Finite Group Databasebeta

← #10592425

Subgroups — #10592425

Order 576.

570 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 125 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×C4×C3⋊D3C211yesmaximal, characteristic
2C2×#3743C212yesmaximal, complemented
3C32×C22⋊C8C211yesmaximal, characteristic, complemented
4#63657C211yesmaximal, characteristic, complemented
5#63647C212yesmaximal, complemented
6#4726—34nomaximal
7C23×C3⋊D3C411yescharacteristic
8C22×C32×C4C2211yescharacteristic
9C22×C3⋊Dic3C411yescharacteristic
10C2×C4×C3⋊D3—24no—
11C2×C4×C3⋊D3C411yescharacteristic
12C2×C4×C3⋊D3C411yescharacteristic
13C2×C4×C3⋊D3C2211yescharacteristic
14C2×C4×C3⋊D3C2211yescharacteristic
15C2×C32×C8C2212yescomplemented
16C2×C3⋊(C3⋊C8)C2212yescomplemented
17#3743—24no—
18C22×C4×D3—34no—
19C2×ΓL(1,𝔽25)—38no—
20C3×C22⋊C8D314yescomplemented
21#830—34no—
22(C2×D3)⋊C8—38no—
23C23×C32C2×C411yescharacteristic
24C22×C3⋊D3—22no—
25C22×C3⋊D3—22no—
26C22×C3⋊D3—22no—
27C22×C3⋊D3C2×C411yescharacteristic
28C22×C3⋊D3C2×C411yescharacteristic
29C2×C32×C4—22no—
30C2×C32×C4C2×C411yescharacteristic
31C2×C32×C4C2311yescharacteristic
32C2×C3⋊Dic3—22no—
33C2×C3⋊Dic3C2×C411yescharacteristic
34C2×C3⋊Dic3C2×C411yescharacteristic
35C4×C3⋊D3—22no—
36C4×C3⋊D3—22no—
37C4×C3⋊D3—22no—
38C4×C3⋊D3D414yes—
39C32×C8—22no—
40C3⋊(C3⋊C8)—22no—
41#407—91noSylow
42C23×D3—34no—
43C22×C3×C4C2×D314yes—
44C22×Dic3—34no—
45C2×C4×D3—34no—
46C2×C4×D3—34no—
47C2×C4×D3—34no—
48C2×C4×D3—34no—
49C2×C4×D3—616no—
50C2×C3×C8C2×D318yescomplemented
51C2×C3⋊C8—38no—
52ΓL(1,𝔽25)—616no—
53C22×C32—21no—
54C22×C32—21no—
55C22×C32M1612yescomplemented
56C22×C32C22×C411yescharacteristic
57C2×C3⋊D3—28no—
58C2×C3⋊D3—21no—
59C2×C3⋊D3—21no—
60C2×C3⋊D3—21no—
61C2×C3⋊D3—21no—
62C2×C3⋊D3C22⋊C412yes—
63C2×C3⋊D3M1612yescomplemented
64C32×C4—21no—
65C32×C4C2×D412yes—
66C3⋊Dic3—21no—
67C3⋊Dic3C22⋊C412yes—
68C23×C4—91no—
69C2×M16—92no—
70C22⋊C8—91no—
71C22⋊C8—92no—
72C22⋊C8C3⋊D311yescharacteristic, complemented
73C23×C3C4×D314yes—
74C22×D3—34no—
75C22×D3—34no—
76C22×D3—68no—
77C22×D3—68no—
78C22×D3—68no—
79C2×C3×C4—28no—
80C2×C3×C4C4×D314yes—
81C2×C3×C4C22×D314yes—
82C2×Dic3—34no—
83C2×Dic3—34no—
84C2×Dic3—68no—
85C4×D3—316no—
86C4×D3—68no—
87C4×D3—68no—
88C4×D3—68no—
89C3×C8—28no—
90C3⋊C8—68no—
91C2×C32—22no—
92C2×C32C2×C22⋊C411yescharacteristic
93C2×C32C2×M1611yescharacteristic
94C2×C32C2×M1611yescharacteristic
95C3⋊D3—22no—
96C3⋊D3—22no—
97C24—91no—
98C22×C4—91no—
99C22×C4—91no—
100C22×C4—91no—
101C22×C4—91no—
102C22×C4—91no—
103C22×C4—184no—
104C22×C4C2×C3⋊D311yescharacteristic
105M16—184no—
106C2×C8—92no—
107C2×C8C2×C3⋊D312yescomplemented
108C22×C3—24no—
109C22×C3—24no—
110C22×C3ΓL(1,𝔽25)18yescomplemented
111C22×C3C2×C4×D314yes—
112C2×D3—38no—
113C2×D3—38no—
114C2×D3—632no—
115C2×D3—64no—
116C2×D3—64no—
117C2×D3—64no—
118C2×D3—64no—
119C3×C4—24no—
120C3×C4D3×D418yes—
121Dic3—38no—
122Dic3—64no—
123C32#40711yescharacteristic, complemented, Sylow
124C23—91no—
125C23—91no—
126C23—182no—
127C23—182no—
128C23—182no—
129C23C4×C3⋊D311yescharacteristic
130C2×C4—22no—
131C2×C4—91no—
132C2×C4—94no—
133C2×C4—91no—
134C2×C4—182no—
135C2×C4—182no—
136C2×C4—182no—
137C2×C4—182no—
138C2×C4C4×C3⋊D311yescharacteristic
139C2×C4C22×C3⋊D311yescharacteristic
140C8—22no—
141C8—182no—
142C2×C3—28no—
143C2×C3D3×C22⋊C414yes—
144C2×C3C2×ΓL(1,𝔽25)14yes—
145C2×C3D3×M1614yes—
146D3—68no—
147D3—68no—
148C22—21no—
149C22—21no—
150C22—92no—
151C22—92no—
152C22—188no—
153C22—181no—
154C22—181no—
155C22—181no—
156C22—181no—
157C22#374312yescomplemented
158C22C2×C4×C3⋊D311yescharacteristic
159C4—21no—
160C4—92no—
161C4—181no—
162C4D4×C3⋊D312yes—
163C3#472614yescomplemented
164C2—22no—
165C2—182no—
166C2—182no—
167C2C22⋊C4×C3⋊D311yescharacteristic
168C2C2×#374311yescharacteristic
169C2M16×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×C3⋊D3
commutator, socleC22×C32C22×C4
FittingC32×C22⋊C8C2
prime radicalC23×C3⋊D3C4
cosocleC2×C32×C4C23
hypercenterC22⋊C8C3⋊D3
hypocenterC32#407
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#3743C2semidirect
3. C32×C22⋊C8C2semidirect
4. #63657C2semidirect
5. #63647C2semidirect
15. C2×C32×C8C22semidirect
16. C2×C3⋊(C3⋊C8)C22semidirect
20. C3×C22⋊C8D3semidirect
50. C2×C3×C8C2×D3semidirect
55. C22×C32M16semidirect
63. C2×C3⋊D3M16semidirect
72. C22⋊C8C3⋊D3semidirect
107. C2×C8C2×C3⋊D3semidirect
110. C22×C3ΓL(1,𝔽25)semidirect
123. C32#407semidirect
157. C22#3743semidirect
163. C3#4726semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊D32
C2×#37432
C32×C22⋊C82
#636572
#636472
#47263