Finite Group Databasebeta

← #30606537

Subgroups — #30606537

Order 15616.

584 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 102 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×#16840668C212yesmaximal
2#21461564C212yesmaximal
3#21460913C212yesmaximal
4#21448362C211yesmaximal, characteristic
5C2×#14519836C2214yes—
6C4×C22⋊#6923—22no—
7#16840668C418yescomplemented
8#16840640—22no—
9C2×#14519766C2211yescharacteristic
10C2×Q8×Dic61C411yescharacteristic, complemented
11C2×#14519739C2211yescharacteristic
12#16839664C2211yescharacteristic
13#16839654—22no—
14#16839627C411yescharacteristic, complemented
15#16839609—22no—
16C42×Dic61C411yescharacteristic, complemented
17C61×#391C411yescharacteristic, complemented
18C2×C22⋊#6923—24no—
19C2×C4×#6923—42no—
20#14519836—24no—
21#14519836C2×C418yes—
22(C2×#6923)⋊C4—42no—
23C22×D244C2311yescharacteristic
24C22×C4×D61D412yes—
25#14519766—22no—
26#14519766C2×C414yes—
27Q8×Dic61—41no—
28Q8×Dic61—41no—
29#14519739—41no—
30#14519739—41no—
31C2×C4⋊Dic61—21no—
32C2×C4⋊Dic61C2×C411yescharacteristic
33C2×#11758198—22no—
34C2×#11758198—22no—
35C2×#11758198D412yes—
36C2×C4×Dic61—21no—
37C2×C4×Dic61—21no—
38C2×C4×Dic61—21no—
39C2×C4×Dic61C2×C411yescharacteristic
40C2×C4×Dic61C2×C411yescharacteristic
41#14519684—22no—
42C22×Q8×C61C2×C411yescharacteristic
43C2×C4⋊C4×C61C2×C411yescharacteristic
44C2×C42×C61C2×C411yescharacteristic
45#95×C61—22no—
46C23×D61C2×D412yes—
47C22×#6923—44no—
48C2×Q8×C61—21no—
49C2×Q8×C61—41no—
50C2×Q8×C61—41no—
51C2×Q8×C61C4212yescomplemented
52C22×C4×C61—21no—
53C22×C4×C61—21no—
54C22×C4×C61C22⋊C412yes—
55C22×C4×C61C22×C411yescharacteristic
56C22×Dic61—21no—
57C22×Dic61—21no—
58C22×Dic61C22⋊C411yescharacteristic
59C22×Dic61C22⋊C411yescharacteristic
60C2×D244—41no—
61C2×D244—41no—
62C2×D244C22×C412yes—
63C2×D244#4112yes—
64C2×C4×D61—22no—
65C2×C4×D61—22no—
66C2×C4×D61—44no—
67C22⋊#6923—28no—
68C22⋊#6923—44no—
69C4×#6923—84no—
70C4⋊C4×C61—41no—
71C42×C61—41no—
72#11758198—28no—
73#11758198—42no—
74#11758198—44no—
75C4⋊Dic61—42no—
76C4⋊Dic61—41no—
77C4×Dic61—21no—
78C4×Dic61—21no—
79C4×Dic61—21no—
80C4×Dic61—42no—
81C4×Dic61—42no—
82C4×Dic61—41no—
83C4×Dic61—41no—
84C4×Dic61C4212yescomplemented
85C23×C61C2×C22⋊C411yescharacteristic
86C22×D61—22no—
87C22×D61—22no—
88C22×D61—42no—
89C22×D61—42no—
90C22×D61C4×D414yes—
91C2×#6923—48no—
92C2×#6923—84no—
93C2×#6923—84no—
94Q8×C61—42no—
95Q8×C61—42no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—41no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—42no—
106C2×C4×C61C2×C4211yescharacteristic
107C2×C4×C61#11711yescharacteristic
108C2×Dic61—22no—
109C2×Dic61—22no—
110C2×Dic61—21no—
111C2×Dic61—41no—
112C2×Dic61—41no—
113C2×Dic61—41no—
114C2×Dic61—42no—
115C2×Dic61—41no—
116C2×Dic61#9911yescharacteristic
117C2×Dic61#9911yescharacteristic
118D244—42no—
119D244—42no—
120C4×D61—44no—
121C4×D61—82no—
122C4×#354—611nomaximal, Sylow
123C22×C61—21no—
124C22×C61—21no—
125C22×C61C4×C22⋊C411yescharacteristic
126C22×C61C2×#9911yescharacteristic
127C22×C61#41011yescharacteristic
128C2×D61—24no—
129C2×D61—42no—
130C2×D61—42no—
131C2×D61—44no—
132C2×D61—82no—
133#6923—88no—
134C4×C61—21no—
135C4×C61—21no—
136C4×C61—42no—
137C4×C61—42no—
138C4×C61—42no—
139Dic61—21no—
140Dic61—42no—
141Dic61—41no—
142Dic61—81no—
143Dic61#35412yescomplemented
144C4×#124—611no—
145C2×#354—612no—
146C4×#99—612no—
147#1393—612no—
148C2×C61—21no—
149C2×C61—21no—
150C2×C61C4×#9911yescharacteristic
151C2×C61C2×#35411yescharacteristic
152C2×C61#212611yescharacteristic
153D61—44no—
154C2×#124—611no—
155C2×C4×Q8—611no—
156C2×C4×D4—611no—
157C2×#99—614no—
158#391#692311yescharacteristic, complemented
159#390—611no—
160#388—1222no—
161#382—611no—
162C4×C22⋊C4—1222no—
163C4×C22⋊C4—1222no—
164C43—611no—
165#354—618no—
166#342—1222no—
167C61C4×#35411yescharacteristic, complemented, Sylow
168C22×Q8C2×#692311yescharacteristic
169C22×D4—611no—
170C23×C4—612no—
171#124—614no—
172#124—1222no—
173C4×Q8—2441no—
174C4×Q8—2441no—
175C4×D4—2441no—
176C4×D4—2441no—
177C2×C4⋊C4—611no—
178C2×C4⋊C4—1221no—
179C2×C4⋊C4C2×#692311yescharacteristic
180C2×C22⋊C4—612no—
181C2×C22⋊C4—1222no—
182C2×C22⋊C4—1224no—
183C2×C22⋊C4—1222no—
184C2×C42—611no—
185C2×C42—611no—
186C2×C42—1221no—
187C2×C42—1221no—
188C2×C42—1221no—
189C2×C42—2442no—
190C2×C42C2×#692311yescharacteristic
191#99—618no—
192#99—1224no—
193#95—22no—
194#95—1222no—
195#95—2442no—
196C24—612no—
197C2×Q8—21no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8C4×#692312yescomplemented
201C2×D4—612no—
202C2×D4—612no—
203C2×D4—2441no—
204C2×D4—2441no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—611no—
208C22×C4—611no—
209C22×C4—1222no—
210C22×C4—1222no—
211C22×C4—1221no—
212C22×C4—1221no—
213C22×C4—2444no—
214C22×C4—2444no—
215C22×C4C22⋊#692312yes—
216C22×C4C22×#692311yescharacteristic
217C4⋊C4—41no—
218C4⋊C4—2441no—
219C4⋊C4—2442no—
220C22⋊C4—1228no—
221C22⋊C4—1228no—
222C22⋊C4—2442no—
223C22⋊C4—2444no—
224C22⋊C4—2444no—
225C42—41no—
226C42—612no—
227C42—1221no—
228C42—1221no—
229C42—1221no—
230C42—4884no—
231C42—2442no—
232C42—2441no—
233C42—2441no—
234C42—2442no—
235C23—614no—
236C23—1222no—
237C23—1222no—
238C23—2442no—
239C23—2442no—
240C23C2×C22⋊#692311yescharacteristic
241Q8—42no—
242Q8—42no—
243D4—2442no—
244D4—2442no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—611no—
256C2×C4—611no—
257C2×C4—1222no—
258C2×C4—1221no—
259C2×C4—1222no—
260C2×C4—4882no—
261C2×C4—4884no—
262C2×C4—4884no—
263C2×C4—2444no—
264C2×C4—2442no—
265C2×C4—2448no—
266C2×C4—2441no—
267C2×C4—2441no—
268C2×C4—2441no—
269C2×C4—2441no—
270C2×C4C2×C4×#692311yescharacteristic
271C2×C4#1451985611yescharacteristic
272C22—21no—
273C22—21no—
274C22—1224no—
275C22—4882no—
276C22—2444no—
277C22—2442no—
278C22—2442no—
279C22C4×C22⋊#692311yescharacteristic
280C22D61⋊#9911yescharacteristic
281C22#1684074211yescharacteristic
282C4—21no—
283C4—21no—
284C4—42no—
285C4—42no—
286C4—42no—
287C4—612no—
288C4—1221no—
289C4—4888no—
290C4—4881no—
291C4—2442no—
292C4—2441no—
293C2—21no—
294C2—21no—
295C2—2444no—
296C2#2146156611yescharacteristic
297C2D61⋊#35411yescharacteristic
298C2#2146209311yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC22×C61C2×#99
prime radical, cosocleC22×D244C23
hypercenter#391#6923
hypocenterC61C4×#354
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #391

Complements

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

NH ≅ G/Ntype
7. #16840668C4semidirect
10. C2×Q8×Dic61C4semidirect
14. #16839627C4semidirect
16. C42×Dic61C4semidirect
17. C61×#391C4semidirect
51. C2×Q8×C61C42semidirect
84. C4×Dic61C42semidirect
143. Dic61#354semidirect
158. #391#6923semidirect
167. C61C4×#354semidirect
200. C2×Q8C4×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168406682
#214615642
#214609132
#214483622
C4×#35461