Finite Group Databasebeta

← #11724326

Subgroups — #11724326

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
1#89046C211yesmaximal, characteristic
2C2×C3×#627C211yesmaximal, characteristic, complemented
3#88993C211yesmaximal, characteristic, complemented
4#88938C212yesmaximal, complemented
5#88936C212yesmaximal, complemented
6#64239—31nomaximal
7C3×#638—21no—
8C2×C3×C3⋊Dic3C2211yescharacteristic
9C2×C32×Dic3—21no—
10#6296C2211yescharacteristic
11#6295C2211yescharacteristic
12#6294—21no—
13C3×#627—21no—
14C3×#627C2212yes—
15C3⋊#622C2212yes—
16C2×C3×Dic6—61no—
17C2×#627D311yescharacteristic, complemented
18#3805—31no—
19#3804—61no—
20#3792—31no—
21#3776—32no—
22#3775—32no—
23C22×C33D411yescharacteristic
24C2×C3×C3⋊D3—21no—
25C3×C3⋊Dic3—21no—
26C3×C3⋊Dic3D411yescharacteristic
27C3×C3⋊Dic3C2311yescharacteristic
28C32×Dic3—22no—
29#942—91nomaximal
30C2×C32×C4—61no—
31#638—21no—
32C2×C3⋊Dic3C2×D311yescharacteristic
33C3×#67—61no—
34C3×#67—61no—
35C3×#67—121no—
36C2×C3×Dic3—21no—
37C2×C3×Dic3—21no—
38C2×C3×Dic3—41no—
39C2×C3×Dic3—61no—
40C2×C3×Dic3—61no—
41C3×Dic6—62no—
42C3×Dic6—62no—
43#627—21no—
44#627—31no—
45#627—61no—
46#627C2×D312yes—
47#626—61no—
48#625—31no—
49#624—61no—
50D3×Dic3—61no—
51D3×Dic3—61no—
52#622—32no—
53C2×C33—21no—
54C2×C33C2×D411yescharacteristic
55C3×C3⋊D3—41no—
56C2×C3×Q8—91no—
57#237—181no—
58#235—91no—
59C2×Dic6—61no—
60#216—92no—
61#215—92no—
62#208—91no—
63C22×C32—21no—
64C22×C32—21no—
65C22×C32—41no—
66C22×C32—41no—
67C22×C32#6711yescharacteristic
68C2×C3⋊D3—21no—
69C2×C3×D3—61no—
70C2×C3×D3—61no—
71C2×C3×D3—121no—
72C32×C4—62no—
73C3⋊Dic3—21no—
74C3⋊Dic3#6711yescharacteristic
75C3⋊Dic3C22×D311yescharacteristic
76C3×Dic3—22no—
77C3×Dic3—22no—
78C3×Dic3—42no—
79C3×Dic3—61no—
80C3×Dic3—61no—
81C3×Dic3—61no—
82C3×Dic3—61no—
83C3×Dic3—61no—
84C3×Dic3—61no—
85C3×Dic3—121no—
86#137—271noSylow
87C33#13711yescharacteristic, complemented, Sylow
88C3×Q8—92no—
89C3×Q8—181no—
90C3×D4—181no—
91C2×C3×C4—61no—
92C2×C3×C4—61no—
93C2×C3×C4—91no—
94C2×C3×C4—121no—
95#67—61no—
96#67—61no—
97#67—121no—
98#67—181no—
99C2×Dic3—21no—
100C2×Dic3—61no—
101C2×Dic3—61no—
102C2×Dic3—181no—
103D12—91no—
104C4×D3—181no—
105C4×D3—181no—
106Dic6—62no—
107Dic6—62no—
108Dic6—91no—
109Dic6—181no—
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×#6711yescharacteristic
121C3⋊D3—41no—
122C3×D3—121no—
123C3×D3—121no—
124C3×D3—241no—
125#41—271no—
126C2×Q8—91no—
127Q16—272no—
128SD16—272no—
129M16—271no—
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—62no—
140C3×C4—62no—
141C3×C4—91no—
142C3×C4—91no—
143C3×C4—122no—
144C3×C4—181no—
145Dic3—22no—
146Dic3—61no—
147Dic3—61no—
148Dic3—61no—
149Dic3—61no—
150Dic3—61no—
151Dic3—61no—
152Dic3—121no—
153C32—21no—
154C32—21no—
155C32—41no—
156C32—41no—
157C32#94211yescharacteristic, complemented
158Q8—92no—
159Q8—181no—
160Q8—271no—
161D4—181no—
162D4—271no—
163C2×C4—61no—
164C2×C4—91no—
165C2×C4—541no—
166C8—272no—
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—
182C22D32⋊D311yescharacteristic
183C4—62no—
184C4—91no—
185C4—91no—
186C4—181no—
187C3—21no—
188C3—21no—
189C3—41no—
190C3—41no—
191C3#6423911yescharacteristic, complemented
192C2—21no—
193C2—361no—
194C2C2×D32⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32⋊D3
commutator, cosocleC3×C3⋊Dic3C23
FittingC22×C33D4
socleC2×C33C2×D4
prime radical#89046C2
hypercenterC22D32⋊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
2. C2×C3×#627C2semidirect
3. #88993C2semidirect
4. #88938C2semidirect
5. #88936C2semidirect
17. C2×#627D3semidirect
87. C33#137semidirect
157. C32#942semidirect
191. C3#64239semidirect

Maximal subgroups

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

subgroupindex
#890462
C2×C3×#6272
#889932
#889382
#889362
#642393
#9429