Finite Group Databasebeta

← #13173051

Subgroups — #13173051

Order 1728.

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

normal subgroups only

This group has 752 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))C211yesmaximal, characteristic
2#11724463C211yesmaximal, characteristic, complemented
3C32×C8×A4C211yesmaximal, characteristic, complemented
4#10593635—312nomaximal
5#10592813—31nomaximal
6C2×C3⋊(C3⋊S4)—21no—
7C32×C4×A4C2211yescharacteristic
8C3⋊(C3⋊SL(2,ℤ/4))—21no—
9#88851—41nomaximal
10C4⋊(C3⋊S4)—312no—
11#64259—312no—
12C22×C32×C8D311yescharacteristic, complemented
13#64139—31no—
14#64133—31no—
15C2×#3744—31no—
16C3×C8×A4D3112yescomplemented
17#63648—31no—
18#63644—31no—
19#63643—31no—
20C2×C32×A4D411yescharacteristic
21C3⋊(C3⋊S4)—41no—
22C4⋊(C3⋊(C3⋊D3))—41no—
23#6308—41no—
24C33×C8—41no—
25#5401—99no—
26#5114—94no—
27C2×C3⋊S4—612no—
28C22×C32×C4C2×D311yescharacteristic
29C2×#638—61no—
30C2×C4⋊(C3⋊D3)—31no—
31C2×#634—31no—
32C3×C4×A4C2×D3112yes—
33C3⋊SL(2,ℤ/4)—612no—
34C2×C32×C8—31no—
35C2×C32×C8—31no—
36#3757—61no—
37#3752—61no—
38C4⋊(C3⋊Dic3)—31no—
39C4⋊(C3⋊Dic3)—31no—
40#3750—61no—
41#3744—61no—
42#3744—61no—
43#3744—1212no—
44C2×C3⋊(C3⋊D3)—81no—
45C32×A4SD1611yescharacteristic, complemented
46C33×C4—41no—
47C3⋊(C3⋊Dic3)—81no—
48C4⋊S4—99no—
49A4⋊Q8—99no—
50C22×C3×C8C3⋊D314yescomplemented
51#930—94no—
52#924—94no—
53C2×#204—94no—
54C8×A4C3⋊D319yescomplemented
55#821—94no—
56#817—94no—
57#816—94no—
58C23×C32D1211yescharacteristic
59C22×C3⋊D3—61no—
60C2×C3×A4D12112yes—
61C3⋊S4—1212no—
62C2×C32×C4—31no—
63C2×C32×C4—31no—
64#638—121no—
65#638—121no—
66C2×C3⋊Dic3—61no—
67C2×C3⋊Dic3—61no—
68C2×C3⋊Dic3—61no—
69C4⋊(C3⋊D3)—61no—
70C4⋊(C3⋊D3)—61no—
71C4⋊(C3⋊D3)—1212no—
72#634—61no—
73#634—61no—
74#634—1212no—
75C32×C8—31no—
76C32×C8—412no—
77C32×C8S411yescharacteristic, complemented
78#465—271noSylow
79C2×C33—41no—
80C3⋊(C3⋊D3)—161no—
81C2×S4—189no—
82C22×C3×C4C2×C3⋊D314yes—
83C2×#67—184no—
84C2×D12—94no—
85C2×Dic6—94no—
86C4×A4C2×C3⋊D319yes—
87SL(2,ℤ/4)—189no—
88C2×C3×C8—34no—
89C2×C3×C8—34no—
90#217—184no—
91(C2×D3)⋊C4—184no—
92C4⋊Dic3—94no—
93C4⋊Dic3—94no—
94Dic3⋊C4—184no—
95#204—184no—
96#204—184no—
97#204—369no—
98C22×C32—31no—
99C22×C32—31no—
100C22×C32#20411yescharacteristic, complemented
101C2×C3⋊D3—121no—
102C2×C3⋊D3—121no—
103C2×C3⋊D3—121no—
104C2×C3⋊D3—2412no—
105C3×A4#204112yescomplemented
106C32×C4—31no—
107C32×C4—412no—
108C32×C4C2×S411yescharacteristic
109C3⋊Dic3—121no—
110C3⋊Dic3—121no—
111C3⋊Dic3—121no—
112C3⋊Dic3—2412no—
113C2×SD16—271no—
114C22×C8C3⋊(C3⋊D3)11yescharacteristic, complemented
115C22⋊Q8—271no—
116#121—271no—
117#106—271no—
118#103—271no—
119#102—271no—
120C33—41noSylow
121C23×C3C4⋊(C3⋊D3)14yes—
122C22×D3—184no—
123C2×A4C4⋊(C3⋊D3)19yes—
124S4—369no—
125C2×C3×C4—34no—
126C2×C3×C4—34no—
127#67—364no—
128#67—364no—
129C2×Dic3—184no—
130C2×Dic3—184no—
131C2×Dic3—184no—
132D12—184no—
133D12—184no—
134D12—369no—
135Dic6—184no—
136Dic6—184no—
137Dic6—369no—
138C3×C8—34no—
139C3×C8—49no—
140C3×C8C3⋊S414yescomplemented
141C2×C32—31no—
142C2×C32—31no—
143C2×C32—412no—
144C2×C32C4⋊S411yescharacteristic
145C3⋊D3—241no—
146C3⋊D3—4812no—
147C2×Q8—271no—
148C2×D4—271no—
149C2×D4—541no—
150C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
151SD16—541no—
152SD16—541no—
153C2×C8—31no—
154C2×C8—31no—
155C4⋊C4—271no—
156C4⋊C4—271no—
157C4⋊C4—541no—
158C22⋊C4—541no—
159C22⋊C4—541no—
160C22×C3—34no—
161C22×C3—34no—
162C22×C3#374414yescomplemented
163C2×D3—364no—
164C2×D3—364no—
165C2×D3—364no—
166C2×D3—729no—
167A4#374419yescomplemented
168C3×C4—34no—
169C3×C4—49no—
170C3×C4C2×C3⋊S414yes—
171Dic3—364no—
172Dic3—364no—
173Dic3—364no—
174Dic3—729no—
175C32—412no—
176C32#540111yescharacteristic, complemented
177C23—541no—
178C23C4⋊(C3⋊(C3⋊D3))11yescharacteristic
179Q8—541no—
180Q8—541no—
181D4—541no—
182D4—541no—
183D4—1081no—
184D4—1081no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—541no—
188C2×C4—541no—
189C2×C4—541no—
190C8—31no—
191C8C3⋊(C3⋊S4)11yescharacteristic, complemented
192C2×C3—34no—
193C2×C3—34no—
194C2×C3—49no—
195C2×C3C4⋊(C3⋊S4)14yes—
196D3—724no—
197D3—1449no—
198C22—31no—
199C22—31no—
200C22—1081no—
201C22—1081no—
202C22—1081no—
203C22#8885111yescharacteristic, complemented
204C4—31no—
205C4—1081no—
206C4—1081no—
207C4—1081no—
208C4C2×C3⋊(C3⋊S4)11yescharacteristic
209C3—49no—
210C3#1059363514yescomplemented
211C2—31no—
212C2—31no—
213C2—2161no—
214C2C4⋊(C3⋊(C3⋊S4))11yescharacteristic
subgroupG/H
centerC2C4⋊(C3⋊(C3⋊S4))
commutator, cosocleC32×C4×A4C22
FrattiniC4C2×C3⋊(C3⋊S4)
FittingC22×C32×C8D3
socleC23×C32D12
prime radicalC4⋊(C3⋊(C3⋊S4))C2
hypercenterC8C3⋊(C3⋊S4)
hypocenterC32×A4SD16
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
2. #11724463C2semidirect
3. C32×C8×A4C2semidirect
12. C22×C32×C8D3semidirect
16. C3×C8×A4D3semidirect
45. C32×A4SD16semidirect
50. C22×C3×C8C3⋊D3semidirect
54. C8×A4C3⋊D3semidirect
77. C32×C8S4semidirect
100. C22×C32#204semidirect
105. C3×A4#204semidirect
114. C22×C8C3⋊(C3⋊D3)semidirect
140. C3×C8C3⋊S4semidirect
162. C22×C3#3744semidirect
167. A4#3744semidirect
176. C32#5401semidirect
191. C8C3⋊(C3⋊S4)semidirect
203. C22#88851semidirect
210. C3#10593635semidirect

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
#117244632
C32×C8×A42
#105936353
#105928133
#888514