Finite Group Databasebeta

← C3×#64239 #11724294

Subgroups — C3×#64239 #11724294

Order 864.

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

normal subgroups only

This group has 230 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
1C2×C3×#627C211yesmaximal, characteristic, complemented
2C3×#3805C211yesmaximal, characteristic
3C3×#3792C211yesmaximal, characteristic, complemented
4C3×#3776C212yesmaximal, complemented
5C3×#3775C212yesmaximal, complemented
6#64239C311yesmaximal, characteristic, direct product factor
7C2×C3×C3⋊Dic3C2211yescharacteristic
8C32×#67—21no—
9C2×C32×Dic3—21no—
10C3×#627—21no—
11C3×#627C2212yes—
12C3×#627C2211yescharacteristic
13C3×#625C2211yescharacteristic
14C3×D3×Dic3—21no—
15C3×#622C2212yes—
16C3×#237—61no—
17C2×C3×Dic6—61no—
18C2×#627C2×C311yescharacteristic, complemented
19#3805C2×C311yescharacteristic
20#3792C2×C311yescharacteristic, complemented
21#3776C2×C312yescomplemented
22#3775C2×C312yescomplemented
23C22×C33D411yescharacteristic
24C2×C32×D3—21no—
25C3×C3⋊Dic3D411yescharacteristic
26C3×C3⋊Dic3C2311yescharacteristic
27C32×Dic3—21no—
28C32×Dic3—22no—
29C3×#137—91nomaximal
30C32×D4—61no—
31C2×C32×C4—61no—
32C2×C3⋊Dic3C22×C311yescharacteristic
33C3×#67—21no—
34C3×#67—21no—
35C3×#67—41no—
36C3×#67—61no—
37C2×C3×Dic3—21no—
38C2×C3×Dic3—21no—
39C2×C3×Dic3—41no—
40C2×C3×Dic3—61no—
41C2×C3×Dic3—61no—
42C2×C3×Dic3—61no—
43C3×C4×D3—61no—
44C3×Dic6—62no—
45C3×Dic6—61no—
46C3×Dic6—62no—
47#627—21no—
48#627C22×C312yes—
49#627C22×C311yescharacteristic
50#625C22×C311yescharacteristic
51D3×Dic3—21no—
52#622C22×C312yes—
53C2×C33—21no—
54C2×C33C2×D411yescharacteristic
55C32×D3—41no—
56C3×#41—91no—
57C2×C3×Q8—91no—
58#237—61no—
59C2×Dic6—61no—
60C3×Q16—92no—
61C3×SD16—92no—
62C3×M16—91no—
63C22×C32—21no—
64C22×C32—21no—
65C22×C32—41no—
66C22×C32—41no—
67C22×C32—61no—
68C22×C32C3×D411yescharacteristic
69C2×C3×D3—21no—
70C2×C3×D3—21no—
71C2×C3×D3—41no—
72C32×C4—62no—
73C32×C4—61no—
74C3⋊Dic3C3×D411yescharacteristic
75C3⋊Dic3C23×C311yescharacteristic
76C3×Dic3—22no—
77C3×Dic3—21no—
78C3×Dic3—21no—
79C3×Dic3—22no—
80C3×Dic3—42no—
81C3×Dic3—41no—
82C3×Dic3—61no—
83C3×Dic3—61no—
84C3×Dic3—61no—
85C3×Dic3—61no—
86#137—91noSylow
87C33#13711yescharacteristic, complemented, Sylow
88C3×Q8—92no—
89C3×Q8—91no—
90C3×Q8—181no—
91C3×D4—61no—
92C3×D4—61no—
93C3×D4—91no—
94C3×D4—121no—
95C2×C3×C4—61no—
96C2×C3×C4—61no—
97C2×C3×C4—91no—
98C2×C3×C4—121no—
99C2×C3×C4—181no—
100#67—21no—
101#67—61no—
102C2×Dic3—21no—
103C2×Dic3—61no—
104C2×Dic3—61no—
105C2×Dic3—61no—
106C4×D3—61no—
107Dic6—62no—
108Dic6—61no—
109Dic6—62no—
110C3×C8—92no—
111C2×C32—22no—
112C2×C32—21no—
113C2×C32—21no—
114C2×C32—21no—
115C2×C32—41no—
116C2×C32—41no—
117C2×C32—41no—
118C2×C32—42no—
119C2×C32—81no—
120C2×C32—121no—
121C2×C32C2×C3×D411yescharacteristic
122C3×D3—41no—
123C3×D3—41no—
124C3×D3—81no—
125#41—91no—
126C2×Q8—91no—
127Q16—92no—
128SD16—92no—
129M16—91no—
130C22×C3—21no—
131C22×C3—21no—
132C22×C3—41no—
133C22×C3—41no—
134C22×C3—61no—
135C22×C3—61no—
136C22×C3—121no—
137C22×C3GO+(4,𝔽2)11yescharacteristic
138C2×D3—21no—
139C3×C4—62no—
140C3×C4—61no—
141C3×C4—62no—
142C3×C4—61no—
143C3×C4—91no—
144C3×C4—91no—
145C3×C4—122no—
146C3×C4—121no—
147Dic3—22no—
148Dic3—21no—
149Dic3—61no—
150Dic3—61no—
151Dic3—61no—
152Dic3—61no—
153C32—21no—
154C32—21no—
155C32—41no—
156C32—41no—
157C32C3×#13711yescharacteristic, complemented
158Q8—92no—
159Q8—91no—
160Q8—181no—
161D4—61no—
162D4—91no—
163C2×C4—61no—
164C2×C4—91no—
165C2×C4—181no—
166C8—92no—
167C2×C3—22no—
168C2×C3—21no—
169C2×C3—21no—
170C2×C3—21no—
171C2×C3—41no—
172C2×C3—42no—
173C2×C3—41no—
174C2×C3—41no—
175C2×C3—81no—
176C2×C3—121no—
177C2×C3—121no—
178C2×C3—241no—
179C2×C3C2×GO+(4,𝔽2)11yescharacteristic
180D3—41no—
181C22—61no—
182C22C3×GO+(4,𝔽2)11yescharacteristic
183C4—62no—
184C4—61no—
185C4—91no—
186C4—91no—
187C3—21no—
188C3—21no—
189C3—41no—
190C3—41no—
191C3#6423911yescharacteristic, direct product factor
192C2—21no—
193C2—121no—
194C2C2×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C3C2×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C23×C3
FrattiniC2C2×C3×GO+(4,𝔽2)
FittingC22×C33D4
socleC2×C33C2×D4
prime radicalC3×#3805C2
hypercenterC22×C3GO+(4,𝔽2)
hypocenterC32C3×#137
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×#627C2semidirect
3. C3×#3792C2semidirect
4. C3×#3776C2semidirect
5. C3×#3775C2semidirect
6. #64239C3direct (N × H)
18. C2×#627C2×C3semidirect
20. #3792C2×C3semidirect
21. #3776C2×C3semidirect
22. #3775C2×C3semidirect
87. C33#137semidirect
157. C32C3×#137semidirect
191. C3#64239direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#6272
C3×#38052
C3×#37922
C3×#37762
C3×#37752
#642393
C3×#1379