Finite Group Databasebeta

← #30605689

Subgroups — #30605689

Order 15616.

312 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 76 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
1D61⋊(C4⋊C16)C211yesmaximal, characteristic, complemented
2#21461382C211yesmaximal, characteristic, complemented
3#21460836C212yesmaximal, complemented
4#21460831C212yesmaximal, complemented
5#21456387C211yesmaximal, characteristic
6C2×D61⋊C16—21no—
7#16840784—21no—
8#16840683C2212yes—
9C4⋊#11758187—21no—
10#16840626—21no—
11#16840624C2212yes—
12#16840294—21no—
13#16840163C2211yescharacteristic
14#16840145C2211yescharacteristic
15#16840115—21no—
16#16840004C411yescharacteristic
17#16839995C411yescharacteristic
18#16839990—21no—
19C4×Dic244C411yescharacteristic
20C4×#11758200—21no—
21C4×C8×D61C2211yescharacteristic
22C4×D8×C61C411yescharacteristic
23D61⋊C16—42no—
24C2×#11758187—22no—
25C4⋊#92420—22no—
26#14519834—41no—
27#14519832—41no—
28C2×Dic244C2×C411yescharacteristic
29C2×#11758200—21no—
30C2×C8×D61C2×C411yescharacteristic
31C2×C8×D61D411yescharacteristic
32#14519770—41no—
33#14519768—21no—
34#14519764—21no—
35#14519759—41no—
36#14519757—21no—
37#14519748—21no—
38C4×Dic122—21no—
39C4×C22⋊D61—41no—
40#14519736—41no—
41C42×D61D411yescharacteristic
42#14519711C2×C411yescharacteristic
43#14519708—21no—
44#14519702—21no—
45#14519700—21no—
46#14519698—21no—
47C4×#92418D411yescharacteristic
48C8×Dic61D411yescharacteristic
49C8×Dic61C2311yescharacteristic
50C2×D8×C61C2×C411yescharacteristic
51C4×D4×C61—21no—
52#107×C61C2×C411yescharacteristic
53#102×C61—21no—
54C4×C8×C61C2×C411yescharacteristic
55C2×D4×C61—21no—
56C22×C4×C61—41no—
57C2×C22⋊D61—41no—
58C2×C4×D61C22⋊C411yescharacteristic
59C2×C4×D61D811yescharacteristic
60C2×C4×D61SD1611yescharacteristic
61C2×Dic122—21no—
62C2×#92420—42no—
63D8×C61—41no—
64C2×C8×C61C22⋊C411yescharacteristic
65C2×C8×C61C22×C411yescharacteristic
66C4⋊C4×C61—21no—
67C22⋊C4×C61—41no—
68C42×C61C22⋊C411yescharacteristic
69C22⋊Dic61—41no—
70C22⋊Dic61—41no—
71C22⋊Dic61—41no—
72#11758200—41no—
73#11758200—41no—
74#11758198—41no—
75C4⋊Dic61—21no—
76C4⋊Dic61—41no—
77C4⋊Dic61—41no—
78#11758196—41no—
79#11758196—41no—
80C4×Dic61—41no—
81C4×Dic61D811yescharacteristic
82C4×Dic61SD1611yescharacteristic
83C4×Dic61C2×D411yescharacteristic
84C2×#92418C22⋊C411yescharacteristic
85C2×#92418C2×D411yescharacteristic
86Dic244—41no—
87C8×D61—21no—
88C8×D61—21no—
89C8×D61—41no—
90#11758187—42no—
91C23×C61—41no—
92C22×D61#10211yescharacteristic
93D4×C61—41no—
94D4×C61—41no—
95C2×C4×C61—41no—
96C2×C4×C61—82no—
97C2×C4×C61#10211yescharacteristic
98C2×C4×C61#10211yescharacteristic
99C2×C4×C61C2×C22⋊C411yescharacteristic
100C22⋊D61—82no—
101C2×Dic61—41no—
102C2×Dic61—41no—
103C2×Dic61#10211yescharacteristic
104C2×Dic61C2×D811yescharacteristic
105C2×Dic61C2×SD1611yescharacteristic
106C4×D61—22no—
107C4×D61—21no—
108C4×D61—21no—
109C4×D61—41no—
110Dic122—41no—
111Dic122—41no—
112#92420—82no—
113C8×C61—21no—
114C8×C61—21no—
115#92418—21no—
116#92418AGL(1,ℤ/8)11yescharacteristic
117#92418#13711yescharacteristic
118#11978—611nomaximal, Sylow
119C22×C61—81no—
120C22×C61—81no—
121C22×C61—81no—
122C22×C61C2×#10211yescharacteristic
123C2×D61—21no—
124C2×D61—21no—
125C2×D61#35912yes—
126C4×C61—21no—
127C4×C61—21no—
128C4×C61—81no—
129C4×C61#41311yescharacteristic
130C4×C61#41711yescharacteristic
131Dic61—21no—
132Dic61—21no—
133Dic61—21no—
134Dic61—81no—
135Dic61—81no—
136#3168—611no—
137C2×C4⋊C16—611no—
138#1568—611no—
139#1363—612no—
140#1360—612no—
141C2×C61—161no—
142C2×C61#154711yescharacteristic
143C2×C61C2×#35911yescharacteristic
144C2×C61#214511yescharacteristic
145D61—22no—
146#529—1221no—
147C22×C16—1221no—
148#499—611no—
149#495—611no—
150#488—1221no—
151#467—611no—
152#466—611no—
153#465—1221no—
154C4×Q16—611no—
155C4×SD16—1221no—
156C4×D8#692311yescharacteristic
157#432—1221no—
158C2×C4×C8—611no—
159C4⋊C16—612no—
160C4⋊C16—1221no—
161#335—1221no—
162#334—612no—
163C61#1197811yescharacteristic, complemented, Sylow
164C2×Q16—611no—
165C2×SD16—1221no—
166C2×D8C2×#692311yescharacteristic
167C22×C8—611no—
168C22×C8—611no—
169#126—2441no—
170#125—1221no—
171#124—1221no—
172#123—2441no—
173C22⋊Q8—1221no—
174#121—1221no—
175C4×Q8—1221no—
176C4×D4—21no—
177C4×D4—2441no—
178#117—2441no—
179C2×C42—611no—
180C2×C16—1222no—
181C2×C16—2442no—
182#107—611no—
183#107C2×#692311yescharacteristic
184#106—1221no—
185C4⋊C8—1222no—
186#103—1221no—
187#103—1221no—
188#102—21no—
189#102—1221no—
190C22⋊C8—2441no—
191#97—2441no—
192C4×C8—611no—
193C4×C8—611no—
194C4×C8—611no—
195C4×C8C2×#692311yescharacteristic
196C2×Q8—1221no—
197C2×D4—21no—
198C2×D4—2441no—
199C22×C4—41no—
200C22×C4—611no—
201C22×C4—611no—
202C22×C4—611no—
203Q16—2441no—
204SD16—2441no—
205SD16—2441no—
206D8—41no—
207C2×C8—611no—
208C2×C8—611no—
209C2×C8—1221no—
210C2×C8—1221no—
211C2×C8—2441no—
212C2×C8—2442no—
213C2×C8C22⋊#692311yescharacteristic
214C2×C8C22×#692311yescharacteristic
215C4⋊C4—21no—
216C4⋊C4—1221no—
217C4⋊C4—2441no—
218C4⋊C4—2441no—
219C4⋊C4—2441no—
220C4⋊C4—2441no—
221C22⋊C4—41no—
222C22⋊C4—2441no—
223C22⋊C4—2441no—
224C22⋊C4—2441no—
225C22⋊C4—2441no—
226C42—611no—
227C42—611no—
228C42—611no—
229C42—2441no—
230C42C22⋊#692311yescharacteristic
231C16—2442no—
232C23—41no—
233C23—611no—
234Q8—2441no—
235Q8—2441no—
236D4—41no—
237D4—41no—
238D4—4882no—
239C2×C4—41no—
240C2×C4—82no—
241C2×C4—611no—
242C2×C4—611no—
243C2×C4—611no—
244C2×C4—1222no—
245C2×C4—1221no—
246C2×C4—1221no—
247C2×C4—2441no—
248C2×C4—2441no—
249C2×C4—2441no—
250C2×C4#1451983911yescharacteristic
251C2×C4#1451983911yescharacteristic
252C2×C4C2×C22⋊#692311yescharacteristic
253C8—21no—
254C8—21no—
255C8—611no—
256C8—611no—
257C8—1221no—
258C8—4882no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22—612no—
263C22—1221no—
264C22—1221no—
265C22C2×#1451983911yescharacteristic
266C4—21no—
267C4—21no—
268C4—81no—
269C4—1221no—
270C4—1221no—
271C4—1221no—
272C4—4881no—
273C4—4881no—
274C4#1684074811yescharacteristic
275C4#1684075311yescharacteristic
276C2—161no—
277C2—1222no—
278C2#2146133011yescharacteristic
279C2D61⋊#35911yescharacteristic
280C2#2146214011yescharacteristic
subgroupG/H
centerC22C2×#14519839
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC4×D8×C61C4
socleC22×C61C2×#102
prime radical#16839995C4
cosocleC8×Dic61C23
hypercenterC4×D8#6923
hypocenterC61#11978
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. D61⋊(C4⋊C16)C2semidirect
2. #21461382C2semidirect
3. #21460836C2semidirect
4. #21460831C2semidirect
163. C61#11978semidirect

Maximal subgroups

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

subgroupindex
D61⋊(C4⋊C16)2
#214613822
#214608362
#214608312
#214563872
#1197861