Finite Group Databasebeta

← #30595022

Subgroups — #30595022

Order 15616.

224 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 36 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#21460913C211yesmaximal, characteristic
2#21460748C211yesmaximal, characteristic
3#21447351C211yesmaximal, characteristic
4C2×#14519836—21no—
5#16840607—21no—
6C2×#14519709—21no—
7C2×#14519699—21no—
8#16839664C2211yescharacteristic
9#16839632C411yescharacteristic, complemented
10C61×#336C411yescharacteristic, complemented
11C2×C22⋊#6923—41no—
12#14519836—22no—
13#14519836—41no—
14C22⋊#92420—41no—
15C22×D244D411yescharacteristic
16C2×C4⋊Dic61—21no—
17C2×C4⋊Dic61—41no—
18C2×C4⋊Dic61D411yescharacteristic
19C2×#11758198—41no—
20#14519709—42no—
21#14519699—42no—
22C22×C8×C61—21no—
23C2×C42×C61C2×C411yescharacteristic
24C23×D61—21no—
25C22×#6923—81no—
26C22×C4×C61—21no—
27C22×C4×C61C22⋊C411yescharacteristic
28C22×Dic61—21no—
29C22×Dic61—41no—
30C2×D244—21no—
31C2×D244—41no—
32C2×D244—41no—
33C2×D244SD1611yescharacteristic
34C2×D244Q1611yescharacteristic
35C22⋊#6923—42no—
36C22⋊#6923—81no—
37C2×#92420—81no—
38C2×C8×C61—22no—
39C2×C8×C61—41no—
40C2×C8×C61—41no—
41C42×C61—41no—
42#11758198—82no—
43C4⋊Dic61—41no—
44C4⋊Dic61—41no—
45C4⋊Dic61—41no—
46C4⋊Dic61—82no—
47C23×C61#9911yescharacteristic
48C22×D61—21no—
49C22×D61—21no—
50C22×D61—41no—
51C22×D61—41no—
52C22×D61—81no—
53C22×D61—81no—
54C2×#6923—82no—
55C2×#6923—161no—
56C2×#6923—161no—
57C2×C4×C61—21no—
58C2×C4×C61—21no—
59C2×C4×C61—22no—
60C2×C4×C61—41no—
61C2×C4×C61—41no—
62C2×C4×C61#10311yescharacteristic
63C2×C4×C61#10411yescharacteristic
64C2×Dic61—41no—
65C2×Dic61—81no—
66C2×Dic61—82no—
67C2×Dic61—81no—
68D244—41no—
69D244—41no—
70D244—81no—
71#92420—161no—
72C8×C61—42no—
73#7435—611nomaximal, Sylow
74C22×C61—21no—
75C22×C61—21no—
76C22×C61#32811yescharacteristic
77C22×C61#35211yescharacteristic
78C22×C61#35311yescharacteristic
79C2×D61—41no—
80C2×D61—41no—
81C2×D61—81no—
82C2×D61—81no—
83C2×D61—81no—
84C2×D61—81no—
85C2×D61—161no—
86#6923—162no—
87C4×C61—21no—
88C4×C61—21no—
89C4×C61—42no—
90Dic61—81no—
91Dic61—161no—
92#2094—611no—
93#1393—611no—
94#1318—611no—
95C2×C61—21no—
96C2×C61—21no—
97C2×C61#135211yescharacteristic
98C2×C61#140611yescharacteristic
99C2×C61#140911yescharacteristic
100D61—81no—
101D61—81no—
102C2×#106—1221no—
103C2×#102—1221no—
104C2×#99—1221no—
105#390—611no—
106#384—611no—
107#336#692311yescharacteristic, complemented
108#324—1221no—
109C61#743511yescharacteristic, complemented, Sylow
110C22×D4—611no—
111C22×C8—21no—
112C2×C4⋊C4—611no—
113C2×C4⋊C4—1221no—
114C2×C4⋊C4—2441no—
115C2×C22⋊C4—2441no—
116C2×C22⋊C4—2441no—
117C2×C42C2×#692311yescharacteristic
118#106—2442no—
119#102—2442no—
120#99—1222no—
121#99—2441no—
122C22⋊C8—2441no—
123C24—1221no—
124C2×D4—611no—
125C2×D4—611no—
126C2×D4—1221no—
127C2×D4—2441no—
128C2×D4—2441no—
129C22×C4—21no—
130C22×C4—1221no—
131C22×C4—4881no—
132C22×C4—2441no—
133C22×C4C22⋊#692311yescharacteristic
134C2×C8—22no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—4881no—
138C4⋊C4—4882no—
139C4⋊C4—2441no—
140C4⋊C4—2441no—
141C4⋊C4—2441no—
142C22⋊C4—4882no—
143C22⋊C4—4881no—
144C22⋊C4—2442no—
145C42—41no—
146C23—1221no—
147C23—1221no—
148C23—4881no—
149C23—4881no—
150C23—2441no—
151C23—2441no—
152C23#1451983611yescharacteristic
153D4—4881no—
154D4—2441no—
155D4—2441no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—9761no—
162C2×C4—9761no—
163C2×C4—4882no—
164C2×C4—4882no—
165C2×C4—4881no—
166C2×C4—4881no—
167C2×C4—2441no—
168C2×C4#1451984011yescharacteristic
169C2×C4#1451984111yescharacteristic
170C8—42no—
171C8—9761no—
172C22—21no—
173C22—21no—
174C22—9761no—
175C22—4881no—
176C22—4881no—
177C22—4881no—
178C22—4881no—
179C22—2441no—
180C22—2441no—
181C22#1684061511yescharacteristic
182C22#1684066511yescharacteristic
183C22#1684066711yescharacteristic
184C4—21no—
185C4—21no—
186C4—42no—
187C4—9762no—
188C4—9761no—
189C4—4881no—
190C2—21no—
191C2—21no—
192C2—4881no—
193C2—4881no—
194C2#2146081511yescharacteristic
195C2#2146094711yescharacteristic
196C2#2146095311yescharacteristic
subgroupG/H
centerC22#16840667
commutatorC2×C42×C61C2×C4
FrattiniC2×C42C2×#6923
FittingC61×#336C4
socleC22×C61#353
prime radicalC22×D244D4
cosocle#16839664C22
hypercenter#336#6923
hypocenterC61#7435
LCSG ▷ C2×C42×C61 ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #336

Complements

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

NH ≅ G/Ntype
9. #16839632C4semidirect
10. C61×#336C4semidirect
107. #336#6923semidirect
109. C61#7435semidirect

Maximal subgroups

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

subgroupindex
#214609132
#214607482
#214473512
#743561