Finite Group Databasebeta

← #13136935

Subgroups — #13136935

Order 1728.

301 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 41 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⋊#1084C211yesmaximal, characteristic
2C22⋊#6188C212yesmaximal
3C22×C4×H(3,𝔽3)C411yescharacteristic
4C22×#1084C411yescharacteristic
5C22×#1084C2211yescharacteristic
6C2×#6188—22no—
7C4⋊#1084—22no—
8C2×C3×C4⋊Dic3—62no—
9C23×H(3,𝔽3)C2×C411yescharacteristic
10C2×C4×H(3,𝔽3)—21no—
11C2×C4×H(3,𝔽3)—21no—
12C2×C4×H(3,𝔽3)C812yescomplemented
13C2×#1084—21no—
14C2×#1084—41no—
15C2×#1084—41no—
16C2×#1084D412yes—
17#6188—42no—
18C3×#324—91nomaximal
19C22×C32×C4—22no—
20C22×C3×Dic3—62no—
21C22×C3×Dic3—62no—
22C3×C4⋊Dic3—68no—
23C22×H(3,𝔽3)—21no—
24C22×H(3,𝔽3)—21no—
25C22×H(3,𝔽3)C22⋊C411yescharacteristic
26C22×H(3,𝔽3)C2×C811yescharacteristic
27C22×H(3,𝔽3)M1611yescharacteristic
28C4×H(3,𝔽3)—22no—
29#1084—22no—
30#1084—41no—
31C2×C3×C4⋊C4—91no—
32C2×C4⋊Dic3—62no—
33C3×C22⋊C8—92no—
34C23×C32—22no—
35C2×C32×C4—24no—
36C2×C32×C4—24no—
37C2×C32×C4—24no—
38C2×C3×Dic3—64no—
39C2×C3×Dic3—64no—
40C2×C3×Dic3—124no—
41C2×C3×Dic3—124no—
42#324—91noSylow
43C2×H(3,𝔽3)—21no—
44C2×H(3,𝔽3)—21no—
45C2×H(3,𝔽3)C22⋊C811yescharacteristic
46C2×H(3,𝔽3)#9911yescharacteristic
47C2×H(3,𝔽3)#10111yescharacteristic
48C22×C3×C4—62no—
49C22×C3×C4—91no—
50C22×C3×C4—91no—
51C22×C3×C4#15711yescharacteristic
52C22×Dic3—62no—
53C22×Dic3—62no—
54C2×C3×C8—182no—
55C3×C4⋊C4—182no—
56C4⋊Dic3—68no—
57C22×C32—24no—
58C22×C32—24no—
59C22×C32—22no—
60C22×C32—22no—
61C22×C32—22no—
62C32×C4—28no—
63C3×Dic3—124no—
64C3×Dic3—124no—
65C2×C4⋊C4—91no—
66C22⋊C8—92no—
67H(3,𝔽3)#32411yescharacteristic, complemented, Sylow
68C23×C3—62no—
69C23×C3C2×#15711yescharacteristic
70C2×C3×C4—21no—
71C2×C3×C4—21no—
72C2×C3×C4—64no—
73C2×C3×C4—64no—
74C2×C3×C4—64no—
75C2×C3×C4—92no—
76C2×C3×C4—181no—
77C2×C3×C4—361no—
78C2×C3×C4—361no—
79C2×C3×C4#62212yes—
80C2×Dic3—64no—
81C2×Dic3—64no—
82C2×Dic3—124no—
83C2×Dic3—124no—
84C3×C8—362no—
85C2×C32—24no—
86C2×C32—24no—
87C2×C32—22no—
88C2×C32—22no—
89C2×C32—22no—
90C22×C4—91no—
91C22×C4—91no—
92C22×C4#108811yescharacteristic
93C2×C8—182no—
94C4⋊C4—182no—
95C22×C3—21no—
96C22×C3—21no—
97C22×C3—64no—
98C22×C3—64no—
99C22×C3—62no—
100C22×C3—62no—
101C22×C3—62no—
102C22×C3C2×#62211yescharacteristic
103C22×C3#379211yescharacteristic
104C22×C3C22⋊#15711yescharacteristic
105C3×C4—22no—
106C3×C4—68no—
107C3×C4—182no—
108C3×C4—361no—
109Dic3—124no—
110Dic3—124no—
111C32—22no—
112C23C2×#108811yescharacteristic
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—92no—
116C2×C4—181no—
117C2×C4—361no—
118C2×C4—361no—
119C2×C4#618812yescomplemented
120C8—362no—
121C2×C3—21no—
122C2×C3—21no—
123C2×C3—64no—
124C2×C3—64no—
125C2×C3—62no—
126C2×C3—62no—
127C2×C3—62no—
128C2×C3#6377411yescharacteristic
129C2×C3#6378011yescharacteristic
130C2×C3C22⋊#62211yescharacteristic
131C22—21no—
132C22—21no—
133C22C2×#618811yescharacteristic
134C22#8863111yescharacteristic
135C22C22⋊#108811yescharacteristic
136C4—22no—
137C4—182no—
138C4—361no—
139C3—62no—
140C3#1059020211yescharacteristic
141C2—21no—
142C2—21no—
143C2#1172078011yescharacteristic
144C2#1172078611yescharacteristic
145C2C22⋊#618811yescharacteristic
subgroupG/H
center, socleC22×C3C22⋊#157
commutatorC22×H(3,𝔽3)C2×C8
FrattiniC23×C3C2×#157
FittingC22×C4×H(3,𝔽3)C4
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC22×#1084C22
hypercenterC22×C3×C4#157
hypocenterH(3,𝔽3)#324
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
12. C2×C4×H(3,𝔽3)C8semidirect
67. H(3,𝔽3)#324semidirect
119. C2×C4#6188semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10842
C22⋊#61882
C3×#3249