Finite Group Databasebeta

← C3×#64224 #11724279

Subgroups — C3×#64224 #11724279

Order 864.

262 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 46 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, complemented
2C3×#3798C211yesmaximal, characteristic, complemented
3C3×#3788C211yesmaximal, characteristic, complemented
4C3×#3775C212yesmaximal, complemented
5C3×#3774C212yesmaximal, complemented
6#64224C311yesmaximal, characteristic, direct product factor
7C2×C3×D32—21no—
8C3×C4×C3⋊D3C2211yescharacteristic
9C32×D12—21no—
10C32×C4×D3—21no—
11C3×#627C2211yescharacteristic, complemented
12C3×#626—21no—
13C3×#625C2212yes—
14C3×#625C2211yescharacteristic, complemented
15C3×#622C2212yescomplemented
16C3×D3×D4—61no—
17C3×#235—61no—
18#3802C2×C311yescharacteristic, complemented
19#3798C2×C311yescharacteristic, complemented
20#3788C2×C311yescharacteristic, complemented
21#3775C2×C312yescomplemented
22#3774C2×C312yescomplemented
23C2×C3×C3⋊D3D411yescharacteristic
24C2×C32×D3—22no—
25C2×C32×D3—21no—
26C3×D32—42no—
27C33×C4D411yescharacteristic
28C3×C3⋊Dic3C2311yescharacteristic
29C32×Dic3—21no—
30C3×AGL(1,ℤ/8)—91nomaximal
31C22×C3×D3—62no—
32C2×D32—21no—
33C32×D4—61no—
34C2×C32×C4—61no—
35C4×C3⋊D3C22×C311yescharacteristic
36C3×#67—62no—
37C3×#67—61no—
38C3×#67—61no—
39C3×D12—21no—
40C3×D12—21no—
41C3×D12—41no—
42C3×D12—61no—
43C3×C4×D3—21no—
44C3×C4×D3—21no—
45C3×C4×D3—41no—
46C3×C4×D3—61no—
47C3×C4×D3—61no—
48C3×Dic6—61no—
49#627C22×C311yescharacteristic, complemented
50#626—21no—
51#625C22×C312yes—
52#625C22×C311yescharacteristic, complemented
53#622C22×C312yescomplemented
54C2×C33C2×D411yescharacteristic
55C3×C3⋊D3—21no—
56C32×D3—41no—
57C32×D3—42no—
58C3×#41—91no—
59C2×C3×D4—91no—
60D3×D4—61no—
61#235—61no—
62C3×SD16—92no—
63C3×D8—92no—
64C3×M16—91no—
65C22×C32—61no—
66C22×C32—62no—
67C2×C3⋊D3C3×D411yescharacteristic
68C2×C3×D3—21no—
69C2×C3×D3—22no—
70C2×C3×D3—22no—
71C2×C3×D3—21no—
72C2×C3×D3—41no—
73C2×C3×D3—42no—
74C2×C3×D3—61no—
75C2×C3×D3—61no—
76C2×C3×D3—124no—
77D32—42no—
78C32×C4—21no—
79C32×C4—21no—
80C32×C4—41no—
81C32×C4—41no—
82C32×C4—61no—
83C32×C4C3×D411yescharacteristic
84C3⋊Dic3C23×C311yescharacteristic
85C3×Dic3—21no—
86C3×Dic3—21no—
87C3×Dic3—41no—
88C3×Dic3—61no—
89C3×Dic3—61no—
90AGL(1,ℤ/8)—91noSylow
91C33AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
92C23×C3—181no—
93C22×D3—62no—
94C3×Q8—91no—
95C3×D4—61no—
96C3×D4—61no—
97C3×D4—91no—
98C3×D4—92no—
99C3×D4—121no—
100C3×D4—181no—
101C2×C3×C4—61no—
102C2×C3×C4—61no—
103C2×C3×C4—91no—
104C2×C3×C4—121no—
105#67—62no—
106#67—61no—
107#67—61no—
108D12—21no—
109D12—61no—
110C4×D3—21no—
111C4×D3—61no—
112C4×D3—61no—
113Dic6—61no—
114C3×C8—92no—
115C2×C32—21no—
116C2×C32—21no—
117C2×C32—41no—
118C2×C32—41no—
119C2×C32—121no—
120C2×C32—122no—
121C2×C32C2×C3×D411yescharacteristic
122C3⋊D3—21no—
123C3×D3—41no—
124C3×D3—42no—
125C3×D3—42no—
126C3×D3—42no—
127C3×D3—41no—
128C3×D3—62no—
129C3×D3—82no—
130C3×D3—121no—
131#41—91no—
132C2×D4—91no—
133SD16—92no—
134D8—92no—
135M16—91no—
136C22×C3—61no—
137C22×C3—62no—
138C22×C3—61no—
139C22×C3—62no—
140C22×C3—91no—
141C22×C3—121no—
142C22×C3—122no—
143C22×C3—362no—
144C2×D3—21no—
145C2×D3—22no—
146C2×D3—61no—
147C2×D3—61no—
148C2×D3—124no—
149C3×C4—21no—
150C3×C4—21no—
151C3×C4—41no—
152C3×C4—41no—
153C3×C4—61no—
154C3×C4—61no—
155C3×C4—91no—
156C3×C4—121no—
157C3×C4GO+(4,𝔽2)11yescharacteristic
158Dic3—21no—
159Dic3—61no—
160Dic3—61no—
161C32—21no—
162C32—21no—
163C32—41no—
164C32—41no—
165C32C3×AGL(1,ℤ/8)11yescharacteristic, complemented
166C23—181no—
167Q8—91no—
168D4—61no—
169D4—91no—
170D4—92no—
171D4—181no—
172C2×C4—61no—
173C2×C4—91no—
174C8—92no—
175C2×C3—21no—
176C2×C3—21no—
177C2×C3—41no—
178C2×C3—41no—
179C2×C3—122no—
180C2×C3—121no—
181C2×C3—122no—
182C2×C3—122no—
183C2×C3—121no—
184C2×C3—181no—
185C2×C3—242no—
186C2×C3C2×GO+(4,𝔽2)11yescharacteristic
187D3—42no—
188D3—41no—
189D3—62no—
190D3—121no—
191C22—61no—
192C22—62no—
193C22—91no—
194C22—362no—
195C4—61no—
196C4—91no—
197C4C3×GO+(4,𝔽2)11yescharacteristic
198C3—21no—
199C3—21no—
200C3—41no—
201C3—41no—
202C3#6422411yescharacteristic, direct product factor
203C2—122no—
204C2—121no—
205C2—181no—
206C2C2×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
hypercenterC3×C4GO+(4,𝔽2)
hypocenterC32C3×AGL(1,ℤ/8)
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
1. C3×#3802C2semidirect
2. C3×#3798C2semidirect
3. C3×#3788C2semidirect
4. C3×#3775C2semidirect
5. C3×#3774C2semidirect
6. #64224C3direct (N × H)
11. C3×#627C22semidirect
14. C3×#625C22semidirect
15. C3×#622C22semidirect
18. #3802C2×C3semidirect
19. #3798C2×C3semidirect
20. #3788C2×C3semidirect
21. #3775C2×C3semidirect
22. #3774C2×C3semidirect
49. #627C22×C3semidirect
52. #625C22×C3semidirect
53. #622C22×C3semidirect
91. C33AGL(1,ℤ/8)semidirect
165. C32C3×AGL(1,ℤ/8)semidirect
202. C3#64224direct (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×#37982
C3×#37882
C3×#37752
C3×#37742
#642243
C3×AGL(1,ℤ/8)9