Finite Group Databasebeta

← C3×#64227 #11724282

Subgroups — C3×#64227 #11724282

Order 864.

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

normal subgroups only

This group has 256 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
1C3×#3802C211yesmaximal, characteristic
2C3×#3797C211yesmaximal, characteristic, complemented
3C3×#3787C211yesmaximal, characteristic, complemented
4C3×#3775C214yesmaximal, complemented
5#64227C311yesmaximal, characteristic, direct product factor
6C2×C3×D32—21no—
7C3×C4×C3⋊D3C2211yescharacteristic
8C32×D12—21no—
9C32×Dic6—21no—
10C3×#627C2212yescomplemented
11C3×#625C2212yes—
12C3×#624—21no—
13C3×#622C2212yescomplemented
14C3×D3×Q8—61no—
15C3×D3×D4—61no—
16#3802C2×C311yescharacteristic
17#3797C2×C311yescharacteristic, complemented
18#3787C2×C311yescharacteristic, complemented
19#3775C2×C314yescomplemented
20C2×C3×C3⋊D3D411yescharacteristic
21C2×C32×D3—22no—
22C3×D32—42no—
23C33×C4D411yescharacteristic
24C3×C3⋊Dic3C2311yescharacteristic
25C32×Dic3—22no—
26C2×C3×SD16—91nomaximal
27C22×C3×D3—62no—
28C2×D32—21no—
29C32×Q8—61no—
30C32×D4—61no—
31C4×C3⋊D3C22×C311yescharacteristic
32C3×#67—62no—
33C3×D12—21no—
34C3×D12—21no—
35C3×D12—41no—
36C3×C4×D3—61no—
37C3×C4×D3—61no—
38C3×C4×D3—62no—
39C3×Dic6—21no—
40C3×Dic6—21no—
41C3×Dic6—41no—
42C3×Dic6—62no—
43#627C22×C312yescomplemented
44#625C22×C312yes—
45#624—21no—
46#622C22×C312yescomplemented
47C2×C33C2×D411yescharacteristic
48C3×C3⋊D3SD1612yescomplemented
49C32×D3—42no—
50C2×C3×Q8—91no—
51C2×C3×D4—91no—
52D3×Q8—61no—
53D3×D4—61no—
54C3×SD16—94no—
55C2×C3×C8—91no—
56C22×C32—62no—
57C2×C3⋊D3C3×D411yescharacteristic
58C2×C3×D3—22no—
59C2×C3×D3—22no—
60C2×C3×D3—42no—
61C2×C3×D3—61no—
62C2×C3×D3—61no—
63C2×C3×D3—124no—
64D32—42no—
65C32×C4—21no—
66C32×C4—21no—
67C32×C4—41no—
68C32×C4—41no—
69C32×C4—62no—
70C32×C4C3×D411yescharacteristic
71C3⋊Dic3C23×C311yescharacteristic
72C3×Dic3—22no—
73C3×Dic3—22no—
74C3×Dic3—42no—
75C3×Dic3—61no—
76C3×Dic3—61no—
77C2×SD16—91noSylow
78C33C2×SD1611yescharacteristic, complemented, Sylow
79C23×C3—181no—
80C22×D3—62no—
81C3×Q8—61no—
82C3×Q8—61no—
83C3×Q8—92no—
84C3×Q8—121no—
85C3×D4—61no—
86C3×D4—61no—
87C3×D4—92no—
88C3×D4—121no—
89C2×C3×C4—91no—
90C2×C3×C4—181no—
91#67—62no—
92D12—21no—
93C4×D3—61no—
94C4×D3—61no—
95C4×D3—62no—
96Dic6—21no—
97Dic6—62no—
98C3×C8—92no—
99C2×C32—21no—
100C2×C32—21no—
101C2×C32—41no—
102C2×C32—41no—
103C2×C32—122no—
104C2×C32C2×C3×D411yescharacteristic
105C3⋊D3C3×SD1612yescomplemented
106C3×D3—42no—
107C3×D3—42no—
108C3×D3—62no—
109C3×D3—62no—
110C3×D3—82no—
111C2×Q8—91no—
112C2×D4—91no—
113SD16—94no—
114C2×C8—91no—
115C22×C3—62no—
116C22×C3—62no—
117C22×C3—91no—
118C22×C3—122no—
119C22×C3—362no—
120C2×D3—22no—
121C2×D3—61no—
122C2×D3—61no—
123C2×D3—124no—
124C3×C4—21no—
125C3×C4—21no—
126C3×C4—41no—
127C3×C4—41no—
128C3×C4—62no—
129C3×C4—62no—
130C3×C4—91no—
131C3×C4—122no—
132C3×C4GO+(4,𝔽2)11yescharacteristic
133Dic3—22no—
134Dic3—61no—
135Dic3—61no—
136C32—21no—
137C32—21no—
138C32—41no—
139C32—41no—
140C32C2×C3×SD1611yescharacteristic, complemented
141C23—181no—
142Q8—61no—
143Q8—92no—
144D4—61no—
145D4—92no—
146C2×C4—91no—
147C2×C4—181no—
148C8—92no—
149C2×C3—21no—
150C2×C3—21no—
151C2×C3—41no—
152C2×C3—41no—
153C2×C3—92no—
154C2×C3—122no—
155C2×C3—122no—
156C2×C3—242no—
157C2×C3C2×GO+(4,𝔽2)11yescharacteristic
158D3—42no—
159D3—62no—
160D3—62no—
161C22—62no—
162C22—91no—
163C22—362no—
164C4—62no—
165C4—91no—
166C4C3×GO+(4,𝔽2)11yescharacteristic
167C3—21no—
168C3—21no—
169C3—41no—
170C3—41no—
171C3#6422711yescharacteristic, direct product factor
172C2—92no—
173C2—122no—
174C2C2×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C3C2×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C23×C3
FrattiniC2C2×C3×GO+(4,𝔽2)
FittingC33×C4D4
socleC2×C33C2×D4
prime radicalC3×#3802C2
hypercenterC3×C4GO+(4,𝔽2)
hypocenterC32C2×C3×SD16
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#3797C2semidirect
3. C3×#3787C2semidirect
4. C3×#3775C2semidirect
5. #64227C3direct (N × H)
10. C3×#627C22semidirect
13. C3×#622C22semidirect
17. #3797C2×C3semidirect
18. #3787C2×C3semidirect
19. #3775C2×C3semidirect
43. #627C22×C3semidirect
46. #622C22×C3semidirect
48. C3×C3⋊D3SD16semidirect
78. C33C2×SD16semidirect
105. C3⋊D3C3×SD16semidirect
140. C32C2×C3×SD16semidirect
171. C3#64227direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#38022
C3×#37972
C3×#37872
C3×#37752
#642273
C2×C3×SD169