Finite Group Databasebeta

← #30595648

Subgroups — #30595648

Order 15616.

468 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 90 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#21461747C212yesmaximal, complemented
2#21460640C212yesmaximal, complemented
3#21460636C212yesmaximal
4#21448358C211yesmaximal, characteristic
5#16840628—22no—
6#16840628—22no—
7#16840628C2214yes—
8C2×#14519765C411yescharacteristic
9C2×#14519742C411yescharacteristic
10C2×#14519739C411yescharacteristic
11#16839667C2211yescharacteristic
12#16839663C2211yescharacteristic
13#16839651—22no—
14#16839609—22no—
15C42×Dic61C2211yescharacteristic
16C61×#382C411yescharacteristic
17C22×#92420—24no—
18C22×Dic122C2×C411yescharacteristic
19C22×D244C2×C411yescharacteristic
20C22×C4×D61—21no—
21#14519765—22no—
22#14519765—22no—
23#14519742—42no—
24#14519739—42no—
25C2×#11758196—21no—
26C2×#11758198—22no—
27C2×#11758198—21no—
28C2×#11758198—22no—
29C2×C4×Dic61—21no—
30C2×C4×Dic61C2×C411yescharacteristic
31C2×C4×Dic61D412yes—
32C2×C4×Dic61D412yes—
33C2×C4×Dic61C2311yescharacteristic
34#14519684—22no—
35C2×C4⋊C4×C61C2×C411yescharacteristic
36C2×C4⋊C4×C61C2×C411yescharacteristic
37C2×C42×C61C2×C411yescharacteristic
38#95×C61—22no—
39C23×D61—21no—
40C22×C4×C61—21no—
41C22×C4×C61—21no—
42C22×C4×C61C22⋊C412yes—
43C22×C4×C61C22×C411yescharacteristic
44C22×Dic61—21no—
45C22×Dic61C22⋊C411yescharacteristic
46C22×Dic61C22⋊C411yescharacteristic
47C22×Dic61C2×D412yes—
48C2×D244—21no—
49C2×D244—41no—
50C2×D244—41no—
51C2×D244M1612yes—
52C2×C4×D61—44no—
53C2×C4×D61—42no—
54C2×Dic122—21no—
55C2×Dic122—41no—
56C2×Dic122—41no—
57C2×Dic122M1612yes—
58C2×#92420—44no—
59C2×#92420—44no—
60C2×#92420—44no—
61C4⋊C4×C61—41no—
62C4⋊C4×C61—41no—
63C42×C61—21no—
64C42×C61—21no—
65#11758198—28no—
66#11758198—42no—
67#11758198—44no—
68#11758196—42no—
69C4×Dic61—22no—
70C4×Dic61—24no—
71C4×Dic61—42no—
72C4×Dic61—41no—
73C4×Dic61—42no—
74C23×C61C2×C22⋊C411yescharacteristic
75C22×D61—22no—
76C22×D61—41no—
77C22×D61—41no—
78C22×D61—42no—
79C22×D61—42no—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—22no—
83C2×C4×C61—22no—
84C2×C4×C61—22no—
85C2×C4×C61—41no—
86C2×C4×C61—41no—
87C2×C4×C61—42no—
88C2×C4×C61—42no—
89C2×C4×C61C2×M1611yescharacteristic
90C2×C4×C61C2×M1611yescharacteristic
91C2×Dic61—22no—
92C2×Dic61—22no—
93C2×Dic61—22no—
94C2×Dic61—22no—
95C2×Dic61—21no—
96C2×Dic61—22no—
97C2×Dic61—41no—
98C2×Dic61—41no—
99C2×Dic61#10012yes—
100C2×Dic61#13514yes—
101D244—42no—
102D244—42no—
103C4×D61—84no—
104Dic122—42no—
105Dic122—42no—
106#92420—84no—
107#7649—611nomaximal, Sylow
108C22×C61#40711yescharacteristic
109C22×C61#41011yescharacteristic
110C22×C61C2×#10011yescharacteristic
111C22×C61#41612yes—
112C22×C61#42112yes—
113C2×D61—42no—
114C2×D61—44no—
115C2×D61—81no—
116C2×D61—82no—
117C2×D61—81no—
118C4×C61—21no—
119C4×C61—21no—
120C4×C61—42no—
121C4×C61—81no—
122C4×C61—81no—
123Dic61—24no—
124Dic61—22no—
125Dic61—41no—
126Dic61—42no—
127C4×#124—611no—
128#1839—612no—
129#1272—612no—
130#1270—612no—
131C2×C61#146611yescharacteristic
132C2×C61#148812yes—
133C2×C61#152112yes—
134C2×C61#153311yescharacteristic
135C2×C61#154311yescharacteristic
136D61—82no—
137C2×#124—611no—
138C2×C4×Q8—611no—
139C2×C4×D4—611no—
140#391—611no—
141#390—611no—
142#388—1222no—
143#382#692311yescharacteristic
144C4×C22⋊C4—1222no—
145C43—611no—
146#336—614no—
147#336—1222no—
148#336—1222no—
149C61#764911yescharacteristic, complemented, Sylow
150C22×Q8—611no—
151C22×D4—611no—
152C23×C4—1221no—
153C22×C8—1224no—
154#124—1222no—
155#124—1222no—
156C4×Q8—2442no—
157C4×D4—2442no—
158C2×C4⋊C4—1221no—
159C2×C4⋊C4C2×#692311yescharacteristic
160C2×C4⋊C4C2×#692311yescharacteristic
161C2×C22⋊C4—1222no—
162C2×C22⋊C4—1221no—
163C2×C22⋊C4—1222no—
164C2×C42—612no—
165C2×C42—611no—
166C2×C42—612no—
167C2×C42—611no—
168C2×C42—1221no—
169C2×C42C2×#692311yescharacteristic
170#95—22no—
171#95—1222no—
172C24—1221no—
173C2×Q8—612no—
174C2×Q8—1221no—
175C2×Q8—2441no—
176C2×Q8—2441no—
177C2×D4—612no—
178C2×D4—1221no—
179C2×D4—2441no—
180C2×D4—2441no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—612no—
184C22×C4—611no—
185C22×C4—611no—
186C22×C4—1221no—
187C22×C4—2442no—
188C22×C4—2444no—
189C22×C4C22⋊#692312yes—
190C22×C4C22×#692311yescharacteristic
191C2×C8—2444no—
192C2×C8—2444no—
193C2×C8—2444no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—2442no—
197C22⋊C4—1228no—
198C22⋊C4—2444no—
199C22⋊C4—2442no—
200C42—21no—
201C42—21no—
202C42—1224no—
203C42—1222no—
204C42—2441no—
205C42—2442no—
206C42—2442no—
207C23—1222no—
208C23—2441no—
209C23—2441no—
210C23—2442no—
211C23—2442no—
212C23C2×C22⋊#692311yescharacteristic
213Q8—2442no—
214Q8—2442no—
215D4—2442no—
216D4—2442no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—612no—
227C2×C4—614no—
228C2×C4—1222no—
229C2×C4—1222no—
230C2×C4—1222no—
231C2×C4—1222no—
232C2×C4—1221no—
233C2×C4—1222no—
234C2×C4—4884no—
235C2×C4—2441no—
236C2×C4—2441no—
237C2×C4D61⋊M1611yescharacteristic
238C2×C4D61⋊M1611yescharacteristic
239C8—4884no—
240C22—4882no—
241C22—4881no—
242C22—4881no—
243C22—2442no—
244C22—2444no—
245C22#1684073411yescharacteristic
246C22#1684074311yescharacteristic
247C22D61⋊#10011yescharacteristic
248C22#1684075212yes—
249C22#1684075912yes—
250C4—21no—
251C4—21no—
252C4—42no—
253C4—81no—
254C4—81no—
255C4—1224no—
256C4—1222no—
257C4—2441no—
258C4—2442no—
259C2—4882no—
260C2#2146108011yescharacteristic
261C2#2146115212yes—
262C2#2146126212yes—
263C2#2146129911yescharacteristic
264C2#2146132311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#382C4
socleC23×C61C2×C22⋊C4
prime radicalC22×D244C2×C4
cosocleC2×C4×Dic61C23
hypercenter#382#6923
hypocenterC61#7649
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
1. #21461747C2semidirect
2. #21460640C2semidirect
149. C61#7649semidirect

Maximal subgroups

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

subgroupindex
#214617472
#214606402
#214606362
#214483582
#764961