Finite Group Databasebeta

← #11723335

Subgroups — #11723335

Order 864.

452 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 92 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#89022C211yesmaximal, characteristic, complemented
2#89021C211yesmaximal, characteristic, complemented
3#88851C211yesmaximal, characteristic, complemented
4C3×#3744C211yesmaximal, characteristic, complemented
5C32×D3×C8C211yesmaximal, characteristic, complemented
6#88800C211yesmaximal, characteristic, complemented
7#88791C211yesmaximal, characteristic, complemented
8#64113—31nomaximal
9#63806—34nomaximal
10C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11#6308C2211yescharacteristic
12C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C3×#634C2211yescharacteristic
14C32×C4×D3C2211yescharacteristic
15#6291—21no—
16#6290—21no—
17#6289—21no—
18Dic3×C3⋊D3—21no—
19C33×C8C2211yescharacteristic
20C32×C3⋊C8C2211yescharacteristic
21#3828—31no—
22#3828—31no—
23#3799—34no—
24#3795—34no—
25C2×C32×C8—31no—
26#3746—31no—
27#3745—31no—
28#3744—31no—
29#3744—34no—
30#3744—64no—
31#3744D311yescharacteristic, complemented
32C3×#204—34no—
33C3×D3×C8D314yescomplemented
34#3719—34no—
35#3714—34no—
36C2×C3⋊(C3⋊D3)—21no—
37C2×C3×C3⋊D3—21no—
38C2×C32×D3D411yescharacteristic
39C33×C4C2311yescharacteristic
40C3⋊(C3⋊Dic3)—21no—
41C3×C3⋊Dic3—21no—
42C32×Dic3D411yescharacteristic
43#916—91no—
44#904—94no—
45C2×C32×C4—31no—
46#638—61no—
47#638—61no—
48C4⋊(C3⋊D3)—31no—
49C4⋊(C3⋊D3)—34no—
50C4⋊(C3⋊D3)—64no—
51C4⋊(C3⋊D3)C2×D311yescharacteristic
52C4×C3⋊D3—61no—
53C4×C3⋊D3—61no—
54#634—31no—
55#634—34no—
56#634—64no—
57#634C2×D311yescharacteristic
58C3×D12—34no—
59C3×C4×D3C2×D314yes—
60C3×Dic6—34no—
61#626—64no—
62#625—64no—
63#624—64no—
64D3×Dic3—64no—
65C32×C8—24no—
66C32×C8—31no—
67C32×C8C2×D314yes—
68C32×C8C2×D311yescharacteristic
69C3×C3⋊C8C2×D314yes—
70C2×C33C2×D411yescharacteristic
71C3⋊(C3⋊D3)—41no—
72C3×C3⋊D3—41no—
73C32×D3—21no—
74#239—91no—
75#237—91no—
76#235—94no—
77#235—94no—
78C3×SD16—91no—
79C2×C3×C8—34no—
80#216—91no—
81#213—91no—
82Dic12—94no—
83D24—94no—
84#204—34no—
85#204—94no—
86#204—91no—
87#204—184no—
88D3×C8C3⋊D311yescharacteristic, complemented
89C22×C32—31no—
90C2×C3⋊D3—21no—
91C2×C3⋊D3—61no—
92C2×C3⋊D3—64no—
93C2×C3⋊D3—124no—
94C2×C3×D3—64no—
95C2×C3×D3D1214yes—
96C32×C4—24no—
97C32×C4—31no—
98C32×C4C22×D314yes—
99C32×C4C22×D311yescharacteristic
100C3⋊Dic3—21no—
101C3⋊Dic3—61no—
102C3⋊Dic3—64no—
103C3⋊Dic3—124no—
104C3×Dic3—64no—
105C3×Dic3D1214yes—
106#135—271noSylow
107C33#13511yescharacteristic, complemented, Sylow
108C3×Q8—91no—
109C3×D4—91no—
110C2×C3×C4—34no—
111#67—184no—
112#67—184no—
113#67—181no—
114C2×Dic3—181no—
115D12—34no—
116D12—94no—
117D12—91no—
118D12—184no—
119D12—181no—
120C4×D3—184no—
121C4×D3—184no—
122C4×D3—181no—
123C4×D3C2×C3⋊D311yescharacteristic
124Dic6—34no—
125Dic6—94no—
126Dic6—91no—
127Dic6—184no—
128C3×C8—24no—
129C3×C8—34no—
130C3×C8D3214yes—
131C3×C8C2×C3⋊D311yescharacteristic
132C3⋊C8C2×C3⋊D311yescharacteristic
133C2×C32—24no—
134C2×C32—61no—
135C2×C32C2×D1214yes—
136C2×C32D3×D411yescharacteristic
137C3⋊D3—41no—
138C3⋊D3—121no—
139C3⋊D3—124no—
140C3⋊D3—244no—
141C3×D3—24no—
142C3×D3—124no—
143#41—271no—
144#41—271no—
145Q16—271no—
146SD16—91no—
147SD16—271no—
148D8—271no—
149C2×C8—31no—
150C22×C3—34no—
151C22×C3—181no—
152C2×D3—64no—
153C2×D3—184no—
154C2×D3—181no—
155C2×D3—364no—
156C2×D3C4⋊(C3⋊D3)11yescharacteristic
157C3×C4—24no—
158C3×C4—34no—
159C3×C4—181no—
160C3×C4C2×D3214yes—
161C3×C4C22×C3⋊D311yescharacteristic
162Dic3—64no—
163Dic3—184no—
164Dic3—181no—
165Dic3—364no—
166Dic3C4⋊(C3⋊D3)11yescharacteristic
167C32—24no—
168C32#90414yescomplemented
169C32#91611yescharacteristic, complemented
170Q8—91no—
171Q8—271no—
172D4—91no—
173D4—271no—
174D4—541no—
175D4—541no—
176C2×C4—31no—
177C2×C4—541no—
178C2×C4—541no—
179C8—31no—
180C8D3×C3⋊D311yescharacteristic
181C2×C3—24no—
182C2×C3—64no—
183C2×C3—361no—
184C2×C3D3×D1214yes—
185C2×C3C2×C4⋊(C3⋊D3)11yescharacteristic
186D3—21no—
187D3—124no—
188D3—364no—
189D3—361no—
190D3—724no—
191C22—31no—
192C22—181no—
193C22—541no—
194C4—31no—
195C4—181no—
196C4—541no—
197C4C2×D3×C3⋊D311yescharacteristic
198C3—24no—
199C3#6380614yescomplemented
200C3#6411311yescharacteristic, complemented
201C2—61no—
202C2—361no—
203C2—1081no—
204C2D3×C4⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC2D3×C4⋊(C3⋊D3)
commutator, cosocleC33×C4C23
FrattiniC4C2×D3×C3⋊D3
FittingC33×C8C22
socleC2×C33C2×D4
hypercenterC8D3×C3⋊D3
hypocenterC33#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #89022C2semidirect
2. #89021C2semidirect
3. #88851C2semidirect
4. C3×#3744C2semidirect
5. C32×D3×C8C2semidirect
6. #88800C2semidirect
7. #88791C2semidirect
31. #3744D3semidirect
33. C3×D3×C8D3semidirect
88. D3×C8C3⋊D3semidirect
107. C33#135semidirect
168. C32#904semidirect
169. C32#916semidirect
199. C3#63806semidirect
200. C3#64113semidirect

Maximal subgroups

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

subgroupindex
#890222
#890212
#888512
C3×#37442
C32×D3×C82
#888002
#887912
#641133
#638063