Finite Group Databasebeta

← #13143775

Subgroups — #13143775

Order 1728.

528 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 117 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#11721718C211yesmaximal, characteristic
2C4×C8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#11720553C212yesmaximal
4#11720547C212yesmaximal, complemented
5#11720520C211yesmaximal, characteristic, complemented
6C3×#4690—34nomaximal
7C2×C4×#272C2211yescharacteristic
8C2×C8×H(3,𝔽3)C2212yes—
9C42×H(3,𝔽3)C2211yescharacteristic
10#88542C412yes—
11#88540C412yes—
12C4×#1084C2211yescharacteristic
13C2×#6180C2212yes—
14C3×#872—34no—
15C32×C4×C8D314yescomplemented
16C3×(C2×D3)⋊C8—38no—
17C3×Dic3⋊C8—38no—
18C3×#803—34no—
19C22×#272C2×C411yescharacteristic
20C2×C4×H(3,𝔽3)C2×C412yes—
21C2×C4×H(3,𝔽3)C2311yescharacteristic
22C2×#1084C2×C411yescharacteristic
23C2×#1084C2×C412yes—
24C4×#272—22no—
25C8×H(3,𝔽3)—22no—
26#6180—22no—
27C3×#433—91no—
28#4690—34no—
29C2×C3×C4×D3—34no—
30C2×C32×C8C2×D318yes—
31C32×C42C2×D314yes—
32C3×(C2×D3)⋊C4—38no—
33C3×Dic3⋊C4—38no—
34C3×C4×Dic3—34no—
35C2×C3×C3⋊C8—38no—
36C22×H(3,𝔽3)C22×C411yescharacteristic
37C2×#272—21no—
38C2×#272—22no—
39C4×H(3,𝔽3)—22no—
40C4×H(3,𝔽3)#4112yes—
41#1084—21no—
42#1084—22no—
43C3×#117—91no—
44#872—34no—
45C3×C4⋊C8—92no—
46C3×C22⋊C8—92no—
47C3×#97—91no—
48C3×C4×C8—34no—
49C3×C4×C8C3⋊D311yescharacteristic
50(C2×D3)⋊C8—38no—
51Dic3⋊C8—38no—
52#803—34no—
53C22×C3×D3—34no—
54C2×C32×C4C4×D318yes—
55C2×C32×C4C22×D314yes—
56C2×C3×Dic3—38no—
57C2×C3×Dic3—34no—
58C3×C4×D3—68no—
59C32×C8—28no—
60C3×C3⋊C8—68no—
61#433—91noSylow
62C2×H(3,𝔽3)#11711yescharacteristic
63C2×H(3,𝔽3)#13112yes—
64#272—41no—
65C22×C3×C4—91no—
66C2×C4×D3—34no—
67C2×C3×C8—38no—
68C2×C3×C8—92no—
69C2×C3×C8C2×C3⋊D312yes—
70C3×C4⋊C4—92no—
71C3×C22⋊C4—92no—
72C3×C42—34no—
73C3×C42—91no—
74C3×C42C2×C3⋊D311yescharacteristic
75(C2×D3)⋊C4—38no—
76Dic3⋊C4—38no—
77C4×Dic3—34no—
78C2×C3⋊C8—38no—
79C22×C32C2×C4×D314yes—
80C2×C3×D3—64no—
81C2×C3×D3—68no—
82C32×C4—28no—
83C32×C4#23518yes—
84C3×Dic3—64no—
85C3×Dic3—68no—
86#117—91no—
87C4⋊C8—92no—
88C22⋊C8—92no—
89#97—91no—
90C4×C8#27211yescharacteristic, complemented
91H(3,𝔽3)#43311yescharacteristic, complemented, Sylow
92C23×C3—91no—
93C22×D3—34no—
94C2×C3×C4—38no—
95C2×C3×C4—34no—
96C2×C3×C4—92no—
97C2×C3×C4—91no—
98C2×C3×C4—182no—
99C2×C3×C4C4×C3⋊D312yes—
100C2×C3×C4C22×C3⋊D311yescharacteristic
101C2×Dic3—38no—
102C2×Dic3—34no—
103C4×D3—68no—
104C3×C8—22no—
105C3×C8—68no—
106C3×C8—182no—
107C3⋊C8—68no—
108C2×C32#87214yes—
109C2×C32#90118yes—
110C3×D3—124no—
111C22×C4—91no—
112C2×C8—92no—
113C2×C8C2×#27212yes—
114C4⋊C4—92no—
115C22⋊C4—92no—
116C42—91no—
117C42C2×#27211yescharacteristic
118C22×C3—34no—
119C22×C3—181no—
120C22×C3—182no—
121C22×C3C2×C4×C3⋊D311yescharacteristic
122C2×D3—64no—
123C2×D3—68no—
124C3×C4—22no—
125C3×C4—38no—
126C3×C4—68no—
127C3×C4—181no—
128C3×C4—182no—
129C3×C4#382812yes—
130Dic3—64no—
131Dic3—68no—
132C32#469014yescomplemented
133C23—91no—
134C2×C4—92no—
135C2×C4—91no—
136C2×C4—182no—
137C2×C4C4×#27212yes—
138C2×C4C22×#27211yescharacteristic
139C8—22no—
140C8—182no—
141C2×C3—38no—
142C2×C3—34no—
143C2×C3—361no—
144C2×C3#6408111yescharacteristic
145C2×C3#6411012yes—
146D3—124no—
147C22—182no—
148C22—181no—
149C22C2×C4×#27211yescharacteristic
150C4—22no—
151C4—182no—
152C4—181no—
153C4#8874012yes—
154C3—34no—
155C3#1059238911yescharacteristic
156C2—361no—
157C2#1172171811yescharacteristic
158C2#1172177612yes—
subgroupG/H
center, FrattiniC2×C3×C4C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FittingC4×C8×H(3,𝔽3)C2
socleC22×C3C2×C4×C3⋊D3
prime radicalC22×#272C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×C4×C8C3⋊D3
hypocenterH(3,𝔽3)#433
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C3×C4×C8

Complements

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

NH ≅ G/Ntype
2. C4×C8×H(3,𝔽3)C2semidirect
4. #11720547C2semidirect
5. #11720520C2semidirect
15. C32×C4×C8D3semidirect
90. C4×C8#272semidirect
91. H(3,𝔽3)#433semidirect
132. C32#4690semidirect

Maximal subgroups

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

subgroupindex
#117217182
C4×C8×H(3,𝔽3)2
#117205532
#117205472
#117205202
C3×#46903