Finite Group Databasebeta

← C4×Dic3×#157 #13166410

Subgroups — C4×Dic3×#157 #13166410

Order 1728.

792 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 206 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×C4×C3⋊#157C211yesmaximal, characteristic
2C2×C3×C4×#157C211yesmaximal, characteristic
3C2×Dic3×#157C214yesmaximal
4C4×Dic3×C3⋊D3C211yesmaximal, characteristic
5C42×#157—31nomaximal
6C22×C3⋊#157C2212yes—
7C22×C3×#157C2212yes—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×Dic3×C3⋊D3C2212yes—
10C4×C3⋊#157C414yescomplemented
11C3×C4×#157C414yescomplemented
12Dic3×#157C4116yesdirect product factor
13C4×C3⋊(C3⋊Dic3)C411yescharacteristic, complemented
14C32×C4×Dic3C411yescharacteristic, complemented
15Dic3×C3⋊Dic3C412yescomplemented
16C2×C4×#157—31no—
17C2×C4×#157—34no—
18C2×C4×#157D311yescharacteristic
19C42×C3⋊D3—31no—
20C4×D3×Dic3—62no—
21C22×C3×C3⋊D3C2311yescharacteristic
22C2×C3⋊#157C2×C418yes—
23C2×C3⋊#157C2×C414yes—
24C2×C3×#157C2×C418yes—
25C2×C3×#157C2×C414yes—
26C2×C33×C4C2×C411yescharacteristic
27C2×C3⋊(C3⋊Dic3)C2×C412yes—
28C2×C3×C3⋊Dic3C2×C411yescharacteristic
29C3×C4×C3⋊D3C2×C412yes—
30C3×C4×C3⋊D3C2×C412yes—
31C2×C32×Dic3C2×C412yes—
32Dic3×C3⋊D3C2×C418yes—
33C42×Dic3—91nomaximal
34C22×#157—32no—
35C22×#157C2×D312yes—
36C2×C4×C3⋊D3—32no—
37C2×C4×C3⋊D3C2×D311yescharacteristic
38C2×C3×C4×D3—62no—
39C2×D3×Dic3—64no—
40C4×#157—316no—
41C4×#157—34no—
42C4×#157Dic314yesdirect product factor
43C32×C42—31no—
44C4×C3⋊Dic3—31no—
45C4×C3⋊Dic3—32no—
46C4×C3⋊Dic3—62no—
47C4×C3⋊Dic3—122no—
48C3×C4×Dic3—22no—
49Dic32—64no—
50C22×C33C22×C411yescharacteristic
51C2×C3×C3⋊D3C22×C414yes—
52C2×C3×C3⋊D3C22×C411yescharacteristic
53C2×C3×C3⋊D3C22×C411yescharacteristic
54C3⋊#157C4218yescomplemented
55C3×#157C4218yescomplemented
56C33×C4C4212yescomplemented
57C3⋊(C3⋊Dic3)C4214yescomplemented
58C3×C3⋊Dic3C4212yescomplemented
59C32×Dic3C4214yescomplemented
60C2×C3×C42—91no—
61C2×C4×Dic3—91no—
62C2×C4×Dic3—94no—
63C2×C4×Dic3—91no—
64C42×D3—182no—
65C22×C3⋊D3C22×D311yescharacteristic
66C22×C3×D3—62no—
67C2×#157—34no—
68C2×#157—38no—
69C2×#157C4×D318yes—
70C2×#157C2×Dic314yes—
71C2×C32×C4—22no—
72C2×C32×C4—32no—
73C2×C32×C4—42no—
74C2×C32×C4C4×D311yescharacteristic
75C2×C3⋊Dic3—32no—
76C2×C3⋊Dic3—64no—
77C2×C3⋊Dic3—124no—
78C2×C3⋊Dic3C4×D311yescharacteristic
79C4×C3⋊D3—38no—
80C4×C3⋊D3C4×D312yes—
81C4×C3⋊D3C2×Dic312yes—
82C2×C3×Dic3—24no—
83C2×C3×Dic3—62no—
84C3×C4×D3—64no—
85C3×C4×D3—64no—
86D3×Dic3—616no—
87C43—271noSylow
88C2×C33C2×C4212yes—
89C2×C33C2×C4211yescharacteristic
90C3×C3⋊D3C2×C4212yes—
91C3×C3⋊D3C2×C4212yes—
92C22×C3×C4—92no—
93C22×C3×C4—91no—
94C22×Dic3—92no—
95C22×Dic3—92no—
96C2×C4×D3—62no—
97C2×C4×D3—184no—
98C3×C42—62no—
99C3×C42—94no—
100C4×Dic3—916no—
101C4×Dic3—94no—
102C4×Dic3—91no—
103C4×Dic3—92no—
104C4×Dic3—184no—
105C4×Dic3—182no—
106C4×Dic3—362no—
107C4×Dic3#15711yescharacteristic, direct product factor
108C22×C32—22no—
109C22×C32—42no—
110C22×C32C2×C4×D311yescharacteristic
111C2×C3⋊D3C2×C4×D311yescharacteristic
112C2×C3⋊D3C2×C4×D314yes—
113C2×C3⋊D3C22×Dic311yescharacteristic
114C2×C3×D3—68no—
115C2×C3×D3—62no—
116C2×C3×D3—62no—
117#157—38no—
118#157C4×Dic318yesdirect product factor
119C32×C4—24no—
120C32×C4—34no—
121C32×C4—44no—
122C32×C4C4×Dic312yescomplemented
123C3⋊Dic3—34no—
124C3⋊Dic3—68no—
125C3⋊Dic3—128no—
126C3⋊Dic3C4×Dic312yescomplemented
127C3×Dic3—28no—
128C3×Dic3—64no—
129C2×C42—91no—
130C2×C42—271no—
131C2×C42—274no—
132C2×C42—271no—
133C33C4311yescharacteristic, complemented, Sylow
134C23×C3—91no—
135C22×D3—62no—
136C2×C3×C4—22no—
137C2×C3×C4—42no—
138C2×C3×C4—64no—
139C2×C3×C4—98no—
140C2×C3×C4—94no—
141C2×C3×C4—91no—
142C2×C3×C4—92no—
143C2×C3×C4—92no—
144C2×C3×C4C2×#15711yescharacteristic
145C2×Dic3—62no—
146C2×Dic3—98no—
147C2×Dic3—94no—
148C2×Dic3—92no—
149C2×Dic3—98no—
150C2×Dic3—184no—
151C2×Dic3—364no—
152C2×Dic3C2×#15712yes—
153C4×D3—64no—
154C4×D3—64no—
155C4×D3—1816no—
156C2×C32—24no—
157C2×C32—22no—
158C2×C32—44no—
159C2×C32—42no—
160C2×C32C42×D312yes—
161C2×C32C2×C4×Dic311yescharacteristic
162C3⋊D3C42×D312yes—
163C3⋊D3C2×C4×Dic312yes—
164C3×D3—64no—
165C3×D3—64no—
166C22×C4—92no—
167C22×C4—91no—
168C22×C4—272no—
169C22×C4—272no—
170C42—31no—
171C42—94no—
172C42—272no—
173C42—2716no—
174C42—274no—
175C42—271no—
176C22×C3—22no—
177C22×C3—42no—
178C22×C3—94no—
179C22×C3—91no—
180C22×C3—91no—
181C22×C3C22×#15711yescharacteristic
182C2×D3—62no—
183C2×D3—62no—
184C2×D3—68no—
185C3×C4—24no—
186C3×C4—44no—
187C3×C4—68no—
188C3×C4—98no—
189C3×C4—92no—
190C3×C4C4×#15712yescomplemented
191Dic3—64no—
192Dic3—98no—
193Dic3—94no—
194Dic3—188no—
195Dic3—368no—
196Dic3C4×#15714yesdirect product factor
197C32—22no—
198C32—42no—
199C32C42×Dic311yescharacteristic, complemented
200C23—91no—
201C2×C4—32no—
202C2×C4—94no—
203C2×C4—98no—
204C2×C4—91no—
205C2×C4—92no—
206C2×C4—92no—
207C2×C4—278no—
208C2×C4—278no—
209C2×C4—274no—
210C2×C4—272no—
211C2×C4D3×#15711yescharacteristic
212C2×C3—24no—
213C2×C3—22no—
214C2×C3—44no—
215C2×C3—42no—
216C2×C3—92no—
217C2×C3—92no—
218C2×C3C2×C4×#15712yes—
219C2×C3C2×C4×#15711yescharacteristic
220D3—64no—
221D3—64no—
222C22—94no—
223C22—91no—
224C22—91no—
225C22C2×D3×#15711yescharacteristic
226C4—34no—
227C4—98no—
228C4—92no—
229C4—278no—
230C4—274no—
231C4Dic3×#15712yesdirect product factor
232C3—22no—
233C3—42no—
234C3C42×#15711yescharacteristic, complemented
235C2—92no—
236C2—92no—
237C2C4×D3×#15712yes—
238C2C2×Dic3×#15711yescharacteristic
subgroupG/H
center, hypercenterC2×C4D3×#157
commutator, hypocenterC33C43
FrattiniC22C2×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C22×C4
prime radical, cosocleC22×C3×C3⋊D3C23
LCSG ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
10. C4×C3⋊#157C4semidirect
11. C3×C4×#157C4semidirect
12. Dic3×#157C4direct (N × H)
13. C4×C3⋊(C3⋊Dic3)C4semidirect
14. C32×C4×Dic3C4semidirect
15. Dic3×C3⋊Dic3C4semidirect
42. C4×#157Dic3direct (N × H)
54. C3⋊#157C42semidirect
55. C3×#157C42semidirect
56. C33×C4C42semidirect
57. C3⋊(C3⋊Dic3)C42semidirect
58. C3×C3⋊Dic3C42semidirect
59. C32×Dic3C42semidirect
107. C4×Dic3#157direct (N × H)
118. #157C4×Dic3direct (N × H)
122. C32×C4C4×Dic3semidirect
126. C3⋊Dic3C4×Dic3semidirect
133. C33C43semidirect
190. C3×C4C4×#157semidirect
196. Dic3C4×#157direct (N × H)
199. C32C42×Dic3semidirect
231. C4Dic3×#157direct (N × H)
234. C3C42×#157semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊#1572
C2×C3×C4×#1572
C2×Dic3×#1572
C4×Dic3×C3⋊D32
C42×#1573
C42×Dic39