Finite Group Databasebeta

← C3×#88988 #13049583

Subgroups — C3×#88988 #13049583

Order 1296.

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

normal subgroups only

This group has 332 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
1C32×C4×C3⋊D3C211yesmaximal, characteristic
2C3×C3⋊#622C212yesmaximal, complemented
3#88988C311yesmaximal, characteristic, direct product factor
4C3×#3787—31nomaximal
5C2×C32×C3⋊D3C411yescharacteristic
6C34×C4C411yescharacteristic
7C32×C3⋊Dic3C2211yescharacteristic
8C3×C4×C3⋊D3—21no—
9C3×C4×C3⋊D3D311yescharacteristic
10C3×C4×C3⋊D3C2×C311yescharacteristic
11C32×C4×D3—62no—
12C3⋊#622C2×C312yescomplemented
13C3×#622—32no—
14C2×C34C2×C411yescharacteristic
15C32×C3⋊D3C812yescomplemented
16#3787—31no—
17C2×C3×C3⋊C8—91nomaximal
18C2×C3×C3⋊D3—21no—
19C2×C3×C3⋊D3Dic311yescharacteristic
20C2×C3×C3⋊D3C3×C411yescharacteristic
21C2×C32×D3—62no—
22C33×C4—21no—
23C33×C4—22no—
24C33×C4—44no—
25C33×C4—42no—
26C33×C4—42no—
27C33×C4Dic311yescharacteristic
28C33×C4C3×C411yescharacteristic
29C3×C3⋊Dic3—21no—
30C3×C3⋊Dic3C2×D311yescharacteristic
31C3×C3⋊Dic3C22×C311yescharacteristic
32C32×Dic3—62no—
33C34C2×C811yescharacteristic, complemented, Sylow
34C2×C32×C4—91no—
35C4×C3⋊D3C3×D311yescharacteristic
36C3×C4×D3—62no—
37C3×C4×D3—62no—
38C3×C4×D3—64no—
39#622—32no—
40C3×C3⋊C8—92no—
41C2×C33—21no—
42C2×C33—22no—
43C2×C33—44no—
44C2×C33—42no—
45C2×C33—42no—
46C2×C33C2×Dic311yescharacteristic
47C2×C33C2×C3×C411yescharacteristic
48C3×C3⋊D3—22no—
49C3×C3⋊D3C3⋊C812yescomplemented
50C3×C3⋊D3C3×C812yescomplemented
51C32×D3—64no—
52C2×C3×C8—271no—
53C2×C3⋊C8—91no—
54C22×C32—91no—
55C2×C3⋊D3C3×Dic311yescharacteristic
56C2×C3×D3—62no—
57C2×C3×D3—62no—
58C2×C3×D3—64no—
59C32×C4—24no—
60C32×C4—22no—
61C32×C4—22no—
62C32×C4—44no—
63C32×C4—44no—
64C32×C4—42no—
65C32×C4—42no—
66C32×C4—48no—
67C32×C4—42no—
68C32×C4—42no—
69C32×C4—44no—
70C32×C4—91no—
71C32×C4C3×Dic311yescharacteristic
72C32×C4#15711yescharacteristic
73C3⋊Dic3C2×C3×D311yescharacteristic
74C3×Dic3—62no—
75C3×Dic3—62no—
76C3×Dic3—64no—
77C33—21no—
78C33—22no—
79C33—44no—
80C33—42no—
81C33—42no—
82C33C2×C3⋊C811yescharacteristic, complemented
83C33C2×C3×C811yescharacteristic, complemented
84C2×C3×C4—91no—
85C2×C3×C4—91no—
86C2×C3×C4—181no—
87C4×D3—62no—
88C3×C8—272no—
89C3⋊C8—92no—
90C2×C32—24no—
91C2×C32—22no—
92C2×C32—22no—
93C2×C32—44no—
94C2×C32—44no—
95C2×C32—42no—
96C2×C32—42no—
97C2×C32—48no—
98C2×C32—42no—
99C2×C32—42no—
100C2×C32—44no—
101C2×C32—92no—
102C2×C32C2×C3×Dic311yescharacteristic
103C2×C32C2×#15711yescharacteristic
104C3⋊D3C3×C3⋊C812yescomplemented
105C3×D3—64no—
106C3×D3—64no—
107C3×D3—68no—
108C2×C8—271noSylow
109C22×C3—91no—
110C22×C3—91no—
111C22×C3—181no—
112C2×D3—62no—
113C3×C4—22no—
114C3×C4—21no—
115C3×C4—44no—
116C3×C4—42no—
117C3×C4—42no—
118C3×C4—91no—
119C3×C4—91no—
120C3×C4—181no—
121C3×C4C3×#15711yescharacteristic
122C3×C4C3⋊#15711yescharacteristic
123Dic3—62no—
124C32—24no—
125C32—22no—
126C32—22no—
127C32—42no—
128C32—42no—
129C32—44no—
130C32—44no—
131C32—44no—
132C32—48no—
133C32—42no—
134C32—42no—
135C32C2×C3×C3⋊C811yescharacteristic, complemented
136C32#378711yescharacteristic, complemented
137C2×C4—91no—
138C8—272no—
139C2×C3—22no—
140C2×C3—21no—
141C2×C3—44no—
142C2×C3—42no—
143C2×C3—42no—
144C2×C3—92no—
145C2×C3—92no—
146C2×C3—182no—
147C2×C3C2×C3×#15711yescharacteristic
148C2×C3C2×C3⋊#15711yescharacteristic
149D3—64no—
150C22—91no—
151C4—91no—
152C4C3×C3⋊#15711yescharacteristic
153C3—22no—
154C3—21no—
155C3—44no—
156C3—42no—
157C3—42no—
158C3C3×#378711yescharacteristic, complemented
159C3#8898811yescharacteristic, direct product factor
160C2—92no—
161C2C2×C3×C3⋊#15711yescharacteristic
subgroupG/H
center, hypercenterC3×C4C3⋊#157
commutator, hypocenterC33C2×C3×C8
FrattiniC2C2×C3×C3⋊#157
FittingC34×C4C4
socleC2×C34C2×C4
prime radicalC2×C32×C3⋊D3C4
cosocleC3×C3⋊Dic3C22×C3
LCSG ▷ C33
UCSC1 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×C3⋊#622C2semidirect
3. #88988C3direct (N × H)
12. C3⋊#622C2×C3semidirect
15. C32×C3⋊D3C8semidirect
33. C34C2×C8semidirect
49. C3×C3⋊D3C3⋊C8semidirect
50. C3×C3⋊D3C3×C8semidirect
82. C33C2×C3⋊C8semidirect
83. C33C2×C3×C8semidirect
104. C3⋊D3C3×C3⋊C8semidirect
135. C32C2×C3×C3⋊C8semidirect
136. C32#3787semidirect
158. C3C3×#3787semidirect
159. C3#88988direct (N × H)

Maximal subgroups

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

subgroupindex
C32×C4×C3⋊D32
C3×C3⋊#6222
#889883
C3×#37873
C2×C3×C3⋊C89