Finite Group Databasebeta

← C3×#10592851 #13163187

Subgroups — C3×#10592851 #13163187

Order 1728.

1096 conjugacy classes of subgroups; only the 290 normal subgroups are shown.

normal subgroups only

This group has 1096 conjugacy classes of subgroups — too many to draw. Only its 290 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×C33×D8C211yesmaximal, characteristic
2C3×D4×C3⋊Dic3C212yesmaximal, complemented
3C3×#63659C212yesmaximal, complemented
4C3×#63644C211yesmaximal, characteristic, complemented
5C3×#63642C211yesmaximal, characteristic, complemented
6#10592851C311yesmaximal, characteristic, direct product factor
7C3×#5152—34nomaximal
8C2×C33×D4C2212yes—
9C22×C3×C3⋊Dic3—22no—
10C33×D8C414yescomplemented
11C2×C33×C8C2211yescharacteristic
12C3×#3757—22no—
13C3×C4⋊(C3⋊Dic3)C2212yes—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C32×D8—24no—
17C2×C32×D8D314yes—
18C2×C32×D8C2×C311yescharacteristic
19D4×C3⋊Dic3C2×C312yescomplemented
20C3×D4×Dic3—38no—
21#63659C2×C312yescomplemented
22#63644C2×C311yescharacteristic, complemented
23#63642C2×C311yescharacteristic, complemented
24C3×#832—38no—
25C3×#817—34no—
26C3×#815—34no—
27C23×C33—22no—
28C33×D4—22no—
29C33×D4C2×C414yescomplemented
30C2×C33×C4C2311yescharacteristic
31C2×C3×C3⋊Dic3—22no—
32C2×C3×C3⋊Dic3—44no—
33C2×C3×C3⋊Dic3D412yes—
34C33×C8C2×C412yescomplemented
35C3×C3⋊(C3⋊C8)—21no—
36C3×#442—91no—
37#5152—34no—
38C2×C32×D4—28no—
39C2×C32×D4C2×D318yes—
40C2×C32×D4C22×C312yes—
41C22×C3⋊Dic3—22no—
42C22×C3×Dic3—68no—
43C32×D8—216no—
44C32×D8Dic3116yescomplemented
45C32×D8C3×C414yescomplemented
46C2×C32×C8—24no—
47C2×C32×C8C2×D314yes—
48C2×C32×C8C22×C311yescharacteristic
49#3757—22no—
50C4⋊(C3⋊Dic3)C22×C312yes—
51C4×C3⋊Dic3C22×C311yescharacteristic
52C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
53C3×#217—68no—
54C3×C4⋊Dic3—38no—
55C3×C4×Dic3—34no—
56C2×C3×C3⋊C8—34no—
57C22×C33—24no—
58C22×C33—42no—
59C22×C33—42no—
60C22×C33C2×D411yescharacteristic
61C33×C4C22×C411yescharacteristic
62C33×C4#4111yescharacteristic
63C3×C3⋊Dic3—22no—
64C3×C3⋊Dic3—42no—
65C2×C3×D8—24no—
66C2×C3×D8C3×D314yes—
67C2×C3×D8C3⋊D311yescharacteristic
68C3×C4×D4—92no—
69D4×Dic3—38no—
70C3×#106—91no—
71C3×#102—92no—
72C3×#97—91no—
73#832—38no—
74#817—34no—
75#815—34no—
76C23×C32—22no—
77C23×C32—28no—
78C23×C32—48no—
79C32×D4—22no—
80C32×D4—216no—
81C32×D4—28no—
82C32×D4—48no—
83C32×D4C2×Dic3116yescomplemented
84C32×D4C2×C3×C414yescomplemented
85C2×C32×C4—24no—
86C2×C32×C4C22×D314yes—
87C2×C32×C4C23×C311yescharacteristic
88C2×C3⋊Dic3—22no—
89C2×C3⋊Dic3—44no—
90C2×C3⋊Dic3C3×D412yes—
91C2×C3×Dic3—38no—
92C2×C3×Dic3—68no—
93C2×C3×Dic3—1216no—
94C32×C8—28no—
95C32×C8C2×Dic318yescomplemented
96C32×C8C2×C3×C412yescomplemented
97C3⋊(C3⋊C8)—21no—
98C3×C3⋊C8—64no—
99#442—91noSylow
100C2×C33—44no—
101C2×C33C4×D411yescharacteristic
102C2×C33AGL(1,ℤ/8)12yes—
103C2×C3×D4—28no—
104C2×C3×D4C2×C3×D318yes—
105C2×C3×D4C2×C3⋊D312yes—
106C22×C3×C4—182no—
107C22×Dic3—68no—
108C3×D8—216no—
109C3×D8C3×Dic3116yescomplemented
110C3×D8C3⋊Dic314yescomplemented
111C2×C3×C8—24no—
112C2×C3×C8—91no—
113C2×C3×C8C2×C3×D314yes—
114C2×C3×C8C2×C3⋊D311yescharacteristic
115C3×C4⋊C4—92no—
116C3×C22⋊C4—182no—
117C3×C42—91no—
118#217—68no—
119C4⋊Dic3—38no—
120C4×Dic3—34no—
121C2×C3⋊C8—34no—
122C22×C32—24no—
123C22×C32—24no—
124C22×C32—216no—
125C22×C32—416no—
126C22×C32—42no—
127C22×C32—42no—
128C22×C32—48no—
129C22×C32—48no—
130C22×C32—88no—
131C22×C32—88no—
132C22×C32D3×D414yes—
133C22×C32C2×C3×D411yescharacteristic
134C32×C4—24no—
135C32×C4—24no—
136C32×C4#23714yes—
137C32×C4C22×Dic314yes—
138C32×C4C22×C3×C411yescharacteristic
139C32×C4C3×#4111yescharacteristic
140C3⋊Dic3—22no—
141C3⋊Dic3—42no—
142C3×Dic3—68no—
143C3×Dic3—128no—
144C2×D8C3×C3⋊D311yescharacteristic
145C4×D4—92no—
146#106—91no—
147#102—92no—
148#97—91no—
149C33#44211yescharacteristic, complemented, Sylow
150C23×C3—28no—
151C23×C3—22no—
152C23×C3—48no—
153C3×D4—28no—
154C3×D4—22no—
155C3×D4—216no—
156C3×D4—48no—
157C3×D4C2×C3×Dic3116yescomplemented
158C3×D4C2×C3⋊Dic314yescomplemented
159C2×C3×C4—24no—
160C2×C3×C4—92no—
161C2×C3×C4—182no—
162C2×C3×C4—364no—
163C2×C3×C4C22×C3×D314yes—
164C2×C3×C4C22×C3⋊D311yescharacteristic
165C2×Dic3—38no—
166C2×Dic3—68no—
167C2×Dic3—1216no—
168C3×C8—28no—
169C3×C8—181no—
170C3×C8C2×C3×Dic318yescomplemented
171C3×C8C2×C3⋊Dic312yescomplemented
172C3⋊C8—64no—
173C2×C32—28no—
174C2×C32—24no—
175C2×C32—44no—
176C2×C32—416no—
177C2×C32—816no—
178C2×C32#91118yes—
179C2×C32D4×Dic314yes—
180C2×C32C3×C4×D411yescharacteristic
181C2×C32C3×AGL(1,ℤ/8)12yes—
182C2×D4C2×C3×C3⋊D312yes—
183C22×C4—182no—
184D8C3×C3⋊Dic314yescomplemented
185C2×C8—91no—
186C2×C8C2×C3×C3⋊D311yescharacteristic
187C4⋊C4—92no—
188C22⋊C4—182no—
189C42—91no—
190C22×C3—216no—
191C22×C3—24no—
192C22×C3—24no—
193C22×C3—416no—
194C22×C3—48no—
195C22×C3—48no—
196C22×C3—42no—
197C22×C3—42no—
198C22×C3—88no—
199C22×C3—88no—
200C22×C3C3×D3×D414yes—
201C22×C3D4×C3⋊D311yescharacteristic
202C3×C4—24no—
203C3×C4—24no—
204C3×C4—182no—
205C3×C4—362no—
206C3×C4C3×#23714yes—
207C3×C4C22×C3×Dic314yes—
208C3×C4#383011yescharacteristic
209C3×C4C22×C3⋊Dic311yescharacteristic
210Dic3—68no—
211Dic3—128no—
212C32—24no—
213C32#515214yescomplemented
214C32C3×#44211yescharacteristic, complemented
215C23—22no—
216D4—22no—
217D4C2×C3×C3⋊Dic314yescomplemented
218C2×C4—92no—
219C2×C4—182no—
220C2×C4—364no—
221C2×C4C22×C3×C3⋊D311yescharacteristic
222C8—181no—
223C8C2×C3×C3⋊Dic312yescomplemented
224C2×C3—28no—
225C2×C3—24no—
226C2×C3—44no—
227C2×C3—416no—
228C2×C3—816no—
229C2×C3C3×#91118yes—
230C2×C3C3×D4×Dic314yes—
231C2×C3#6412012yes—
232C2×C3D4×C3⋊Dic311yescharacteristic
233C22—24no—
234C22—42no—
235C22—42no—
236C22C3×D4×C3⋊D311yescharacteristic
237C4—182no—
238C4—362no—
239C4C3×#383011yescharacteristic
240C4C22×C3×C3⋊Dic311yescharacteristic
241C3—24no—
242C3C3×#515214yescomplemented
243C3#1059285111yescharacteristic, direct product factor
244C2—44no—
245C2C3×#6412012yes—
246C2C3×D4×C3⋊Dic311yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
Fitting, prime radicalC2×C33×D8C2
socleC22×C33C2×D4
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×D8C3⋊D3
hypocenterC32C3×#442
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D8

Complements

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

NH ≅ G/Ntype
2. C3×D4×C3⋊Dic3C2semidirect
3. C3×#63659C2semidirect
4. C3×#63644C2semidirect
5. C3×#63642C2semidirect
6. #10592851C3direct (N × H)
10. C33×D8C4semidirect
19. D4×C3⋊Dic3C2×C3semidirect
21. #63659C2×C3semidirect
22. #63644C2×C3semidirect
23. #63642C2×C3semidirect
29. C33×D4C2×C4semidirect
34. C33×C8C2×C4semidirect
44. C32×D8Dic3semidirect
45. C32×D8C3×C4semidirect
83. C32×D4C2×Dic3semidirect
84. C32×D4C2×C3×C4semidirect
95. C32×C8C2×Dic3semidirect
96. C32×C8C2×C3×C4semidirect
109. C3×D8C3×Dic3semidirect
110. C3×D8C3⋊Dic3semidirect
149. C33#442semidirect
157. C3×D4C2×C3×Dic3semidirect
158. C3×D4C2×C3⋊Dic3semidirect
170. C3×C8C2×C3×Dic3semidirect
171. C3×C8C2×C3⋊Dic3semidirect
184. D8C3×C3⋊Dic3semidirect
213. C32#5152semidirect
214. C32C3×#442semidirect
217. D4C2×C3×C3⋊Dic3semidirect
223. C8C2×C3×C3⋊Dic3semidirect
242. C3C3×#5152semidirect
243. C3#10592851direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×D82
C3×D4×C3⋊Dic32
C3×#636592
C3×#636442
C3×#636422
#105928513
C3×#51523