Finite Group Databasebeta

← #88671

Subgroups — #88671

Order 432.

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

normal subgroups only

This group has 156 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#6235C211yesmaximal, characteristic, complemented
2#638⋊C3C211yesmaximal, characteristic, complemented
3C2×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4#6200C211yesmaximal, characteristic, complemented
5#6199C211yesmaximal, characteristic, complemented
6((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
7#6196C211yesmaximal, characteristic, complemented
8#3805—31nomaximal
9#3804—31nomaximal
10C22×H(3,𝔽3)C2211yescharacteristic
11C2×#272C2211yescharacteristic
12C2×(C3⋊D3)⋊C3C2211yescharacteristic
13#1084C2211yescharacteristic
14(C3⋊Dic3)⋊C3C2211yescharacteristic
15(C3⋊Dic3)⋊C3C2211yescharacteristic
16(C3⋊Dic3)⋊C3C2211yescharacteristic
17#638D311yescharacteristic, complemented
18C2×C3⋊Dic3D311yescharacteristic, complemented
19C3×#67—31no—
20C3×#67—31no—
21C3×#67—31no—
22C3×#67—61no—
23C2×C3×Dic3—31no—
24#627—31no—
25#627—31no—
26#626—31no—
27#625—31no—
28#624—31no—
29D3×Dic3—31no—
30D3×Dic3—31no—
31D3×Dic3—31no—
32C2×H(3,𝔽3)—21no—
33C2×H(3,𝔽3)C2311yescharacteristic
34#272—21no—
35(C3⋊D3)⋊C3—21no—
36#237—91no—
37#237—91no—
38#235—91no—
39C22×C32—21no—
40C22×C32C2×D311yescharacteristic
41C22×C32C2×D311yescharacteristic
42C2×C3⋊D3C2×D311yescharacteristic
43C2×C3×D3—31no—
44C2×C3×D3—31no—
45C2×C3×D3—31no—
46C2×C3×D3—61no—
47C3⋊Dic3C2×D311yescharacteristic
48C3⋊Dic3C2×D311yescharacteristic
49C3⋊Dic3C2×D311yescharacteristic
50C3×Dic3—31no—
51C3×Dic3—31no—
52C3×Dic3—31no—
53C3×Dic3—31no—
54C3×Dic3—31no—
55C3×Dic3—61no—
56H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
57C3×D4—91no—
58C3×D4—91no—
59C2×C3×C4—91no—
60#67—31no—
61#67—31no—
62#67—31no—
63#67—61no—
64#67—91no—
65#67—91no—
66#67—91no—
67C2×Dic3—31no—
68C2×Dic3—91no—
69C2×Dic3—91no—
70C2×Dic3—91no—
71D12—91no—
72C4×D3—91no—
73C4×D3—91no—
74C4×D3—91no—
75C4×D3—91no—
76Dic6—91no—
77Dic6—91no—
78Dic6—91no—
79C2×C32—21no—
80C2×C32—21no—
81C2×C32—21no—
82C2×C32—41no—
83C2×C32C22×D311yescharacteristic
84C2×C32C22×D311yescharacteristic
85C3⋊D3—21no—
86C3×D3—61no—
87C3×D3—61no—
88C3×D3—61no—
89C3×D3—121no—
90#41—271noSylow
91C22×C3—31no—
92C22×C3—31no—
93C22×C3—61no—
94C22×C3—91no—
95C22×C3—91no—
96C22×C3D3211yescharacteristic
97C2×D3—31no—
98C2×D3—31no—
99C2×D3—31no—
100C2×D3—61no—
101C2×D3—91no—
102C3×C4—91no—
103C3×C4—91no—
104C3×C4—91no—
105C3×C4—91no—
106Dic3—31no—
107Dic3—31no—
108Dic3—31no—
109Dic3—31no—
110Dic3—31no—
111Dic3—61no—
112Dic3—91no—
113Dic3—91no—
114Dic3—91no—
115C32—21no—
116C32#23511yescharacteristic, complemented
117C32#23711yescharacteristic, complemented
118Q8—271no—
119D4—91no—
120D4—91no—
121D4—271no—
122C2×C4—91no—
123C2×C4—271no—
124C2×C4—271no—
125C2×C3—21no—
126C2×C3—31no—
127C2×C3—31no—
128C2×C3—61no—
129C2×C3—61no—
130C2×C3—61no—
131C2×C3—121no—
132C2×C3—181no—
133C2×C3—181no—
134C2×C3C2×D3211yescharacteristic
135D3—61no—
136D3—61no—
137D3—61no—
138D3—121no—
139D3—181no—
140C22—91no—
141C22—91no—
142C22((C3⋊D3)⋊C3)⋊C211yescharacteristic
143C4—91no—
144C4—91no—
145C4—91no—
146C4—91no—
147C3—31no—
148C3—31no—
149C3—61no—
150C3#380411yescharacteristic
151C2—21no—
152C2—181no—
153C2—181no—
154C2C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC2C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C23
Frattini, socleC2×C3C2×D32
FittingC22×H(3,𝔽3)C22
hypercenterC22((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#41
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #6235C2semidirect
2. #638⋊C3C2semidirect
3. C2×(C3⋊Dic3)⋊C3C2semidirect
4. #6200C2semidirect
5. #6199C2semidirect
6. ((C3⋊D3)⋊C3)⋊C4C2semidirect
7. #6196C2semidirect
17. #638D3semidirect
18. C2×C3⋊Dic3D3semidirect
56. H(3,𝔽3)#41semidirect
116. C32#235semidirect
117. C32#237semidirect

Maximal subgroups

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

subgroupindex
#62352
#638⋊C32
C2×(C3⋊Dic3)⋊C32
#62002
#61992
((C3⋊D3)⋊C3)⋊C42
#61962
#38053
#38043