Finite Group Databasebeta

← C3×C4⋊(C3⋊#157) #13049586

Subgroups — C3×C4⋊(C3⋊#157) #13049586

Order 1296.

348 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 50 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×C3×C3⋊#157C212yesmaximal
2C32×C4×C3⋊D3C211yesmaximal, characteristic
3C4⋊(C3⋊#157)C311yesmaximal, characteristic, direct product factor
4C3×C4⋊#157—31nomaximal
5C2×C32×C3⋊D3C2211yescharacteristic
6C3×C3⋊#157—22no—
7C34×C4C411yescharacteristic, complemented
8C32×C3⋊Dic3C411yescharacteristic, complemented
9C2×C3⋊#157C2×C312yes—
10C2×C3×#157—32no—
11C3×C4×C3⋊D3—21no—
12C3×C4×C3⋊D3D311yescharacteristic
13C3×C4×C3⋊D3C2×C311yescharacteristic
14C32×C4×D3—62no—
15C2×C34C2×C411yescharacteristic
16C32×C3⋊D3D411yescharacteristic
17C32×C3⋊D3Q811yescharacteristic
18C4⋊#157—31no—
19C3×C4⋊Dic3—91nomaximal
20C2×C3×C3⋊D3—21no—
21C2×C3×C3⋊D3C2×D311yescharacteristic
22C2×C3×C3⋊D3C22×C311yescharacteristic
23C2×C32×D3—62no—
24C3⋊#157—22no—
25C3×#157—62no—
26C33×C4—21no—
27C33×C4—22no—
28C33×C4—44no—
29C33×C4—42no—
30C33×C4—42no—
31C33×C4Dic311yescharacteristic, complemented
32C33×C4C3×C411yescharacteristic, complemented
33C3×C3⋊Dic3—21no—
34C3×C3⋊Dic3Dic311yescharacteristic, complemented
35C3×C3⋊Dic3C3×C411yescharacteristic, complemented
36C32×Dic3—62no—
37C34C4⋊C411yescharacteristic, complemented, Sylow
38C2×#157—32no—
39C2×C32×C4—91no—
40C4×C3⋊D3C3×D311yescharacteristic
41C2×C3×Dic3—92no—
42C3×C4×D3—62no—
43C3×C4×D3—62no—
44C3×C4×D3—64no—
45C2×C33—21no—
46C2×C33—22no—
47C2×C33—44no—
48C2×C33—42no—
49C2×C33—42no—
50C2×C33C2×Dic311yescharacteristic
51C2×C33C2×C3×C411yescharacteristic
52C3×C3⋊D3—21no—
53C3×C3⋊D3—21no—
54C3×C3⋊D3Dic611yescharacteristic
55C3×C3⋊D3D1211yescharacteristic
56C3×C3⋊D3C3×D411yescharacteristic
57C3×C3⋊D3C3×Q811yescharacteristic
58C32×D3—62no—
59C32×D3—62no—
60C3×C4⋊C4—271no—
61C4⋊Dic3—91no—
62C22×C32—91no—
63C2×C3⋊D3C2×C3×D311yescharacteristic
64C2×C3×D3—62no—
65C2×C3×D3—62no—
66C2×C3×D3—64no—
67#157—62no—
68C32×C4—24no—
69C32×C4—22no—
70C32×C4—22no—
71C32×C4—44no—
72C32×C4—44no—
73C32×C4—42no—
74C32×C4—42no—
75C32×C4—48no—
76C32×C4—42no—
77C32×C4—42no—
78C32×C4—44no—
79C32×C4—91no—
80C32×C4C3×Dic311yescharacteristic, complemented
81C32×C4#15711yescharacteristic, complemented
82C3⋊Dic3C3×Dic311yescharacteristic, complemented
83C3×Dic3—62no—
84C3×Dic3—62no—
85C3×Dic3—64no—
86C3×Dic3—182no—
87C33—21no—
88C33—22no—
89C33—44no—
90C33—42no—
91C33—42no—
92C33C4⋊Dic311yescharacteristic, complemented
93C33C3×C4⋊C411yescharacteristic, complemented
94C2×C3×C4—91no—
95C2×C3×C4—91no—
96C2×C3×C4—181no—
97C2×C3×C4—272no—
98C2×Dic3—92no—
99C4×D3—62no—
100C2×C32—24no—
101C2×C32—22no—
102C2×C32—22no—
103C2×C32—44no—
104C2×C32—44no—
105C2×C32—42no—
106C2×C32—42no—
107C2×C32—48no—
108C2×C32—42no—
109C2×C32—42no—
110C2×C32—44no—
111C2×C32—91no—
112C2×C32—91no—
113C2×C32C2×C3×Dic311yescharacteristic
114C2×C32C2×#15711yescharacteristic
115C3⋊D3C3×Dic611yescharacteristic
116C3⋊D3C3×D1211yescharacteristic
117C3×D3—62no—
118C3×D3—62no—
119C3×D3—62no—
120C3×D3—62no—
121C3×D3—64no—
122C3×D3—64no—
123C4⋊C4—271noSylow
124C22×C3—91no—
125C22×C3—91no—
126C22×C3—181no—
127C2×D3—62no—
128C3×C4—22no—
129C3×C4—21no—
130C3×C4—44no—
131C3×C4—42no—
132C3×C4—42no—
133C3×C4—91no—
134C3×C4—91no—
135C3×C4—181no—
136C3×C4—542no—
137C3×C4C3×#15711yescharacteristic, complemented
138C3×C4C3⋊#15711yescharacteristic, complemented
139Dic3—62no—
140Dic3—182no—
141C32—24no—
142C32—22no—
143C32—22no—
144C32—42no—
145C32—42no—
146C32—44no—
147C32—44no—
148C32—44no—
149C32—48no—
150C32—42no—
151C32—42no—
152C32C3×C4⋊Dic311yescharacteristic, complemented
153C32C4⋊#15711yescharacteristic, complemented
154C2×C4—91no—
155C2×C4—272no—
156C2×C3—22no—
157C2×C3—21no—
158C2×C3—44no—
159C2×C3—42no—
160C2×C3—42no—
161C2×C3—91no—
162C2×C3—91no—
163C2×C3—91no—
164C2×C3—91no—
165C2×C3—181no—
166C2×C3—181no—
167C2×C3C2×C3×#15711yescharacteristic
168C2×C3C2×C3⋊#15711yescharacteristic
169D3—62no—
170D3—62no—
171C22—91no—
172C4—91no—
173C4—542no—
174C4C3×C3⋊#15711yescharacteristic, complemented
175C3—22no—
176C3—21no—
177C3—44no—
178C3—42no—
179C3—42no—
180C3C3×C4⋊#15711yescharacteristic, complemented
181C3C4⋊(C3⋊#157)11yescharacteristic, direct product factor
182C2—91no—
183C2—91no—
184C2C2×C3×C3⋊#15711yescharacteristic
subgroupG/H
centerC2×C3C2×C3⋊#157
commutatorC2×C33C2×C3×C4
FrattiniC2C2×C3×C3⋊#157
FittingC34×C4C4
socleC2×C34C2×C4
prime radicalC2×C32×C3⋊D3C22
cosocleC2×C3×C3⋊D3C22×C3
hypercenterC3×C4C3⋊#157
hypocenterC33C3×C4⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C3 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
3. C4⋊(C3⋊#157)C3direct (N × H)
7. C34×C4C4semidirect
8. C32×C3⋊Dic3C4semidirect
31. C33×C4Dic3semidirect
32. C33×C4C3×C4semidirect
34. C3×C3⋊Dic3Dic3semidirect
35. C3×C3⋊Dic3C3×C4semidirect
37. C34C4⋊C4semidirect
80. C32×C4C3×Dic3semidirect
81. C32×C4#157semidirect
82. C3⋊Dic3C3×Dic3semidirect
92. C33C4⋊Dic3semidirect
93. C33C3×C4⋊C4semidirect
137. C3×C4C3×#157semidirect
138. C3×C4C3⋊#157semidirect
152. C32C3×C4⋊Dic3semidirect
153. C32C4⋊#157semidirect
174. C4C3×C3⋊#157semidirect
180. C3C3×C4⋊#157semidirect
181. C3C4⋊(C3⋊#157)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊#1572
C32×C4×C3⋊D32
C4⋊(C3⋊#157)3
C3×C4⋊#1573
C3×C4⋊Dic39