—1221no—
147C4×Q16—1221no—
148#431—611no—
149C2×#107—1221no—
150C4×M16—611no—
151C2×C4×C8—611no—
152C4×C4⋊C4—612no—
153#340—1221no—
154#339—1224no—
155#328—1224no—
156#322—1221no—
157C82—1221no—
158C61#971811yescharacteristic, complemented, Sylow
159C2×Q16—1221no—
160C2×Q16—1221no—
161C2×M16—612no—
162C22×C8—611no—
163C22×C8—611no—
164C4⋊Q8—612no—
165C4⋊Q8C2×#692312yes—
166#125—1222no—
167C22⋊Q8—1222no—
168C22⋊Q8—1222no—
169C4×Q8—1222no—
170C4×Q8—1222no—
171#117—1222no—
172C2×C4⋊C4—1222no—
173C2×C42—611no—
174C2×C42—1222no—
175#107—21no—
176#107—21no—
177#107—1224no—
178C4⋊C8—1221no—
179#103—1222no—
180#103—1222no—
181C22⋊C8—612no—
182#97—1221no—
183C4×C8—611no—
184C4×C8—611no—
185C4×C8—611no—
186C4×C8—1221no—
187C4×C8—1221no—
188C4×C8C2×#692311yescharacteristic
189#95—1222no—
190C2×Q8—22no—
191C2×Q8—1222no—
192C22×C4—611no—
193C22×C4—611no—
194C22×C4—611no—
195C22×C4—2442no—
196Q16—2442no—
197Q16—2442no—
198M16—614no—
199M16—1222no—
200C2×C8—611no—
201C2×C8—611no—
202C2×C8—1222no—
203C2×C8—1222no—
204C2×C8—1222no—
205C2×C8—2441no—
206C2×C8C22⋊#692311yescharacteristic
207C2×C8C22⋊#692311yescharacteristic
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—1224no—
211C4⋊C4—1222no—
212C4⋊C4—1222no—
213C4⋊C4—2442no—
214C4⋊C4—2442no—
215C4⋊C4—2442no—
216C22⋊C4—2442no—
217C42—611no—
218C42—611no—
219C42—611no—
220C42—2442no—
221C42—2444no—
222C42C22×#692311yescharacteristic
223C23—611no—
224Q8—82no—
225Q8—2442no—
226Q8—2442no—
227C2×C4—42no—
228C2×C4—611no—
229C2×C4—612no—
230C2×C4—612no—
231C2×C4—611no—
232C2×C4—611no—
233C2×C4—1221no—
234C2×C4—1221no—
235C2×C4—4882no—
236C2×C4—4882no—
237C2×C4—2441no—
238C2×C4—2444no—
239C2×C4—2442no—
240C2×C4C2×C22⋊#692311yescharacteristic
241C2×C4D4×#692311yescharacteristic
242C2×C4D4×#692311yescharacteristic
243C8—21no—
244C8—1224no—
245C8—1221no—
246C8—1221no—
247C8—2442no—
248C8#1451984212yes—
249C22—611no—
250C22—611no—
251C22—1221no—
252C22—1221no—
253C22#1684071411yescharacteristic
254C4—21no—
255C4—82no—
256C4—612no—
257C4—1221no—
258C4—1221no—
259C4—4884no—
260C4—4882no—
261C4C2×#1451984211yescharacteristic
262C4#1684076111yescharacteristic
263C4Q16×#692312yes—
264C2—1221no—
265C2—1221no—
266C2#2146196511yescharacteristic
267C2#2146197611yescharacteristic
268C2#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#9718
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
67. #11758213Q16semidirect
141. #500#6923semidirect
158. C61#9718semidirect

Maximal subgroups

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

subgroupindex
#214619942
C4×#145198462
#214612032
C8×#117582132
#214564062
#971861