Finite Group Databasebeta

← #10591069

Subgroups — #10591069

Order 576.

414 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 120 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
1C22×C3×C4×D3C211yesmaximal, characteristic
2C2×#3748C211yesmaximal, characteristic, complemented
3C2×C3×#208C211yesmaximal, characteristic, complemented
4(C2×D3)⋊(C3⋊C8)C214yesmaximal, complemented
5#5207—31nomaximal
6#5126—31nomaximal
7C23×C3×D3C411yescharacteristic
8C22×C32×C4C2211yescharacteristic
9C22×C3×Dic3C411yescharacteristic
10C2×C3×C4×D3—22no—
11C2×C3×C4×D3—22no—
12C2×C3×C4×D3C412yes—
13C2×C3×C4×D3C2212yes—
14#3748—22no—
15C2×C3⋊(C3⋊C8)C2212yescomplemented
16C3×#208—22no—
17C2×C3×C3⋊C8C2212yescomplemented
18C23×C3×C4—31no—
19C22×C4×D3D311yescharacteristic
20C2×C3×M16—31no—
21C2×#208—31no—
22C2×#208—31no—
23C2×#208—61no—
24C2×#208D311yescharacteristic, complemented
25#830—34no—
26(C2×D3)⋊C8—34no—
27C23×C32C2×C411yescharacteristic
28C22×C3×D3—22no—
29C22×C3×D3—24no—
30C22×C3×D3C2×C412yes—
31C2×C32×C4C2×C411yescharacteristic
32C2×C32×C4D412yes—
33C2×C32×C4D412yes—
34C2×C32×C4C2311yescharacteristic
35C2×C3×Dic3—22no—
36C2×C3×Dic3C2×C412yes—
37C3×C4×D3—24no—
38C3×C4×D3—22no—
39C3×C4×D3—22no—
40C3⋊(C3⋊C8)—22no—
41C3×C3⋊C8—22no—
42#407—91noSylow
43C24×C3—31no—
44C23×D3Dic311yescharacteristic
45C22×C3×C4—21no—
46C22×C3×C4—31no—
47C22×C3×C4—32no—
48C22×C3×C4—32no—
49C22×C3×C4—62no—
50C22×C3×C4—62no—
51C22×C3×C4C2×D311yescharacteristic
52C22×C3×C4C2×D311yescharacteristic
53C22×Dic3Dic311yescharacteristic
54C2×C4×D3—22no—
55C2×C4×D3—22no—
56C2×C4×D3Dic312yes—
57C2×C4×D3C2×D312yes—
58C3×M16—62no—
59C2×C3×C8—32no—
60#208—22no—
61#208—64no—
62#208—62no—
63#208—62no—
64C2×C3⋊C8—32no—
65C2×C3⋊C8—32no—
66C2×C3⋊C8—62no—
67C2×C3⋊C8C2×D312yescomplemented
68C22×C32—22no—
69C22×C32C22⋊C412yes—
70C22×C32C22×C411yescharacteristic
71C2×C3×D3—28no—
72C2×C3×D3—22no—
73C2×C3×D3—22no—
74C2×C3×D3M1614yescomplemented
75C32×C4C22⋊C412yes—
76C32×C4C2×D411yescharacteristic
77C32×C4C2×D411yescharacteristic
78C3×Dic3—22no—
79C23×C4—31no—
80C2×M16—31no—
81C2×M16—91no—
82C22⋊C8—94no—
83C23×C3—21no—
84C23×C3—32no—
85C23×C3—64no—
86C23×C3—62no—
87C23×C3C4×D311yescharacteristic
88C23×C3C2×Dic311yescharacteristic
89C22×D3—22no—
90C22×D3—24no—
91C22×D3C2×Dic312yes—
92C2×C3×C4—21no—
93C2×C3×C4—22no—
94C2×C3×C4—22no—
95C2×C3×C4—21no—
96C2×C3×C4—32no—
97C2×C3×C4—62no—
98C2×C3×C4—64no—
99C2×C3×C4—62no—
100C2×C3×C4—62no—
101C2×C3×C4C4×D311yescharacteristic
102C2×C3×C4D1212yes—
103C2×C3×C4C2×Dic311yescharacteristic
104C2×C3×C4#6712yes—
105C2×C3×C4#6712yes—
106C2×C3×C4#6712yes—
107C2×C3×C4C22×D311yescharacteristic
108C2×C3×C4C22×D311yescharacteristic
109C2×Dic3—22no—
110C2×Dic3C2×Dic312yes—
111C4×D3—24no—
112C4×D3—22no—
113C4×D3—22no—
114C3×C8—62no—
115C3⋊C8—22no—
116C3⋊C8—64no—
117C3⋊C8—62no—
118C3⋊C8—62no—
119C2×C32—22no—
120C2×C32C2×C22⋊C411yescharacteristic
121C2×C32C2×M1612yes—
122C3×D3—24no—
123C24—31no—
124C22×C4—31no—
125C22×C4—32no—
126C22×C4—32no—
127C22×C4—62no—
128C22×C4—62no—
129C22×C4D3211yescharacteristic
130M16—62no—
131M16—182no—
132C2×C8—32no—
133C2×C8—92no—
134C22×C3—22no—
135C22×C3—22no—
136C22×C3—22no—
137C22×C3—21no—
138C22×C3—34no—
139C22×C3—42no—
140C22×C3—68no—
141C22×C3—62no—
142C22×C3—62no—
143C22×C3(C2×D3)⋊C412yes—
144C22×C3#21712yes—
145C22×C3C2×C4×D311yescharacteristic
146C22×C3C22×Dic311yescharacteristic
147C2×D3—28no—
148C2×D3—22no—
149C2×D3—22no—
150C2×D3#20814yescomplemented
151C3×C4—22no—
152C3×C4—21no—
153C3×C4—21no—
154C3×C4—62no—
155C3×C4(C2×D3)⋊C412yes—
156C3×C4#21712yes—
157C3×C4C2×D1211yescharacteristic
158C3×C4C2×#6711yescharacteristic
159C3×C4C2×#6711yescharacteristic
160C3×C4C2×#6711yescharacteristic
161Dic3—22no—
162C32#40711yescharacteristic, complemented, Sylow
163C23—32no—
164C23—64no—
165C23—62no—
166C23D3×Dic311yescharacteristic
167C2×C4—32no—
168C2×C4—62no—
169C2×C4—64no—
170C2×C4—62no—
171C2×C4—62no—
172C2×C4D3×Dic311yescharacteristic
173C2×C4#62512yes—
174C2×C4#62612yes—
175C2×C4C2×D3211yescharacteristic
176C8—62no—
177C8—182no—
178C2×C3—22no—
179C2×C3—22no—
180C2×C3—22no—
181C2×C3—21no—
182C2×C3—42no—
183C2×C3—64no—
184C2×C3D3×M1612yes—
185C2×C3C2×#20812yes—
186C2×C3C2×(C2×D3)⋊C411yescharacteristic
187C2×C3C2×#21711yescharacteristic
188D3—24no—
189C22—22no—
190C22—34no—
191C22—68no—
192C22—62no—
193C22—62no—
194C22(C2×D3)⋊Dic312yes—
195C22C2×D3×Dic311yescharacteristic
196C4—62no—
197C4(C2×D3)⋊Dic312yes—
198C4C2×#62511yescharacteristic
199C4C2×#62611yescharacteristic
200C3—21no—
201C3#512611yescharacteristic, complemented
202C3#520711yescharacteristic, complemented
203C2—22no—
204C2—64no—
205C2D3×#20812yes—
206C2C2×(C2×D3)⋊Dic311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D32
commutator, socleC22×C32C22×C4
FittingC22×C32×C4C22
prime radicalC23×C3×D3C4
cosocleC2×C32×C4C23
hypercenterC22×C4D32
hypocenterC32#407
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#3748C2semidirect
3. C2×C3×#208C2semidirect
4. (C2×D3)⋊(C3⋊C8)C2semidirect
15. C2×C3⋊(C3⋊C8)C22semidirect
17. C2×C3×C3⋊C8C22semidirect
24. C2×#208D3semidirect
67. C2×C3⋊C8C2×D3semidirect
74. C2×C3×D3M16semidirect
150. C2×D3#208semidirect
162. C32#407semidirect
201. C3#5126semidirect
202. C3#5207semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D32
C2×#37482
C2×C3×#2082
(C2×D3)⋊(C3⋊C8)2
#52073
#51263