Finite Group Databasebeta

← #13145733

Subgroups — #13145733

Order 1728.

602 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 155 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×C3×#3748C213yesmaximal
2#10589159—33nomaximal
3C22×C33×C4C2211yescharacteristic
4C3×#3748—26no—
5C2×C3×C3⋊(C3⋊C8)C416yes—
6C2×#3748D313yes—
7C2×C3×#208—36no—
8C2×C3×#208—63no—
9C23×C33Q811yescharacteristic
10C2×C33×C4C2×C413yes—
11C2×C33×C4D413yes—
12C3×C3⋊(C3⋊C8)—26no—
13C3⋊#338—93no—
14C22×C32×C4—23no—
15C22×C32×C4—41no—
16C22×C32×C4C2×D313yes—
17#3748—26no—
18C2×C3⋊(C3⋊C8)Dic316yes—
19C3×#208—612no—
20C3×#208—66no—
21C3×#208—66no—
22C2×C3×C3⋊C8—312no—
23C2×C3×C3⋊C8—66no—
24C22×C33—41no—
25C22×C33C4⋊C413yes—
26C33×C4C4211yescharacteristic
27C33×C4C22⋊C413yes—
28C2×C3×M16—93no—
29C2×#208—36no—
30C2×#208—63no—
31C23×C32—23no—
32C23×C32—41no—
33C23×C32Dic613yes—
34C2×C32×C4—26no—
35C2×C32×C4—23no—
36C2×C32×C4—23no—
37C2×C32×C4—26no—
38C2×C32×C4—43no—
39C2×C32×C4—43no—
40C2×C32×C4C4×D316yes—
41C2×C32×C4D1213yes—
42C2×C32×C4C2×Dic313yes—
43C2×C32×C4#6716yes—
44C3⋊(C3⋊C8)—26no—
45C3×C3⋊C8—612no—
46C3×C3⋊C8—612no—
47#338—271noSylow
48C2×C33—23no—
49C2×C33#9511yescharacteristic
50C22×C3×C4—23no—
51C22×C3×C4—41no—
52C22×C3×C4D3213yes—
53C3×M16—186no—
54C2×C3×C8—96no—
55#208—612no—
56#208—66no—
57#208—66no—
58C2×C3⋊C8—312no—
59C2×C3⋊C8—66no—
60C22×C32—26no—
61C22×C32—23no—
62C22×C32—43no—
63C22×C32—43no—
64C22×C32—46no—
65C22×C32—44no—
66C22×C32Dic3⋊C416yes—
67C22×C32C4⋊Dic313yes—
68C32×C4—23no—
69C32×C4—26no—
70C32×C4—23no—
71C32×C4—41no—
72C32×C4—43no—
73C32×C4C4×Dic313yes—
74C32×C4(C2×D3)⋊C416yes—
75C32×C4#21713yes—
76C2×M16—93no—
77C33#33811yescharacteristic, complemented, Sylow
78C23×C3—23no—
79C23×C3—41no—
80C23×C3#62713yes—
81C2×C3×C4—26no—
82C2×C3×C4—23no—
83C2×C3×C4—23no—
84C2×C3×C4—26no—
85C2×C3×C4—43no—
86C2×C3×C4—43no—
87C2×C3×C4D3×Dic316yes—
88C2×C3×C4#62413yes—
89C2×C3×C4#62513yes—
90C2×C3×C4#62616yes—
91C3×C8—186no—
92C3⋊C8—612no—
93C3⋊C8—612no—
94C2×C32—23no—
95C2×C32—26no—
96C2×C32—26no—
97C2×C32—23no—
98C2×C32—41no—
99C2×C32—46no—
100C2×C32—46no—
101C2×C32C3⋊#9513yes—
102C22×C4#111311yescharacteristic
103M16—186no—
104C2×C8—96no—
105C22×C3—23no—
106C22×C3—26no—
107C22×C3—43no—
108C22×C3—46no—
109C22×C3—44no—
110C22×C3—43no—
111C22×C3Dic3⋊Dic316yes—
112C22×C3#372413yes—
113C3×C4—23no—
114C3×C4—23no—
115C3×C4—26no—
116C3×C4—41no—
117C3×C4—43no—
118C3×C4Dic3213yes—
119C3×C4(C2×D3)⋊Dic316yes—
120C3×C4#372213yes—
121C32—23no—
122C32—41no—
123C32C3⋊#33813yescomplemented
124C23#629611yescharacteristic
125C2×C4#629413yes—
126C2×C4#629513yes—
127C8—186no—
128C2×C3—26no—
129C2×C3—26no—
130C2×C3—23no—
131C2×C3—23no—
132C2×C3—46no—
133C2×C3—46no—
134C2×C3—41no—
135C2×C3#6357913yes—
136C22—41no—
137C22#8881513yes—
138C4#8881311yescharacteristic
139C4#8881413yes—
140C3—23no—
141C3—41no—
142C3#1058915913yescomplemented
143C2—23no—
144C2#1172208211yescharacteristic
subgroupG/H
centerC4#88813
commutatorC33×C4C42
Frattini, hypercenterC22×C4#1113
Fitting, cosocleC22×C33×C4C22
socleC2×C33#95
prime radicalC23×C33Q8
hypocenterC33#338
LCSG ▷ C33×C4 ▷ C33
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
77. C33#338semidirect
123. C32C3⋊#338semidirect
142. C3#10589159semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37482
#105891593