Finite Group Databasebeta

← #13163644

Subgroups — #13163644

Order 1728.

1484 conjugacy classes of subgroups; only the 403 normal subgroups are shown.

normal subgroups only

This group has 1484 conjugacy classes of subgroups — too many to draw. Only its 403 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
1C33×#117C211yesmaximal, characteristic, complemented
2#11723971C211yesmaximal, characteristic
3#11722200C211yesmaximal, characteristic, complemented
4#11722179C211yesmaximal, characteristic, complemented
5#11722179C211yesmaximal, characteristic, complemented
6#11722178C211yesmaximal, characteristic, complemented
7#11722177C211yesmaximal, characteristic, complemented
8#10592702—313nomaximal
9C22×C33×C4C2211yescharacteristic
10C2×#6312—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C33×C22⋊C4—21no—
15C33×C42—21no—
16#88859—21no—
17C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
18C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C32×#117D3113yescomplemented
21#64139—313no—
22#63657—313no—
23#63636—313no—
24#63636—313no—
25#63635—313no—
26#63634—313no—
27C23×C33D411yescharacteristic
28C22×C3⋊(C3⋊D3)—21no—
29C2×C33×C4—21no—
30C2×C33×C4—21no—
31C2×C33×C4—21no—
32C2×C33×C4—21no—
33C2×C33×C4D411yescharacteristic
34C2×C33×C4C2311yescharacteristic
35#6312—42no—
36C2×C3⋊(C3⋊Dic3)—21no—
37C4⋊(C3⋊(C3⋊D3))—21no—
38C4⋊(C3⋊(C3⋊D3))—21no—
39C3⋊(C3⋊(C3⋊C8))—21no—
40C3⋊(C3⋊(C3⋊C8))—21no—
41#5003—913no—
42C22×C32×C4C2×D3113yes—
43C2×#638—613no—
44C2×C4⋊(C3⋊D3)—313no—
45C32×C4⋊C4C2×D3113yes—
46C32×C4⋊C4C2×D3113yes—
47C32×C22⋊C4—213no—
48C32×C42—213no—
49#3752—613no—
50C4⋊(C3⋊Dic3)—313no—
51C2×C3⋊(C3⋊C8)—313no—
52C2×C3⋊(C3⋊C8)—313no—
53C22×C33—21no—
54C22×C33—21no—
55C22×C33—21no—
56C22×C33C2×D411yescharacteristic
57C2×C3⋊(C3⋊D3)—41no—
58C2×C3⋊(C3⋊D3)—41no—
59C2×C3⋊(C3⋊D3)—41no—
60C33×C4—21no—
61C33×C4—41no—
62C33×C4—41no—
63C33×C4#4111yescharacteristic
64C33×C4#4111yescharacteristic
65C3⋊(C3⋊Dic3)—41no—
66C3×#117C3⋊D3113yescomplemented
67#930—913no—
68#830—913no—
69#809—913no—
70#809—913no—
71#808—913no—
72#807—913no—
73C23×C32#67113yes—
74C22×C3⋊D3—613no—
75C2×C32×C4—213no—
76C2×C32×C4—213no—
77C2×C32×C4—213no—
78C2×C32×C4—213no—
79C2×C32×C4#67113yes—
80C2×C32×C4C22×D3113yes—
81#638—1226no—
82C2×C3⋊Dic3—613no—
83C4⋊(C3⋊D3)—613no—
84C4⋊(C3⋊D3)—613no—
85C3⋊(C3⋊C8)—613no—
86C3⋊(C3⋊C8)—613no—
87#482—271noSylow
88C2×C33—41no—
89C2×C33#12311yescharacteristic
90C2×C33#13511yescharacteristic
91C2×C33AGL(1,ℤ/8)11yescharacteristic
92C3⋊(C3⋊D3)—81no—
93C22×C3×C4C2×C3⋊D3113yes—
94C2×#67—1813no—
95C2×D12—913no—
96C3×C4⋊C4C2×C3⋊D3113yes—
97C3×C4⋊C4C2×C3⋊D3113yes—
98C3×C22⋊C4—213no—
99C3×C42—213no—
100(C2×D3)⋊C4—1813no—
101C4⋊Dic3—913no—
102C2×C3⋊C8—913no—
103C2×C3⋊C8—913no—
104C22×C32—213no—
105C22×C32—213no—
106C22×C32—213no—
107C22×C32C2×#67113yes—
108C2×C3⋊D3—1213no—
109C2×C3⋊D3—1213no—
110C2×C3⋊D3—1213no—
111C32×C4—213no—
112C32×C4—413no—
113C32×C4—413no—
114C32×C4#235113yes—
115C32×C4#235113yes—
116C3⋊Dic3—1213no—
117#121—271no—
118#117C3⋊(C3⋊D3)11yescharacteristic, complemented
119#107—271no—
120#106—271no—
121#102—271no—
122#102—271no—
123C22⋊C8—271no—
124C33#48211yescharacteristic, complemented, Sylow
125C23×C3#638113yes—
126C22×D3—1813no—
127C2×C3×C4—213no—
128C2×C3×C4—213no—
129C2×C3×C4—213no—
130C2×C3×C4—213no—
131C2×C3×C4#638113yes—
132C2×C3×C4C22×C3⋊D3113yes—
133#67—3626no—
134C2×Dic3—1813no—
135D12—1813no—
136D12—1813no—
137C3⋊C8—1813no—
138C3⋊C8—1813no—
139C2×C32—413no—
140C2×C32#929113yes—
141C2×C32#949113yes—
142C2×C32#950113yes—
143C3⋊D3—2413no—
144C2×D4—271no—
145C2×D4—541no—
146C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
147C2×C8—271no—
148C2×C8—271no—
149C4⋊C4—271no—
150C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
151C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—541no—
154C42—21no—
155C22×C3—213no—
156C22×C3—213no—
157C22×C3—213no—
158C22×C3C2×#638113yes—
159C2×D3—3613no—
160C2×D3—3613no—
161C2×D3—3613no—
162C3×C4—213no—
163C3×C4—413no—
164C3×C4—413no—
165C3×C4#3828113yes—
166C3×C4#3828113yes—
167Dic3—3613no—
168C32#5003113yescomplemented
169C23—541no—
170C23#631211yescharacteristic
171D4—541no—
172D4—541no—
173D4—1082no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—541no—
179C2×C4#631211yescharacteristic
180C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
181C8—541no—
182C8—541no—
183C2×C3—413no—
184C2×C3#64138113yes—
185C2×C3#64158113yes—
186C2×C3#64159113yes—
187D3—7213no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—1081no—
192C22—1081no—
193C22—1081no—
194C22C2×#631211yescharacteristic
195C4—21no—
196C4—41no—
197C4—41no—
198C4—1081no—
199C4#8907611yescharacteristic
200C4#8907611yescharacteristic
201C3#10592702113yescomplemented
202C2—41no—
203C2—2161no—
204C2#1172397011yescharacteristic
205C2#1172399011yescharacteristic
206C2#1172399111yescharacteristic
subgroupG/H
centerC22C2×#6312
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#117C2
socleC22×C33C2×D4
prime radical#11723971C2
hypercenter#117C3⋊(C3⋊D3)
hypocenterC33#482
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C33×#117C2semidirect
3. #11722200C2semidirect
4. #11722179C2semidirect
5. #11722179C2semidirect
6. #11722178C2semidirect
7. #11722177C2semidirect
20. C32×#117D3semidirect
66. C3×#117C3⋊D3semidirect
118. #117C3⋊(C3⋊D3)semidirect
124. C33#482semidirect
168. C32#5003semidirect
201. C3#10592702semidirect

Maximal subgroups

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

subgroupindex
C33×#1172
#117239712
#117222002
#117221792
#117221782
#117221772
#105927023