Finite Group Databasebeta

← C22×Dic3×#157 #13180994

Subgroups — C22×Dic3×#157 #13180994

Order 1728.

1576 conjugacy classes of subgroups; only the 410 normal subgroups are shown.

normal subgroups only

This group has 1576 conjugacy classes of subgroups — too many to draw. Only its 410 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×C3⋊#157C211yesmaximal, characteristic
2C23×C3×#157C211yesmaximal, characteristic
3C22×Dic3×C3⋊D3C211yesmaximal, characteristic
4C2×Dic3×#157C2112yesmaximal, direct product factor
5C22×C4×#157—31nomaximal
6C23×C3×C3⋊D3C2211yescharacteristic
7C22×C3⋊#157C418yescomplemented
8C22×C3⋊#157C2216yes—
9C22×C3×#157C418yescomplemented
10C22×C3×#157C2216yes—
11C22×C3⋊(C3⋊Dic3)C411yescharacteristic, complemented
12C22×C32×Dic3C411yescharacteristic, complemented
13C2×Dic3×C3⋊D3C416yescomplemented
14C2×Dic3×C3⋊D3C2216yes—
15Dic3×#157C22116yesdirect product factor
16C23×#157—31no—
17C23×#157D311yescharacteristic
18C22×C4×C3⋊D3—31no—
19C22×D3×Dic3—62no—
20C2×C4×#157—312no—
21C23×C33C2×C411yescharacteristic
22C22×C3×C3⋊D3C2×C413yes—
23C22×C3×C3⋊D3C2×C414yes—
24C22×C3×C3⋊D3C2×C414yes—
25C22×C3×C3⋊D3C2313yes—
26C2×C3⋊#157C2×C4124yescomplemented
27C2×C3⋊#157C2314yes—
28C2×C3×#157C2×C4124yescomplemented
29C2×C3×#157C2314yes—
30C2×C3⋊(C3⋊Dic3)C2×C416yescomplemented
31C2×C32×Dic3C2×C416yescomplemented
32Dic3×C3⋊D3C2×C4112yescomplemented
33Dic3×C3⋊D3C2314yes—
34C22×C4×Dic3—91nomaximal
35C23×C3⋊D3C2×D311yescharacteristic
36C23×C3×D3—62no—
37C22×#157—38no—
38C22×#157—36no—
39C22×#157Dic318yesdirect product factor
40C22×#157C2×D316yes—
41C22×C32×C4—31no—
42C22×C3⋊Dic3—31no—
43C22×C3⋊Dic3—62no—
44C22×C3⋊Dic3—122no—
45C2×C4×C3⋊D3—36no—
46C2×C4×C3⋊D3—36no—
47C22×C3×Dic3—22no—
48C2×D3×Dic3—612no—
49C2×D3×Dic3—612no—
50C4×#157—316no—
51C22×C33C4214yescomplemented
52C22×C33C22×C413yes—
53C2×C3×C3⋊D3C42112yescomplemented
54C2×C3×C3⋊D3C22×C413yes—
55C2×C3×C3⋊D3C22×C416yes—
56C2×C3×C3⋊D3C22×C416yes—
57C2×C3×C3⋊D3C2411yescharacteristic
58C3⋊#157C22×C418yescomplemented
59C3×#157C22×C418yescomplemented
60C3⋊(C3⋊Dic3)C22×C414yescomplemented
61C32×Dic3C22×C414yescomplemented
62C23×C3×C4—91no—
63C23×Dic3—91no—
64C23×Dic3—91no—
65C22×C4×D3—182no—
66C2×C4×Dic3—912no—
67C23×C32—22no—
68C23×C32—42no—
69C23×C32C4×D311yescharacteristic
70C22×C3⋊D3C4×D313yes—
71C22×C3⋊D3C4×D314yes—
72C22×C3⋊D3C2×Dic314yes—
73C22×C3⋊D3C22×D313yes—
74C22×C3×D3—68no—
75C22×C3×D3—68no—
76C22×C3×D3—66no—
77C22×C3×D3—66no—
78C2×#157—324no—
79C2×#157—34no—
80C2×#157C2×Dic3124yesdirect product factor
81C2×#157C22×D314yes—
82C2×C32×C4—36no—
83C2×C3⋊Dic3—36no—
84C2×C3⋊Dic3—612no—
85C2×C3⋊Dic3—1212no—
86C4×C3⋊D3—312no—
87C4×C3⋊D3—34no—
88C2×C3×Dic3—212no—
89D3×Dic3—624no—
90D3×Dic3—68no—
91C22×C42—271noSylow
92C2×C33C2×C4216yescomplemented
93C2×C33C23×C411yescharacteristic
94C3×C3⋊D3C2×C4216yescomplemented
95C3×C3⋊D3C23×C411yescharacteristic
96C3×C3⋊D3C23×C411yescharacteristic
97C24×C3—91no—
98C23×D3—62no—
99C22×C3×C4—62no—
100C22×C3×C4—98no—
101C22×C3×C4—96no—
102C22×Dic3—98no—
103C22×Dic3—96no—
104C22×Dic3—91no—
105C22×Dic3—96no—
106C22×Dic3—96no—
107C22×Dic3—182no—
108C22×Dic3—362no—
109C22×Dic3#15711yescharacteristic, direct product factor
110C2×C4×D3—1812no—
111C2×C4×D3—1812no—
112C4×Dic3—916no—
113C22×C32—26no—
114C22×C32—28no—
115C22×C32—48no—
116C22×C32—46no—
117C22×C32C4×Dic314yescomplemented
118C22×C32C2×C4×D313yes—
119C2×C3⋊D3C4×Dic3112yescomplemented
120C2×C3⋊D3C2×C4×D313yes—
121C2×C3⋊D3C2×C4×D316yes—
122C2×C3⋊D3C22×Dic316yes—
123C2×C3⋊D3C23×D311yescharacteristic
124C2×C3×D3—624no—
125C2×C3×D3—66no—
126C2×C3×D3—612no—
127C2×C3×D3—612no—
128C2×C3×D3—62no—
129#157—38no—
130#157C22×Dic318yesdirect product factor
131C32×C4—34no—
132C3⋊Dic3—34no—
133C3⋊Dic3—68no—
134C3⋊Dic3—128no—
135C3×Dic3—28no—
136C23×C4—91no—
137C23×C4—271no—
138C23×C4—271no—
139C2×C42—2712no—
140C33C22×C4211yescharacteristic, complemented, Sylow
141C23×C3—22no—
142C23×C3—42no—
143C23×C3—93no—
144C23×C3—93no—
145C23×C3—94no—
146C23×C3—94no—
147C23×C3C2×#15711yescharacteristic
148C22×D3—66no—
149C22×D3—68no—
150C22×D3—68no—
151C22×D3—66no—
152C2×C3×C4—612no—
153C2×C3×C4—924no—
154C2×C3×C4—94no—
155C2×Dic3—924no—
156C2×Dic3—94no—
157C2×Dic3—96no—
158C2×Dic3—94no—
159C2×Dic3—912no—
160C2×Dic3—1812no—
161C2×Dic3—3612no—
162C2×Dic3C2×#15716yesdirect product factor
163C4×D3—188no—
164C4×D3—1824no—
165C2×C32—212no—
166C2×C32—22no—
167C2×C32—412no—
168C2×C32—42no—
169C2×C32C2×C4×Dic316yescomplemented
170C2×C32C22×C4×D311yescharacteristic
171C3⋊D3C2×C4×Dic316yescomplemented
172C3⋊D3C22×C4×D311yescharacteristic
173C3⋊D3C23×Dic311yescharacteristic
174C3×D3—612no—
175C3×D3—62no—
176C3×D3—62no—
177C24—91no—
178C22×C4—31no—
179C22×C4—98no—
180C22×C4—96no—
181C22×C4—276no—
182C22×C4—276no—
183C22×C4—276no—
184C22×C4—278no—
185C22×C4—271no—
186C42—2716no—
187C22×C3—26no—
188C22×C3—28no—
189C22×C3—46no—
190C22×C3—48no—
191C22×C3—96no—
192C22×C3—96no—
193C22×C3—912no—
194C22×C3—91no—
195C22×C3—93no—
196C22×C3C4×#15714yescomplemented
197C22×C3C22×#15713yes—
198C2×D3—66no—
199C2×D3—62no—
200C2×D3—624no—
201C2×D3—612no—
202C2×D3—612no—
203C3×C4—68no—
204C3×C4—98no—
205Dic3—98no—
206Dic3—94no—
207Dic3—188no—
208Dic3—368no—
209Dic3C22×#15714yesdirect product factor
210C32—22no—
211C32—42no—
212C32C22×C4×Dic311yescharacteristic, complemented
213C23—93no—
214C23—93no—
215C23—94no—
216C23—94no—
217C23D3×#15711yescharacteristic
218C2×C4—36no—
219C2×C4—94no—
220C2×C4—924no—
221C2×C4—274no—
222C2×C4—2712no—
223C2×C4—274no—
224C2×C4—2724no—
225C2×C4—276no—
226C2×C3—22no—
227C2×C3—212no—
228C2×C3—42no—
229C2×C3—412no—
230C2×C3—91no—
231C2×C3—91no—
232C2×C3—96no—
233C2×C3C2×C4×#15716yescomplemented
234C2×C3C23×#15711yescharacteristic
235D3—612no—
236D3—62no—
237D3—62no—
238C22—91no—
239C22—93no—
240C22—96no—
241C22—96no—
242C22—912no—
243C22Dic3×#15714yesdirect product factor
244C22C2×D3×#15713yes—
245C4—34no—
246C4—98no—
247C4—278no—
248C4—274no—
249C3—22no—
250C3—42no—
251C3C22×C4×#15711yescharacteristic, complemented
252C2—91no—
253C2—91no—
254C2—96no—
255C2C2×Dic3×#15716yesdirect product factor
256C2C22×D3×#15711yescharacteristic
subgroupG/H
center, hypercenterC23D3×#157
commutator, hypocenterC33C22×C42
FrattiniC2C22×D3×#157
Fitting, socleC23×C33C2×C4
prime radicalC23×C3×C3⋊D3C22
cosocleC2×C3×C3⋊D3C24
LCSG ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
4. C2×Dic3×#157C2direct (N × H)
7. C22×C3⋊#157C4semidirect
9. C22×C3×#157C4semidirect
11. C22×C3⋊(C3⋊Dic3)C4semidirect
12. C22×C32×Dic3C4semidirect
13. C2×Dic3×C3⋊D3C4semidirect
15. Dic3×#157C22direct (N × H)
26. C2×C3⋊#157C2×C4semidirect
28. C2×C3×#157C2×C4semidirect
30. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
31. C2×C32×Dic3C2×C4semidirect
32. Dic3×C3⋊D3C2×C4semidirect
39. C22×#157Dic3direct (N × H)
51. C22×C33C42semidirect
53. C2×C3×C3⋊D3C42semidirect
58. C3⋊#157C22×C4semidirect
59. C3×#157C22×C4semidirect
60. C3⋊(C3⋊Dic3)C22×C4semidirect
61. C32×Dic3C22×C4semidirect
80. C2×#157C2×Dic3direct (N × H)
92. C2×C33C2×C42semidirect
94. C3×C3⋊D3C2×C42semidirect
109. C22×Dic3#157direct (N × H)
117. C22×C32C4×Dic3semidirect
119. C2×C3⋊D3C4×Dic3semidirect
130. #157C22×Dic3direct (N × H)
140. C33C22×C42semidirect
162. C2×Dic3C2×#157direct (N × H)
169. C2×C32C2×C4×Dic3semidirect
171. C3⋊D3C2×C4×Dic3semidirect
196. C22×C3C4×#157semidirect
209. Dic3C22×#157direct (N × H)
212. C32C22×C4×Dic3semidirect
233. C2×C3C2×C4×#157semidirect
243. C22Dic3×#157direct (N × H)
251. C3C22×C4×#157semidirect
255. C2C2×Dic3×#157direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3⋊#1572
C23×C3×#1572
C22×Dic3×C3⋊D32
C2×Dic3×#1572
C22×C4×#1573
C22×C4×Dic39