Finite Group Databasebeta

← #13137393

Subgroups — #13137393

Order 1728.

611 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 611 conjugacy classes of subgroups — too many to draw. Only its 99 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#11721385C212yesmaximal, complemented
2C2×#88432C214yesmaximal, complemented
3C4×#6180C211yesmaximal, characteristic, complemented
4#10590350—32nomaximal
5C2×C4⋊((C3⋊D3)⋊C3)—24no—
6C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
7C42×H(3,𝔽3)C2211yescharacteristic
8C2×#6180C2212yes—
9#88432—28no—
10#64083D312yescomplemented
11C3×#874—32no—
12C2×#3714—38no—
13C3×C4×C3⋊C8—32no—
14C3×C4×C3⋊C8—61no—
15C22×(C3⋊D3)⋊C3—24no—
16C2×C4×H(3,𝔽3)D412yes—
17C2×C4×H(3,𝔽3)C2311yescharacteristic
18C4⋊((C3⋊D3)⋊C3)—28no—
19C4⋊((C3⋊D3)⋊C3)—48no—
20#6180D414yescomplemented
21#5073—92no—
22#4695—91no—
23C2×C4⋊(C3⋊D3)—24no—
24C2×C4⋊(C3⋊D3)C2×D314yes—
25C2×C3×D12—34no—
26C2×C3×D12—64no—
27C32×C42—21no—
28C32×C42C2×D312yes—
29C2×C3×C3⋊C8—34no—
30C2×C3×C3⋊C8—62no—
31#3714—616no—
32C22×H(3,𝔽3)C2×D411yescharacteristic
33C2×(C3⋊D3)⋊C3—44no—
34C2×(C3⋊D3)⋊C3—48no—
35C4×H(3,𝔽3)D814yescomplemented
36C4×H(3,𝔽3)C2×D412yes—
37C3×#127—92no—
38C2×#213—98no—
39C2×D24—94no—
40#874—32no—
41#874—92no—
42C3×C4×C8—91no—
43C4×C3⋊C8—32no—
44C4×C3⋊C8—61no—
45C22×C3⋊D3—24no—
46C22×C3×D3—64no—
47C2×C32×C4—22no—
48C2×C32×C4—21no—
49C2×C32×C4#6714yes—
50C2×C32×C4C22×D312yes—
51C4⋊(C3⋊D3)—28no—
52C4⋊(C3⋊D3)—48no—
53C3×D12—68no—
54C3×D12—128no—
55C3×C3⋊C8—38no—
56C3×C3⋊C8—64no—
57#493—271noSylow
58C2×H(3,𝔽3)#12711yescharacteristic
59C2×H(3,𝔽3)C2×D812yes—
60(C3⋊D3)⋊C3—84no—
61C2×C3×D4—94no—
62C2×C3×D4—184no—
63C2×D12—34no—
64C2×D12—64no—
65C2×D12—94no—
66C2×D12—184no—
67C2×C3×C8—92no—
68C3×C42—32no—
69C3×C42—61no—
70C3×C42D3211yescharacteristic
71#213—1816no—
72C2×C3⋊C8—34no—
73C2×C3⋊C8—62no—
74D24—188no—
75C22×C32—21no—
76C22×C32C2×#6712yes—
77C2×C3⋊D3—48no—
78C2×C3⋊D3—44no—
79C2×C3×D3—128no—
80C2×C3×D3—124no—
81C32×C4—24no—
82C32×C4—22no—
83C32×C4#21318yescomplemented
84C32×C4D3×D414yes—
85C2×D8—274no—
86#127—92no—
87C4×C8—91no—
88H(3,𝔽3)#49311yescharacteristic, complemented, Sylow
89C23×C3—184no—
90C22×D3—64no—
91C22×D3—184no—
92C3×D4—188no—
93C3×D4—368no—
94C2×C3×C4—34no—
95C2×C3×C4—32no—
96C2×C3×C4—62no—
97C2×C3×C4—61no—
98C2×C3×C4#62512yes—
99C2×C3×C4C2×D3211yescharacteristic
100D12—68no—
101D12—128no—
102D12—188no—
103D12—368no—
104C3×C8—94no—
105C3⋊C8—38no—
106C3⋊C8—64no—
107C2×C32—22no—
108C2×C32—21no—
109C2×C32C2×#21314yes—
110C2×C32#94012yes—
111C3⋊D3—84no—
112C3×D3—244no—
113C2×D4—94no—
114C2×D4—184no—
115D8—548no—
116C2×C8—92no—
117C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
118C22×C3—32no—
119C22×C3—61no—
120C22×C3—364no—
121C22×C3—368no—
122C22×C3C2×#62511yescharacteristic
123C2×D3—128no—
124C2×D3—124no—
125C2×D3—364no—
126C2×D3—368no—
127C3×C4—34no—
128C3×C4—38no—
129C3×C4—62no—
130C3×C4—64no—
131C3×C4#371314yes—
132C3×C4#380212yes—
133C32—21no—
134C32#507312yescomplemented
135C23—184no—
136D4—188no—
137D4—368no—
138C2×C4#619812yes—
139C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
140C8—94no—
141C2×C3—34no—
142C2×C3—32no—
143C2×C3—62no—
144C2×C3—61no—
145C2×C3—724no—
146C2×C3C2×#371312yes—
147C2×C3#6391411yescharacteristic
148D3—244no—
149D3—724no—
150C22—368no—
151C22—364no—
152C22C2×#619811yescharacteristic
153C4#8843214yescomplemented
154C4#8865512yes—
155C3—32no—
156C3—61no—
157C3#1059035211yescharacteristic
158C2—724no—
159C2C2×#8843212yes—
160C2#1172114411yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#625
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#493
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721385C2semidirect
2. C2×#88432C2semidirect
3. C4×#6180C2semidirect
10. #64083D3semidirect
20. #6180D4semidirect
35. C4×H(3,𝔽3)D8semidirect
83. C32×C4#213semidirect
88. H(3,𝔽3)#493semidirect
134. C32#5073semidirect
153. C4#88432semidirect

Maximal subgroups

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

subgroupindex
#117213852
C2×#884322
C4×#61802
#105903503