Finite Group Databasebeta

← #13143782

Subgroups — #13143782

Order 1728.

528 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 109 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#11721721C211yesmaximal, characteristic
2#11721716C211yesmaximal, characteristic, complemented
3C4×C8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#11720554C212yesmaximal
5#11720549C212yesmaximal, complemented
6C3×#4697—34nomaximal
7C2×C4⋊#272C2211yescharacteristic
8C2×#6229C2211yescharacteristic
9C2×C8×H(3,𝔽3)C2212yes—
10C42×H(3,𝔽3)C2211yescharacteristic
11#88542—22no—
12C4⋊#1084C2212yes—
13#88540—22no—
14C3×#875—34no—
15C3×#870—34no—
16C32×C4×C8D314yescomplemented
17C3×#821—38no—
18C3×#816—38no—
19C22×#272—21no—
20C2×C4×H(3,𝔽3)D412yes—
21C2×C4×H(3,𝔽3)C2311yescharacteristic
22C2×#1084—21no—
23C2×#1084—22no—
24C4⋊#272—22no—
25#6229—22no—
26C8×H(3,𝔽3)—22no—
27C3×#488—91no—
28#4697—34no—
29C2×C3×D12—34no—
30C2×C3×Dic6—34no—
31C2×C32×C8C2×D318yes—
32C32×C42C2×D314yes—
33C3×(C2×D3)⋊C4—68no—
34C3×C4⋊Dic3—38no—
35C3×Dic3⋊C4—68no—
36C22×H(3,𝔽3)C2×D411yescharacteristic
37C2×#272—41no—
38C2×#272—42no—
39C4×H(3,𝔽3)—22no—
40C4×H(3,𝔽3)#4112yes—
41#1084—41no—
42#1084—42no—
43C3×#125—91no—
44C3×#124—91no—
45#875—34no—
46#870—34no—
47C3×#103—92no—
48C3×#102—92no—
49C3×C4×C8—34no—
50C3×C4×C8C3⋊D311yescharacteristic
51#821—38no—
52#816—38no—
53C22×C3×D3—64no—
54C2×C32×C4D1218yes—
55C2×C32×C4C22×D314yes—
56C2×C3×Dic3—68no—
57C2×C3×Dic3—64no—
58C3×D12—68no—
59C3×Dic6—68no—
60C32×C8—28no—
61#488—91noSylow
62C2×H(3,𝔽3)#12411yescharacteristic
63C2×H(3,𝔽3)#13512yes—
64#272—81no—
65C2×C3×Q8—91no—
66C2×C3×D4—91no—
67C2×D12—34no—
68C2×Dic6—34no—
69C2×C3×C8—38no—
70C2×C3×C8C2×C3⋊D312yes—
71C3×C4⋊C4—92no—
72C3×C4⋊C4—182no—
73C3×C22⋊C4—182no—
74C3×C42—34no—
75C3×C42C2×C3⋊D311yescharacteristic
76(C2×D3)⋊C4—68no—
77C4⋊Dic3—38no—
78Dic3⋊C4—68no—
79C22×C32C2×D1214yes—
80C2×C3×D3—124no—
81C2×C3×D3—128no—
82C32×C4—28no—
83C32×C4#23518yes—
84C3×Dic3—124no—
85C3×Dic3—128no—
86#125—91no—
87#124—91no—
88#103—92no—
89#102—92no—
90C4×C8#27211yescharacteristic, complemented
91H(3,𝔽3)#48811yescharacteristic, complemented, Sylow
92C23×C3—181no—
93C22×D3—64no—
94C3×Q8—182no—
95C3×D4—182no—
96C2×C3×C4—38no—
97C2×C3×C4—34no—
98C2×C3×C4—181no—
99C2×C3×C4—182no—
100C2×C3×C4C4⋊(C3⋊D3)12yes—
101C2×C3×C4C22×C3⋊D311yescharacteristic
102C2×Dic3—68no—
103C2×Dic3—64no—
104D12—68no—
105Dic6—68no—
106C3×C8—22no—
107C3×C8—68no—
108C2×C32#87514yes—
109C2×C32#90418yes—
110C3×D3—244no—
111C2×Q8—91no—
112C2×D4—91no—
113C2×C8C2×#27212yes—
114C4⋊C4—92no—
115C4⋊C4—182no—
116C22⋊C4—182no—
117C42C2×#27211yescharacteristic
118C22×C3—34no—
119C22×C3—361no—
120C22×C3—362no—
121C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
122C2×D3—124no—
123C2×D3—128no—
124C3×C4—22no—
125C3×C4—38no—
126C3×C4—68no—
127C3×C4—361no—
128C3×C4—362no—
129C3×C4#382812yes—
130Dic3—124no—
131Dic3—128no—
132C32#469714yescomplemented
133C23—181no—
134Q8—182no—
135D4—182no—
136C2×C4—181no—
137C2×C4—182no—
138C2×C4C4⋊#27212yes—
139C2×C4C22×#27211yescharacteristic
140C8—22no—
141C2×C3—38no—
142C2×C3—34no—
143C2×C3—721no—
144C2×C3#6408411yescharacteristic
145C2×C3#6411312yes—
146D3—244no—
147C22—362no—
148C22—361no—
149C22C2×C4⋊#27211yescharacteristic
150C4—22no—
151C4—362no—
152C4—361no—
153C4#8874012yes—
154C3—34no—
155C3#1059239611yescharacteristic
156C2—721no—
157C2#1172172111yescharacteristic
158C2#1172177912yes—
subgroupG/H
center, socleC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC4×C8×H(3,𝔽3)C2
prime radicalC2×C4⋊#272C22
hypercenterC3×C4×C8C3⋊D3
hypocenterH(3,𝔽3)#488
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×C8

Complements

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

NH ≅ G/Ntype
2. #11721716C2semidirect
3. C4×C8×H(3,𝔽3)C2semidirect
5. #11720549C2semidirect
16. C32×C4×C8D3semidirect
90. C4×C8#272semidirect
91. H(3,𝔽3)#488semidirect
132. C32#4697semidirect

Maximal subgroups

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

subgroupindex
#117217212
#117217162
C4×C8×H(3,𝔽3)2
#117205542
#117205492
C3×#46973