Finite Group Databasebeta

← #13158840

Subgroups — #13158840

Order 1728.

1092 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1092 conjugacy classes of subgroups — too many to draw. Only its 164 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#11723915C211yesmaximal, characteristic
2C3×#64079C211yesmaximal, characteristic, complemented
3C32×#803C211yesmaximal, characteristic, complemented
4#11722036C214yesmaximal, complemented
5#10592412—31nomaximal
6#10590282—34nomaximal
7C2×C4⋊(C3⋊(C3⋊D3))—22no—
8C2×C4⋊(C3⋊(C3⋊D3))C2212yes—
9C33×C42C2211yescharacteristic
10C3×C4⋊(C3⋊Dic3)C2212yes—
11C3×#3750—22no—
12C2×C32×C3⋊C8C2212yes—
13#64083—31no—
14#64083—34no—
15#64083—64no—
16#64079D311yescharacteristic, complemented
17C3×#870—34no—
18C32×#97—31no—
19#63648—34no—
20C3×#803D314yescomplemented
21#63564—316no—
22C22×C3⋊(C3⋊D3)—22no—
23C2×C33×C4D412yes—
24C2×C33×C4C2311yescharacteristic
25C4⋊(C3⋊(C3⋊D3))—24no—
26C4⋊(C3⋊(C3⋊D3))—44no—
27C2×C3×C3⋊Dic3—22no—
28C32×C3⋊C8—22no—
29#5067—91no—
30#4713—94no—
31C2×C4⋊(C3⋊D3)—32no—
32C2×C4⋊(C3⋊D3)—38no—
33C2×C4⋊(C3⋊D3)—68no—
34C2×C4⋊(C3⋊D3)—616no—
35C2×C4⋊(C3⋊D3)—68no—
36C2×C4⋊(C3⋊D3)—62no—
37C2×C32×C8—32no—
38C32×C42—24no—
39C32×C42C2×D314yes—
40C32×C42C2×D311yescharacteristic
41C4⋊(C3⋊Dic3)C2×D312yes—
42#3750—22no—
43C3×C4⋊Dic3—38no—
44C3×Dic3⋊C4—68no—
45C2×C3×C3⋊C8C2×D318yes—
46C22×C33C2×D411yescharacteristic
47C2×C3⋊(C3⋊D3)—42no—
48C2×C3⋊(C3⋊D3)—44no—
49C33×C4—22no—
50C33×C4#4112yes—
51C3×C3⋊Dic3—42no—
52C3×#125—91no—
53#874—94no—
54#874—91no—
55#874—184no—
56#870—34no—
57C3×#97—34no—
58#821—916no—
59#809—94no—
60#803C3⋊D311yescharacteristic, complemented
61C22×C3⋊D3—616no—
62C22×C3⋊D3—68no—
63C22×C3⋊D3—62no—
64C2×C32×C4—28no—
65C2×C32×C4—24no—
66C2×C32×C4D1218yes—
67C2×C32×C4#6712yes—
68C2×C32×C4C22×D314yes—
69C2×C32×C4C22×D311yescharacteristic
70C2×C3⋊Dic3—22no—
71C4⋊(C3⋊D3)—64no—
72C4⋊(C3⋊D3)—616no—
73C4⋊(C3⋊D3)—1216no—
74C4⋊(C3⋊D3)—1232no—
75C4⋊(C3⋊D3)—1216no—
76C4⋊(C3⋊D3)—124no—
77C2×C3×Dic3—68no—
78C32×C8—62no—
79C3×C3⋊C8—28no—
80#490—271noSylow
81C2×C33#12411yescharacteristic
82C2×C33AGL(1,ℤ/8)12yes—
83C3⋊(C3⋊D3)—82no—
84C2×D12—98no—
85C2×D12—92no—
86C2×D12—188no—
87C2×D12—1816no—
88C2×D12—188no—
89C2×D12—182no—
90C2×C3×C8—38no—
91C3×C4⋊C4—92no—
92C3×C4⋊C4—182no—
93C3×C42—24no—
94C3×C42D3214yes—
95C3×C42C2×C3⋊D311yescharacteristic
96C4⋊Dic3—38no—
97Dic3⋊C4—68no—
98C2×C3⋊C8C2×C3⋊D312yes—
99C22×C32—24no—
100C22×C32C2×D1214yes—
101C22×C32C2×#6711yescharacteristic
102C2×C3⋊D3—1232no—
103C2×C3⋊D3—1216no—
104C2×C3⋊D3—128no—
105C2×C3⋊D3—1216no—
106C2×C3⋊D3—122no—
107C2×C3⋊D3—124no—
108C32×C4—216no—
109C32×C4—28no—
110C32×C4—22no—
111C32×C4—28no—
112C32×C4#23518yes—
113C32×C4#23912yes—
114C3⋊Dic3—42no—
115C3×Dic3—128no—
116#127—271no—
117#125—91no—
118#102—274no—
119#97—31no—
120C33#49011yescharacteristic, complemented, Sylow
121C22×D3—1816no—
122C22×D3—188no—
123C22×D3—182no—
124C2×C3×C4—28no—
125C2×C3×C4—24no—
126C2×C3×C4—182no—
127C2×C3×C4#62618yes—
128C2×C3×C4C4⋊(C3⋊D3)12yes—
129C2×C3×C4C2×D3214yes—
130C2×C3×C4C22×C3⋊D311yescharacteristic
131C2×Dic3—68no—
132D12—1816no—
133D12—184no—
134D12—3616no—
135D12—3632no—
136D12—3616no—
137D12—364no—
138C3×C8—68no—
139C3⋊C8—22no—
140C2×C32—28no—
141C2×C32—24no—
142C2×C32#87514yes—
143C2×C32#90818yes—
144C2×C32#94711yescharacteristic
145C2×C32#94912yes—
146C3⋊D3—2416no—
147C3⋊D3—248no—
148C3⋊D3—242no—
149C2×D4—272no—
150C2×D4—542no—
151C2×C8—32no—
152C4⋊C4—92no—
153C4⋊C4—182no—
154C42D3×C3⋊D311yescharacteristic
155C22×C3—24no—
156C22×C3C2×#62614yes—
157C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
158C2×D3—3632no—
159C2×D3—3616no—
160C2×D3—3616no—
161C2×D3—362no—
162C2×D3—364no—
163C2×D3—368no—
164C3×C4—216no—
165C3×C4—28no—
166C3×C4—22no—
167C3×C4—28no—
168C3×C4—362no—
169C3×C4#379918yes—
170C3×C4#382812yes—
171Dic3—128no—
172C32—24no—
173C32#471314yescomplemented
174C32#506711yescharacteristic, complemented
175C23—542no—
176D4—544no—
177D4—1084no—
178C2×C4—182no—
179C2×C4#629212yes—
180C2×C4C2×D3×C3⋊D311yescharacteristic
181C8—62no—
182C2×C3—28no—
183C2×C3—24no—
184C2×C3#6382518yes—
185C2×C3#6386414yes—
186C2×C3#6408411yescharacteristic
187C2×C3#6411712yes—
188D3—7216no—
189D3—728no—
190D3—722no—
191C22—1084no—
192C22—1082no—
193C22C2×#629211yescharacteristic
194C4—22no—
195C4—362no—
196C4#8902212yes—
197C3—24no—
198C3#1059028214yescomplemented
199C3#1059241211yescharacteristic, complemented
200C2—2162no—
201C2#1172337112yes—
202C2#1172344111yescharacteristic
subgroupG/H
centerC22C2×#6292
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radical#11723915C2
hypercenterC42D3×C3⋊D3
hypocenterC33#490
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×#64079C2semidirect
3. C32×#803C2semidirect
4. #11722036C2semidirect
16. #64079D3semidirect
20. C3×#803D3semidirect
60. #803C3⋊D3semidirect
120. C33#490semidirect
173. C32#4713semidirect
174. C32#5067semidirect
198. C3#10590282semidirect
199. C3#10592412semidirect

Maximal subgroups

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

subgroupindex
#117239152
C3×#640792
C32×#8032
#117220362
#105924123
#105902823