Finite Group Databasebeta

← #29572528

Subgroups — #29572528

Order 15616.

690 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 131 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#21445902C211yesmaximal, characteristic
2#21445896C211yesmaximal, characteristic
3#21444352C214yesmaximal
4C2×C61×#541C211yesmaximal, characteristic
5#16839726C2212yes—
6#16839625C2212yes—
7#16839531—22no—
8#16839529—22no—
9#16839499—28no—
10C61×#541—28no—
11C2×C4⋊Q8×C61C411yescharacteristic, complemented
12C2×#126×C61—22no—
13C2×#125×C61—22no—
14C2×#125×C61C411yescharacteristic, complemented
15C2×C22⋊Q8×C61C2212yes—
16C2×C4×Q8×C61C412yescomplemented
17C2×#117×C61C2211yescharacteristic
18C2×#11758194—22no—
19C2×C22⋊Dic61—22no—
20C22⋊#92418—24no—
21#14519684—24no—
22C22×Q8×C61C2×C412yes—
23C23×C4×C61C2311yescharacteristic
24C4⋊Q8×C61—24no—
25#126×C61—216no—
26#125×C61—216no—
27#125×C61—24no—
28C22⋊Q8×C61—24no—
29C22⋊Q8×C61D418yes—
30C4×Q8×C61—24no—
31C4×Q8×C61—24no—
32#117×C61—24no—
33C2×C4⋊C4×C61—22no—
34C2×C4⋊C4×C61—22no—
35C2×C4⋊C4×C61—22no—
36C2×C4⋊C4×C61—22no—
37C2×C4⋊C4×C61C2×C412yes—
38C2×C4⋊C4×C61D412yes—
39C2×C22⋊C4×C61—22no—
40C2×C22⋊C4×C61D412yes—
41C2×C42×C61—22no—
42C2×C42×C61C2×C411yescharacteristic
43C2×C42×C61C2×C411yescharacteristic
44C24×C61C2×D411yescharacteristic
45C2×Q8×C61—22no—
46C2×Q8×C61—44no—
47C2×Q8×C61C22⋊C414yes—
48C22×C4×C61—22no—
49C22×C4×C61—22no—
50C22×C4×C61—42no—
51C22×C4×C61C22⋊C412yes—
52C22×C4×C61C22⋊C412yes—
53C22×C4×C61C22×C411yescharacteristic
54C22×C4×C61C2×D411yescharacteristic
55C22×C4×C61C2×D414yes—
56C22×Dic61—42no—
57C4⋊C4×C61—28no—
58C4⋊C4×C61—28no—
59C4⋊C4×C61—28no—
60C4⋊C4×C61—22no—
61C4⋊C4×C61—28no—
62C4⋊C4×C61—24no—
63C4⋊C4×C61C22⋊C414yes—
64C22⋊C4×C61—28no—
65C22⋊C4×C61—24no—
66C42×C61—28no—
67C42×C61—22no—
68C42×C61—22no—
69C22⋊Dic61—44no—
70#11758194—44no—
71C2×#92418—42no—
72C23×C61—22no—
73C23×C61—42no—
74C23×C61C2×C22⋊C411yescharacteristic
75C23×C61#12012yes—
76Q8×C61—44no—
77Q8×C61—44no—
78C2×C4×C61—22no—
79C2×C4×C61—22no—
80C2×C4×C61—22no—
81C2×C4×C61—22no—
82C2×C4×C61—24no—
83C2×C4×C61—24no—
84C2×C4×C61—44no—
85C2×C4×C61—44no—
86C2×C4×C61—44no—
87C2×C4×C61—42no—
88C2×C4×C61—42no—
89C2×C4×C61C2×C22⋊C412yes—
90C2×C4×C61#12012yes—
91C2×Dic61—82no—
92C2×Dic61—84no—
93#92418—82no—
94#8521—611nomaximal, Sylow
95C22×C61—22no—
96C22×C61—24no—
97C22×C61—22no—
98C22×C61—82no—
99C22×C61#37911yescharacteristic
100C22×C61#41012yes—
101C4×C61—22no—
102C4×C61—22no—
103C4×C61—42no—
104C4×C61—42no—
105C4×C61—44no—
106C4×C61—44no—
107Dic61—162no—
108C2×#541D6111yescharacteristic
109#1822—611no—
110#1820—611no—
111#1511—614no—
112C2×C61—22no—
113C2×C61—42no—
114C2×C61#178211yescharacteristic
115C2×C61#179612yes—
116#541—28no—
117C2×C4⋊Q8Dic6111yescharacteristic, complemented
118C2×#126—22no—
119C2×#125—22no—
120C2×#125Dic6111yescharacteristic, complemented
121C2×C22⋊Q8C2×D6112yes—
122C2×C4×Q8Dic6112yescomplemented
123C2×#117C2×D6111yescharacteristic
124#407—612no—
125#381—612no—
126#343—1222no—
127#342—1222no—
128#328—1228no—
129C61#852111yescharacteristic, complemented, Sylow
130C22×Q8C2×Dic6112yes—
131C23×C4C22×D6111yescharacteristic
132C2×M16—1222no—
133C4⋊Q8—24no—
134#126—216no—
135#125—216no—
136#125—24no—
137C22⋊Q8—24no—
138C22⋊Q8C22⋊D6118yes—
139C4×Q8—24no—
140C4×Q8—24no—
141#117—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4C2×Dic6112yes—
147C2×C4⋊C4C22⋊D6112yes—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—1222no—
150C2×C22⋊C4C22⋊D6112yes—
151C2×C42—22no—
152C2×C42C2×Dic6111yescharacteristic
153C2×C42C2×Dic6111yescharacteristic
154C22⋊C8—1224no—
155#95—1224no—
156C24C2×C22⋊D6111yescharacteristic
157C2×Q8—22no—
158C2×Q8—44no—
159C2×Q8C22⋊Dic6114yes—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—42no—
163C22×C4—2442no—
164C22×C4C22⋊Dic6112yes—
165C22×C4C22⋊Dic6112yes—
166C22×C4C22×Dic6111yescharacteristic
167C22×C4C2×C22⋊D6111yescharacteristic
168C22×C4C2×C22⋊D6114yes—
169M16—2444no—
170C2×C8—2442no—
171C4⋊C4—28no—
172C4⋊C4—28no—
173C4⋊C4—28no—
174C4⋊C4—22no—
175C4⋊C4—28no—
176C4⋊C4—24no—
177C4⋊C4C22⋊Dic6114yes—
178C22⋊C4—28no—
179C22⋊C4—24no—
180C22⋊C4—2444no—
181C42—28no—
182C42—22no—
183C42—22no—
184C23—22no—
185C23—42no—
186C23C2×C22⋊Dic6111yescharacteristic
187C23#1451974612yes—
188Q8—44no—
189Q8—44no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—4882no—
202C2×C4—4884no—
203C2×C4C2×C22⋊Dic6112yes—
204C2×C4#1451974612yes—
205C8—4882no—
206C22—22no—
207C22—22no—
208C22—24no—
209C22—82no—
210C22#1683961811yescharacteristic
211C22#1683973612yes—
212C4—22no—
213C4—22no—
214C4—44no—
215C4—42no—
216C4—42no—
217C4—44no—
218C4—9762no—
219C2—22no—
220C2—42no—
221C2#2144573711yescharacteristic
222C2#2144579412yes—
subgroupG/H
centerC22#16839618
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#541C2
socleC22×C61#379
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenterC2×#541D61
hypocenterC61#8521
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
11. C2×C4⋊Q8×C61C4semidirect
14. C2×#125×C61C4semidirect
16. C2×C4×Q8×C61C4semidirect
117. C2×C4⋊Q8Dic61semidirect
120. C2×#125Dic61semidirect
122. C2×C4×Q8Dic61semidirect
129. C61#8521semidirect

Maximal subgroups

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

subgroupindex
#214459022
#214458962
#214443522
C2×C61×#5412
#852161