Finite Group Databasebeta

← #30606539

Subgroups — #30606539

Order 15616.

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

normal subgroups only

This group has 564 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
2C4×#14519836C212yesmaximal
3#21460914C212yesmaximal
4#21448526C211yesmaximal, characteristic
5C2×#14519836C2214yes—
6C4×C22⋊#6923—22no—
7#16840668C418yescomplemented
8#16840640—22no—
9C2×#14519766C2211yescharacteristic
10C2×C4×D244C2211yescharacteristic
11#16839669C411yescharacteristic, complemented
12#16839663C2211yescharacteristic
13#16839653—22no—
14#16839633—22no—
15#16839631C411yescharacteristic, complemented
16#16839608C411yescharacteristic, complemented
17C2×C4×Q8×C61C411yescharacteristic, 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—
27C4×D244—42no—
28C2×C4⋊Dic61C2×C411yescharacteristic
29C2×#11758196—22no—
30C2×#11758198—22no—
31C2×#11758198—22no—
32C2×#11758198D412yes—
33C2×C4×Dic61C2×C411yescharacteristic
34C2×C4×Dic61C2×C411yescharacteristic
35#14519684—21no—
36#14519684—21no—
37#14519684—22no—
38C22×Q8×C61C2×C411yescharacteristic
39C4×Q8×C61—42no—
40C2×C4⋊C4×C61—21no—
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C42×C61—21no—
43C2×C42×C61C2×C411yescharacteristic
44C23×D61C2×D412yes—
45C22×#6923—44no—
46C2×Q8×C61—21no—
47C2×Q8×C61—41no—
48C2×Q8×C61—41no—
49C2×Q8×C61C4212yescomplemented
50C22×C4×C61—21no—
51C22×C4×C61—21no—
52C22×C4×C61C22⋊C411yescharacteristic
53C22×C4×C61C22⋊C411yescharacteristic
54C22×C4×C61C22×C411yescharacteristic
55C22×Dic61—21no—
56C22×Dic61—21no—
57C22×Dic61C22⋊C412yes—
58C2×D244—41no—
59C2×D244—41no—
60C2×D244C22×C412yes—
61C2×D244#4112yes—
62C2×C4×D61—22no—
63C2×C4×D61—22no—
64C2×C4×D61—82no—
65C22⋊#6923—28no—
66C22⋊#6923—44no—
67C4×#6923—84no—
68C4⋊C4×C61—41no—
69C4⋊C4×C61—42no—
70C42×C61—22no—
71C42×C61—42no—
72#11758198—28no—
73#11758198—42no—
74#11758198—82no—
75C4⋊Dic61—41no—
76#11758196—82no—
77C4×Dic61—21no—
78C4×Dic61—41no—
79C4×Dic61C4212yescomplemented
80C23×C61C2×C22⋊C411yescharacteristic
81C22×D61—22no—
82C22×D61—22no—
83C22×D61—42no—
84C22×D61—42no—
85C22×D61C4×D414yes—
86C2×#6923—48no—
87C2×#6923—84no—
88C2×#6923—84no—
89Q8×C61—42no—
90Q8×C61—42no—
91C2×C4×C61—21no—
92C2×C4×C61—21no—
93C2×C4×C61—22no—
94C2×C4×C61—21no—
95C2×C4×C61—41no—
96C2×C4×C61—41no—
97C2×C4×C61—41no—
98C2×C4×C61—41no—
99C2×C4×C61—42no—
100C2×C4×C61—41no—
101C2×C4×C61#9912yes—
102C2×C4×C61C2×C4211yescharacteristic
103C2×C4×C61#11711yescharacteristic
104C2×Dic61—22no—
105C2×Dic61—22no—
106C2×Dic61—41no—
107C2×Dic61—41no—
108C2×Dic61—42no—
109C2×Dic61—41no—
110C2×Dic61—41no—
111C2×Dic61—41no—
112D244—42no—
113D244—42no—
114C4×D61—44no—
115C4×D61—82no—
116#12627—611nomaximal, Sylow
117C22×C61—21no—
118C22×C61—21no—
119C22×C61C4×C22⋊C411yescharacteristic
120C22×C61C2×#9911yescharacteristic
121C22×C61#41011yescharacteristic
122C2×D61—24no—
123C2×D61—42no—
124C2×D61—42no—
125C2×D61—44no—
126C2×D61—82no—
127#6923—88no—
128C4×C61—22no—
129C4×C61—21no—
130C4×C61—21no—
131C4×C61—41no—
132C4×C61—42no—
133C4×C61—81no—
134Dic61—42no—
135Dic61—42no—
136Dic61—81no—
137#2326—611no—
138C2×#354—612no—
139C4×#99—612no—
140#1393—612no—
141C2×C61—21no—
142C2×C61—21no—
143C2×C61C4×#9911yescharacteristic
144C2×C61#212711yescharacteristic
145C2×C61#212911yescharacteristic
146D61—44no—
147C2×#124—611no—
148C2×C4×Q8#692311yescharacteristic, complemented
149C2×C4×D4—611no—
150C2×#99—614no—
151#391—611no—
152#390—611no—
153#388—1222no—
154#385—1222no—
155#384—611no—
156C4×C22⋊C4—1222no—
157#376—611no—
158#354—618no—
159#342—1222no—
160C61#1262711yescharacteristic, complemented, Sylow
161C22×Q8C2×#692311yescharacteristic
162C22×D4—611no—
163C23×C4—612no—
164#124—614no—
165#124—1222no—
166C4×Q8—42no—
167C4×D4—2442no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—611no—
170C2×C4⋊C4—1222no—
171C2×C4⋊C4C2×#692311yescharacteristic
172C2×C22⋊C4—612no—
173C2×C22⋊C4—1222no—
174C2×C22⋊C4—1224no—
175C2×C22⋊C4—1222no—
176C2×C42—21no—
177C2×C42—611no—
178C2×C42—611no—
179C2×C42—2442no—
180C2×C42C2×#692311yescharacteristic
181#99—618no—
182#99—1224no—
183#95—1222no—
184#95—1221no—
185#95—1221no—
186#95—2442no—
187C24—612no—
188C2×Q8—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8C4×#692312yescomplemented
192C2×D4—612no—
193C2×D4—612no—
194C2×D4—2441no—
195C2×D4—2441no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—612no—
199C22×C4—1221no—
200C22×C4—1222no—
201C22×C4—1222no—
202C22×C4—1221no—
203C22×C4—4882no—
204C22×C4—2444no—
205C22×C4C22⋊#692311yescharacteristic
206C22×C4C22⋊#692311yescharacteristic
207C22×C4C22×#692311yescharacteristic
208C4⋊C4—41no—
209C4⋊C4—42no—
210C4⋊C4—4882no—
211C4⋊C4—2441no—
212C22⋊C4—1228no—
213C22⋊C4—1228no—
214C22⋊C4—4882no—
215C22⋊C4—2442no—
216C22⋊C4—2444no—
217C42—22no—
218C42—42no—
219C42—612no—
220C42—1221no—
221C42—4884no—
222C42—2441no—
223C23—614no—
224C23—1222no—
225C23—1222no—
226C23—2442no—
227C23—2442no—
228C23C2×C22⋊#692311yescharacteristic
229Q8—42no—
230Q8—42no—
231D4—2442no—
232D4—2442no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—1222no—
244C2×C4—1222no—
245C2×C4—4882no—
246C2×C4—4884no—
247C2×C4—4884no—
248C2×C4—2444no—
249C2×C4—2441no—
250C2×C4—2441no—
251C2×C4—2448no—
252C2×C4—2441no—
253C2×C4—2441no—
254C2×C4—2441no—
255C2×C4—2442no—
256C2×C4#1451983512yes—
257C2×C4C2×C4×#692311yescharacteristic
258C2×C4#1451985611yescharacteristic
259C22—21no—
260C22—21no—
261C22—1224no—
262C22—4882no—
263C22—2444no—
264C22—2442no—
265C22—2442no—
266C22C4×C22⋊#692311yescharacteristic
267C22C2×#1451983511yescharacteristic
268C22#1684074111yescharacteristic
269C4—21no—
270C4—21no—
271C4—22no—
272C4—42no—
273C4—41no—
274C4—81no—
275C4—4881no—
276C4—4888no—
277C4—2442no—
278C4—2442no—
279C2—21no—
280C2—21no—
281C2—2444no—
282C2C4×#1451983511yescharacteristic
283C2#2146209411yescharacteristic
284C2#2146210111yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61C2×#99
prime radical, cosocleC22×D244C23
hypercenterC2×C4×Q8#6923
hypocenterC61#12627
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
7. #16840668C4semidirect
11. #16839669C4semidirect
15. #16839631C4semidirect
16. #16839608C4semidirect
17. C2×C4×Q8×C61C4semidirect
49. C2×Q8×C61C42semidirect
79. C4×Dic61C42semidirect
148. C2×C4×Q8#6923semidirect
160. C61#12627semidirect
191. 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
C4×#145198362
#214609142
#214485262
#1262761