Finite Group Databasebeta

← #30604095

Subgroups — #30604095

Order 15616.

598 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 88 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#21461067C212yesmaximal
2#21460749C214yesmaximal, complemented
3C2×#16839669C211yesmaximal, characteristic
4#16840732—22no—
5#16840607—22no—
6#16840607C2214yes—
7C22×C4⋊Dic61C2211yescharacteristic
8C22×C4×Dic61C411yescharacteristic
9#16839669—22no—
10#16839669C412yes—
11#16839669C2212yes—
12C2×#14519684—22no—
13C23×Q8×C61C411yescharacteristic
14C2×#11758217—42no—
15C22⋊#92420—24no—
16C22⋊#92420—42no—
17C23×Dic61—21no—
18C23×Dic61D412yes—
19C2×C4⋊Dic61—41no—
20C2×C4⋊Dic61—41no—
21C2×C4⋊Dic61C2×C411yescharacteristic
22C2×C4⋊Dic61D412yes—
23C2×C4⋊Dic61C2311yescharacteristic
24C2×C4×Dic61—21no—
25C2×C4×Dic61—42no—
26C2×C4×Dic61C2×C412yes—
27#14519684—24no—
28#14519684—24no—
29C22×Q8×C61—24no—
30C22×Q8×C61—21no—
31C22×Q8×C61—42no—
32C22×Q8×C61—41no—
33C22×Q8×C61—41no—
34C22×Q8×C61C2×C412yes—
35C23×C4×C61—21no—
36C23×C4×C61C2×C411yescharacteristic
37C24×C61C22⋊C411yescharacteristic
38C2×Q8×C61—24no—
39C2×Q8×C61—22no—
40C2×Q8×C61—48no—
41C2×Q8×C61—48no—
42C2×Q8×C61—42no—
43C2×Q8×C61—42no—
44C2×Q8×C61—42no—
45C2×Q8×C61—42no—
46C2×Q8×C61M1614yescomplemented
47C22×C4×C61—21no—
48C22×C4×C61—21no—
49C22×C4×C61—22no—
50C22×C4×C61—22no—
51C22×C4×C61—24no—
52C22×C4×C61—22no—
53C22×C4×C61—42no—
54C22×C4×C61—41no—
55C22×C4×C61—41no—
56C22×C4×C61C22⋊C411yescharacteristic
57C22×C4×C61C22×C411yescharacteristic
58C22×Dic61—24no—
59C22×Dic61—22no—
60C22×Dic61—22no—
61C22×Dic61—22no—
62C22×Dic61—42no—
63C22×Dic61—42no—
64C22×Dic61—44no—
65C22×Dic61C22⋊C412yes—
66C22×Dic61C2×D412yes—
67#11758217—84no—
68C2×#92420—44no—
69C4⋊Dic61—41no—
70C4⋊Dic61—42no—
71C4⋊Dic61—41no—
72C4×Dic61—42no—
73C4×Dic61—42no—
74C23×C61—22no—
75C23×C61—22no—
76C23×C61—21no—
77C23×C61—21no—
78C23×C61#9911yescharacteristic
79C23×C61#9911yescharacteristic
80C23×C61C2×C22⋊C411yescharacteristic
81Q8×C61—48no—
82Q8×C61—44no—
83Q8×C61—44no—
84C2×C4×C61—24no—
85C2×C4×C61—24no—
86C2×C4×C61—21no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—21no—
90C2×C4×C61—24no—
91C2×C4×C61—44no—
92C2×C4×C61—44no—
93C2×C4×C61—42no—
94C2×C4×C61—42no—
95C2×C4×C61#10012yes—
96C2×C4×C61C2×M1612yes—
97C2×Dic61—24no—
98C2×Dic61—22no—
99C2×Dic61—22no—
100C2×Dic61—48no—
101C2×Dic61—42no—
102C2×Dic61—44no—
103C2×Dic61—42no—
104C2×Dic61—42no—
105C2×Dic61—42no—
106C2×Dic61—82no—
107#92420—84no—
108#11506—611nomaximal, Sylow
109C22×C61—24no—
110C22×C61—24no—
111C22×C61—21no—
112C22×C61—21no—
113C22×C61—21no—
114C22×C61—21no—
115C22×C61—21no—
116C22×C61—21no—
117C22×C61#35412yes—
118C22×C61#35612yes—
119C22×C61#40711yescharacteristic
120C22×C61C2×#9911yescharacteristic
121C22×C61C2×#10011yescharacteristic
122C4×C61—22no—
123C4×C61—22no—
124C4×C61—44no—
125Dic61—44no—
126Dic61—42no—
127Dic61—42no—
128C2×#391—611no—
129#1463—612no—
130#1319—614no—
131C2×C61—21no—
132C2×C61—22no—
133C2×C61—22no—
134C2×C61—21no—
135C2×C61#146211yescharacteristic
136C2×C61C2×#35411yescharacteristic
137C2×C61C2×#35611yescharacteristic
138C23×Q8#692311yescharacteristic
139C22×C4⋊C4—611no—
140C22×C42—611no—
141#407—1222no—
142#391—612no—
143#391—612no—
144#391—1222no—
145C2×#95—1222no—
146#324—614no—
147#324—1222no—
148C61#1150611yescharacteristic, complemented, Sylow
149C22×Q8—24no—
150C22×Q8—21no—
151C22×Q8—42no—
152C22×Q8—41no—
153C22×Q8—41no—
154C22×Q8C2×#692312yes—
155C23×C4—21no—
156C23×C4—612no—
157C23×C4—1221no—
158C23×C4C2×#692311yescharacteristic
159C2×M16—2442no—
160C2×C4⋊C4—612no—
161C2×C4⋊C4—611no—
162C2×C4⋊C4—611no—
163C2×C4⋊C4—2441no—
164C2×C4⋊C4—2441no—
165C2×C42—612no—
166C2×C42—1221no—
167C2×C42—2442no—
168C22⋊C8—1224no—
169C22⋊C8—2442no—
170#95—1224no—
171#95—1224no—
172C24C22⋊#692311yescharacteristic
173C2×Q8—24no—
174C2×Q8—22no—
175C2×Q8—48no—
176C2×Q8—48no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8#1175821714yescomplemented
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—612no—
192C22×C4—612no—
193C22×C4—1222no—
194C22×C4—1224no—
195C22×C4—1222no—
196C22×C4—1222no—
197C22×C4—2444no—
198C22×C4—2442no—
199C22×C4—2442no—
200C22×C4C22⋊#692311yescharacteristic
201C22×C4C22×#692311yescharacteristic
202M16—4884no—
203C2×C8—2444no—
204C4⋊C4—2442no—
205C4⋊C4—2441no—
206C4⋊C4—2441no—
207C42—2442no—
208C42—2442no—
209C23—21no—
210C23—21no—
211C23—22no—
212C23—22no—
213C23#1451983511yescharacteristic
214C23#1451983511yescharacteristic
215C23C2×C22⋊#692311yescharacteristic
216Q8—44no—
217Q8—48no—
218Q8—44no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—24no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—1224no—
231C2×C4—1222no—
232C2×C4—1222no—
233C2×C4—4882no—
234C2×C4—2448no—
235C2×C4—2442no—
236C2×C4—2442no—
237C2×C4—2444no—
238C2×C4—2442no—
239C2×C4—2442no—
240C2×C4#1451983712yes—
241C2×C4C2×#1175821712yes—
242C8—4884no—
243C22—24no—
244C22—24no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22#1684066812yes—
252C22#1684067212yes—
253C22#1684073211yescharacteristic
254C22C2×#1451983511yescharacteristic
255C22C2×#1451983711yescharacteristic
256C4—22no—
257C4—22no—
258C4—44no—
259C4—2444no—
260C4—2442no—
261C4—2442no—
262C2—21no—
263C2—21no—
264C2—22no—
265C2—22no—
266C2#2146106411yescharacteristic
267C2C2×#1684066811yescharacteristic
268C2C2×#1684067211yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC23×Q8×C61C4
socleC22×C61C2×#99
prime radicalC24×C61C22⋊C4
cosocleC2×C4⋊Dic61C23
hypercenterC23×Q8#6923
hypocenterC61#11506
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
2. #21460749C2semidirect
46. C2×Q8×C61M16semidirect
148. C61#11506semidirect
181. C2×Q8#11758217semidirect

Maximal subgroups

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

subgroupindex
#214610672
#214607492
C2×#168396692
#1150661