Finite Group Databasebeta

← #13049666

Subgroups — #13049666

Order 1296.

1744 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1744 conjugacy classes of subgroups — too many to draw. Only its 266 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
1C4×C3⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
2C3×C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C33×C4×D3C211yesmaximal, characteristic, complemented
4#10620148C211yesmaximal, characteristic, complemented
5#10620147C211yesmaximal, characteristic, complemented
6#10620146C211yesmaximal, characteristic, complemented
7#10620145C211yesmaximal, characteristic, complemented
8#89076—31nomaximal
9#89020—313nomaximal
10C2×C3⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C33×D3C2211yescharacteristic
13C34×C4C2211yescharacteristic
14C3⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C33×Dic3C2211yescharacteristic
17C2×C33×C4—31no—
18#6312—31no—
19#6312—31no—
20C4⋊(C3⋊(C3⋊D3))—31no—
21C4×C3⋊(C3⋊D3)—31no—
22C4×C3⋊(C3⋊D3)—313no—
23C4×C3⋊(C3⋊D3)—613no—
24C4×C3⋊(C3⋊D3)D311yescharacteristic, complemented
25#6308—31no—
26C3×C4×C3⋊D3—313no—
27C32×C4×D3D3113yescomplemented
28#6293—313no—
29#6292—313no—
30#6291—313no—
31#6290—313no—
32C2×C34C2311yescharacteristic
33C3⋊(C3⋊(C3⋊D3))—21no—
34C3×C3⋊(C3⋊D3)—21no—
35C33×D3—21no—
36#3828—913no—
37#3798—913no—
38C22×C33—31no—
39C2×C3⋊(C3⋊D3)—31no—
40C2×C3⋊(C3⋊D3)—313no—
41C2×C3⋊(C3⋊D3)—613no—
42C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
43C2×C3×C3⋊D3—313no—
44C2×C32×D3C2×D3113yes—
45C33×C4—213no—
46C33×C4—31no—
47C33×C4C2×D3113yes—
48C33×C4C2×D311yescharacteristic
49C3⋊(C3⋊Dic3)—31no—
50C3⋊(C3⋊Dic3)—313no—
51C3⋊(C3⋊Dic3)—613no—
52C3⋊(C3⋊Dic3)C2×D311yescharacteristic
53C3×C3⋊Dic3—313no—
54C32×Dic3C2×D3113yes—
55C34#4111yescharacteristic, complemented, Sylow
56C2×C32×C4—313no—
57#638—913no—
58#638—913no—
59C4⋊(C3⋊D3)—913no—
60C4×C3⋊D3—313no—
61C4×C3⋊D3—913no—
62C4×C3⋊D3—913no—
63C4×C3⋊D3—1852no—
64#634—913no—
65C3×C4×D3—913no—
66C3×C4×D3C3⋊D3113yescomplemented
67#627—913no—
68#626—913no—
69#626—913no—
70#625—913no—
71C2×C33—213no—
72C2×C33—61no—
73C2×C33C22×D3113yes—
74C2×C33C22×D311yescharacteristic
75C3⋊(C3⋊D3)—21no—
76C3⋊(C3⋊D3)—626no—
77C3⋊(C3⋊D3)—613no—
78C3⋊(C3⋊D3)—61no—
79C3×C3⋊D3—613no—
80C32×D3—213no—
81#235—2713no—
82#235—271no—
83C22×C32—313no—
84C2×C3⋊D3—313no—
85C2×C3⋊D3—913no—
86C2×C3⋊D3—913no—
87C2×C3⋊D3—1852no—
88C2×C3×D3—913no—
89C2×C3×D3C2×C3⋊D3113yes—
90C32×C4—252no—
91C32×C4—313no—
92C32×C4D32113yes—
93C32×C4C2×C3⋊D3113yes—
94C3⋊Dic3—313no—
95C3⋊Dic3—913no—
96C3⋊Dic3—913no—
97C3⋊Dic3—1852no—
98C3×Dic3—913no—
99C3×Dic3C2×C3⋊D3113yes—
100C33—213no—
101C33#235113yescomplemented
102C33#23511yescharacteristic, complemented
103C2×C3×C4—313no—
104C2×C3×C4—271no—
105#67—2713no—
106#67—2713no—
107#67—271no—
108#67—271no—
109D12—2713no—
110D12—271no—
111C4×D3—913no—
112C4×D3—2713no—
113C4×D3—271no—
114C4×D3—5413no—
115C4×D3C3⋊(C3⋊D3)11yescharacteristic, complemented
116Dic6—2713no—
117Dic6—271no—
118C2×C32—252no—
119C2×C32—613no—
120C2×C32C2×D32113yes—
121C2×C32C22×C3⋊D3113yes—
122C3⋊D3—613no—
123C3⋊D3—18104no—
124C3⋊D3—1813no—
125C3⋊D3—1813no—
126C3×D3—213no—
127C3×D3—1813no—
128#41—811noSylow
129C22×C3—313no—
130C22×C3—271no—
131C2×D3—913no—
132C2×D3—2713no—
133C2×D3—271no—
134C2×D3—5413no—
135C2×D3C2×C3⋊(C3⋊D3)11yescharacteristic
136C3×C4—213no—
137C3×C4—313no—
138C3×C4—271no—
139C3×C4D3×C3⋊D3113yes—
140C3×C4C2×C3⋊(C3⋊D3)11yescharacteristic
141Dic3—913no—
142Dic3—2713no—
143Dic3—271no—
144Dic3—5413no—
145Dic3C2×C3⋊(C3⋊D3)11yescharacteristic
146C32—252no—
147C32#3798113yescomplemented
148C32#3828113yescomplemented
149Q8—811no—
150D4—811no—
151D4—811no—
152D4—811no—
153C2×C4—31no—
154C2×C4—271no—
155C2×C4—811no—
156C2×C3—213no—
157C2×C3—613no—
158C2×C3—541no—
159C2×C3C2×D3×C3⋊D3113yes—
160C2×C3C22×C3⋊(C3⋊D3)11yescharacteristic
161D3—21no—
162D3—1813no—
163D3—5426no—
164D3—5413no—
165D3—541no—
166C22—31no—
167C22—271no—
168C22—811no—
169C4—31no—
170C4—271no—
171C4—811no—
172C4D3×C3⋊(C3⋊D3)11yescharacteristic
173C3—213no—
174C3#89020113yescomplemented
175C3#8907611yescharacteristic, complemented
176C2—61no—
177C2—541no—
178C2—1621no—
179C2C2×D3×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
center, hypercenterC4D3×C3⋊(C3⋊D3)
commutator, socle, cosocleC2×C34C23
FrattiniC2C2×D3×C3⋊(C3⋊D3)
FittingC34×C4C22
hypocenterC34#41
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C4×C3⋊(C3⋊(C3⋊D3))C2semidirect
2. C3×C4×C3⋊(C3⋊D3)C2semidirect
3. C33×C4×D3C2semidirect
4. #10620148C2semidirect
5. #10620147C2semidirect
6. #10620146C2semidirect
7. #10620145C2semidirect
24. C4×C3⋊(C3⋊D3)D3semidirect
27. C32×C4×D3D3semidirect
55. C34#41semidirect
66. C3×C4×D3C3⋊D3semidirect
101. C33#235semidirect
102. C33#235semidirect
115. C4×D3C3⋊(C3⋊D3)semidirect
147. C32#3798semidirect
148. C32#3828semidirect
174. C3#89020semidirect
175. C3#89076semidirect

Maximal subgroups

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

subgroupindex
C4×C3⋊(C3⋊(C3⋊D3))2
C3×C4×C3⋊(C3⋊D3)2
C33×C4×D32
#106201482
#106201472
#106201462
#106201452
#890763
#890203