Finite Group Databasebeta

← #30607253

Subgroups — #30607253

Order 15616.

642 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 80 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#21461648C212yesmaximal
2#21460914C214yesmaximal, complemented
3#21448412C211yesmaximal, characteristic
4C2×#14519836C2214yes—
5C2×#14519835—22no—
6#16840705—22no—
7C22×#11758222C2211yescharacteristic
8C2×#14519736C411yescharacteristic, complemented
9#16839667C412yescomplemented
10#16839663C2212yes—
11#16839641—22no—
12#16839619—22no—
13C22×C4⋊C4×C61C411yescharacteristic, complemented
14C2×C22⋊#6923—24no—
15#14519836—28no—
16#14519835—44no—
17(C2×#6923)⋊C4—44no—
18C2×#11758222—24no—
19C2×#11758222—42no—
20C2×#11758222—42no—
21C22×Dic122C2×C411yescharacteristic
22C22×D244C2311yescharacteristic
23C22×C22⋊D61D412yes—
24C22×C4×D61D412yes—
25#14519736—42no—
26C2×C4⋊Dic61—21no—
27C2×C22⋊Dic61—21no—
28C2×#11758198—24no—
29C2×C4×Dic61C2×C412yes—
30#14519684—24no—
31C23×C4×C61—21no—
32C23×C4×C61C2×C411yescharacteristic
33C2×C4⋊C4×C61—21no—
34C2×C4⋊C4×C61—42no—
35C2×C4⋊C4×C61C2×C412yes—
36C24×C61C22⋊C411yescharacteristic
37C23×D61C2×D412yes—
38C22×#6923—44no—
39C22×C4×C61—22no—
40C22×C4×C61—22no—
41C22×C4×C61—22no—
42C22×C4×C61—22no—
43C22×C4×C61—42no—
44C22×C4×C61—82no—
45C22×C4×C61C22⋊C411yescharacteristic
46C22×C4×C61C22×C411yescharacteristic
47C2×C22⋊D61—24no—
48C2×C22⋊D61—24no—
49C2×C22⋊D61—42no—
50C22×Dic61—22no—
51C22×Dic61C22⋊C412yes—
52#11758222—416no—
53C2×D244—41no—
54C2×D244—41no—
55C2×D244#4114yes—
56C2×C4×D61—24no—
57C2×C4×D61—24no—
58C2×C4×D61—42no—
59C2×Dic122—22no—
60C2×Dic122—41no—
61C2×Dic122—41no—
62C22⋊#6923—44no—
63C22⋊#6923—44no—
64C4⋊C4×C61—42no—
65C4⋊C4×C61—42no—
66C22⋊Dic61—42no—
67#11758198—84no—
68C4⋊Dic61—42no—
69C4×Dic61—42no—
70C23×C61—21no—
71C23×C61—21no—
72C23×C61—41no—
73C23×C61—41no—
74C23×C61#9912yes—
75C23×C61C2×C22⋊C411yescharacteristic
76C22×D61—22no—
77C22×D61—22no—
78C22×D61—42no—
79C22×D61—42no—
80C22×D61#12314yes—
81C2×#6923—84no—
82C2×#6923—84no—
83C2×#6923—84no—
84C2×C4×C61—28no—
85C2×C4×C61—21no—
86C2×C4×C61—21no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—44no—
90C2×C4×C61—42no—
91C2×C4×C61—84no—
92C2×C4×C61#9912yes—
93C2×C4×C61#11712yes—
94C22⋊D61—48no—
95C2×Dic61—22no—
96C2×Dic61—22no—
97C2×Dic61—22no—
98C2×Dic61—42no—
99C2×Dic61—42no—
100C2×Dic61—42no—
101D244—44no—
102C4×D61—48no—
103Dic122—44no—
104#12859—611nomaximal, Sylow
105C22×C61—21no—
106C22×C61—21no—
107C22×C61—22no—
108C22×C61—42no—
109C22×C61—42no—
110C22×C61—41no—
111C22×C61—41no—
112C22×C61#38111yescharacteristic
113C22×C61C2×#9911yescharacteristic
114C22×C61C2×#9911yescharacteristic
115C2×D61—24no—
116C2×D61—42no—
117C2×D61—42no—
118C2×D61—44no—
119C2×D61—82no—
120#6923—164no—
121C4×C61—22no—
122C4×C61—22no—
123C4×C61—82no—
124Dic61—42no—
125Dic61—82no—
126#2311—611no—
127#1782—612no—
128#1393—614no—
129C2×C61—21no—
130C2×C61—21no—
131C2×C61—42no—
132C2×C61#181911yescharacteristic
133C2×C61#212811yescharacteristic
134C2×C61#212911yescharacteristic
135D61—44no—
136C22×#41—611no—
137C2×#117—611no—
138C22×C4⋊C4#692311yescharacteristic, complemented
139C2×#99—614no—
140C2×#99—1222no—
141#391—612no—
142#390—612no—
143#386—1222no—
144#381—1222no—
145#380—1222no—
146C61#1285911yescharacteristic, complemented, Sylow
147C2×#41—1224no—
148C2×#41—2442no—
149C2×#41—2442no—
150C22×Q8—611no—
151C22×D4—612no—
152C22×D4—611no—
153C23×C4—21no—
154C23×C4—612no—
155C23×C4C2×#692311yescharacteristic
156#117—2442no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—1221no—
160C2×C4⋊C4C2×#692312yes—
161C2×C22⋊C4—1224no—
162C2×C22⋊C4—1221no—
163C2×C22⋊C4—1224no—
164C2×C42—612no—
165#99—1228no—
166#99—2444no—
167#95—1224no—
168#95—2444no—
169C24—612no—
170C24C22⋊#692311yescharacteristic
171#41—24416no—
172C2×Q8—1222no—
173C2×Q8—2441no—
174C2×Q8—2441no—
175C2×D4—614no—
176C2×D4—1224no—
177C2×D4—1224no—
178C2×D4—2441no—
179C2×D4—2441no—
180C2×D4—2442no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—42no—
186C22×C4—82no—
187C22×C4—612no—
188C22×C4—1224no—
189C22×C4—1224no—
190C22×C4—1222no—
191C22×C4—2442no—
192C22×C4—2444no—
193C22×C4C22⋊#692311yescharacteristic
194C22×C4C22×#692311yescharacteristic
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—2442no—
198C22⋊C4—4884no—
199C22⋊C4—2444no—
200C22⋊C4—2444no—
201C22⋊C4—2442no—
202C42—2442no—
203C23—21no—
204C23—21no—
205C23—41no—
206C23—41no—
207C23—614no—
208C23—1222no—
209C23—1222no—
210C23—2442no—
211C23—2442no—
212C23#1451983612yes—
213C23C2×C22⋊#692311yescharacteristic
214Q8—2444no—
215D4—2444no—
216D4—2448no—
217C2×C4—28no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—84no—
225C2×C4—1222no—
226C2×C4—1222no—
227C2×C4—1222no—
228C2×C4—4884no—
229C2×C4—4884no—
230C2×C4—4884no—
231C2×C4—2448no—
232C2×C4—2442no—
233C2×C4—2442no—
234C2×C4—2442no—
235C2×C4#1451983512yes—
236C2×C4#1451985612yes—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—42no—
242C22—41no—
243C22—41no—
244C22—1224no—
245C22—4882no—
246C22—2444no—
247C22—2442no—
248C22—2442no—
249C22#1684070511yescharacteristic
250C22C2×#1451983511yescharacteristic
251C22C2×#1451983611yescharacteristic
252C4—22no—
253C4—22no—
254C4—82no—
255C4—9764no—
256C4—4882no—
257C4—2442no—
258C2—21no—
259C2—21no—
260C2—42no—
261C2—2444no—
262C2#2146171411yescharacteristic
263C2#2146209811yescharacteristic
264C2#2146210111yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×C4⋊C4×C61C4
socleC22×C61C2×#99
prime radicalC22×#11758222C22
cosocleC22×D244C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#12859
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #21460914C2semidirect
8. C2×#14519736C4semidirect
9. #16839667C4semidirect
13. C22×C4⋊C4×C61C4semidirect
138. C22×C4⋊C4#6923semidirect
146. C61#12859semidirect

Maximal subgroups

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

subgroupindex
#214616482
#214609142
#214484122
#1285961