Finite Group Databasebeta

← C4×C3⋊(C3⋊SL(2,ℤ/4)) #13173060

Subgroups — C4×C3⋊(C3⋊SL(2,ℤ/4)) #13173060

Order 1728.

1230 conjugacy classes of subgroups; only the 279 normal subgroups are shown.

normal subgroups only

This group has 1230 conjugacy classes of subgroups — too many to draw. Only its 279 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
1C2×C32×C4×A4C211yesmaximal, characteristic
2C2×C3⋊(C3⋊SL(2,ℤ/4))C212yesmaximal
3C4×C3⋊SL(2,ℤ/4)—312nomaximal
4C4×#3757—31nomaximal
5C22×C32×A4C2211yescharacteristic
6C32×C4×A4C412yescomplemented
7C3⋊(C3⋊SL(2,ℤ/4))C414yesdirect product factor
8C4×C3⋊(C3⋊Dic3)—41nomaximal
9C23×C32×C4D311yescharacteristic
10C2×C3×C4×A4D3112yes—
11C2×C3⋊SL(2,ℤ/4)—324no—
12C2×#3757—32no—
13C2×C4×C3⋊Dic3—31no—
14C2×C4×C3⋊Dic3—31no—
15#63658—32no—
16C2×C32×A4C2×C412yes—
17C2×C32×A4C2×C411yescharacteristic
18C2×C33×C4—41no—
19C2×C3⋊(C3⋊Dic3)—42no—
20C4×SL(2,ℤ/4)—99no—
21C4×#217—94no—
22C24×C32C2×D311yescharacteristic
23C22×C3×A4C2×D3112yes—
24C22×C32×C4—31no—
25C22×C32×C4—32no—
26C22×C32×C4—31no—
27C22×C32×C4Dic312yescomplemented
28C22×C3⋊Dic3—32no—
29C22×C3⋊Dic3—32no—
30C3×C4×A4Dic3124yescomplemented
31C3⋊SL(2,ℤ/4)—348no—
32#3757—34no—
33#3757—34no—
34C4×C3⋊Dic3—62no—
35C4×C3⋊Dic3—61no—
36C4×C3⋊Dic3—61no—
37C4×C3⋊Dic3—1212no—
38C22×C33—41no—
39C32×A4C4211yescharacteristic, complemented
40C33×C4—42no—
41C3⋊(C3⋊Dic3)—44no—
42C23×C3×C4C3⋊D314yes—
43C2×C4×A4C3⋊D319yes—
44C2×SL(2,ℤ/4)—918no—
45C2×#217—98no—
46C2×C4×Dic3—94no—
47C2×C4×Dic3—94no—
48C3⋊#95—98no—
49C23×C32—31no—
50C23×C32—32no—
51C23×C32—31no—
52C23×C32C4×D312yes—
53C23×C32C2×Dic311yescharacteristic
54C2×C3×A4C4×D3124yes—
55C2×C3×A4C2×Dic3112yes—
56C2×C32×C4—31no—
57C2×C32×C4—32no—
58C2×C32×C4—32no—
59C2×C32×C4—32no—
60C2×C32×C4—412no—
61C2×C32×C4—61no—
62C2×C32×C4S411yescharacteristic
63C2×C3⋊Dic3—34no—
64C2×C3⋊Dic3—34no—
65C2×C3⋊Dic3—62no—
66C2×C3⋊Dic3—62no—
67C2×C3⋊Dic3—62no—
68C2×C3⋊Dic3—62no—
69C2×C3⋊Dic3—1224no—
70C4×C22⋊C4—271noSylow
71C2×C33—42no—
72C2×C33—41no—
73C24×C3C2×C3⋊D314yes—
74C22×A4C2×C3⋊D319yes—
75C22×C3×C4—34no—
76C22×C3×C4—34no—
77C22×C3×C4—38no—
78C22×C3×C4C3⋊Dic318yescomplemented
79C22×Dic3—98no—
80C22×Dic3—98no—
81C4×A4C3⋊Dic3118yescomplemented
82SL(2,ℤ/4)—936no—
83#217—916no—
84#217—916no—
85C4×Dic3—188no—
86C4×Dic3—184no—
87C4×Dic3—184no—
88C4×Dic3—369no—
89C22×C32—32no—
90C22×C32—31no—
91C22×C32—32no—
92C22×C32—32no—
93C22×C32—31no—
94C22×C32—31no—
95C22×C32—412no—
96C22×C32—61no—
97C22×C32C4×Dic311yescharacteristic, complemented
98C22×C32C2×S411yescharacteristic
99C3×A4C4×Dic3112yescomplemented
100C32×C4—32no—
101C32×C4—424no—
102C32×C4SL(2,ℤ/4)12yescomplemented
103C3⋊Dic3—64no—
104C3⋊Dic3—64no—
105C3⋊Dic3—1248no—
106C23×C4C3⋊(C3⋊D3)11yescharacteristic
107C2×C22⋊C4—272no—
108C2×C42—271no—
109C2×C42—271no—
110#95—272no—
111C33—41noSylow
112C23×C3—34no—
113C23×C3—38no—
114C23×C3—34no—
115C23×C3C4×C3⋊D318yes—
116C23×C3C2×C3⋊Dic314yes—
117C2×A4C4×C3⋊D3118yes—
118C2×A4C2×C3⋊Dic319yes—
119C2×C3×C4—34no—
120C2×C3×C4—38no—
121C2×C3×C4—38no—
122C2×C3×C4—38no—
123C2×C3×C4—49no—
124C2×C3×C4—64no—
125C2×C3×C4C3⋊S414yes—
126C2×Dic3—916no—
127C2×Dic3—916no—
128C2×Dic3—188no—
129C2×Dic3—188no—
130C2×Dic3—188no—
131C2×Dic3—188no—
132C2×Dic3—3618no—
133C2×C32—32no—
134C2×C32—31no—
135C2×C32—31no—
136C2×C32—424no—
137C2×C32—412no—
138C2×C32C4×S412yes—
139C2×C32C2×SL(2,ℤ/4)11yescharacteristic
140C24C2×C3⋊(C3⋊D3)11yescharacteristic
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—32no—
144C22×C4—272no—
145C22×C4—272no—
146C22×C4C3⋊(C3⋊Dic3)12yescomplemented
147C22⋊C4—274no—
148C22⋊C4—274no—
149C42—541no—
150C42—542no—
151C42—541no—
152C22×C3—38no—
153C22×C3—34no—
154C22×C3—38no—
155C22×C3—38no—
156C22×C3—34no—
157C22×C3—34no—
158C22×C3—49no—
159C22×C3—64no—
160C22×C3C4×C3⋊Dic314yescomplemented
161C22×C3C2×C3⋊S414yes—
162A4C4×C3⋊Dic319yescomplemented
163C3×C4—38no—
164C3×C4—418no—
165C3×C4C3⋊SL(2,ℤ/4)18yescomplemented
166Dic3—1816no—
167Dic3—1816no—
168Dic3—3636no—
169C32—412no—
170C32C4×SL(2,ℤ/4)11yescharacteristic, complemented
171C23—31no—
172C23—32no—
173C23—31no—
174C23C4×C3⋊(C3⋊D3)12yes—
175C23C2×C3⋊(C3⋊Dic3)11yescharacteristic
176C2×C4—31no—
177C2×C4—32no—
178C2×C4—32no—
179C2×C4—32no—
180C2×C4—61no—
181C2×C4—274no—
182C2×C4—274no—
183C2×C4—542no—
184C2×C4—542no—
185C2×C4—542no—
186C2×C4—542no—
187C2×C4C3⋊(C3⋊S4)11yescharacteristic
188C2×C3—38no—
189C2×C3—34no—
190C2×C3—34no—
191C2×C3—418no—
192C2×C3—49no—
193C2×C3C4×C3⋊S418yes—
194C2×C3C2×C3⋊SL(2,ℤ/4)14yes—
195C22—32no—
196C22—31no—
197C22—32no—
198C22—32no—
199C22—31no—
200C22—31no—
201C22—61no—
202C22C4×C3⋊(C3⋊Dic3)11yescharacteristic, complemented
203C22C2×C3⋊(C3⋊S4)11yescharacteristic
204C4—32no—
205C4—544no—
206C4—544no—
207C4C3⋊(C3⋊SL(2,ℤ/4))12yesdirect product factor
208C3—49no—
209C3C4×C3⋊SL(2,ℤ/4)14yescomplemented
210C2—32no—
211C2—31no—
212C2—31no—
213C2C4×C3⋊(C3⋊S4)12yes—
214C2C2×C3⋊(C3⋊SL(2,ℤ/4))11yescharacteristic
subgroupG/H
center, hypercenterC2×C4C3⋊(C3⋊S4)
commutator, hypocenterC32×A4C42
FrattiniC22C2×C3⋊(C3⋊S4)
FittingC23×C32×C4D3
socleC24×C32C2×D3
prime radical, cosocleC22×C32×A4C22
LCSG ▷ C32×A4
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
6. C32×C4×A4C4semidirect
7. C3⋊(C3⋊SL(2,ℤ/4))C4direct (N × H)
27. C22×C32×C4Dic3semidirect
30. C3×C4×A4Dic3semidirect
39. C32×A4C42semidirect
78. C22×C3×C4C3⋊Dic3semidirect
81. C4×A4C3⋊Dic3semidirect
97. C22×C32C4×Dic3semidirect
99. C3×A4C4×Dic3semidirect
102. C32×C4SL(2,ℤ/4)semidirect
146. C22×C4C3⋊(C3⋊Dic3)semidirect
160. C22×C3C4×C3⋊Dic3semidirect
162. A4C4×C3⋊Dic3semidirect
165. C3×C4C3⋊SL(2,ℤ/4)semidirect
170. C32C4×SL(2,ℤ/4)semidirect
202. C22C4×C3⋊(C3⋊Dic3)semidirect
207. C4C3⋊(C3⋊SL(2,ℤ/4))direct (N × H)
209. C3C4×C3⋊SL(2,ℤ/4)semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×A42
C2×C3⋊(C3⋊SL(2,ℤ/4))2
C4×C3⋊SL(2,ℤ/4)3
C4×#37573
C4×C3⋊(C3⋊Dic3)4