Finite Group Databasebeta

← C2×C3×#3743 #11723855

Subgroups — C2×C3×#3743 #11723855

Order 864.

496 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 170 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
1C2×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×C33×C8C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C3×#3743C214yesmaximal, direct product factor
5C2×#3743C311yesmaximal, characteristic, direct product factor
6C2×C3×ΓL(1,𝔽25)—34nomaximal
7C22×C3×C3⋊D3C411yescharacteristic
8C2×C33×C4C2211yescharacteristic
9C2×C3×C3⋊Dic3C411yescharacteristic
10C3×C4×C3⋊D3C412yes—
11C3×C4×C3⋊D3C2212yes—
12C33×C8C2212yescomplemented
13C3×C3⋊(C3⋊C8)C2212yescomplemented
14C2×C4×C3⋊D3C2×C311yescharacteristic
15C2×C3×C4×D3—34no—
16C2×C32×C8—24no—
17C2×C32×C8D314yescomplemented
18C2×C32×C8C2×C311yescharacteristic, complemented
19C2×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
20#3743C2×C314yesdirect product factor
21C2×C3×C3⋊C8—34no—
22C3×ΓL(1,𝔽25)—316no—
23C22×C33C2×C411yescharacteristic
24C2×C3×C3⋊D3—22no—
25C2×C3×C3⋊D3C2×C412yes—
26C33×C4C2×C411yescharacteristic
27C33×C4C2311yescharacteristic
28C3×C3⋊Dic3C2×C412yes—
29C2×C3×M16—91no—
30C2×ΓL(1,𝔽25)—34no—
31C22×C3⋊D3C3×C411yescharacteristic
32C22×C3×D3—34no—
33C2×C32×C4—24no—
34C2×C32×C4C2×D314yes—
35C2×C32×C4C22×C311yescharacteristic
36C2×C3⋊Dic3C3×C411yescharacteristic
37C4×C3⋊D3C3×C412yes—
38C4×C3⋊D3C22×C312yes—
39C2×C3×Dic3—34no—
40C3×C4×D3—38no—
41C3×C4×D3—38no—
42C32×C8—28no—
43C32×C8C2×D318yescomplemented
44C32×C8C22×C312yescomplemented
45C3⋊(C3⋊C8)C22×C312yescomplemented
46C3×C3⋊C8—38no—
47C2×C33M1612yescomplemented
48C2×C33C22×C411yescharacteristic
49C3×C3⋊D3—22no—
50C22×C3×C4—91no—
51C2×C4×D3—34no—
52C3×M16—94no—
53C2×C3×C8—24no—
54C2×C3×C8—91no—
55C2×C3×C8C3×D314yescomplemented
56C2×C3×C8C3⋊D311yescharacteristic, complemented
57C2×C3⋊C8—34no—
58ΓL(1,𝔽25)—316no—
59C22×C32—24no—
60C22×C32C4×D314yes—
61C22×C32C2×C3×C411yescharacteristic
62C2×C3⋊D3—22no—
63C2×C3⋊D3C2×C3×C412yes—
64C2×C3×D3—38no—
65C2×C3×D3—68no—
66C32×C4—24no—
67C32×C4—24no—
68C32×C4C4×D314yes—
69C32×C4C2×C3×C411yescharacteristic
70C32×C4C22×D314yes—
71C32×C4C23×C311yescharacteristic
72C3⋊Dic3C2×C3×C412yes—
73C3×Dic3—38no—
74C2×M16—91noSylow
75C33C2×M1611yescharacteristic, complemented, Sylow
76C23×C3—91no—
77C22×D3—34no—
78C2×C3×C4—24no—
79C2×C3×C4—91no—
80C2×C3×C4—92no—
81C2×C3×C4—92no—
82C2×C3×C4C2×C3×D314yes—
83C2×C3×C4C2×C3⋊D311yescharacteristic
84C2×Dic3—34no—
85C4×D3—38no—
86C4×D3—38no—
87C3×C8—28no—
88C3×C8—92no—
89C3×C8C2×C3×D318yescomplemented
90C3×C8C2×C3⋊D312yescomplemented
91C3⋊C8—38no—
92C2×C32—28no—
93C2×C32—24no—
94C2×C32ΓL(1,𝔽25)18yescomplemented
95C2×C32C3×M1612yescomplemented
96C2×C32C2×C4×D314yes—
97C2×C32C22×C3×C411yescharacteristic
98C3⋊D3—22no—
99C3×D3—68no—
100C22×C4—91no—
101M16—94no—
102C2×C8—91no—
103C2×C8C3×C3⋊D311yescharacteristic, complemented
104C22×C3—24no—
105C22×C3—92no—
106C22×C3—182no—
107C22×C3C3×C4×D314yes—
108C22×C3C4×C3⋊D311yescharacteristic
109C2×D3—38no—
110C2×D3—68no—
111C3×C4—24no—
112C3×C4—24no—
113C3×C4—92no—
114C3×C4C3×C4×D314yes—
115C3×C4C4×C3⋊D311yescharacteristic
116C3×C4C22×C3×D314yes—
117C3×C4C22×C3⋊D311yescharacteristic
118Dic3—38no—
119C32—24no—
120C32C2×ΓL(1,𝔽25)14yescomplemented
121C32C2×C3×M1611yescharacteristic, complemented
122C23—91no—
123C2×C4—91no—
124C2×C4—92no—
125C2×C4—92no—
126C2×C4C2×C3×C3⋊D311yescharacteristic
127C8—92no—
128C8C2×C3×C3⋊D312yescomplemented
129C2×C3—28no—
130C2×C3—24no—
131C2×C3—182no—
132C2×C3C3×ΓL(1,𝔽25)18yescomplemented
133C2×C3#374312yesdirect product factor
134C2×C3C2×C3×C4×D314yes—
135C2×C3C2×C4×C3⋊D311yescharacteristic
136D3—68no—
137C22—92no—
138C22—182no—
139C22C3×C4×C3⋊D311yescharacteristic
140C4—92no—
141C4C3×C4×C3⋊D311yescharacteristic
142C4C22×C3×C3⋊D311yescharacteristic
143C3—24no—
144C3C2×C3×ΓL(1,𝔽25)14yescomplemented
145C3C2×#374311yescharacteristic, direct product factor
146C2—182no—
147C2C3×#374312yesdirect product factor
148C2C2×C3×C4×C3⋊D311yescharacteristic
subgroupG/H
centerC2×C3×C4C2×C3⋊D3
commutatorC2×C32C22×C3×C4
FrattiniC4C22×C3×C3⋊D3
FittingC2×C33×C8C2
socleC22×C33C2×C4
prime radicalC22×C3×C3⋊D3C4
cosocleC32×C4C23×C3
hypercenterC2×C3×C8C3⋊D3
hypocenterC32C2×C3×M16
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
2. C2×C33×C8C2semidirect
3. C2×C3×C3⋊(C3⋊C8)C2semidirect
4. C3×#3743C2direct (N × H)
5. C2×#3743C3direct (N × H)
12. C33×C8C22semidirect
13. C3×C3⋊(C3⋊C8)C22semidirect
17. C2×C32×C8D3semidirect
18. C2×C32×C8C2×C3semidirect
19. C2×C3⋊(C3⋊C8)C2×C3semidirect
20. #3743C2×C3direct (N × H)
43. C32×C8C2×D3semidirect
44. C32×C8C22×C3semidirect
45. C3⋊(C3⋊C8)C22×C3semidirect
47. C2×C33M16semidirect
55. C2×C3×C8C3×D3semidirect
56. C2×C3×C8C3⋊D3semidirect
75. C33C2×M16semidirect
89. C3×C8C2×C3×D3semidirect
90. C3×C8C2×C3⋊D3semidirect
94. C2×C32ΓL(1,𝔽25)semidirect
95. C2×C32C3×M16semidirect
103. C2×C8C3×C3⋊D3semidirect
120. C32C2×ΓL(1,𝔽25)semidirect
121. C32C2×C3×M16semidirect
128. C8C2×C3×C3⋊D3semidirect
132. C2×C3C3×ΓL(1,𝔽25)semidirect
133. C2×C3#3743direct (N × H)
144. C3C2×C3×ΓL(1,𝔽25)semidirect
145. C3C2×#3743direct (N × H)
147. C2C3×#3743direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊D32
C2×C33×C82
C2×C3×C3⋊(C3⋊C8)2
C3×#37432
C2×#37433
C2×C3×ΓL(1,𝔽25)3