Finite Group Databasebeta

← #13147319

Subgroups — #13147319

Order 1728.

483 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 483 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×C4×#1088C212yesmaximal
2C42×#272C211yesmaximal, characteristic
3#11720787C214yesmaximal
4C22×#1088C2214yes—
5C2×C4×#272C2212yes—
6C2×C4×#272C2211yescharacteristic
7C4×#1088C418yescomplemented
8C42×H(3,𝔽3)C411yescharacteristic, complemented
9C4×#1084C411yescharacteristic, complemented
10C4×#1084C412yescomplemented
11C3×C42×D3—62no—
12C22×#272C2311yescharacteristic
13C2×#1088—28no—
14C2×#1088C2×C418yes—
15C2×C4×H(3,𝔽3)C2×C412yes—
16C2×C4×H(3,𝔽3)C2×C411yescharacteristic
17C2×#1084C2×C412yes—
18C2×#1084C2×C411yescharacteristic
19C4×#272—24no—
20C4×#272C2×C414yes—
21C3×#376—91nomaximal
22C2×C3×C4×D3—62no—
23C2×C3×C4×D3—64no—
24C32×C42—22no—
25C3×C4×Dic3—64no—
26C3×C4×Dic3—62no—
27C22×H(3,𝔽3)C22×C411yescharacteristic
28C2×#272C22×C412yes—
29C2×#272#4114yes—
30#1088—28no—
31C4×H(3,𝔽3)—22no—
32C4×H(3,𝔽3)C4212yes—
33#1084—22no—
34#1084C4212yes—
35C2×C3×C42—92no—
36C2×C3×C42—91no—
37C42×D3—62no—
38C3×#95—94no—
39C22×C3×D3—62no—
40C2×C32×C4—24no—
41C2×C32×C4—22no—
42C2×C3×Dic3—62no—
43C2×C3×Dic3—64no—
44C3×C4×D3—68no—
45C3×C4×D3—616no—
46#376—91noSylow
47C2×H(3,𝔽3)C2×C4211yescharacteristic
48C2×H(3,𝔽3)#11712yes—
49#272#11714yes—
50C22×C3×C4—91no—
51C22×C3×C4—94no—
52C22×C3×C4—92no—
53C2×C4×D3—62no—
54C2×C4×D3—64no—
55C3×C42—62no—
56C3×C42—98no—
57C3×C42—92no—
58C3×C42—91no—
59C3×C42#15711yescharacteristic
60C4×Dic3—64no—
61C4×Dic3—62no—
62C22×C32—22no—
63C2×C3×D3—64no—
64C2×C3×D3—68no—
65C32×C4—28no—
66C32×C4—24no—
67C3×Dic3—64no—
68C3×Dic3—68no—
69C2×C42—92no—
70C2×C42—91no—
71#95—94no—
72H(3,𝔽3)#37611yescharacteristic, complemented, Sylow
73C23×C3—91no—
74C22×D3—62no—
75C2×C3×C4—64no—
76C2×C3×C4—62no—
77C2×C3×C4—91no—
78C2×C3×C4—94no—
79C2×C3×C4—98no—
80C2×C3×C4—92no—
81C2×C3×C4—184no—
82C2×C3×C4—188no—
83C2×C3×C4C2×#15712yes—
84C2×C3×C4C2×#15711yescharacteristic
85C2×Dic3—62no—
86C2×Dic3—64no—
87C4×D3—68no—
88C4×D3—616no—
89C2×C32—22no—
90C2×C32—24no—
91C3×D3—68no—
92C22×C4—91no—
93C22×C4—94no—
94C22×C4—92no—
95C42—98no—
96C42—92no—
97C42—91no—
98C42#108811yescharacteristic, complemented
99C22×C3—62no—
100C22×C3—92no—
101C22×C3—94no—
102C22×C3C22×#15711yescharacteristic
103C2×D3—64no—
104C2×D3—68no—
105C3×C4—22no—
106C3×C4—68no—
107C3×C4—64no—
108C3×C4—92no—
109C3×C4—188no—
110C3×C4—182no—
111C3×C4C4×#15712yes—
112Dic3—64no—
113Dic3—68no—
114C32—22no—
115C23—91no—
116C2×C4—91no—
117C2×C4—94no—
118C2×C4—98no—
119C2×C4—92no—
120C2×C4—184no—
121C2×C4—188no—
122C2×C4C2×#108812yes—
123C2×C4C2×#108811yescharacteristic
124C2×C3—62no—
125C2×C3—64no—
126C2×C3—94no—
127C2×C3C2×C4×#15711yescharacteristic
128C2×C3#6428612yes—
129D3—68no—
130C22—92no—
131C22—94no—
132C22C22×#108811yescharacteristic
133C4—22no—
134C4—92no—
135C4—188no—
136C4—182no—
137C4C4×#108812yes—
138C3—62no—
139C3#1059373311yescharacteristic
140C2—94no—
141C2C2×C4×#108811yescharacteristic
142C2#1172244812yes—
subgroupG/H
centerC2×C3×C4C2×#157
commutatorC2×H(3,𝔽3)C2×C42
Frattini, socleC22×C3C22×#157
FittingC42×H(3,𝔽3)C4
prime radical, cosocleC22×#272C23
hypercenterC3×C42#157
hypocenterH(3,𝔽3)#376
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
7. C4×#1088C4semidirect
8. C42×H(3,𝔽3)C4semidirect
9. C4×#1084C4semidirect
10. C4×#1084C4semidirect
72. H(3,𝔽3)#376semidirect
98. C42#1088semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10882
C42×#2722
#117207872
C3×#3769