Finite Group Databasebeta

← #30598703

Subgroups — #30598703

Order 15616.

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

normal subgroups only

This group has 818 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, complemented
2#21461716C211yesmaximal, characteristic, complemented
3#21461224C214yesmaximal, complemented
4C2×#16840366C211yesmaximal, characteristic
5#16840734C2212yes—
6#16840705C2212yes—
7#16840642—22no—
8#16840640—22no—
9#16840613—28no—
10#16840366—28no—
11C2×#14519769C411yescharacteristic, complemented
12C2×#14519758—22no—
13C2×#14519749—22no—
14C2×#14519748C2212yes—
15C2×D4×Dic61C412yescomplemented
16C2×#14519734C2211yescharacteristic
17C2×#127×C61C411yescharacteristic, complemented
18C2×#11758217—22no—
19C2×C22⋊#6923—22no—
20#14519834—24no—
21(C2×#6923)⋊C4—24no—
22C22×C22⋊D61—22no—
23C23×Dic61—22no—
24C22×C4×D61C2311yescharacteristic
25#14519769—24no—
26#14519758—216no—
27#14519749—216no—
28#14519748—24no—
29#14519748D418yes—
30D4×Dic61—24no—
31D4×Dic61—24no—
32#14519734—24no—
33C2×C4⋊Dic61C2×C412yes—
34C2×#11758196—22no—
35C2×#11758196D412yes—
36C2×C22⋊Dic61—22no—
37C2×C22⋊Dic61—22no—
38C2×#11758198D412yes—
39C2×C4×Dic61C2×C411yescharacteristic
40C22×D4×C61—22no—
41C22×D4×C61C2×C412yes—
42#127×C61—24no—
43C2×C42×C61C2×C411yescharacteristic
44C24×C61—22no—
45C23×D61C2×D411yescharacteristic
46C22×#6923—42no—
47C2×D4×C61—28no—
48C2×D4×C61—22no—
49C2×D4×C61—44no—
50C2×D4×C61—84no—
51C2×D4×C61C22⋊C414yes—
52C22×C4×C61C22⋊C412yes—
53C22×C4×C61C22×C411yescharacteristic
54C2×C22⋊D61—28no—
55C2×C22⋊D61—84no—
56C22×Dic61—22no—
57C22×Dic61—28no—
58C22×Dic61—84no—
59C22×Dic61C22⋊C412yes—
60C22×Dic61C2×D411yescharacteristic
61C2×C4×D61—42no—
62C2×C4×D61C2×D414yes—
63C22⋊#6923—44no—
64#11758217—44no—
65C2×#92420—42no—
66C42×C61—22no—
67C22⋊Dic61—28no—
68C22⋊Dic61—28no—
69#11758198—24no—
70C4⋊Dic61—22no—
71C4⋊Dic61C22⋊C414yes—
72#11758196—28no—
73#11758196—24no—
74C4×Dic61—22no—
75C23×C61—24no—
76C23×C61—44no—
77C23×C61—44no—
78C23×C61—44no—
79C23×C61C2×C22⋊C411yescharacteristic
80C22×D61—22no—
81C22×D61—42no—
82C22×D61#12012yes—
83C2×#6923—82no—
84C2×#6923—84no—
85D4×C61—44no—
86D4×C61—44no—
87D4×C61—88no—
88C2×C4×C61—22no—
89C2×C4×C61—22no—
90C2×C4×C61—42no—
91C2×C4×C61C2×C22⋊C412yes—
92C22⋊D61—88no—
93C2×Dic61—22no—
94C2×Dic61—22no—
95C2×Dic61—24no—
96C2×Dic61—44no—
97C2×Dic61—42no—
98C2×Dic61—88no—
99C2×Dic61#12012yes—
100C4×D61—44no—
101#92420—82no—
102#8518—611nomaximal, Sylow
103C22×C61—22no—
104C22×C61—44no—
105C22×C61—44no—
106C22×C61—44no—
107C22×C61—88no—
108C22×C61#37911yescharacteristic
109C22×C61#41012yes—
110C2×D61—24no—
111C2×D61—22no—
112C2×D61—82no—
113#6923—162no—
114C4×C61—22no—
115C4×C61—42no—
116Dic61—22no—
117Dic61—44no—
118Dic61—42no—
119C2×#540—611no—
120#1822—611no—
121#1820—611no—
122#1510—614no—
123C2×C61—22no—
124C2×C61—84no—
125C2×C61#178211yescharacteristic
126C2×C61#179512yes—
127D61—42no—
128#540—1228no—
129C2×#127#692311yescharacteristic, complemented
130C2×#125—611no—
131C2×#123—1222no—
132C2×#121—612no—
133C2×#121—1222no—
134C2×C4×D4—612no—
135C2×#117—611no—
136#407—612no—
137#381—612no—
138#343—1222no—
139#342—1222no—
140#327—1228no—
141C61#851811yescharacteristic, complemented, Sylow
142C22×D4—22no—
143C22×D4—1222no—
144C22×D4C2×#692312yes—
145C23×C4—611no—
146C23×C4—1222no—
147C2×M16—1222no—
148#127—24no—
149#125—1224no—
150#123—12216no—
151#121—618no—
152#121—1224no—
153#121—12216no—
154C4×D4—1224no—
155C4×D4—1224no—
156#117—1224no—
157C2×C4⋊C4—612no—
158C2×C4⋊C4—612no—
159C2×C4⋊C4—1222no—
160C2×C22⋊C4—612no—
161C2×C22⋊C4—1222no—
162C2×C22⋊C4—1222no—
163C2×C22⋊C4—1222no—
164C2×C42—611no—
165C2×C42C2×#692311yescharacteristic
166C22⋊C8—1224no—
167#95—1224no—
168C24—22no—
169C24—611no—
170C2×D4—28no—
171C2×D4—22no—
172C2×D4—44no—
173C2×D4—84no—
174C2×D4—1228no—
175C2×D4—4884no—
176C2×D4C22⋊#692314yes—
177C22×C4—611no—
178C22×C4—614no—
179C22×C4—612no—
180C22×C4—1228no—
181C22×C4—1222no—
182C22×C4—4884no—
183C22×C4—2442no—
184C22×C4—2442no—
185C22×C4C22⋊#692312yes—
186C22×C4C22×#692311yescharacteristic
187M16—2444no—
188C2×C8—2442no—
189C4⋊C4—614no—
190C4⋊C4—1224no—
191C4⋊C4—1228no—
192C4⋊C4—1222no—
193C22⋊C4—1224no—
194C22⋊C4—1228no—
195C22⋊C4—1228no—
196C22⋊C4—2444no—
197C42—22no—
198C42—1222no—
199C23—24no—
200C23—44no—
201C23—44no—
202C23—44no—
203C23—612no—
204C23—1222no—
205C23—2442no—
206C23C2×C22⋊#692311yescharacteristic
207D4—44no—
208D4—44no—
209D4—88no—
210D4—4888no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—612no—
215C2×C4—1222no—
216C2×C4—1222no—
217C2×C4—1224no—
218C2×C4—4882no—
219C2×C4—4884no—
220C2×C4—4888no—
221C2×C4—2444no—
222C2×C4—2444no—
223C2×C4—2442no—
224C2×C4C2×C22⋊#692312yes—
225C8—4882no—
226C22—22no—
227C22—44no—
228C22—44no—
229C22—44no—
230C22—88no—
231C22—1222no—
232C22—1224no—
233C22—4882no—
234C22#1684070311yescharacteristic
235C22#1684074212yes—
236C4—22no—
237C4—42no—
238C4—1222no—
239C4—9762no—
240C4—2442no—
241C4—2444no—
242C2—22no—
243C2—84no—
244C2—2442no—
245C2#2146164911yescharacteristic
246C2#2146166912yes—
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#127×C61C4
socleC22×C61#379
prime radicalC2×#16840366C2
cosocleC22×C4×D61C23
hypercenterC2×#127#6923
hypocenterC61#8518
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #21461720C2semidirect
2. #21461716C2semidirect
3. #21461224C2semidirect
11. C2×#14519769C4semidirect
15. C2×D4×Dic61C4semidirect
17. C2×#127×C61C4semidirect
129. C2×#127#6923semidirect
141. C61#8518semidirect

Maximal subgroups

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

subgroupindex
#214617202
#214617162
#214612242
C2×#168403662
#851861