Finite Group Databasebeta

← C3×#10592408 #13162748

Subgroups — C3×#10592408 #13162748

Order 1728.

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

normal subgroups only

This group has 736 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
1C3×#64081C211yesmaximal, characteristic
2C33×#97C211yesmaximal, characteristic, complemented
3C3×#63647C212yesmaximal
4C3×#63641C212yesmaximal, complemented
5C3×C4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#10592408C311yesmaximal, characteristic, direct product factor
7C3×#4709—34nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C33×C8C2212yes—
10C33×C42C2211yescharacteristic
11C3×#3752C412yes—
12C3×#3750C412yes—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2212yes—
15#64081C2×C311yescharacteristic
16C3×#872—34no—
17C32×#97—24no—
18C32×#97D314yescomplemented
19C32×#97C2×C311yescharacteristic, complemented
20#63647C2×C312yes—
21#63641C2×C312yescomplemented
22C4×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
23C3×(C2×D3)⋊C8—38no—
24C3×Dic3⋊C8—38no—
25C3×C4×C3⋊C8—34no—
26C22×C3×C3⋊D3C2×C411yescharacteristic
27C2×C33×C4C2×C412yes—
28C2×C33×C4C2311yescharacteristic
29C2×C3×C3⋊Dic3C2×C412yes—
30C2×C3×C3⋊Dic3C2×C411yescharacteristic
31C3×C4×C3⋊D3—22no—
32C33×C8—22no—
33C3×C3⋊(C3⋊C8)—22no—
34C3×#433—91no—
35#4709—34no—
36C2×C4×C3⋊D3C22×C311yescharacteristic
37C2×C3×C4×D3—34no—
38C2×C32×C8—28no—
39C2×C32×C8C2×D318yes—
40C2×C32×C8C22×C312yes—
41C32×C42—24no—
42C32×C42C2×D314yes—
43C32×C42C22×C311yescharacteristic
44#3752C3×C412yes—
45#3750C3×C412yes—
46C4×C3⋊Dic3C22×C311yescharacteristic
47C2×C3⋊(C3⋊C8)C22×C312yes—
48C3×(C2×D3)⋊C4—38no—
49C3×Dic3⋊C4—38no—
50C3×C4×Dic3—34no—
51C2×C3×C3⋊C8—38no—
52C22×C33C22×C411yescharacteristic
53C2×C3×C3⋊D3—21no—
54C2×C3×C3⋊D3—22no—
55C33×C4—22no—
56C33×C4#4112yes—
57C3×C3⋊Dic3—21no—
58C3×C3⋊Dic3—22no—
59C3×#117—91no—
60#872—34no—
61C3×C4⋊C8—92no—
62C3×C22⋊C8—92no—
63C3×#97—24no—
64C3×#97C3×D314yescomplemented
65C3×#97C3⋊D311yescharacteristic, complemented
66C3×C4×C8—91no—
67(C2×D3)⋊C8—38no—
68Dic3⋊C8—38no—
69C4×C3⋊C8—34no—
70C22×C3⋊D3C2×C3×C411yescharacteristic
71C22×C3×D3—34no—
72C2×C32×C4—28no—
73C2×C32×C4—24no—
74C2×C32×C4C4×D318yes—
75C2×C32×C4C2×C3×C412yes—
76C2×C32×C4C22×D314yes—
77C2×C32×C4C23×C311yescharacteristic
78C2×C3⋊Dic3C2×C3×C412yes—
79C2×C3⋊Dic3C2×C3×C411yescharacteristic
80C4×C3⋊D3—22no—
81C2×C3×Dic3—38no—
82C2×C3×Dic3—34no—
83C3×C4×D3—68no—
84C32×C8—216no—
85C32×C8—28no—
86C32×C8—22no—
87C3⋊(C3⋊C8)—22no—
88C3×C3⋊C8—68no—
89#433—91noSylow
90C2×C33#11711yescharacteristic
91C2×C33#13112yes—
92C3×C3⋊D3—41no—
93C22×C3×C4—91no—
94C2×C4×D3—34no—
95C2×C3×C8—28no—
96C2×C3×C8—92no—
97C2×C3×C8C2×C3×D318yes—
98C2×C3×C8C2×C3⋊D312yes—
99C3×C4⋊C4—92no—
100C3×C22⋊C4—92no—
101C3×C42—24no—
102C3×C42—91no—
103C3×C42C2×C3×D314yes—
104C3×C42C2×C3⋊D311yescharacteristic
105(C2×D3)⋊C4—38no—
106Dic3⋊C4—38no—
107C4×Dic3—34no—
108C2×C3⋊C8—38no—
109C22×C32—24no—
110C22×C32C2×C4×D314yes—
111C22×C32C22×C3×C411yescharacteristic
112C2×C3⋊D3—21no—
113C2×C3⋊D3—22no—
114C2×C3×D3—64no—
115C2×C3×D3—68no—
116C32×C4—22no—
117C32×C4—28no—
118C32×C4—28no—
119C32×C4—48no—
120C32×C4#23518yes—
121C32×C4C3×#4112yes—
122C3⋊Dic3—21no—
123C3⋊Dic3—22no—
124C3×Dic3—64no—
125C3×Dic3—68no—
126#117—91no—
127C4⋊C8—92no—
128C22⋊C8—92no—
129#97C3×C3⋊D311yescharacteristic, complemented
130C4×C8—91no—
131C33#43311yescharacteristic, complemented, Sylow
132C23×C3—91no—
133C22×D3—34no—
134C2×C3×C4—28no—
135C2×C3×C4—24no—
136C2×C3×C4—92no—
137C2×C3×C4—91no—
138C2×C3×C4—182no—
139C2×C3×C4C3×C4×D318yes—
140C2×C3×C4C4×C3⋊D312yes—
141C2×C3×C4C22×C3×D314yes—
142C2×C3×C4C22×C3⋊D311yescharacteristic
143C2×Dic3—38no—
144C2×Dic3—34no—
145C4×D3—68no—
146C3×C8—216no—
147C3×C8—28no—
148C3×C8—22no—
149C3×C8—182no—
150C3⋊C8—68no—
151C2×C32—28no—
152C2×C32—24no—
153C2×C32#87214yes—
154C2×C32#90718yes—
155C2×C32C3×#11711yescharacteristic
156C2×C32C3×#13112yes—
157C3⋊D3—41no—
158C3×D3—124no—
159C22×C4—91no—
160C2×C8—92no—
161C2×C8C2×C3×C3⋊D312yes—
162C4⋊C4—92no—
163C22⋊C4—92no—
164C42—91no—
165C42C2×C3×C3⋊D311yescharacteristic
166C22×C3—24no—
167C22×C3—181no—
168C22×C3—182no—
169C22×C3C2×C3×C4×D314yes—
170C22×C3C2×C4×C3⋊D311yescharacteristic
171C2×D3—68no—
172C2×D3—64no—
173C3×C4—28no—
174C3×C4—22no—
175C3×C4—28no—
176C3×C4—48no—
177C3×C4—181no—
178C3×C4—182no—
179C3×C4C3×#23518yes—
180C3×C4#382812yes—
181Dic3—68no—
182Dic3—64no—
183C32—24no—
184C32#470914yescomplemented
185C32C3×#43311yescharacteristic, complemented
186C23—91no—
187C2×C4—92no—
188C2×C4—91no—
189C2×C4—182no—
190C2×C4C3×C4×C3⋊D312yes—
191C2×C4C22×C3×C3⋊D311yescharacteristic
192C8—22no—
193C8—182no—
194C2×C3—24no—
195C2×C3—28no—
196C2×C3—361no—
197C2×C3C3×#87214yes—
198C2×C3C3×#90718yes—
199C2×C3#6408111yescharacteristic
200C2×C3#6411612yes—
201D3—124no—
202C22—181no—
203C22—182no—
204C22C2×C3×C4×C3⋊D311yescharacteristic
205C4—22no—
206C4—181no—
207C4—182no—
208C4C3×#382812yes—
209C3—24no—
210C3C3×#470914yescomplemented
211C3#1059240811yescharacteristic, direct product factor
212C2—361no—
213C2C3×#6408111yescharacteristic
214C2C3×#6411612yes—
subgroupG/H
centerC2×C3×C4C22×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#97C2
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC3×#97C3⋊D3
hypocenterC32C3×#433
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×#97

Complements

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

NH ≅ G/Ntype
2. C33×#97C2semidirect
4. C3×#63641C2semidirect
5. C3×C4×C3⋊(C3⋊C8)C2semidirect
6. #10592408C3direct (N × H)
18. C32×#97D3semidirect
19. C32×#97C2×C3semidirect
21. #63641C2×C3semidirect
22. C4×C3⋊(C3⋊C8)C2×C3semidirect
64. C3×#97C3×D3semidirect
65. C3×#97C3⋊D3semidirect
129. #97C3×C3⋊D3semidirect
131. C33#433semidirect
184. C32#4709semidirect
185. C32C3×#433semidirect
210. C3C3×#4709semidirect
211. C3#10592408direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#640812
C33×#972
C3×#636472
C3×#636412
C3×C4×C3⋊(C3⋊C8)2
#105924083
C3×#47093