Finite Group Databasebeta

← #13135761

Subgroups — #13135761

Order 1728.

330 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 75 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#11721716C211yesmaximal, characteristic
2H(3,𝔽3)×#97C211yesmaximal, characteristic
3#11720520C211yesmaximal, characteristic
4C3×#4465—34nomaximal
5C2×C8×H(3,𝔽3)—21no—
6C42×H(3,𝔽3)C2211yescharacteristic
7C4⋊#1084C411yescharacteristic
8C4⋊#1084C411yescharacteristic
9#88540—21no—
10#88540—21no—
11C2×#6180—21no—
12C3×#870—34no—
13C32×#97D314yes—
14C3×#803—34no—
15C2×C4×H(3,𝔽3)C2×C411yescharacteristic
16C2×C4×H(3,𝔽3)D411yescharacteristic
17C2×C4×H(3,𝔽3)D411yescharacteristic
18C2×#1084—21no—
19C2×#1084—21no—
20C8×H(3,𝔽3)—22no—
21#6180—22no—
22C3×#330—91no—
23#4465—34no—
24C2×C32×C8—24no—
25C32×C42C2×D314yes—
26C3×C4⋊Dic3—34no—
27C3×C4⋊Dic3—34no—
28C3×Dic3⋊C4—64no—
29C3×Dic3⋊C4—64no—
30C2×C3×C3⋊C8—64no—
31C22×H(3,𝔽3)C22⋊C411yescharacteristic
32C4×H(3,𝔽3)—21no—
33C4×H(3,𝔽3)—21no—
34C4×H(3,𝔽3)—21no—
35#1084—41no—
36#1084—41no—
37C3×#125—91no—
38#870—34no—
39C3×#97—34no—
40C3×#97—91no—
41C3×#97C3⋊D311yescharacteristic
42#803—34no—
43C2×C32×C4C4×D314yes—
44C2×C32×C4D1214yes—
45C2×C32×C4#6714yes—
46C2×C3×Dic3—64no—
47C2×C3×Dic3—64no—
48C32×C8—28no—
49C3×C3⋊C8—68no—
50#330—91noSylow
51C2×H(3,𝔽3)#10111yescharacteristic
52C2×H(3,𝔽3)#10411yescharacteristic
53C2×H(3,𝔽3)#10411yescharacteristic
54C2×C3×C8—21no—
55C2×C3×C8—64no—
56C2×C3×C8—181no—
57C3×C4⋊C4—91no—
58C3×C4⋊C4—91no—
59C3×C4⋊C4—181no—
60C3×C4⋊C4—181no—
61C3×C42—34no—
62C3×C42C2×C3⋊D311yescharacteristic
63C4⋊Dic3—34no—
64C4⋊Dic3—34no—
65Dic3⋊C4—64no—
66Dic3⋊C4—64no—
67C2×C3⋊C8—64no—
68C22×C32(C2×D3)⋊C414yes—
69C32×C4—24no—
70C32×C4—24no—
71C32×C4—24no—
72C3×Dic3—124no—
73C3×Dic3—124no—
74#125—91no—
75#97—91no—
76#97#27211yescharacteristic
77H(3,𝔽3)#33011yescharacteristic, complemented, Sylow
78C2×C3×C4—34no—
79C2×C3×C4—34no—
80C2×C3×C4—34no—
81C2×C3×C4—181no—
82C2×C3×C4—181no—
83C2×C3×C4C4×C3⋊D311yescharacteristic
84C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
85C2×C3×C4#63811yescharacteristic
86C2×Dic3—64no—
87C2×Dic3—64no—
88C3×C8—22no—
89C3×C8—68no—
90C3×C8—182no—
91C3⋊C8—68no—
92C2×C32#80514yes—
93C2×C32#82414yes—
94C2×C32#82514yes—
95C2×C8—21no—
96C2×C8—181no—
97C4⋊C4—91no—
98C4⋊C4—91no—
99C4⋊C4—181no—
100C4⋊C4—181no—
101C42C2×#27211yescharacteristic
102C22×C3—34no—
103C22×C3#375211yescharacteristic
104C3×C4—21no—
105C3×C4—21no—
106C3×C4—21no—
107C3×C4—64no—
108C3×C4—64no—
109C3×C4—64no—
110C3×C4—361no—
111C3×C4—361no—
112Dic3—124no—
113Dic3—124no—
114C32#446514yescomplemented
115C2×C4—181no—
116C2×C4—181no—
117C2×C4C4×#27211yescharacteristic
118C2×C4C4⋊#27211yescharacteristic
119C2×C4#623511yescharacteristic
120C8—22no—
121C8—182no—
122C2×C3—34no—
123C2×C3—34no—
124C2×C3—34no—
125C2×C3#6363211yescharacteristic
126C2×C3#6365111yescharacteristic
127C2×C3#6365211yescharacteristic
128C22#8854211yescharacteristic
129C4—21no—
130C4—21no—
131C4—21no—
132C4—361no—
133C4—361no—
134C3—34no—
135C3#1058933111yescharacteristic
136C2#1172052211yescharacteristic
137C2#1172056111yescharacteristic
138C2#1172056211yescharacteristic
subgroupG/H
center, socleC22×C3#3752
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC3×C42C2×C3⋊D3
FittingH(3,𝔽3)×#97C2
prime radicalC22×H(3,𝔽3)C22⋊C4
cosocleC42×H(3,𝔽3)C22
hypercenterC3×#97C3⋊D3
hypocenterH(3,𝔽3)#330
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#97

Complements

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

NH ≅ G/Ntype
77. H(3,𝔽3)#330semidirect
114. C32#4465semidirect

Maximal subgroups

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

subgroupindex
#117217162
H(3,𝔽3)×#972
#117205202
C3×#44653