Finite Group Databasebeta

← #11724352

Subgroups — #11724352

Order 864.

230 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 35 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×#6296C211yesmaximal, characteristic, complemented
2#89046C211yesmaximal, characteristic
3C3×#3792C211yesmaximal, characteristic, complemented
4#88943C212yesmaximal, complemented
5#88942C212yesmaximal, complemented
6#64239—31nomaximal
7C3×#638—21no—
8C2×C3×C3⋊Dic3—21no—
9C2×C3×C3⋊Dic3C2211yescharacteristic
10#6296—21no—
11#6296C2212yes—
12#6296C2211yescharacteristic
13#6295C2211yescharacteristic
14#6294—21no—
15C3×#622C2212yes—
16C2×#627—31no—
17C2×#627—61no—
18#3805—31no—
19#3804—61no—
20#3792D311yescharacteristic, complemented
21#3776—32no—
22#3775—32no—
23C22×C33D411yescharacteristic
24C2×C3×C3⋊D3—21no—
25C3×C3⋊Dic3—21no—
26C3×C3⋊Dic3—22no—
27C3×C3⋊Dic3D411yescharacteristic
28C3×C3⋊Dic3C2311yescharacteristic
29#909—91nomaximal
30#638—21no—
31C2×C3⋊Dic3—21no—
32C2×C3⋊Dic3C2×D311yescharacteristic
33C3×#67—61no—
34C3×#67—61no—
35C3×#67—121no—
36C2×C3×Dic3—61no—
37C2×C3×Dic3—61no—
38C2×C3×Dic3—61no—
39C2×C3×Dic3—61no—
40C2×C3×Dic3—121no—
41#627—32no—
42#627—31no—
43#627—61no—
44#627—62no—
45#627—62no—
46#627—61no—
47#626—61no—
48#625—31no—
49#624—61no—
50D3×Dic3—61no—
51D3×Dic3—61no—
52#622C2×D312yes—
53C2×C33—21no—
54C2×C33C2×D411yescharacteristic
55C3×C3⋊D3—41no—
56#237—181no—
57#235—91no—
58C2×Dic6—91no—
59C2×Dic6—181no—
60C3×M16—91no—
61Dic12—92no—
62#204—92no—
63C22×C32—21no—
64C22×C32—21no—
65C22×C32—41no—
66C22×C32—41no—
67C22×C32D1211yescharacteristic
68C2×C3⋊D3—21no—
69C2×C3×D3—61no—
70C2×C3×D3—61no—
71C2×C3×D3—121no—
72C3⋊Dic3—21no—
73C3⋊Dic3—22no—
74C3⋊Dic3D1211yescharacteristic
75C3⋊Dic3C22×D311yescharacteristic
76C3×Dic3—61no—
77C3×Dic3—61no—
78C3×Dic3—61no—
79C3×Dic3—61no—
80C3×Dic3—61no—
81C3×Dic3—62no—
82C3×Dic3—61no—
83C3×Dic3—62no—
84C3×Dic3—122no—
85C3×Dic3—121no—
86#137—271noSylow
87C33#13711yescharacteristic, complemented, Sylow
88C3×D4—181no—
89C2×C3×C4—91no—
90C2×C3×C4—181no—
91#67—61no—
92#67—61no—
93#67—121no—
94#67—181no—
95C2×Dic3—61no—
96C2×Dic3—61no—
97C2×Dic3—61no—
98C2×Dic3—61no—
99C2×Dic3—121no—
100C2×Dic3—181no—
101D12—91no—
102C4×D3—181no—
103C4×D3—181no—
104Dic6—92no—
105Dic6—91no—
106Dic6—182no—
107Dic6—181no—
108Dic6—181no—
109Dic6—182no—
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×C32C2×D1211yescharacteristic
121C3⋊D3—41no—
122C3×D3—121no—
123C3×D3—121no—
124C3×D3—241no—
125#41—271no—
126C2×Q8—271no—
127Q16—272no—
128SD16—272no—
129M16—91no—
130C22×C3—21no—
131C22×C3—21no—
132C22×C3—41no—
133C22×C3—41no—
134C22×C3—181no—
135C22×C3GO+(4,𝔽2)11yescharacteristic
136C2×D3—61no—
137C2×D3—61no—
138C2×D3—121no—
139C3×C4—91no—
140C3×C4—91no—
141C3×C4—182no—
142C3×C4—181no—
143Dic3—61no—
144Dic3—61no—
145Dic3—61no—
146Dic3—61no—
147Dic3—62no—
148Dic3—61no—
149Dic3—62no—
150Dic3—61no—
151Dic3—122no—
152Dic3—121no—
153C32—21no—
154C32—21no—
155C32—41no—
156C32—41no—
157C32#90911yescharacteristic, complemented
158Q8—272no—
159Q8—271no—
160Q8—541no—
161D4—181no—
162D4—271no—
163C2×C4—91no—
164C2×C4—181no—
165C2×C4—541no—
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—361no—
177C2×C3C2×GO+(4,𝔽2)11yescharacteristic
178D3—121no—
179D3—121no—
180D3—241no—
181C22—181no—
182C22#157⋊D311yescharacteristic
183C4—91no—
184C4—91no—
185C4—182no—
186C4—181no—
187C3—21no—
188C3—21no—
189C3—41no—
190C3—41no—
191C3#6423911yescharacteristic, complemented
192C2—21no—
193C2—361no—
194C2C2×#157⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C2×#157⋊D3
commutator, cosocleC3×C3⋊Dic3C23
FittingC22×C33D4
socleC2×C33C2×D4
prime radical#89046C2
hypercenterC22#157⋊D3
hypocenterC33#137
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#6296C2semidirect
3. C3×#3792C2semidirect
4. #88943C2semidirect
5. #88942C2semidirect
20. #3792D3semidirect
87. C33#137semidirect
157. C32#909semidirect
191. C3#64239semidirect

Maximal subgroups

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

subgroupindex
C2×#62962
#890462
C3×#37922
#889432
#889422
#642393
#9099