Finite Group Databasebeta

← #30604013

Subgroups — #30604013

Order 15616.

460 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 98 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#21461055C212yesmaximal
2#21460761C212yesmaximal, complemented
3#21460749C212yesmaximal, complemented
4#21448532C211yesmaximal, characteristic
5C2×C22⋊#92420—22no—
6#16840607C2214yes—
7#16840605—22no—
8C2×Q8×Dic61C411yescharacteristic
9C2×D4×Dic61C2211yescharacteristic
10#16839669C2211yescharacteristic
11#16839665C411yescharacteristic
12#16839655—22no—
13#16839637—22no—
14#16839632C2211yescharacteristic
15#16839608C411yescharacteristic
16C2×#124×C61C411yescharacteristic
17C22×#92420—42no—
18C22⋊#92420—24no—
19C22⋊#92420—42no—
20C23×Dic61D412yes—
21Q8×Dic61—42no—
22D4×Dic61—42no—
23C2×C4⋊Dic61—21no—
24C2×C4⋊Dic61—22no—
25C2×C4⋊Dic61C2×C411yescharacteristic
26C2×C4⋊Dic61C2311yescharacteristic
27C2×C22⋊Dic61—22no—
28C2×C22⋊Dic61D412yes—
29C2×C4×Dic61—21no—
30C2×C4×Dic61C2×C411yescharacteristic
31C2×C4×Dic61C2×C411yescharacteristic
32#14519684—21no—
33#14519684—21no—
34#14519684—22no—
35C22×Q8×C61C2×C411yescharacteristic
36C22×D4×C61C2×C411yescharacteristic
37#124×C61—22no—
38#124×C61C814yescomplemented
39C2×C22⋊C4×C61—22no—
40C2×C42×C61C2×C411yescharacteristic
41C24×C61C22⋊C412yes—
42C2×Q8×C61—21no—
43C2×Q8×C61—41no—
44C2×Q8×C61—41no—
45C2×Q8×C61C2×C812yescomplemented
46C2×D4×C61—41no—
47C2×D4×C61—41no—
48C2×D4×C61C2×C812yes—
49C2×D4×C61M1612yes—
50C22×C4×C61—21no—
51C22×C4×C61—21no—
52C22×C4×C61C22×C411yescharacteristic
53C22×Dic61—22no—
54C22×Dic61—22no—
55C22×Dic61—21no—
56C22×Dic61—21no—
57C22×Dic61—82no—
58C22×Dic61C22⋊C411yescharacteristic
59C22×Dic61C22⋊C411yescharacteristic
60C22×Dic61C2×D412yes—
61C2×#92420—44no—
62C2×#92420—84no—
63C22⋊C4×C61—28no—
64C42×C61—21no—
65C42×C61C2×C812yescomplemented
66C22⋊Dic61—42no—
67C22⋊Dic61—82no—
68C4⋊Dic61—41no—
69C4⋊Dic61—42no—
70C4⋊Dic61—41no—
71C4⋊Dic61—82no—
72C4×Dic61—22no—
73C4×Dic61—42no—
74C4×Dic61—41no—
75C23×C61—22no—
76C23×C61—22no—
77C23×C61—42no—
78C23×C61—42no—
79C23×C61C22⋊C814yes—
80C23×C61C2×C22⋊C411yescharacteristic
81Q8×C61—42no—
82Q8×C61—42no—
83D4×C61—42no—
84D4×C61—42no—
85C2×C4×C61—21no—
86C2×C4×C61—21no—
87C2×C4×C61—22no—
88C2×C4×C61—22no—
89C2×C4×C61—41no—
90C2×C4×C61—41no—
91C2×C4×C61—41no—
92C2×C4×C61—41no—
93C2×C4×C61C22×C811yescharacteristic
94C2×C4×C61C2×M1611yescharacteristic
95C2×Dic61—21no—
96C2×Dic61—22no—
97C2×Dic61—41no—
98C2×Dic61—44no—
99C2×Dic61—42no—
100C2×Dic61—41no—
101C2×Dic61—42no—
102C2×Dic61—41no—
103C2×Dic61—41no—
104C2×Dic61—41no—
105C2×Dic61—41no—
106C2×Dic61—82no—
107C2×Dic61#9912yes—
108#92420—84no—
109#11481—611nomaximal, Sylow
110C22×C61—21no—
111C22×C61—21no—
112C22×C61—24no—
113C22×C61—44no—
114C22×C61—42no—
115C22×C61—42no—
116C22×C61—82no—
117C22×C61C2×C22⋊C811yescharacteristic
118C22×C61C2×#9911yescharacteristic
119C22×C61#41311yescharacteristic
120C4×C61—21no—
121C4×C61—21no—
122C4×C61—42no—
123C4×C61—42no—
124Dic61—22no—
125Dic61—42no—
126Dic61—41no—
127Dic61—81no—
128Dic61—81no—
129#2326—611no—
130#1459—612no—
131#1325—612no—
132#1319—612no—
133C2×C61—21no—
134C2×C61—21no—
135C2×C61—44no—
136C2×C61#145911yescharacteristic
137C2×C61#212711yescharacteristic
138C2×C61#213511yescharacteristic
139C2×#124#692311yescharacteristic
140C2×C4×Q8—611no—
141C2×C4×D4—611no—
142C2×C22⋊C8—1222no—
143#391—611no—
144#390—611no—
145#388—1222no—
146#385—1222no—
147#384—611no—
148#376—611no—
149#324—614no—
150#323—1222no—
151C61#1148111yescharacteristic, complemented, Sylow
152C22×Q8C2×#692311yescharacteristic
153C22×D4C2×#692311yescharacteristic
154C23×C4—612no—
155C22×C8—2442no—
156#124—22no—
157#124#9242014yescomplemented
158C4×Q8—2442no—
159C4×D4—2442no—
160C2×C4⋊C4—611no—
161C2×C4⋊C4—611no—
162C2×C4⋊C4—1222no—
163C2×C4⋊C4—1221no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—612no—
166C2×C22⋊C4—1222no—
167C2×C42—611no—
168C2×C42—611no—
169C2×C42—1221no—
170C2×C42C2×#692311yescharacteristic
171C22⋊C8—1224no—
172C22⋊C8—2442no—
173#95—1221no—
174#95—1221no—
175#95—1222no—
176C24C22⋊#692312yes—
177C2×Q8—21no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8C2×#9242012yescomplemented
181C2×D4—41no—
182C2×D4—41no—
183C2×D4C2×#9242012yes—
184C2×D4#1175821712yes—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—611no—
188C22×C4—612no—
189C22×C4—611no—
190C22×C4—1222no—
191C22×C4—1222no—
192C22×C4—1221no—
193C22×C4—1221no—
194C22×C4—4882no—
195C22×C4C22×#692311yescharacteristic
196C2×C8—4884no—
197C2×C8—2444no—
198C4⋊C4—4882no—
199C4⋊C4—2441no—
200C4⋊C4—2441no—
201C4⋊C4—2442no—
202C22⋊C4—28no—
203C22⋊C4—4882no—
204C22⋊C4—2442no—
205C42—21no—
206C42—1222no—
207C42—2441no—
208C42—2442no—
209C42C2×#9242012yescomplemented
210C23—22no—
211C23—22no—
212C23—42no—
213C23—42no—
214C23C22⋊#9242014yes—
215C23C2×C22⋊#692311yescharacteristic
216Q8—42no—
217Q8—42no—
218D4—42no—
219D4—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—612no—
229C2×C4—1221no—
230C2×C4—1222no—
231C2×C4—4882no—
232C2×C4—2441no—
233C2×C4—2444no—
234C2×C4—2441no—
235C2×C4—2442no—
236C2×C4—2441no—
237C2×C4—2441no—
238C2×C4—2441no—
239C2×C4—2442no—
240C2×C4—2441no—
241C2×C4C22×#9242011yescharacteristic
242C2×C4C2×#1175821711yescharacteristic
243C8—4884no—
244C22—21no—
245C22—21no—
246C22—24no—
247C22—44no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22C2×C22⋊#9242011yescharacteristic
252C22D61⋊#9911yescharacteristic
253C22#1684074911yescharacteristic
254C4—21no—
255C4—21no—
256C4—42no—
257C4—42no—
258C4—1222no—
259C4—4881no—
260C4—4881no—
261C4—2442no—
262C4—2441no—
263C2—21no—
264C2—21no—
265C2—44no—
266C2#2146105511yescharacteristic
267C2#2146209711yescharacteristic
268C2#2146212211yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC2×#124×C61C4
socleC22×C61C2×#99
prime radicalC22×D4×C61C2×C4
cosocleC2×C4⋊Dic61C23
hypercenterC2×#124#6923
hypocenterC61#11481
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. #21460761C2semidirect
3. #21460749C2semidirect
38. #124×C61C8semidirect
45. C2×Q8×C61C2×C8semidirect
65. C42×C61C2×C8semidirect
151. C61#11481semidirect
157. #124#92420semidirect
180. C2×Q8C2×#92420semidirect
209. C42C2×#92420semidirect

Maximal subgroups

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

subgroupindex
#214610552
#214607612
#214607492
#214485322
#1148161