Finite Group Databasebeta

← #11723339

Subgroups — #11723339

Order 864.

444 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 84 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#89014C212yesmaximal, complemented
2#88853C211yesmaximal, characteristic, complemented
3C3×C8×C3⋊D3C211yesmaximal, characteristic, complemented
4C32×D24C211yesmaximal, characteristic, complemented
5#88794C212yesmaximal, complemented
6#64121—31nomaximal
7#63807—34nomaximal
8#6308C2212yes—
9C3×C4×C3⋊D3C2211yescharacteristic
10C32×D12C2212yes—
11#6290—22no—
12D3×C3⋊Dic3—22no—
13C33×C8C2211yescharacteristic
14C3×C3⋊(C3⋊C8)C2211yescharacteristic
15#3830—32no—
16#3795—38no—
17C32×D8—31no—
18#3754—32no—
19#3746—31no—
20#3746—34no—
21#3746—64no—
22C8×C3⋊D3D311yescharacteristic, complemented
23C3×D24D314yescomplemented
24C3×D3×C8—34no—
25#3716—38no—
26C2×C3×C3⋊D3D411yescharacteristic
27C2×C32×D3—22no—
28C33×C4C2311yescharacteristic
29C3⋊(C3⋊Dic3)—22no—
30C3×C3⋊Dic3D411yescharacteristic
31#912—94no—
32#904—91no—
33C32×D4—32no—
34#638—62no—
35C2×C3⋊Dic3—62no—
36C4×C3⋊D3C2×D311yescharacteristic
37#634—32no—
38#634—38no—
39#634—68no—
40C3×D12C2×D318yes—
41C3×C4×D3—34no—
42#625—68no—
43D3×Dic3—68no—
44C32×C8—24no—
45C32×C8C2×D314yes—
46C32×C8C2×D311yescharacteristic
47C3⋊(C3⋊C8)C2×D311yescharacteristic
48C3×C3⋊C8—34no—
49C2×C33C2×D411yescharacteristic
50C3×C3⋊D3—21no—
51C32×D3—42no—
52#237—98no—
53#235—92no—
54C3×D8—34no—
55C2×C3×C8—91no—
56#214—98no—
57Dic12—94no—
58Dic12—91no—
59Dic12—184no—
60D24C3⋊D311yescharacteristic, complemented
61#204—92no—
62D3×C8—34no—
63C22×C32—62no—
64C2×C3⋊D3D1211yescharacteristic
65C2×C3×D3—28no—
66C2×C3×D3—34no—
67C32×C4—24no—
68C32×C4C22×D314yes—
69C32×C4C22×D311yescharacteristic
70C3⋊Dic3—62no—
71C3⋊Dic3—68no—
72C3⋊Dic3—128no—
73C3⋊Dic3D1211yescharacteristic
74C3×Dic3—34no—
75#135—271noSylow
76C33#13511yescharacteristic, complemented, Sylow
77C3×D4—38no—
78C2×C3×C4—91no—
79#67—182no—
80#67—188no—
81C2×Dic3—188no—
82D12C2×C3⋊D312yes—
83C4×D3—34no—
84C4×D3—182no—
85Dic6—98no—
86Dic6—92no—
87Dic6—188no—
88C3×C8—24no—
89C3×C8—91no—
90C3×C8D3214yes—
91C3×C8C2×C3⋊D311yescharacteristic
92C3⋊C8—34no—
93C2×C32—24no—
94C2×C32—122no—
95C2×C32C2×D1211yescharacteristic
96C2×C32D3×D414yes—
97C3⋊D3—21no—
98C3×D3—48no—
99C3×D3—64no—
100#41—272no—
101Q16—271no—
102SD16—272no—
103D8—31no—
104C2×C8—91no—
105C22×C3—68no—
106C22×C3—91no—
107C2×D3—22no—
108C2×D3—34no—
109C3×C4—24no—
110C3×C4—91no—
111C3×C4C2×D3214yes—
112C3×C4C22×C3⋊D311yescharacteristic
113Dic3—34no—
114Dic3—188no—
115Dic3—182no—
116Dic3—368no—
117C32—24no—
118C32#90411yescharacteristic, complemented
119C32#91214yescomplemented
120Q8—272no—
121D4—32no—
122D4—542no—
123C2×C4—91no—
124C2×C4—542no—
125C8—91no—
126C8D3×C3⋊D311yescharacteristic
127C2×C3—24no—
128C2×C3—128no—
129C2×C3—181no—
130C2×C3D3×D1214yes—
131C2×C3D4×C3⋊D311yescharacteristic
132D3—42no—
133D3—64no—
134C22—62no—
135C22—91no—
136C4—91no—
137C4—542no—
138C4C2×D3×C3⋊D311yescharacteristic
139C3—24no—
140C3#6380714yescomplemented
141C3#6412111yescharacteristic, complemented
142C2—122no—
143C2—181no—
144C2C3⋊D3×D1211yescharacteristic
subgroupG/H
centerC2C3⋊D3×D12
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. #89014C2semidirect
2. #88853C2semidirect
3. C3×C8×C3⋊D3C2semidirect
4. C32×D24C2semidirect
5. #88794C2semidirect
22. C8×C3⋊D3D3semidirect
23. C3×D24D3semidirect
60. D24C3⋊D3semidirect
76. C33#135semidirect
118. C32#904semidirect
119. C32#912semidirect
140. C3#63807semidirect
141. C3#64121semidirect

Maximal subgroups

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

subgroupindex
#890142
#888532
C3×C8×C3⋊D32
C32×D242
#887942
#641213
#638073