Finite Group Databasebeta

← #30598258

Subgroups — #30598258

Order 15616.

802 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 128 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#21461676C212yesmaximal
2C2×#16840621C214yesmaximal, complemented
3C2×#16840318C211yesmaximal, characteristic
4C2×#14519844C2214yes—
5C2×#14519834—22no—
6#16840628—22no—
7#16840621—28no—
8#16840318—28no—
9C2×#14519776C2212yes—
10C2×#14519769C411yescharacteristic
11C2×#14519755—22no—
12C2×#14519742—22no—
13C2×#14519735C412yes—
14C2×C42×D61C2211yescharacteristic
15C2×#125×C61C411yescharacteristic
16C22×#92420—24no—
17#14519844—28no—
18#14519834—44no—
19C22×Dic122—22no—
20C22×C4×D61C2×C411yescharacteristic
21C22×C4×D61D412yes—
22#14519776—24no—
23#14519776D418yes—
24#14519769—24no—
25#14519755—216no—
26#14519742—216no—
27#14519735—24no—
28#14519735—24no—
29C2×C4⋊Dic61C2×C412yes—
30C2×#11758196—22no—
31C2×#11758196—22no—
32C2×#11758198—22no—
33C2×C4×Dic61—22no—
34C2×C4×Dic61D412yes—
35C2×C4×Dic61C2311yescharacteristic
36C42×D61—24no—
37#125×C61—24no—
38C2×C4⋊C4×C61—22no—
39C2×C4⋊C4×C61C2×C412yes—
40C2×C42×C61C2×C411yescharacteristic
41C23×D61C22⋊C411yescharacteristic
42C22×C4×C61—22no—
43C22×C4×C61C22⋊C412yes—
44C22×C4×C61C22×C411yescharacteristic
45C22×Dic61—22no—
46C22×Dic61C22⋊C411yescharacteristic
47C22×Dic61C2×D412yes—
48C2×C4×D61—22no—
49C2×C4×D61—22no—
50C2×C4×D61—22no—
51C2×C4×D61—24no—
52C2×C4×D61—24no—
53C2×C4×D61—22no—
54C2×C4×D61—22no—
55C2×Dic122—28no—
56C2×Dic122—44no—
57C2×Dic122—42no—
58C2×Dic122—42no—
59C2×#92420—44no—
60C2×#92420—44no—
61C2×#92420—44no—
62C4⋊C4×C61—28no—
63C4⋊C4×C61—22no—
64C4⋊C4×C61C22⋊C414yes—
65C42×C61—22no—
66#11758198—28no—
67C4⋊Dic61—22no—
68C4⋊Dic61C22⋊C414yes—
69#11758196—28no—
70#11758196—28no—
71C4×Dic61—28no—
72C4×Dic61—24no—
73C4×Dic61C2×D414yes—
74C23×C61C2×C22⋊C411yescharacteristic
75C22×D61—21no—
76C22×D61—21no—
77C22×D61—22no—
78C22×D61—22no—
79C22×D61#10112yes—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—22no—
83C2×C4×C61—24no—
84C2×C4×C61—22no—
85C2×C4×C61—22no—
86C2×C4×C61—42no—
87C2×C4×C61—42no—
88C2×C4×C61C2×C22⋊C412yes—
89C2×Dic61—21no—
90C2×Dic61—21no—
91C2×Dic61—24no—
92C2×Dic61—42no—
93C2×Dic61—42no—
94C2×Dic61#10112yes—
95C2×Dic61#12014yes—
96C2×Dic61#13514yes—
97C2×Dic61AGL(1,ℤ/8)12yes—
98C2×Dic61#13712yes—
99C4×D61—28no—
100C4×D61—44no—
101C4×D61—44no—
102Dic122—48no—
103Dic122—48no—
104#92420—84no—
105#8345—611nomaximal, Sylow
106C22×C61#37911yescharacteristic
107C22×C61C2×#10111yescharacteristic
108C22×C61C2×#10111yescharacteristic
109C22×C61#41612yes—
110C22×C61#41712yes—
111C2×D61—24no—
112C2×D61—22no—
113C2×D61—41no—
114C2×D61—41no—
115C2×D61—42no—
116C4×C61—22no—
117C4×C61—44no—
118C4×C61—42no—
119Dic61—22no—
120Dic61—44no—
121Dic61#44914yes—
122Dic61#45214yes—
123C2×#533—611no—
124#1797—612no—
125C2×#332—614no—
126C2×C61#154411yescharacteristic
127C2×C61#155212yes—
128C2×C61#156111yescharacteristic
129C2×C61#178411yescharacteristic
130C2×C61#178812yes—
131D61—42no—
132#533—1228no—
133C2×C4⋊Q8—612no—
134C2×#125—611no—
135C2×#125#692311yescharacteristic
136C2×C22⋊Q8—1222no—
137C2×C4×Q8—1222no—
138C2×#117—612no—
139C22×C42—611no—
140C2×C4⋊C8—614no—
141C2×C22⋊C8—1222no—
142#336—1222no—
143#332—1228no—
144C61#834511yescharacteristic, complemented, Sylow
145C22×Q8—1222no—
146C23×C4—612no—
147C23×C4—611no—
148C22×C8—1224no—
149C4⋊Q8—618no—
150C4⋊Q8—1224no—
151#125—24no—
152#125—1224no—
153C22⋊Q8—12216no—
154C4×Q8—12216no—
155#117—1224no—
156#117—1224no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—612no—
159C2×C4⋊C4—1222no—
160C2×C4⋊C4—1222no—
161C2×C4⋊C4C2×#692312yes—
162C2×C22⋊C4—1222no—
163C2×C42—611no—
164C2×C42—612no—
165C2×C42—1224no—
166C2×C42—1222no—
167C2×C42C2×#692311yescharacteristic
168C4⋊C8—1228no—
169C22⋊C8—2444no—
170C24—611no—
171C2×Q8—1228no—
172C2×Q8—2444no—
173C2×Q8—2442no—
174C2×Q8—2442no—
175C22×C4—22no—
176C22×C4—612no—
177C22×C4—611no—
178C22×C4—1222no—
179C22×C4—1222no—
180C22×C4—1224no—
181C22×C4—1224no—
182C22×C4—1222no—
183C22×C4—1222no—
184C22×C4—1222no—
185C22×C4—1222no—
186C22×C4C22⋊#692312yes—
187C22×C4C22×#692311yescharacteristic
188C2×C8—2444no—
189C2×C8—2444no—
190C2×C8—2444no—
191C4⋊C4—28no—
192C4⋊C4—22no—
193C4⋊C4—614no—
194C4⋊C4—1228no—
195C4⋊C4—1228no—
196C4⋊C4—1222no—
197C4⋊C4C22⋊#692314yes—
198C22⋊C4—1228no—
199C42—22no—
200C42—614no—
201C42—1224no—
202C42—1228no—
203C23—612no—
204C23—1221no—
205C23—1222no—
206C23—1222no—
207C23—1221no—
208C23C2×C22⋊#692311yescharacteristic
209Q8—2448no—
210Q8—2448no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—614no—
220C2×C4—612no—
221C2×C4—612no—
222C2×C4—614no—
223C2×C4—612no—
224C2×C4—1228no—
225C2×C4—1221no—
226C2×C4—1221no—
227C2×C4—1224no—
228C2×C4—2444no—
229C2×C4—2444no—
230C2×C4—2442no—
231C2×C4—2442no—
232C2×C4C2×C22⋊#692312yes—
233C8—4884no—
234C22—1224no—
235C22—1222no—
236C22—2441no—
237C22—2441no—
238C22—2442no—
239C22#1684070311yescharacteristic
240C22D61⋊#10111yescharacteristic
241C22D61⋊#10111yescharacteristic
242C22#1684075212yes—
243C22#1684075312yes—
244C4—22no—
245C4—42no—
246C4—44no—
247C4—614no—
248C4—614no—
249C4—1222no—
250C4—2444no—
251C2—2442no—
252C2#2146132511yescharacteristic
253C2#2146134112yes—
254C2#2146135911yescharacteristic
255C2#2146165411yescharacteristic
256C2#2146165912yes—
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#125×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC2×#125#6923
hypocenterC61#8345
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. C2×#16840621C2semidirect
144. C61#8345semidirect

Maximal subgroups

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

subgroupindex
#214616762
C2×#168406212
C2×#168403182
#834561