Finite Group Databasebeta

← #13136919

Subgroups — #13136919

Order 1728.

287 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 47 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×#88627C212yesmaximal
2C2×C8×#272C211yesmaximal, characteristic
3C2×C4×#272C2211yescharacteristic
4C2×#6188—22no—
5#88627—22no—
6#88627C414yes—
7C2×C8×H(3,𝔽3)C411yescharacteristic
8C2×#6180C411yescharacteristic
9C8×#272—22no—
10C2×C3×D3×C8—62no—
11C22×#272Q811yescharacteristic
12C2×C4×H(3,𝔽3)C2×C411yescharacteristic
13C2×#1084D411yescharacteristic
14C4×#272C2×C412yes—
15C4×#272D412yes—
16#6188—24no—
17C8×H(3,𝔽3)—21no—
18#6180—21no—
19C3×#341—91nomaximal
20C2×C3×C4×D3—62no—
21C2×C32×C8—22no—
22C2×C3×C3⋊C8—62no—
23C3×D3×C8—68no—
24C22×H(3,𝔽3)C4⋊C411yescharacteristic
25C2×#272—21no—
26C2×#272—21no—
27C2×#272C4⋊C412yes—
28C4×H(3,𝔽3)C4211yescharacteristic
29C4×H(3,𝔽3)C22⋊C411yescharacteristic
30#1084C22⋊C412yes—
31C2×C3×M16—92no—
32C22×C3×C8—91no—
33C2×D3×C8—62no—
34C22×C3×D3—62no—
35C2×C32×C4—22no—
36C2×C3×Dic3—62no—
37C3×C4×D3—64no—
38C3×C4×D3—64no—
39C32×C8—24no—
40C3×C3⋊C8—64no—
41#341—91noSylow
42C2×H(3,𝔽3)#9511yescharacteristic
43C2×H(3,𝔽3)#10811yescharacteristic
44C2×H(3,𝔽3)#10811yescharacteristic
45#272—22no—
46C22×C3×C4—91no—
47C2×C4×D3—62no—
48C3×M16—94no—
49C3×M16—182no—
50C2×C3×C8—62no—
51C2×C3×C8—91no—
52C2×C3×C8—182no—
53C2×C3×C8—182no—
54C2×C3×C8#15711yescharacteristic
55C2×C3⋊C8—62no—
56D3×C8—68no—
57C22×C32—22no—
58C2×C3×D3—64no—
59C2×C3×D3—64no—
60C2×C3×D3—64no—
61C32×C4—22no—
62C32×C4—22no—
63C3×Dic3—64no—
64C2×M16—92no—
65C22×C8—91no—
66H(3,𝔽3)#34111yescharacteristic, complemented, Sylow
67C23×C3—91no—
68C22×D3—62no—
69C2×C3×C4—62no—
70C2×C3×C4—91no—
71C2×C3×C4—92no—
72C2×C3×C4—92no—
73C2×C3×C4C2×#15711yescharacteristic
74C2×Dic3—62no—
75C4×D3—64no—
76C4×D3—64no—
77C3×C8—21no—
78C3×C8—64no—
79C3×C8—184no—
80C3×C8—181no—
81C3⋊C8—64no—
82C2×C32—22no—
83C2×C32—22no—
84C2×C32—22no—
85C3×D3—68no—
86C22×C4—91no—
87M16—94no—
88M16—182no—
89C2×C8—91no—
90C2×C8—182no—
91C2×C8—182no—
92C2×C8#108811yescharacteristic
93C22×C3—62no—
94C22×C3—92no—
95C22×C3—181no—
96C22×C3—181no—
97C22×C3C4⋊#15711yescharacteristic
98C2×D3—64no—
99C2×D3—64no—
100C2×D3—64no—
101C3×C4—62no—
102C3×C4—62no—
103C3×C4—92no—
104C3×C4C4×#15711yescharacteristic
105C3×C4C22⋊#15711yescharacteristic
106Dic3—64no—
107C32—22no—
108C23—91no—
109C2×C4—91no—
110C2×C4—92no—
111C2×C4—92no—
112C2×C4C2×#108811yescharacteristic
113C8—21no—
114C8—184no—
115C8—181no—
116C2×C3—62no—
117C2×C3—62no—
118C2×C3—62no—
119C2×C3—182no—
120C2×C3#6377011yescharacteristic
121C2×C3#6377111yescharacteristic
122C2×C3#6378111yescharacteristic
123D3—68no—
124C22—92no—
125C22—181no—
126C22—181no—
127C22C4⋊#108811yescharacteristic
128C4—92no—
129C4C4×#108811yescharacteristic
130C4C22⋊#108811yescharacteristic
131C3—62no—
132C3#1059018511yescharacteristic
133C2—182no—
134C2#1172077611yescharacteristic
135C2#1172077711yescharacteristic
136C2#1172078711yescharacteristic
subgroupG/H
center, socleC22×C3C4⋊#157
commutatorC4×H(3,𝔽3)C42
FrattiniC2×C3×C4C2×#157
FittingC2×C8×H(3,𝔽3)C4
prime radicalC22×#272Q8
cosocleC2×C4×#272C22
hypercenterC2×C3×C8#157
hypocenterH(3,𝔽3)#341
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
66. H(3,𝔽3)#341semidirect

Maximal subgroups

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

subgroupindex
C2×#886272
C2×C8×#2722
C3×#3419