Finite Group Databasebeta

← #13136700

Subgroups — #13136700

Order 1728.

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

normal subgroups only

This group has 283 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⋊#1088C211yesmaximal, characteristic
2#11721065C212yesmaximal
3C22×#1088C412yes—
4C2×C4×#272C2211yescharacteristic
5C4⋊#1088—22no—
6#3748⋊C3—22no—
7#88435C414yescomplemented
8#88434—22no—
9D3×#208—62no—
10C22×#272C2×C411yescharacteristic
11C2×#1088—22no—
12C2×#1088—42no—
13C2×C4×H(3,𝔽3)D411yescharacteristic
14C2×#1084Q811yescharacteristic
15C4×#272C2×C412yes—
16C4×#272D412yes—
17(C3⋊(C3⋊C8))⋊C3—24no—
18#4530—91nomaximal
19C2×C3×C4×D3—62no—
20#3748—22no—
21C3×#208—62no—
22#3711—64no—
23D3×C3⋊C8—64no—
24C22×H(3,𝔽3)C4⋊C411yescharacteristic
25C2×#272—21no—
26C2×#272—21no—
27C2×#272C4211yescharacteristic
28C2×#272C22⋊C411yescharacteristic
29#1088—42no—
30C4×H(3,𝔽3)C22⋊C412yes—
31#1084C4⋊C412yes—
32C2×C3×C4⋊C4—91no—
33C2×#208—92no—
34D3×M16—182no—
35C22×C3×D3—62no—
36C2×C32×C4—22no—
37C2×C3×Dic3—62no—
38C3×C4×D3—62no—
39C3×C4×D3—62no—
40C3×C4×D3—62no—
41C3×C4×D3—62no—
42C3⋊(C3⋊C8)—24no—
43C3×C3⋊C8—64no—
44#343—271noSylow
45C2×H(3,𝔽3)—21no—
46C2×H(3,𝔽3)#9511yescharacteristic
47#272—21no—
48#272#10011yescharacteristic
49#272#10111yescharacteristic
50C22×C3×C4—91no—
51C22×C3×C4—92no—
52C2×C4×D3—62no—
53C3×M16—182no—
54C3×C4⋊C4—182no—
55#208—62no—
56#208—94no—
57#208—182no—
58C2×C3⋊C8—182no—
59ΓL(1,𝔽25)—184no—
60D3×C8—184no—
61C22×C32—22no—
62C2×C3×D3—62no—
63C2×C3×D3—62no—
64C2×C3×D3—122no—
65C2×C3×D3—122no—
66C32×C4—22no—
67C32×C4—22no—
68C3×Dic3—62no—
69C3×Dic3—62no—
70C2×M16—272no—
71C2×C4⋊C4—91no—
72H(3,𝔽3)#34311yescharacteristic, complemented, Sylow
73C23×C3—91no—
74C22×D3—62no—
75C2×C3×C4—62no—
76C2×C3×C4—91no—
77C2×C3×C4—92no—
78C2×C3×C4—92no—
79C2×C3×C4—182no—
80C2×C3×C4—362no—
81C2×C3×C4GO+(4,𝔽2)11yescharacteristic
82C2×Dic3—62no—
83C4×D3—62no—
84C4×D3—62no—
85C4×D3—62no—
86C4×D3—62no—
87C3×C8—184no—
88C3⋊C8—64no—
89C3⋊C8—184no—
90C2×C32—22no—
91C2×C32—42no—
92C3×D3—62no—
93C3×D3—62no—
94C3×D3—122no—
95C22×C4—91no—
96C22×C4—92no—
97M16—182no—
98M16—274no—
99C2×C8—542no—
100C4⋊C4—182no—
101C22×C3—62no—
102C22×C3—91no—
103C22×C3—91no—
104C22×C3—181no—
105C22×C3—181no—
106C22×C3#377311yescharacteristic
107C2×D3—62no—
108C2×D3—62no—
109C2×D3—122no—
110C2×D3—122no—
111C3×C4—62no—
112C3×C4—62no—
113C3×C4—92no—
114C3×C4—362no—
115C3×C4D32⋊C412yes—
116Dic3—62no—
117Dic3—62no—
118C32—22no—
119C23—91no—
120C2×C4—91no—
121C2×C4—92no—
122C2×C4—92no—
123C2×C4—182no—
124C2×C4—362no—
125C2×C4#1088⋊C211yescharacteristic
126C8—184no—
127C2×C3—21no—
128C2×C3—62no—
129C2×C3—91no—
130C2×C3—91no—
131C2×C3—122no—
132C2×C3—181no—
133C2×C3#6373711yescharacteristic
134D3—62no—
135D3—62no—
136D3—122no—
137C22—91no—
138C22—91no—
139C22—181no—
140C22—181no—
141C22#8859011yescharacteristic
142C4—92no—
143C4—362no—
144C4(((C3⋊D3)⋊C3)⋊C2)⋊C412yes—
145C3—62no—
146C3#1059005311yescharacteristic
147C2—21no—
148C2—91no—
149C2—91no—
150C2—181no—
151C2#1172070711yescharacteristic
subgroupG/H
centerC2#11720707
commutatorC2×#272C42
FrattiniC2×C3×C4GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
socleC2×C3#63737
prime radicalC22×#272C2×C4
cosocleC2×C4×#272C22
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#343
LCSG ▷ C2×#272 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
7. #88435C4semidirect
72. H(3,𝔽3)#343semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10882
#117210652
#45309