Finite Group Databasebeta

← #13138760

Subgroups — #13138760

Order 1728.

345 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 79 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#11721095C212yesmaximal
2C8×#1084C211yesmaximal, characteristic
3#63644⋊C3C212yesmaximal
4#11720315C212yesmaximal
5#10590871—32nomaximal
6C2×C8×H(3,𝔽3)C2211yescharacteristic
7C4×#1084C2211yescharacteristic
8C2×#6180C2211yescharacteristic
9C4⋊((C3⋊Dic3)⋊C3)C2214yes—
10#88448—24no—
11#63869—34no—
12#63644D312yes—
13C3×#817—32no—
14C3×C8×Dic3—32no—
15C3×C8×Dic3—61no—
16#63575—34no—
17C2×C4×H(3,𝔽3)C2311yescharacteristic
18C2×#1084D412yes—
19C2×(C3⋊Dic3)⋊C3—24no—
20C8×H(3,𝔽3)Q812yes—
21#6180Q812yes—
22#4858—92no—
23#4683—91no—
24C2×C32×C8—21no—
25C2×C32×C8C2×D312yes—
26C4⋊(C3⋊Dic3)C2×D314yes—
27C3×C4⋊Dic3—34no—
28C3×C4×Dic3—32no—
29C3×C4×Dic3—61no—
30C2×C3×C3⋊C8—32no—
31C2×C3×C3⋊C8—61no—
32#3724—64no—
33Dic3⋊Dic3—64no—
34C22×H(3,𝔽3)C2×D411yescharacteristic
35C4×H(3,𝔽3)C2×Q811yescharacteristic
36C4×H(3,𝔽3)C2×Q811yescharacteristic
37#1084—22no—
38(C3⋊Dic3)⋊C3—44no—
39#890—94no—
40#870—92no—
41C3×#106—92no—
42C3×C4×C8—91no—
43#818—92no—
44#817—32no—
45#817—92no—
46C8×Dic3—32no—
47C8×Dic3—61no—
48#807—94no—
49C2×C32×C4—21no—
50C2×C32×C4C22×D312yes—
51C2×C3⋊Dic3—24no—
52C2×C3×Dic3—34no—
53C2×C3×Dic3—62no—
54C2×C3×Dic3—64no—
55C32×C8—22no—
56C32×C8Dic614yes—
57C3×C3⋊C8—34no—
58C3×C3⋊C8—62no—
59#499—271noSylow
60C2×H(3,𝔽3)C4⋊Q811yescharacteristic
61C2×H(3,𝔽3)#13512yes—
62C2×C3×C8—32no—
63C2×C3×C8—61no—
64C2×C3×C8—91no—
65C2×C3×C8D3211yescharacteristic
66C3×C4⋊C4—94no—
67C3×C42—91no—
68C4⋊Dic3—34no—
69C4⋊Dic3—94no—
70C4⋊Dic3—184no—
71Dic3⋊C4—184no—
72Dic3⋊C4—184no—
73C4×Dic3—32no—
74C4×Dic3—61no—
75C2×C3⋊C8—32no—
76C2×C3⋊C8—61no—
77C22×C32—21no—
78C22×C32D3×D412yes—
79C32×C4—21no—
80C32×C4—21no—
81C32×C4C2×Dic612yes—
82C32×C4D3×Q812yes—
83C3⋊Dic3—44no—
84C3×Dic3—64no—
85C3×Dic3—64no—
86C3×Dic3—124no—
87#125—272no—
88#107—272no—
89#106—92no—
90C4×C8—91no—
91H(3,𝔽3)#49911yescharacteristic, complemented, Sylow
92C2×C3×C4—32no—
93C2×C3×C4—61no—
94C2×C3×C4—92no—
95C2×C3×C4—184no—
96C2×C3×C4C2×D3211yescharacteristic
97C2×Dic3—34no—
98C2×Dic3—62no—
99C2×Dic3—64no—
100C2×Dic3—184no—
101C3×C8—34no—
102C3×C8—62no—
103C3×C8—92no—
104C3×C8#62712yes—
105C3⋊C8—34no—
106C3⋊C8—62no—
107C2×C32—22no—
108C2×C32—21no—
109C2×C32#88812yes—
110C2×C32#91614yes—
111C2×C8—91no—
112C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
113C4⋊C4—94no—
114C4⋊C4—544no—
115C42—91no—
116C22×C3—32no—
117C22×C3—61no—
118C22×C3#380211yescharacteristic
119C3×C4—32no—
120C3×C4—32no—
121C3×C4—61no—
122C3×C4—61no—
123C3×C4—182no—
124C3×C4—364no—
125C3×C4#379711yescharacteristic
126C3×C4C2×#62711yescharacteristic
127Dic3—64no—
128Dic3—64no—
129Dic3—124no—
130Dic3—364no—
131C32—21no—
132C32#485812yescomplemented
133C2×C4—92no—
134C2×C4—184no—
135C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
136C8—92no—
137C8#619612yes—
138C2×C3—34no—
139C2×C3—32no—
140C2×C3—62no—
141C2×C3—61no—
142C2×C3#6380912yes—
143C2×C3#6391911yescharacteristic
144C22#8865511yescharacteristic
145C4—182no—
146C4—364no—
147C4#8865111yescharacteristic
148C4C2×#619611yescharacteristic
149C3—32no—
150C3—61no—
151C3#1059087411yescharacteristic
152C2#1172084312yes—
153C2#1172111711yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#499
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
91. H(3,𝔽3)#499semidirect
132. C32#4858semidirect

Maximal subgroups

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

subgroupindex
#117210952
C8×#10842
#63644⋊C32
#117203152
#105908713