Finite Group Databasebeta

← #13173052

Subgroups — #13173052

Order 1728.

826 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 139 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⋊S4))C212yesmaximal, complemented
2C32×C8×A4C211yesmaximal, characteristic, complemented
3#10593636—312nomaximal
4#10592814—31nomaximal
5C2×C3⋊(C3⋊S4)—22no—
6C32×C4×A4C2211yescharacteristic
7#88852—41nomaximal
8C4⋊(C3⋊S4)—324no—
9C22×C32×C8D311yescharacteristic, complemented
10#64139—32no—
11C2×#3745—31no—
12C3×C8×A4D3112yescomplemented
13#63648—32no—
14#63645—31no—
15C2×C32×A4D411yescharacteristic
16C3⋊(C3⋊S4)—42no—
17C4⋊(C3⋊(C3⋊D3))—42no—
18C33×C8—41no—
19#5402—99no—
20#5115—94no—
21C2×C3⋊S4—624no—
22C22×C32×C4C2×D311yescharacteristic
23C2×#638—62no—
24C2×C4⋊(C3⋊D3)—32no—
25C3×C4×A4C2×D3112yes—
26C2×C32×C8—31no—
27C2×C32×C8—31no—
28#3752—62no—
29C4⋊(C3⋊Dic3)—32no—
30#3745—61no—
31#3745—61no—
32#3745—1212no—
33C2×C3⋊(C3⋊D3)—82no—
34C32×A4D811yescharacteristic, complemented
35C33×C4—41no—
36C4⋊S4—918no—
37C22×C3×C8C3⋊D314yescomplemented
38#930—98no—
39C2×D24—94no—
40C8×A4C3⋊D319yescomplemented
41#821—98no—
42#818—94no—
43C23×C32D1211yescharacteristic
44C22×C3⋊D3—62no—
45C2×C3×A4D12112yes—
46C3⋊S4—1224no—
47C2×C32×C4—31no—
48C2×C32×C4—31no—
49#638—122no—
50#638—122no—
51C2×C3⋊Dic3—62no—
52C4⋊(C3⋊D3)—62no—
53C4⋊(C3⋊D3)—62no—
54C4⋊(C3⋊D3)—1224no—
55C32×C8—31no—
56C32×C8—412no—
57C32×C8S411yescharacteristic, complemented
58#466—271noSylow
59C2×C33—41no—
60C3⋊(C3⋊D3)—162no—
61C2×S4—1818no—
62C22×C3×C4C2×C3⋊D314yes—
63C2×#67—188no—
64C2×D12—98no—
65C4×A4C2×C3⋊D319yes—
66C2×C3×C8—34no—
67C2×C3×C8—34no—
68(C2×D3)⋊C4—188no—
69C4⋊Dic3—98no—
70D24—184no—
71D24—184no—
72D24—369no—
73C22×C32—31no—
74C22×C32—31no—
75C22×C32D2411yescharacteristic, complemented
76C2×C3⋊D3—122no—
77C2×C3⋊D3—122no—
78C2×C3⋊D3—122no—
79C2×C3⋊D3—2424no—
80C3×A4D24112yescomplemented
81C32×C4—31no—
82C32×C4—412no—
83C32×C4C2×S411yescharacteristic
84C3⋊Dic3—122no—
85C2×D8—271no—
86C22×C8C3⋊(C3⋊D3)11yescharacteristic, complemented
87#121—272no—
88#107—271no—
89#102—272no—
90C33—41noSylow
91C23×C3C4⋊(C3⋊D3)14yes—
92C22×D3—188no—
93C2×A4C4⋊(C3⋊D3)19yes—
94S4—3618no—
95C2×C3×C4—34no—
96C2×C3×C4—34no—
97#67—368no—
98#67—368no—
99C2×Dic3—188no—
100D12—188no—
101D12—188no—
102D12—3618no—
103C3×C8—34no—
104C3×C8—49no—
105C3×C8C3⋊S414yescomplemented
106C2×C32—31no—
107C2×C32—31no—
108C2×C32—412no—
109C2×C32C4⋊S411yescharacteristic
110C3⋊D3—242no—
111C3⋊D3—4824no—
112C2×D4—272no—
113C2×D4—542no—
114C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
115D8—541no—
116D8—541no—
117C2×C8—31no—
118C2×C8—31no—
119C4⋊C4—272no—
120C22⋊C4—542no—
121C22×C3—34no—
122C22×C3—34no—
123C22×C3#374514yescomplemented
124C2×D3—368no—
125C2×D3—368no—
126C2×D3—368no—
127C2×D3—7218no—
128A4#374519yescomplemented
129C3×C4—34no—
130C3×C4—49no—
131C3×C4C2×C3⋊S414yes—
132Dic3—368no—
133C32—412no—
134C32#540211yescharacteristic, complemented
135C23—542no—
136C23C4⋊(C3⋊(C3⋊D3))11yescharacteristic
137D4—542no—
138D4—542no—
139D4—1082no—
140D4—1082no—
141C2×C4—31no—
142C2×C4—31no—
143C2×C4—542no—
144C8—31no—
145C8C3⋊(C3⋊S4)11yescharacteristic, complemented
146C2×C3—34no—
147C2×C3—34no—
148C2×C3—49no—
149C2×C3C4⋊(C3⋊S4)14yes—
150D3—728no—
151D3—14418no—
152C22—31no—
153C22—31no—
154C22—1082no—
155C22—1082no—
156C22—1082no—
157C22#8885211yescharacteristic, complemented
158C4—31no—
159C4—1082no—
160C4C2×C3⋊(C3⋊S4)11yescharacteristic
161C3—49no—
162C3#1059363614yescomplemented
163C2—31no—
164C2—31no—
165C2—2162no—
166C2C4⋊(C3⋊(C3⋊S4))11yescharacteristic
subgroupG/H
centerC2C4⋊(C3⋊(C3⋊S4))
commutator, cosocleC32×C4×A4C22
FrattiniC4C2×C3⋊(C3⋊S4)
FittingC22×C32×C8D3
socleC23×C32D12
hypercenterC8C3⋊(C3⋊S4)
hypocenterC32×A4D8
LCSG ▷ C32×C4×A4 ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. C4⋊(C3⋊(C3⋊S4))C2semidirect
2. C32×C8×A4C2semidirect
9. C22×C32×C8D3semidirect
12. C3×C8×A4D3semidirect
34. C32×A4D8semidirect
37. C22×C3×C8C3⋊D3semidirect
40. C8×A4C3⋊D3semidirect
57. C32×C8S4semidirect
75. C22×C32D24semidirect
80. C3×A4D24semidirect
86. C22×C8C3⋊(C3⋊D3)semidirect
105. C3×C8C3⋊S4semidirect
123. C22×C3#3745semidirect
128. A4#3745semidirect
134. C32#5402semidirect
145. C8C3⋊(C3⋊S4)semidirect
157. C22#88852semidirect
162. C3#10593636semidirect

Maximal subgroups

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

subgroupindex
C4⋊(C3⋊(C3⋊S4))2
C32×C8×A42
#105936363
#105928143
#888524