Finite Group Databasebeta

← C22×C3⋊(C3⋊SL(2,ℤ/4)) #13181412

Subgroups — C22×C3⋊(C3⋊SL(2,ℤ/4)) #13181412

Order 1728.

2526 conjugacy classes of subgroups; only the 555 normal subgroups are shown.

normal subgroups only

This group has 2526 conjugacy classes of subgroups — too many to draw. Only its 555 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
1C23×C32×A4C211yesmaximal, characteristic
2C2×C3⋊(C3⋊SL(2,ℤ/4))C216yesmaximal, direct product factor
3C22×C3⋊SL(2,ℤ/4)—312nomaximal
4C22×#3757—31nomaximal
5C22×C32×A4C414yescomplemented
6C22×C32×A4C2213yes—
7C22×C3⋊(C3⋊Dic3)—41nomaximal
8C3⋊(C3⋊SL(2,ℤ/4))C2214yesdirect product factor
9C25×C32D311yescharacteristic
10C23×C3×A4D3112yes—
11C23×C3⋊Dic3—31no—
12C23×C3⋊Dic3—31no—
13C2×C3⋊SL(2,ℤ/4)—372no—
14C2×#3757—36no—
15C2×#3757—36no—
16C23×C33—41no—
17C2×C32×A4C2×C416yescomplemented
18C2×C32×A4C2311yescharacteristic
19C2×C3⋊(C3⋊Dic3)—46no—
20C22×SL(2,ℤ/4)—99no—
21C22×#217—94no—
22C24×C32—31no—
23C24×C32—34no—
24C24×C32—33no—
25C24×C32Dic314yescomplemented
26C24×C32C2×D313yes—
27C22×C3×A4Dic3148yescomplemented
28C22×C3×A4C2×D3136yes—
29C22×C3⋊Dic3—36no—
30C22×C3⋊Dic3—36no—
31C22×C3⋊Dic3—64no—
32C22×C3⋊Dic3—63no—
33C22×C3⋊Dic3—61no—
34C22×C3⋊Dic3—1212no—
35C3⋊SL(2,ℤ/4)—348no—
36#3757—312no—
37#3757—34no—
38C22×C33—44no—
39C22×C33—43no—
40C32×A4C22×C411yescharacteristic, complemented
41C3⋊(C3⋊Dic3)—44no—
42C25×C3C3⋊D314yes—
43C23×A4C3⋊D319yes—
44C23×Dic3—94no—
45C23×Dic3—94no—
46C2×SL(2,ℤ/4)—954no—
47C2×#217—924no—
48C2×#217—924no—
49C23×C32—34no—
50C23×C32—34no—
51C23×C32—33no—
52C23×C32—33no—
53C23×C32—312no—
54C23×C32—36no—
55C23×C32—33no—
56C23×C32—412no—
57C23×C32—66no—
58C23×C32—61no—
59C23×C32C2×Dic316yescomplemented
60C23×C32S411yescharacteristic
61C23×C32C22×D311yescharacteristic
62C2×C3×A4C2×Dic3172yescomplemented
63C2×C3×A4C22×D3112yes—
64C2×C3⋊Dic3—34no—
65C2×C3⋊Dic3—34no—
66C2×C3⋊Dic3—612no—
67C2×C3⋊Dic3—66no—
68C2×C3⋊Dic3—66no—
69C2×C3⋊Dic3—1272no—
70C22×C22⋊C4—271noSylow
71C2×C33—46no—
72C2×C33—41no—
73C24×C3—34no—
74C24×C3—316no—
75C24×C3—312no—
76C24×C3C3⋊Dic3116yescomplemented
77C24×C3C2×C3⋊D3112yes—
78C22×A4C3⋊Dic3136yescomplemented
79C22×A4C2×C3⋊D3127yes—
80C22×Dic3—924no—
81C22×Dic3—924no—
82C22×Dic3—1816no—
83C22×Dic3—1812no—
84C22×Dic3—184no—
85C22×Dic3—369no—
86SL(2,ℤ/4)—936no—
87#217—948no—
88#217—916no—
89C22×C32—36no—
90C22×C32—31no—
91C22×C32—36no—
92C22×C32—36no—
93C22×C32—312no—
94C22×C32—31no—
95C22×C32—33no—
96C22×C32—436no—
97C22×C32—448no—
98C22×C32—63no—
99C22×C32—64no—
100C22×C32SL(2,ℤ/4)14yescomplemented
101C22×C32C22×Dic311yescharacteristic, complemented
102C22×C32C2×S413yes—
103C3×A4C22×Dic3112yescomplemented
104C3⋊Dic3—64no—
105C3⋊Dic3—64no—
106C3⋊Dic3—1248no—
107C25C3⋊(C3⋊D3)11yescharacteristic
108C23×C4—271no—
109C23×C4—271no—
110C2×C22⋊C4—276no—
111C2×C22⋊C4—276no—
112C33—41noSylow
113C23×C3—312no—
114C23×C3—312no—
115C23×C3—316no—
116C23×C3—316no—
117C23×C3—312no—
118C23×C3—348no—
119C23×C3—324no—
120C23×C3—49no—
121C23×C3—624no—
122C23×C3—64no—
123C23×C3C2×C3⋊Dic3124yescomplemented
124C23×C3C3⋊S414yes—
125C23×C3C22×C3⋊D314yes—
126C2×A4C2×C3⋊Dic3154yescomplemented
127C2×A4C22×C3⋊D319yes—
128C2×Dic3—916no—
129C2×Dic3—916no—
130C2×Dic3—1848no—
131C2×Dic3—1824no—
132C2×Dic3—1824no—
133C2×Dic3—3654no—
134C2×C32—31no—
135C2×C32—31no—
136C2×C32—36no—
137C2×C32—472no—
138C2×C32—412no—
139C2×C32C2×SL(2,ℤ/4)16yescomplemented
140C2×C32C22×S411yescharacteristic
141C24—31no—
142C24—33no—
143C24—34no—
144C24C3⋊(C3⋊Dic3)14yescomplemented
145C24C2×C3⋊(C3⋊D3)13yes—
146C22×C4—276no—
147C22×C4—276no—
148C22×C4—544no—
149C22×C4—543no—
150C22×C4—541no—
151C22⋊C4—2712no—
152C22⋊C4—274no—
153C22×C3—324no—
154C22×C3—34no—
155C22×C3—324no—
156C22×C3—324no—
157C22×C3—348no—
158C22×C3—34no—
159C22×C3—312no—
160C22×C3—427no—
161C22×C3—436no—
162C22×C3—612no—
163C22×C3—616no—
164C22×C3C3⋊SL(2,ℤ/4)116yescomplemented
165C22×C3C22×C3⋊Dic314yescomplemented
166C22×C3C2×C3⋊S4112yes—
167A4C22×C3⋊Dic319yescomplemented
168Dic3—1816no—
169Dic3—1816no—
170Dic3—3636no—
171C32—412no—
172C32C22×SL(2,ℤ/4)11yescharacteristic, complemented
173C23—33no—
174C23—33no—
175C23—34no—
176C23—34no—
177C23—33no—
178C23—312no—
179C23—36no—
180C23—66no—
181C23—61no—
182C23C2×C3⋊(C3⋊Dic3)16yescomplemented
183C23C3⋊(C3⋊S4)11yescharacteristic
184C23C22×C3⋊(C3⋊D3)11yescharacteristic
185C2×C4—274no—
186C2×C4—274no—
187C2×C4—5412no—
188C2×C4—546no—
189C2×C4—546no—
190C2×C3—34no—
191C2×C3—34no—
192C2×C3—324no—
193C2×C3—454no—
194C2×C3—49no—
195C2×C3C2×C3⋊SL(2,ℤ/4)124yescomplemented
196C2×C3C22×C3⋊S414yes—
197C22—36no—
198C22—31no—
199C22—36no—
200C22—36no—
201C22—312no—
202C22—31no—
203C22—33no—
204C22—63no—
205C22—64no—
206C22C3⋊(C3⋊SL(2,ℤ/4))14yesdirect product factor
207C22C22×C3⋊(C3⋊Dic3)11yescharacteristic, complemented
208C22C2×C3⋊(C3⋊S4)13yes—
209C4—544no—
210C4—544no—
211C3—49no—
212C3C22×C3⋊SL(2,ℤ/4)14yescomplemented
213C2—31no—
214C2—31no—
215C2—36no—
216C2C2×C3⋊(C3⋊SL(2,ℤ/4))16yesdirect product factor
217C2C22×C3⋊(C3⋊S4)11yescharacteristic
subgroupG/H
center, hypercenterC23C3⋊(C3⋊S4)
commutator, hypocenterC32×A4C22×C4
FrattiniC2C22×C3⋊(C3⋊S4)
Fitting, socleC25×C32D3
prime radicalC23×C32×A4C2
cosocleC2×C32×A4C23
LCSG ▷ C32×A4
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊(C3⋊SL(2,ℤ/4))C2direct (N × H)
5. C22×C32×A4C4semidirect
8. C3⋊(C3⋊SL(2,ℤ/4))C22direct (N × H)
17. C2×C32×A4C2×C4semidirect
25. C24×C32Dic3semidirect
27. C22×C3×A4Dic3semidirect
40. C32×A4C22×C4semidirect
59. C23×C32C2×Dic3semidirect
62. C2×C3×A4C2×Dic3semidirect
76. C24×C3C3⋊Dic3semidirect
78. C22×A4C3⋊Dic3semidirect
100. C22×C32SL(2,ℤ/4)semidirect
101. C22×C32C22×Dic3semidirect
103. C3×A4C22×Dic3semidirect
123. C23×C3C2×C3⋊Dic3semidirect
126. C2×A4C2×C3⋊Dic3semidirect
139. C2×C32C2×SL(2,ℤ/4)semidirect
144. C24C3⋊(C3⋊Dic3)semidirect
164. C22×C3C3⋊SL(2,ℤ/4)semidirect
165. C22×C3C22×C3⋊Dic3semidirect
167. A4C22×C3⋊Dic3semidirect
172. C32C22×SL(2,ℤ/4)semidirect
182. C23C2×C3⋊(C3⋊Dic3)semidirect
195. C2×C3C2×C3⋊SL(2,ℤ/4)semidirect
206. C22C3⋊(C3⋊SL(2,ℤ/4))direct (N × H)
207. C22C22×C3⋊(C3⋊Dic3)semidirect
212. C3C22×C3⋊SL(2,ℤ/4)semidirect
216. C2C2×C3⋊(C3⋊SL(2,ℤ/4))direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×A42
C2×C3⋊(C3⋊SL(2,ℤ/4))2
C22×C3⋊SL(2,ℤ/4)3
C22×#37573
C22×C3⋊(C3⋊Dic3)4