Finite Group Databasebeta

← #13173050

Subgroups — #13173050

Order 1728.

752 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 141 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
2C32×C8×A4C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊(A4⋊C8))C211yesmaximal, characteristic, complemented
4#10593634—312nomaximal
5#10592810—31nomaximal
6C2×C3⋊(C3⋊S4)C411yescharacteristic
7C32×C4×A4C2211yescharacteristic
8C3⋊(C3⋊SL(2,ℤ/4))C411yescharacteristic
9#88850—41nomaximal
10C4×C3⋊S4—312no—
11C22×C32×C8D311yescharacteristic, complemented
12C4×#638—31no—
13C2×#3743—31no—
14C3×C8×A4D3112yescomplemented
15C3⋊(A4⋊C8)—312no—
16#63657—31no—
17#63647—31no—
18#63642—31no—
19#63641—31no—
20C2×C32×A4C2×C411yescharacteristic
21C3⋊(C3⋊S4)—21no—
22C4×C3⋊(C3⋊D3)—41no—
23C33×C8—41no—
24C3⋊(C3⋊(C3⋊C8))—41no—
25#5400—99no—
26#5111—94no—
27C2×C3⋊S4—312no—
28C22×C32×C4C2×D311yescharacteristic
29C2×#638—31no—
30C2×C4×C3⋊D3—31no—
31C3×C4×A4C2×D3112yes—
32C3⋊SL(2,ℤ/4)—312no—
33C2×C32×C8—31no—
34C2×C32×C8—31no—
35#3757—31no—
36#3752—31no—
37#3750—31no—
38C4×C3⋊Dic3—31no—
39C2×C3⋊(C3⋊C8)—31no—
40C2×C3⋊(C3⋊C8)—31no—
41#3743—61no—
42#3743—61no—
43#3743—1212no—
44C2×C3⋊(C3⋊D3)—41no—
45C32×A4M1611yescharacteristic, complemented
46C33×C4—41no—
47C3⋊(C3⋊Dic3)—41no—
48C4×S4—99no—
49C22×C3×C8C3⋊D314yescomplemented
50C4×#67—94no—
51C2×ΓL(1,𝔽25)—94no—
52C8×A4C3⋊D319yescomplemented
53A4⋊C8—99no—
54#830—94no—
55(C2×D3)⋊C8—94no—
56#815—94no—
57Dic3⋊C8—94no—
58C23×C32C4×D311yescharacteristic
59C22×C3⋊D3—31no—
60C2×C3×A4C4×D3112yes—
61C3⋊S4—612no—
62C2×C32×C4—31no—
63C2×C32×C4—31no—
64#638—61no—
65#638—61no—
66C2×C3⋊Dic3—31no—
67C2×C3⋊Dic3—31no—
68C2×C3⋊Dic3—31no—
69C4×C3⋊D3—61no—
70C4×C3⋊D3—61no—
71C4×C3⋊D3—1212no—
72C32×C8—31no—
73C32×C8—412no—
74C32×C8S411yescharacteristic, complemented
75C3⋊(C3⋊C8)—61no—
76C3⋊(C3⋊C8)—61no—
77C3⋊(C3⋊C8)—1212no—
78#435—271noSylow
79C2×C33—41no—
80C3⋊(C3⋊D3)—81no—
81C2×S4—99no—
82C22×C3×C4C2×C3⋊D314yes—
83C2×#67—94no—
84C2×C4×D3—94no—
85C4×A4C2×C3⋊D319yes—
86SL(2,ℤ/4)—99no—
87C2×C3×C8—34no—
88C2×C3×C8—34no—
89#217—94no—
90(C2×D3)⋊C4—94no—
91Dic3⋊C4—94no—
92C4×Dic3—94no—
93C2×C3⋊C8—94no—
94C2×C3⋊C8—94no—
95ΓL(1,𝔽25)—184no—
96ΓL(1,𝔽25)—184no—
97ΓL(1,𝔽25)—369no—
98C22×C32—31no—
99C22×C32—31no—
100C22×C32ΓL(1,𝔽25)11yescharacteristic, complemented
101C2×C3⋊D3—61no—
102C2×C3⋊D3—61no—
103C2×C3⋊D3—61no—
104C2×C3⋊D3—1212no—
105C3×A4ΓL(1,𝔽25)112yescomplemented
106C32×C4—31no—
107C32×C4—412no—
108C32×C4C2×S411yescharacteristic
109C3⋊Dic3—61no—
110C3⋊Dic3—61no—
111C3⋊Dic3—61no—
112C3⋊Dic3—1212no—
113C2×M16—271no—
114C22×C8C3⋊(C3⋊D3)11yescharacteristic, complemented
115C4×D4—271no—
116C4⋊C8—271no—
117C22⋊C8—271no—
118C22⋊C8—271no—
119#97—271no—
120C33—41noSylow
121C23×C3C4×C3⋊D314yes—
122C22×D3—94no—
123C2×A4C4×C3⋊D319yes—
124S4—189no—
125C2×C3×C4—34no—
126C2×C3×C4—34no—
127#67—184no—
128#67—184no—
129C2×Dic3—94no—
130C2×Dic3—94no—
131C2×Dic3—94no—
132C4×D3—184no—
133C4×D3—184no—
134C4×D3—369no—
135C3×C8—34no—
136C3×C8—49no—
137C3×C8C3⋊S414yescomplemented
138C3⋊C8—184no—
139C3⋊C8—184no—
140C3⋊C8—369no—
141C2×C32—31no—
142C2×C32—31no—
143C2×C32—412no—
144C2×C32C4×S411yescharacteristic
145C3⋊D3—121no—
146C3⋊D3—2412no—
147C2×D4—271no—
148C22×C4—271no—
149C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
150M16—541no—
151M16—541no—
152C2×C8—31no—
153C2×C8—31no—
154C2×C8—271no—
155C2×C8—271no—
156C4⋊C4—271no—
157C22⋊C4—271no—
158C22⋊C4—271no—
159C42—271no—
160C22×C3—34no—
161C22×C3—34no—
162C22×C3#374314yescomplemented
163C2×D3—184no—
164C2×D3—184no—
165C2×D3—184no—
166C2×D3—369no—
167A4#374319yescomplemented
168C3×C4—34no—
169C3×C4—49no—
170C3×C4C2×C3⋊S414yes—
171Dic3—184no—
172Dic3—184no—
173Dic3—184no—
174Dic3—369no—
175C32—412no—
176C32#540011yescharacteristic, complemented
177C23—271no—
178C23C4×C3⋊(C3⋊D3)11yescharacteristic
179D4—541no—
180D4—541no—
181C2×C4—31no—
182C2×C4—31no—
183C2×C4—271no—
184C2×C4—271no—
185C2×C4—271no—
186C2×C4—541no—
187C2×C4—541no—
188C8—31no—
189C8—541no—
190C8—541no—
191C8C3⋊(C3⋊S4)11yescharacteristic, complemented
192C2×C3—34no—
193C2×C3—34no—
194C2×C3—49no—
195C2×C3C4×C3⋊S414yes—
196D3—364no—
197D3—729no—
198C22—31no—
199C22—31no—
200C22—541no—
201C22—541no—
202C22—541no—
203C22#8885011yescharacteristic, complemented
204C4—31no—
205C4—541no—
206C4—541no—
207C4—541no—
208C4C2×C3⋊(C3⋊S4)11yescharacteristic
209C3—49no—
210C3#1059363414yescomplemented
211C2—31no—
212C2—31no—
213C2—1081no—
214C2C4×C3⋊(C3⋊S4)11yescharacteristic
subgroupG/H
center, FrattiniC4C2×C3⋊(C3⋊S4)
commutatorC2×C32×A4C2×C4
FittingC22×C32×C8D3
socleC23×C32C4×D3
prime radicalC2×C3⋊(C3⋊S4)C4
cosocleC32×C4×A4C22
hypercenterC8C3⋊(C3⋊S4)
hypocenterC32×A4M16
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. C32×C8×A4C2semidirect
3. C3⋊(C3⋊(A4⋊C8))C2semidirect
11. C22×C32×C8D3semidirect
14. C3×C8×A4D3semidirect
45. C32×A4M16semidirect
49. C22×C3×C8C3⋊D3semidirect
52. C8×A4C3⋊D3semidirect
74. C32×C8S4semidirect
100. C22×C32ΓL(1,𝔽25)semidirect
105. C3×A4ΓL(1,𝔽25)semidirect
114. C22×C8C3⋊(C3⋊D3)semidirect
137. C3×C8C3⋊S4semidirect
162. C22×C3#3743semidirect
167. A4#3743semidirect
176. C32#5400semidirect
191. C8C3⋊(C3⋊S4)semidirect
203. C22#88850semidirect
210. C3#10593634semidirect

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
C3⋊(C3⋊(A4⋊C8))2
#105936343
#105928103
#888504