Finite Group Databasebeta

← #30604203

Subgroups — #30604203

Order 15616.

440 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 70 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#21461075C212yesmaximal
2#21460762C212yesmaximal, complemented
3#21460750C212yesmaximal, complemented
4#21448529C211yesmaximal, characteristic
5#16840732—22no—
6#16840607C2214yes—
7#16840605—22no—
8C2×#14519764C411yescharacteristic
9C2×#14519742C411yescharacteristic
10C2×D4×Dic61C2211yescharacteristic
11#16839668C2211yescharacteristic
12#16839656—22no—
13#16839634—22no—
14#16839631C2211yescharacteristic
15#16839608C411yescharacteristic
16C61×#390C411yescharacteristic
17C2×#11758217—42no—
18C22⋊#92420—24no—
19C22⋊#92420—42no—
20C22×Dic122C2×C411yescharacteristic
21C23×Dic61D412yes—
22#14519764—22no—
23#14519764—22no—
24#14519742—42no—
25D4×Dic61—42no—
26C2×C4⋊Dic61C2311yescharacteristic
27C2×#11758196—21no—
28C2×#11758196—22no—
29C2×C22⋊Dic61—22no—
30C2×C22⋊Dic61D412yes—
31C2×C4×Dic61—21no—
32C2×C4×Dic61C2×C411yescharacteristic
33C2×C4×Dic61C2×C411yescharacteristic
34#14519684—21no—
35#14519684—21no—
36#14519684—22no—
37C22×D4×C61C2×C411yescharacteristic
38C2×C4⋊C4×C61C2×C411yescharacteristic
39C2×C22⋊C4×C61—22no—
40C2×C42×C61C2×C411yescharacteristic
41C24×C61C22⋊C412yes—
42C2×D4×C61—21no—
43C2×D4×C61—21no—
44C2×D4×C61—41no—
45C2×D4×C61—41no—
46C22×C4×C61—21no—
47C22×C4×C61—21no—
48C22×C4×C61C22×C411yescharacteristic
49C22×Dic61—21no—
50C22×Dic61—21no—
51C22×Dic61—22no—
52C22×Dic61—22no—
53C22×Dic61—82no—
54C22×Dic61C22⋊C411yescharacteristic
55C22×Dic61C22⋊C411yescharacteristic
56C22×Dic61C2×D412yes—
57C2×Dic122—21no—
58C2×Dic122—41no—
59C2×Dic122—41no—
60C2×Dic122M1612yescomplemented
61#11758217—84no—
62C2×#92420—44no—
63C4⋊C4×C61—41no—
64C22⋊C4×C61—82no—
65C42×C61—41no—
66C22⋊Dic61—28no—
67C22⋊Dic61—42no—
68C4⋊Dic61—41no—
69#11758196—42no—
70#11758196—82no—
71C4×Dic61—21no—
72C4×Dic61—22no—
73C4×Dic61—42no—
74C4×Dic61M1612yescomplemented
75C23×C61—22no—
76C23×C61—22no—
77C23×C61—22no—
78C23×C61—42no—
79C23×C61—42no—
80C23×C61C2×C22⋊C411yescharacteristic
81D4×C61—42no—
82D4×C61—42no—
83C2×C4×C61—21no—
84C2×C4×C61—21no—
85C2×C4×C61—42no—
86C2×C4×C61—41no—
87C2×C4×C61—41no—
88C2×C4×C61—41no—
89C2×C4×C61—41no—
90C2×C4×C61C2×M1611yescharacteristic
91C2×C4×C61C2×M1611yescharacteristic
92C2×Dic61—22no—
93C2×Dic61—22no—
94C2×Dic61—21no—
95C2×Dic61—22no—
96C2×Dic61—41no—
97C2×Dic61—44no—
98C2×Dic61—41no—
99C2×Dic61—41no—
100C2×Dic61—41no—
101C2×Dic61—41no—
102C2×Dic61—42no—
103C2×Dic61—41no—
104C2×Dic61—82no—
105C2×Dic61#9912yes—
106Dic122—42no—
107Dic122—42no—
108#92420—84no—
109#11535—611nomaximal, Sylow
110C22×C61—21no—
111C22×C61—21no—
112C22×C61—44no—
113C22×C61—42no—
114C22×C61—42no—
115C22×C61—42no—
116C22×C61—82no—
117C22×C61#40711yescharacteristic
118C22×C61C2×#9911yescharacteristic
119C22×C61#41311yescharacteristic
120C4×C61—21no—
121C4×C61—21no—
122C4×C61—81no—
123C4×C61—81no—
124Dic61—22no—
125Dic61—42no—
126Dic61—42no—
127Dic61—41no—
128Dic61—42no—
129#2326—611no—
130#1465—612no—
131#1325—612no—
132#1319—612no—
133C2×C61—21no—
134C2×C61—21no—
135C2×C61—82no—
136C2×C61#146511yescharacteristic
137C2×C61#212711yescharacteristic
138C2×C61#213511yescharacteristic
139C2×#124—611no—
140C2×C4×Q8—611no—
141C2×C4×D4—611no—
142#407—1222no—
143#391—611no—
144#390#692311yescharacteristic
145#388—1222no—
146#385—1222no—
147#384—611no—
148#376—611no—
149#324—614no—
150#323—1222no—
151C61#1153511yescharacteristic, complemented, Sylow
152C22×Q8—611no—
153C22×D4C2×#692311yescharacteristic
154C23×C4—612no—
155C2×M16—2442no—
156#124—1222no—
157#124—1222no—
158C4×Q8—2442no—
159C4×D4—2442no—
160C2×C4⋊C4—611no—
161C2×C4⋊C4—1222no—
162C2×C4⋊C4—1221no—
163C2×C4⋊C4C2×#692311yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—612no—
166C2×C22⋊C4—1222no—
167C2×C42—611no—
168C2×C42—611no—
169C2×C42—1221no—
170C2×C42C2×#692311yescharacteristic
171C22⋊C8—1224no—
172C22⋊C8—2442no—
173#95—1221no—
174#95—1221no—
175#95—1222no—
176C24C22⋊#692312yes—
177C2×Q8—612no—
178C2×Q8—1221no—
179C2×Q8—2441no—
180C2×Q8—2441no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—41no—
184C2×D4—41no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—611no—
188C22×C4—612no—
189C22×C4—611no—
190C22×C4—1222no—
191C22×C4—1222no—
192C22×C4—1221no—
193C22×C4—1221no—
194C22×C4—4882no—
195C22×C4C22×#692311yescharacteristic
196M16—4884no—
197C2×C8—2444no—
198C4⋊C4—41no—
199C4⋊C4—4882no—
200C4⋊C4—2441no—
201C4⋊C4—2442no—
202C22⋊C4—82no—
203C22⋊C4—1228no—
204C22⋊C4—2442no—
205C42—41no—
206C42—612no—
207C42—1222no—
208C42—1221no—
209C42—2442no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×C22⋊#692311yescharacteristic
216Q8—2442no—
217Q8—2442no—
218D4—42no—
219D4—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—612no—
228C2×C4—1222no—
229C2×C4—1222no—
230C2×C4—1221no—
231C2×C4—1222no—
232C2×C4—4882no—
233C2×C4—2441no—
234C2×C4—2444no—
235C2×C4—2441no—
236C2×C4—2441no—
237C2×C4—2441no—
238C2×C4—2441no—
239C2×C4—2442no—
240C2×C4—2441no—
241C2×C4D61⋊M1611yescharacteristic
242C2×C4D61⋊M1611yescharacteristic
243C8—4884no—
244C22—21no—
245C22—21no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22#1684073411yescharacteristic
252C22D61⋊#9911yescharacteristic
253C22#1684074911yescharacteristic
254C4—21no—
255C4—21no—
256C4—81no—
257C4—81no—
258C4—1222no—
259C4—2442no—
260C4—2442no—
261C4—2442no—
262C4—2441no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#2146107711yescharacteristic
267C2#2146209511yescharacteristic
268C2#2146212211yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC22×C61C2×#99
prime radicalC22×D4×C61C2×C4
cosocleC2×C4⋊Dic61C23
hypercenter#390#6923
hypocenterC61#11535
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #21460762C2semidirect
3. #21460750C2semidirect
60. C2×Dic122M16semidirect
74. C4×Dic61M16semidirect
151. C61#11535semidirect

Maximal subgroups

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

subgroupindex
#214610752
#214607622
#214607502
#214485292
#1153561