Finite Group Databasebeta

← #30604195

Subgroups — #30604195

Order 15616.

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

normal subgroups only

This group has 504 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, complemented
2#21460760C212yesmaximal, complemented
3#21460748C212yesmaximal, complemented
4#21448543C211yesmaximal, characteristic
5#16840732—22no—
6#16840607C2214yes—
7#16840605—22no—
8C2×#14519773C411yescharacteristic
9C2×D4×Dic61C2211yescharacteristic
10C2×#14519739C411yescharacteristic
11#16839664C2211yescharacteristic
12#16839639—22no—
13#16839638—22no—
14#16839627C2211yescharacteristic
15#16839608C411yescharacteristic
16C61×#390C411yescharacteristic
17C2×#11758217—42no—
18C22⋊#92420—24no—
19C22⋊#92420—42no—
20C22×D244C2×C411yescharacteristic
21C22×C22⋊D61—22no—
22C23×Dic61D412yes—
23C22×C4×D61—21no—
24#14519773—22no—
25#14519773—22no—
26D4×Dic61—42no—
27#14519739—42no—
28C2×C4⋊Dic61C2311yescharacteristic
29C2×C22⋊Dic61D412yes—
30C2×#11758198—21no—
31C2×#11758198—22no—
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—
42C23×D61—21no—
43C2×D4×C61—21no—
44C2×D4×C61—21no—
45C2×D4×C61—41no—
46C2×D4×C61—41no—
47C22×C4×C61—21no—
48C22×C4×C61—21no—
49C22×C4×C61C22×C411yescharacteristic
50C2×C22⋊D61—28no—
51C2×C22⋊D61—82no—
52C2×C22⋊D61—82no—
53C22×Dic61—21no—
54C22×Dic61—22no—
55C22×Dic61—22no—
56C22×Dic61—82no—
57C22×Dic61C22⋊C411yescharacteristic
58C22×Dic61C22⋊C411yescharacteristic
59C22×Dic61C2×D412yes—
60C2×D244—21no—
61C2×D244—41no—
62C2×D244—41no—
63C2×D244M1612yescomplemented
64C2×C4×D61—42no—
65C2×C4×D61—82no—
66#11758217—84no—
67C2×#92420—44no—
68C4⋊C4×C61—41no—
69C22⋊C4×C61—82no—
70C42×C61—41no—
71C22⋊Dic61—42no—
72#11758198—42no—
73#11758198—82no—
74C4⋊Dic61—41no—
75C4×Dic61—21no—
76C4×Dic61—22no—
77C4×Dic61M1612yescomplemented
78C23×C61—22no—
79C23×C61—22no—
80C23×C61—22no—
81C23×C61—42no—
82C23×C61—42no—
83C23×C61C2×C22⋊C411yescharacteristic
84C22×D61—22no—
85C22×D61—42no—
86C22×D61—42no—
87C22×D61—41no—
88C22×D61—41no—
89D4×C61—42no—
90D4×C61—42no—
91C2×C4×C61—21no—
92C2×C4×C61—21no—
93C2×C4×C61—41no—
94C2×C4×C61—41no—
95C2×C4×C61—42no—
96C2×C4×C61—41no—
97C2×C4×C61—41no—
98C2×C4×C61C2×M1611yescharacteristic
99C2×C4×C61C2×M1611yescharacteristic
100C22⋊D61—88no—
101C2×Dic61—22no—
102C2×Dic61—21no—
103C2×Dic61—22no—
104C2×Dic61—41no—
105C2×Dic61—44no—
106C2×Dic61—41no—
107C2×Dic61—41no—
108C2×Dic61—42no—
109C2×Dic61—41no—
110C2×Dic61—82no—
111C2×Dic61#9912yes—
112D244—42no—
113D244—42no—
114C4×D61—84no—
115#92420—84no—
116#11533—611nomaximal, Sylow
117C22×C61—21no—
118C22×C61—21no—
119C22×C61—44no—
120C22×C61—42no—
121C22×C61—42no—
122C22×C61—42no—
123C22×C61—82no—
124C22×C61#40711yescharacteristic
125C22×C61C2×#9911yescharacteristic
126C22×C61#41311yescharacteristic
127C2×D61—44no—
128C2×D61—42no—
129C2×D61—82no—
130C2×D61—81no—
131C2×D61—81no—
132C4×C61—21no—
133C4×C61—21no—
134C4×C61—81no—
135C4×C61—81no—
136Dic61—22no—
137Dic61—42no—
138Dic61—42no—
139Dic61—41no—
140#2328—611no—
141#1465—612no—
142#1324—612no—
143#1318—612no—
144C2×C61—21no—
145C2×C61—21no—
146C2×C61—82no—
147C2×C61#146511yescharacteristic
148C2×C61#212711yescharacteristic
149C2×C61#213411yescharacteristic
150D61—82no—
151C2×#127—611no—
152C2×C4×D4—611no—
153C2×C4×D4—611no—
154#407—1222no—
155#390—611no—
156#390#692311yescharacteristic
157#386—1222no—
158#386—1222no—
159#382—611no—
160#376—611no—
161#324—614no—
162#323—1222no—
163C61#1153311yescharacteristic, complemented, Sylow
164C22×D4—611no—
165C22×D4—1222no—
166C22×D4C2×#692311yescharacteristic
167C23×C4—612no—
168C23×C4—1221no—
169C2×M16—2442no—
170#127—1222no—
171#127—1222no—
172C4×D4—2442no—
173C4×D4—2442no—
174C2×C4⋊C4—611no—
175C2×C4⋊C4C2×#692311yescharacteristic
176C2×C22⋊C4—22no—
177C2×C22⋊C4—612no—
178C2×C22⋊C4—1222no—
179C2×C22⋊C4—1221no—
180C2×C42—611no—
181C2×C42—611no—
182C2×C42C2×#692311yescharacteristic
183C22⋊C8—1224no—
184C22⋊C8—2442no—
185#95—1221no—
186#95—1221no—
187#95—1222no—
188C24—1221no—
189C24C22⋊#692312yes—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—612no—
195C2×D4—1228no—
196C2×D4—1221no—
197C2×D4—4882no—
198C2×D4—4882no—
199C2×D4—2441no—
200C2×D4—2441no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—611no—
204C22×C4—612no—
205C22×C4—611no—
206C22×C4—1222no—
207C22×C4—1222no—
208C22×C4—1221no—
209C22×C4—4882no—
210C22×C4—4882no—
211C22×C4—2442no—
212C22×C4C22×#692311yescharacteristic
213M16—4884no—
214C2×C8—2444no—
215C4⋊C4—41no—
216C4⋊C4—2441no—
217C22⋊C4—82no—
218C22⋊C4—4882no—
219C22⋊C4—2442no—
220C22⋊C4—2442no—
221C42—41no—
222C42—612no—
223C42—1222no—
224C42—1221no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—42no—
229C23—42no—
230C23—1222no—
231C23—2442no—
232C23—2442no—
233C23—2441no—
234C23—2441no—
235C23C2×C22⋊#692311yescharacteristic
236D4—42no—
237D4—42no—
238D4—4888no—
239D4—2442no—
240D4—2442no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—612no—
249C2×C4—1222no—
250C2×C4—1221no—
251C2×C4—1222no—
252C2×C4—4884no—
253C2×C4—4882no—
254C2×C4—2441no—
255C2×C4—2444no—
256C2×C4—2441no—
257C2×C4—2441no—
258C2×C4—2442no—
259C2×C4—2441no—
260C2×C4D61⋊M1611yescharacteristic
261C2×C4D61⋊M1611yescharacteristic
262C8—4884no—
263C22—21no—
264C22—21no—
265C22—44no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—82no—
270C22—4882no—
271C22—4881no—
272C22—4881no—
273C22—2444no—
274C22—2442no—
275C22#1684073411yescharacteristic
276C22D61⋊#9911yescharacteristic
277C22#1684074911yescharacteristic
278C4—21no—
279C4—21no—
280C4—81no—
281C4—81no—
282C4—1222no—
283C4—2442no—
284C4—2442no—
285C4—2441no—
286C2—21no—
287C2—21no—
288C2—82no—
289C2—4882no—
290C2#2146107711yescharacteristic
291C2#2146209711yescharacteristic
292C2#2146211911yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC22×C61C2×#99
prime radicalC2×#14519773C4
cosocleC2×C4⋊Dic61C23
hypercenter#390#6923
hypocenterC61#11533
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
1. #21461075C2semidirect
2. #21460760C2semidirect
3. #21460748C2semidirect
63. C2×D244M16semidirect
77. C4×Dic61M16semidirect
163. C61#11533semidirect

Maximal subgroups

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

subgroupindex
#214610752
#214607602
#214607482
#214485432
#1153361