Finite Group Databasebeta

← #30606567

Subgroups — #30606567

Order 15616.

512 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C2×#16840670C212yesmaximal
2#21461572C212yesmaximal, complemented
3#21460916C212yesmaximal, complemented
4#21448529C211yesmaximal, characteristic
5C2×#14519838C2214yes—
6C4×#11758217—22no—
7#16840670C418yescomplemented
8#16840642—22no—
9C2×#14519764C2211yescharacteristic
10C2×#14519742C2211yescharacteristic
11C2×D4×Dic61C411yescharacteristic
12#16839668C2211yescharacteristic
13#16839656—22no—
14#16839634—22no—
15#16839631C411yescharacteristic
16#16839608C411yescharacteristic
17C61×#390C411yescharacteristic
18C2×#11758217—24no—
19#14519838—24no—
20#14519838C2×C418yescomplemented
21#14519832—42no—
22C4×#92420—42no—
23C22×Dic122C2311yescharacteristic
24C23×Dic61—21no—
25#14519764—22no—
26#14519764C2×C414yes—
27#14519742—42no—
28D4×Dic61—42no—
29C2×C4⋊Dic61C2×C411yescharacteristic
30C2×#11758196—22no—
31C2×#11758196D412yes—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61—21no—
34C2×C4×Dic61C2×C411yescharacteristic
35C2×C4×Dic61C2×C411yescharacteristic
36C2×C4×Dic61D412yes—
37#14519684—21no—
38#14519684—21no—
39#14519684—22no—
40C22×D4×C61C2×C411yescharacteristic
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C22⋊C4×C61—22no—
43C2×C42×C61C2×C411yescharacteristic
44C24×C61—21no—
45C2×D4×C61—21no—
46C2×D4×C61—41no—
47C2×D4×C61—41no—
48C2×D4×C61C4212yes—
49C22×C4×C61—21no—
50C22×C4×C61C22⋊C412yes—
51C22×C4×C61C22×C411yescharacteristic
52C22×Dic61—21no—
53C22×Dic61—21no—
54C22×Dic61—42no—
55C22×Dic61—82no—
56C22×Dic61C22⋊C411yescharacteristic
57C22×Dic61C22⋊C411yescharacteristic
58C22×Dic61C2×D412yes—
59C2×Dic122—41no—
60C2×Dic122—41no—
61C2×Dic122C22×C412yes—
62C2×Dic122#4112yes—
63#11758217—28no—
64#11758217—44no—
65C2×#92420—44no—
66C4⋊C4×C61—41no—
67C22⋊C4×C61—82no—
68C42×C61—41no—
69C22⋊Dic61—28no—
70C22⋊Dic61—42no—
71C4⋊Dic61—41no—
72#11758196—42no—
73#11758196—82no—
74C4×Dic61—21no—
75C4×Dic61—22no—
76C4×Dic61—22no—
77C4×Dic61—22no—
78C4×Dic61C4212yes—
79C23×C61—22no—
80C23×C61—42no—
81C23×C61—42no—
82C23×C61—41no—
83C23×C61—41no—
84C23×C61C2×C22⋊C411yescharacteristic
85D4×C61—42no—
86D4×C61—42no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—22no—
90C2×C4×C61—42no—
91C2×C4×C61—41no—
92C2×C4×C61—41no—
93C2×C4×C61—41no—
94C2×C4×C61C2×C4211yescharacteristic
95C2×C4×C61#11711yescharacteristic
96C2×Dic61—22no—
97C2×Dic61—22no—
98C2×Dic61—22no—
99C2×Dic61—21no—
100C2×Dic61—41no—
101C2×Dic61—41no—
102C2×Dic61—41no—
103C2×Dic61—41no—
104C2×Dic61—42no—
105C2×Dic61—41no—
106C2×Dic61—84no—
107C2×Dic61#9912yes—
108C2×Dic61C4×D414yes—
109Dic122—42no—
110Dic122—42no—
111#92420—48no—
112#12632—611nomaximal, Sylow
113C22×C61—21no—
114C22×C61—21no—
115C22×C61—44no—
116C22×C61—42no—
117C22×C61—82no—
118C22×C61—81no—
119C22×C61—81no—
120C22×C61C4×C22⋊C411yescharacteristic
121C22×C61C2×#9911yescharacteristic
122C22×C61#41011yescharacteristic
123C4×C61—21no—
124C4×C61—21no—
125C4×C61—42no—
126C4×C61—81no—
127Dic61—22no—
128Dic61—24no—
129Dic61—42no—
130Dic61—41no—
131Dic61—81no—
132#2326—611no—
133C2×#355—612no—
134C4×#101—612no—
135#1394—612no—
136C2×C61—21no—
137C2×C61—21no—
138C2×C61—82no—
139C2×C61C4×#9911yescharacteristic
140C2×C61#213311yescharacteristic
141C2×C61#213411yescharacteristic
142C2×#124—611no—
143C2×C4×Q8—611no—
144C2×C4×D4—611no—
145C2×#101—614no—
146C4×M16—1222no—
147#391—611no—
148#390#692311yescharacteristic
149#388—1222no—
150#385—1222no—
151#384—611no—
152#376—611no—
153#355—618no—
154#343—1222no—
155C61#1263211yescharacteristic, complemented, Sylow
156C22×Q8—611no—
157C22×D4C2×#692311yescharacteristic
158C23×C4—1221no—
159C2×M16—1224no—
160#124—614no—
161#124—1222no—
162C4×Q8—2442no—
163C4×D4—2442no—
164C2×C4⋊C4—612no—
165C2×C4⋊C4—611no—
166C2×C4⋊C4—1222no—
167C2×C4⋊C4C2×#692311yescharacteristic
168C2×C22⋊C4—22no—
169C2×C22⋊C4—1222no—
170C2×C22⋊C4—1221no—
171C2×C42—611no—
172C2×C42—612no—
173C2×C42—611no—
174C2×C42C2×#692311yescharacteristic
175#101—618no—
176#101—1224no—
177#97—2442no—
178C4×C8—2442no—
179#95—1221no—
180#95—1221no—
181#95—1222no—
182C24—21no—
183C2×Q8—612no—
184C2×Q8—612no—
185C2×Q8—2441no—
186C2×Q8—2441no—
187C2×D4—21no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4C4×#692312yes—
191C22×C4—21no—
192C22×C4—612no—
193C22×C4—611no—
194C22×C4—611no—
195C22×C4—1221no—
196C22×C4—1221no—
197C22×C4—4882no—
198C22×C4—2442no—
199C22×C4C22⋊#692312yes—
200C22×C4C22×#692311yescharacteristic
201M16—1228no—
202M16—2444no—
203C2×C8—2444no—
204C4⋊C4—41no—
205C4⋊C4—4882no—
206C4⋊C4—2442no—
207C4⋊C4—2441no—
208C22⋊C4—82no—
209C22⋊C4—1228no—
210C22⋊C4—2442no—
211C42—41no—
212C42—612no—
213C42—1222no—
214C42—1222no—
215C42—1222no—
216C42—1221no—
217C23—22no—
218C23—42no—
219C23—42no—
220C23—41no—
221C23—41no—
222C23C2×C22⋊#692311yescharacteristic
223Q8—2442no—
224Q8—2442no—
225D4—42no—
226D4—42no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—612no—
235C2×C4—614no—
236C2×C4—1222no—
237C2×C4—1222no—
238C2×C4—1221no—
239C2×C4—1222no—
240C2×C4—4884no—
241C2×C4—2441no—
242C2×C4—2442no—
243C2×C4—2441no—
244C2×C4—2441no—
245C2×C4—2441no—
246C2×C4—2441no—
247C2×C4C2×C4×#692311yescharacteristic
248C2×C4#1451985611yescharacteristic
249C8—2448no—
250C22—21no—
251C22—21no—
252C22—44no—
253C22—42no—
254C22—82no—
255C22—81no—
256C22—81no—
257C22C4×C22⋊#692311yescharacteristic
258C22D61⋊#9911yescharacteristic
259C22#1684074211yescharacteristic
260C4—21no—
261C4—21no—
262C4—42no—
263C4—81no—
264C4—1224no—
265C4—1222no—
266C4—4881no—
267C4—2442no—
268C4—2441no—
269C2—21no—
270C2—21no—
271C2—82no—
272C2#2146156611yescharacteristic
273C2#2146211511yescharacteristic
274C2#2146211911yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC22×C61C2×#99
prime radicalC22×D4×C61C2×C4
cosocleC22×Dic122C23
hypercenter#390#6923
hypocenterC61#12632
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #21461572C2semidirect
3. #21460916C2semidirect
7. #16840670C4semidirect
20. #14519838C2×C4semidirect
155. C61#12632semidirect

Maximal subgroups

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

subgroupindex
C2×#168406702
#214615722
#214609162
#214485292
#1263261