Finite Group Databasebeta

← #13163495

Subgroups — #13163495

Order 1728.

1428 conjugacy classes of subgroups; only the 351 normal subgroups are shown.

normal subgroups only

This group has 1428 conjugacy classes of subgroups — too many to draw. Only its 351 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#11723934C211yesmaximal, characteristic
2#11723931C211yesmaximal, characteristic, complemented
3C33×#106C211yesmaximal, characteristic, complemented
4#11722191C211yesmaximal, characteristic
5#11722184C211yesmaximal, characteristic, complemented
6#11722179C211yesmaximal, characteristic
7#11722177C211yesmaximal, characteristic, complemented
8#10592570—313nomaximal
9C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
10C2×C4×C3⋊(C3⋊D3)—21no—
11C2×C33×C8C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14#88859—21no—
15C4⋊(C3⋊(C3⋊Dic3))—21no—
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17#88857—21no—
18C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20#64102—313no—
21#64099—313no—
22C32×#106D3113yescomplemented
23#63648—313no—
24#63641—313no—
25#63636—313no—
26#63634—313no—
27C22×C3⋊(C3⋊D3)—21no—
28C2×C33×C4—21no—
29C2×C33×C4—21no—
30C2×C33×C4C2311yescharacteristic
31C2×C3⋊(C3⋊Dic3)—21no—
32C2×C3⋊(C3⋊Dic3)D411yescharacteristic
33C2×C3⋊(C3⋊Dic3)D411yescharacteristic
34C4⋊(C3⋊(C3⋊D3))—21no—
35C4⋊(C3⋊(C3⋊D3))Q812yes—
36C4×C3⋊(C3⋊D3)—42no—
37C33×C8—21no—
38C3⋊(C3⋊(C3⋊C8))—21no—
39#4871—913no—
40C2×C4⋊(C3⋊D3)—313no—
41C2×C4×C3⋊D3—613no—
42C2×C32×C8C2×D3113yes—
43C32×C4⋊C4C2×D3113yes—
44C32×C4⋊C4C2×D3113yes—
45#3752—613no—
46C4⋊(C3⋊Dic3)—313no—
47C4⋊(C3⋊Dic3)—613no—
48#3750—613no—
49C4×C3⋊Dic3—313no—
50C2×C3⋊(C3⋊C8)—313no—
51C22×C33C2×D411yescharacteristic
52C2×C3⋊(C3⋊D3)—22no—
53C2×C3⋊(C3⋊D3)—41no—
54C2×C3⋊(C3⋊D3)—41no—
55C33×C4—41no—
56C33×C4—41no—
57C33×C4C2×Q811yescharacteristic
58C33×C4#4111yescharacteristic
59C3⋊(C3⋊Dic3)—21no—
60C3⋊(C3⋊Dic3)—21no—
61C3⋊(C3⋊Dic3)—41no—
62#893—913no—
63#890—913no—
64C3×#106C3⋊D3113yescomplemented
65#821—913no—
66Dic3⋊C8—913no—
67#809—913no—
68#807—913no—
69C22×C3⋊D3—613no—
70C2×C32×C4—213no—
71C2×C32×C4—213no—
72C2×C32×C4C22×D3113yes—
73C2×C3⋊Dic3—313no—
74C2×C3⋊Dic3—313no—
75C2×C3⋊Dic3—613no—
76C4⋊(C3⋊D3)—326no—
77C4⋊(C3⋊D3)—613no—
78C4×C3⋊D3—1226no—
79C32×C8—213no—
80C3⋊(C3⋊C8)—613no—
81#478—271noSylow
82C2×C33C22⋊Q811yescharacteristic
83C2×C33#13511yescharacteristic
84C2×C33AGL(1,ℤ/8)11yescharacteristic
85C3⋊(C3⋊D3)—42no—
86C2×D12—913no—
87C2×C4×D3—1813no—
88C2×C3×C8C2×C3⋊D3113yes—
89C3×C4⋊C4C2×C3⋊D3113yes—
90C3×C4⋊C4C2×C3⋊D3113yes—
91(C2×D3)⋊C4—1813no—
92C4⋊Dic3—913no—
93C4⋊Dic3—1813no—
94Dic3⋊C4—1813no—
95C4×Dic3—913no—
96C2×C3⋊C8—913no—
97C22×C32D3×D4113yes—
98C2×C3⋊D3—626no—
99C2×C3⋊D3—1213no—
100C2×C3⋊D3—1213no—
101C32×C4—413no—
102C32×C4—413no—
103C32×C4#235113yes—
104C32×C4D3×Q8113yes—
105C3⋊Dic3—613no—
106C3⋊Dic3—613no—
107C3⋊Dic3—1213no—
108#125—271no—
109C4×D4—271no—
110#107—271no—
111#106C3⋊(C3⋊D3)11yescharacteristic, complemented
112C4⋊C8—271no—
113#102—271no—
114#102—271no—
115C33#47811yescharacteristic, complemented, Sylow
116C22×D3—1813no—
117C2×C3×C4—213no—
118C2×C3×C4—213no—
119C2×C3×C4C22×C3⋊D3113yes—
120C2×Dic3—913no—
121C2×Dic3—913no—
122C2×Dic3—1813no—
123D12—926no—
124D12—1813no—
125C4×D3—3626no—
126C3×C8—213no—
127C3⋊C8—1813no—
128C2×C32#896113yes—
129C2×C32#914113yes—
130C2×C32#916113yes—
131C3⋊D3—1226no—
132C2×D4—271no—
133C22×C4—541no—
134C2×C8—271no—
135C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
136C4⋊C4—271no—
137C4⋊C4—541no—
138C4⋊C4—541no—
139C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
140C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
141C22⋊C4—541no—
142C42—271no—
143C22×C3D4×C3⋊D3113yes—
144C2×D3—1826no—
145C2×D3—3613no—
146C2×D3—3613no—
147C3×C4—413no—
148C3×C4—413no—
149C3×C4#3828113yes—
150C3×C4Q8×C3⋊D3113yes—
151Dic3—1813no—
152Dic3—1813no—
153Dic3—3613no—
154C32#4871113yescomplemented
155C23—541no—
156D4—272no—
157D4—541no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—271no—
161C2×C4—271no—
162C2×C4—541no—
163C2×C4—1082no—
164C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
165C8—21no—
166C8—541no—
167C2×C3#64105113yes—
168C2×C3#64123113yes—
169C2×C3#64125113yes—
170D3—3626no—
171C22—542no—
172C22—1081no—
173C22—1081no—
174C22D4×C3⋊(C3⋊D3)11yescharacteristic
175C4—41no—
176C4—41no—
177C4—541no—
178C4—541no—
179C4—1081no—
180C4#8907611yescharacteristic
181C4Q8×C3⋊(C3⋊D3)11yescharacteristic
182C3#10592570113yescomplemented
183C2—1082no—
184C2#1172393711yescharacteristic
185C2#1172395511yescharacteristic
186C2#1172395711yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#106C2
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenter#106C3⋊(C3⋊D3)
hypocenterC33#478
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
2. #11723931C2semidirect
3. C33×#106C2semidirect
5. #11722184C2semidirect
7. #11722177C2semidirect
22. C32×#106D3semidirect
64. C3×#106C3⋊D3semidirect
111. #106C3⋊(C3⋊D3)semidirect
115. C33#478semidirect
154. C32#4871semidirect
182. C3#10592570semidirect

Maximal subgroups

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

subgroupindex
#117239342
#117239312
C33×#1062
#117221912
#117221842
#117221792
#117221772
#105925703