Finite Group Databasebeta

← #13143735

Subgroups — #13143735

Order 1728.

1771 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1771 conjugacy classes of subgroups — too many to draw. Only its 199 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
1C22×C4⋊#272C213yesmaximal, complemented
2C2×#88542C213yesmaximal, complemented
3H(3,𝔽3)×#95C211yesmaximal, characteristic, complemented
4C3×#4671—34nomaximal
5C23×#272C2213yes—
6C22×C4×H(3,𝔽3)C2213yes—
7C22×#1084C2211yescharacteristic
8C2×C4⋊#272—212no—
9C2×C4⋊#272—26no—
10#88542—26no—
11C22×C3×D12—312no—
12C2×C3×(C2×D3)⋊C4—312no—
13C32×#95D314yescomplemented
14C23×H(3,𝔽3)C2311yescharacteristic
15C22×#272—26no—
16C22×#272—26no—
17C22×#272—23no—
18C22×#272—23no—
19C22×#272D416yescomplemented
20C2×C4×H(3,𝔽3)—26no—
21C2×C4×H(3,𝔽3)D416yescomplemented
22C2×#1084—23no—
23C4⋊#272—224no—
24C3×#392—91no—
25#4671—34no—
26C23×C3×D3—312no—
27C22×C32×C4C2×D3112yes—
28C22×C3×Dic3—34no—
29C2×C3×D12—648no—
30C2×C3×D12—624no—
31C3×(C2×D3)⋊C4—624no—
32C22×H(3,𝔽3)C2×D413yes—
33C22×H(3,𝔽3)C2×D413yes—
34C22×H(3,𝔽3)#4111yescharacteristic
35C2×#272—212no—
36C2×#272—26no—
37C2×#272—43no—
38C2×#272—46no—
39C2×#272—43no—
40C4×H(3,𝔽3)—26no—
41#1084—41no—
42C22×C3×D4—93no—
43C22×D12—312no—
44C2×C3×C22⋊C4—93no—
45C2×(C2×D3)⋊C4—312no—
46C3×#95—34no—
47C3×#95C3⋊D311yescharacteristic
48C23×C32C22×D314yes—
49C22×C3×D3—324no—
50C22×C3×D3—612no—
51C22×C3×D3—624no—
52C22×C3×D3—624no—
53C22×C3×D3—612no—
54C2×C32×C4—224no—
55C2×C32×C4D12124yescomplemented
56C2×C3×Dic3—612no—
57C3×D12—696no—
58#392—91noSylow
59C2×H(3,𝔽3)#12013yes—
60C2×H(3,𝔽3)#12113yes—
61C2×H(3,𝔽3)#12711yescharacteristic
62#272—46no—
63C24×C3—93no—
64C23×D3—312no—
65C2×C3×D4—1812no—
66C2×C3×D4—186no—
67C22×C3×C4—312no—
68C22×C3×C4—91no—
69C22×C3×C4C2×C3⋊D313yes—
70C22×Dic3—34no—
71C2×D12—624no—
72C2×D12—648no—
73C3×C22⋊C4—186no—
74(C2×D3)⋊C4—624no—
75C22×C32C2×D12112yes—
76C22×C32D3×D4112yes—
77C22×C32#23914yes—
78C2×C3×D3—648no—
79C2×C3×D3—624no—
80C2×C3×D3—1212no—
81C2×C3×D3—1212no—
82C2×C3×D3—1224no—
83C32×C4—224no—
84C3×Dic3—124no—
85C22×D4—93no—
86C2×C22⋊C4—93no—
87#95#27211yescharacteristic, complemented
88H(3,𝔽3)#39211yescharacteristic, complemented, Sylow
89C23×C3—34no—
90C23×C3—96no—
91C23×C3—183no—
92C23×C3—186no—
93C23×C3—186no—
94C23×C3—183no—
95C23×C3C22×C3⋊D311yescharacteristic
96C22×D3—324no—
97C22×D3—612no—
98C22×D3—624no—
99C22×D3—624no—
100C22×D3—612no—
101C3×D4—1824no—
102C2×C3×C4—26no—
103C2×C3×C4—324no—
104C2×C3×C4—624no—
105C2×C3×C4—183no—
106C2×C3×C4C4⋊(C3⋊D3)16yes—
107C2×Dic3—612no—
108D12—696no—
109C2×C32#87414yes—
110C2×C32#882112yes—
111C2×C32#895112yes—
112C3×D3—1224no—
113C24—93no—
114C2×D4—1812no—
115C2×D4—186no—
116C22×C4—91no—
117C22×C4C2×#27213yes—
118C22⋊C4—186no—
119C22×C3—312no—
120C22×C3—312no—
121C22×C3—34no—
122C22×C3—1812no—
123C22×C3—186no—
124C22×C3—363no—
125C22×C3—363no—
126C22×C3—366no—
127C22×C3C2×C4⋊(C3⋊D3)13yes—
128C22×C3D4×C3⋊D313yes—
129C22×C3#383211yescharacteristic
130C2×D3—648no—
131C2×D3—624no—
132C2×D3—1212no—
133C2×D3—1224no—
134C2×D3—1212no—
135C3×C4—26no—
136C3×C4—624no—
137C3×C4—361no—
138Dic3—124no—
139C32#467114yescomplemented
140C23—96no—
141C23—183no—
142C23—186no—
143C23—186no—
144C23—183no—
145C23C22×#27211yescharacteristic
146D4—1824no—
147C2×C4—26no—
148C2×C4—183no—
149C2×C4C4⋊#27216yescomplemented
150C2×C3—312no—
151C2×C3—312no—
152C2×C3—34no—
153C2×C3—366no—
154C2×C3#6408311yescharacteristic
155C2×C3#6409113yes—
156C2×C3#6410413yes—
157D3—1224no—
158C22—1812no—
159C22—186no—
160C22—363no—
161C22—366no—
162C22—363no—
163C22C2×C4⋊#27213yes—
164C22D4×#27213yes—
165C22#8874811yescharacteristic
166C4—26no—
167C4—361no—
168C3—34no—
169C3#1059237011yescharacteristic
170C2—366no—
171C2#1172172011yescharacteristic
172C2#1172173813yes—
173C2#1172176313yes—
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, cosocleC23×H(3,𝔽3)C23
FittingH(3,𝔽3)×#95C2
hypercenterC3×#95C3⋊D3
hypocenterH(3,𝔽3)#392
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C3×#95

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊#272C2semidirect
2. C2×#88542C2semidirect
3. H(3,𝔽3)×#95C2semidirect
13. C32×#95D3semidirect
19. C22×#272D4semidirect
21. C2×C4×H(3,𝔽3)D4semidirect
55. C2×C32×C4D12semidirect
87. #95#272semidirect
88. H(3,𝔽3)#392semidirect
139. C32#4671semidirect
149. C2×C4C4⋊#272semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#2722
C2×#885422
H(3,𝔽3)×#952
C3×#46713