Finite Group Databasebeta

← #13180071

Subgroups — #13180071

Order 1728.

833 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 833 conjugacy classes of subgroups — too many to draw. Only its 27 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
1C3×C22⋊(C22⋊A4)C311yesmaximal, characteristic
2C22×C22⋊(C22⋊C9)C313yesmaximal, complemented
3(C22⋊A4)⋊C9—43nomaximal
4C2×C22⋊(C22⋊C9)—33no—
5C26×C3C3211yescharacteristic
6C22⋊(C22⋊A4)C913yescomplemented
7C3×C22⋊A4—43no—
8C3×C22⋊A4—123no—
9C3×C22⋊A4—123no—
10C22⋊(C22⋊C9)A413yescomplemented
11C22×C22⋊C9—46no—
12C22×C22⋊C9—123no—
13C9×A4—163no—
14C25×C3—33no—
15C25×C3—93no—
16C25×C3—93no—
17C2×C22⋊C9—126no—
18C2×C22⋊C9—129no—
19C26C3×C911yescharacteristic, complemented, Sylow
20C24×C3—33no—
21C24×C3—33no—
22C24×C3—33no—
23C24×C3—93no—
24C24×C3—96no—
25C24×C3—99no—
26C24×C3—99no—
27C24×C3—99no—
28C24×C3—93no—
29C24×C3—93no—
30C24×C3—918no—
31C24×C3—99no—
32C24×C3C3×A413yes—
33C22⋊A4—49no—
34C22⋊A4—129no—
35C22⋊A4—129no—
36C3×A4—163no—
37C3×A4—483no—
38C3×A4—483no—
39C22×C9—163no—
40C22⋊C9—46no—
41C22⋊C9—123no—
42C25—33no—
43C25—93no—
44C25—93no—
45C3×C9—641noSylow
46C23×C3—39no—
47C23×C3—36no—
48C23×C3—93no—
49C23×C3—93no—
50C23×C3—99no—
51C23×C3—96no—
52C23×C3—99no—
53C23×C3—99no—
54C23×C3—99no—
55C23×C3—93no—
56C23×C3—918no—
57C23×C3—99no—
58C23×C3—99no—
59C23×C3—936no—
60C23×C3—96no—
61C23×C3—918no—
62C23×C3—93no—
63C2×C9—483no—
64C24—33no—
65C24—33no—
66C24—33no—
67C24—93no—
68C24—96no—
69C24—99no—
70C24—99no—
71C24—99no—
72C24—93no—
73C24—93no—
74C24—918no—
75C24—99no—
76C24C9×A413yescomplemented
77C22×C3—33no—
78C22×C3—33no—
79C22×C3—33no—
80C22×C3—93no—
81C22×C3—96no—
82C22×C3—99no—
83C22×C3—99no—
84C22×C3—99no—
85C22×C3—93no—
86C22×C3—918no—
87C22×C3—99no—
88C22×C3—93no—
89C22×C3A4213yes—
90A4—169no—
91A4—489no—
92A4—489no—
93C32—641no—
94C9—163no—
95C23—39no—
96C23—36no—
97C23—93no—
98C23—93no—
99C23—99no—
100C23—96no—
101C23—99no—
102C23—99no—
103C23—99no—
104C23—93no—
105C23—918no—
106C23—99no—
107C23—99no—
108C23—936no—
109C23—96no—
110C23—918no—
111C23—93no—
112C2×C3—33no—
113C2×C3—93no—
114C2×C3—93no—
115C22—33no—
116C22—33no—
117C22—33no—
118C22—93no—
119C22—96no—
120C22—99no—
121C22—99no—
122C22—99no—
123C22—93no—
124C22—918no—
125C22—99no—
126C22—93no—
127C22(C22⋊A4)⋊C913yescomplemented
128C3—643no—
129C3#1059678211yescharacteristic
130C2—33no—
131C2—93no—
132C2—93no—
subgroupG/H
center, Frattini, hypercenterC3#10596782
commutator, hypocenterC26C3×C9
Fitting, socle, cosocleC26×C3C32
prime radicalC3×C22⋊(C22⋊A4)C3
LCSG ▷ C26
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
2. C22×C22⋊(C22⋊C9)C3semidirect
6. C22⋊(C22⋊A4)C9semidirect
10. C22⋊(C22⋊C9)A4semidirect
19. C26C3×C9semidirect
76. C24C9×A4semidirect
127. C22(C22⋊A4)⋊C9semidirect

Maximal subgroups

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

subgroupindex
C3×C22⋊(C22⋊A4)3
C22×C22⋊(C22⋊C9)3
(C22⋊A4)⋊C94