Finite Group Databasebeta

← #13180072

Subgroups — #13180072

Order 1728.

922 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 922 conjugacy classes of subgroups — too many to draw. Only its 24 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, complemented
2C22×C3×C22⋊A4C313yesmaximal, complemented
3(C3×A4)⋊A4—43nomaximal
4C2×C3×C22⋊A4—33no—
5C26×C3C3211yescharacteristic
6C22⋊(C22⋊A4)—31no—
7C22×C22⋊A4—33no—
8C3×C22⋊A4—43no—
9C3×C22⋊A4—123no—
10C3×C22⋊A4—123no—
11C3×C22⋊A4A413yescomplemented
12C22×C3×A4—46no—
13C22×C3×A4—123no—
14C22⋊H(3,𝔽3)—163no—
15C25×C3—33no—
16C25×C3—93no—
17C25×C3—93no—
18C2×C22⋊A4—39no—
19C2×C3×A4—126no—
20C2×C3×A4—129no—
21C26H(3,𝔽3)11yescharacteristic, complemented, Sylow
22C24×C3—33no—
23C24×C3—33no—
24C24×C3—33no—
25C24×C3—93no—
26C24×C3—96no—
27C24×C3—99no—
28C24×C3—99no—
29C24×C3—99no—
30C24×C3—93no—
31C24×C3—93no—
32C24×C3—918no—
33C24×C3—99no—
34C24×C3C3×A413yes—
35C22⋊A4—33no—
36C22⋊A4—129no—
37C22⋊A4—129no—
38C22⋊A4—123no—
39C22×A4—129no—
40C22×A4—126no—
41C22×C32—163no—
42C3×A4—46no—
43C3×A4—123no—
44C3×A4—163no—
45C3×A4—483no—
46C3×A4—483no—
47C25—33no—
48C25—93no—
49C25—93no—
50H(3,𝔽3)—641noSylow
51C23×C3—39no—
52C23×C3—36no—
53C23×C3—93no—
54C23×C3—93no—
55C23×C3—99no—
56C23×C3—96no—
57C23×C3—99no—
58C23×C3—99no—
59C23×C3—99no—
60C23×C3—93no—
61C23×C3—918no—
62C23×C3—99no—
63C23×C3—99no—
64C23×C3—936no—
65C23×C3—96no—
66C23×C3—918no—
67C23×C3—93no—
68C2×A4—1227no—
69C2×A4—1218no—
70C2×C32—483no—
71C24—33no—
72C24—33no—
73C24—33no—
74C24—93no—
75C24—96no—
76C24—99no—
77C24—99no—
78C24—99no—
79C24—93no—
80C24—93no—
81C24—918no—
82C24—99no—
83C24C22⋊H(3,𝔽3)13yescomplemented
84C22×C3—33no—
85C22×C3—33no—
86C22×C3—33no—
87C22×C3—93no—
88C22×C3—96no—
89C22×C3—99no—
90C22×C3—99no—
91C22×C3—99no—
92C22×C3—93no—
93C22×C3—918no—
94C22×C3—99no—
95C22×C3—93no—
96C22×C3—483no—
97C22×C3A4213yes—
98A4—129no—
99A4—126no—
100A4—489no—
101A4—489no—
102A4—483no—
103C32—163no—
104C32—641no—
105C23—39no—
106C23—36no—
107C23—93no—
108C23—93no—
109C23—99no—
110C23—96no—
111C23—99no—
112C23—99no—
113C23—99no—
114C23—93no—
115C23—918no—
116C23—99no—
117C23—99no—
118C23—936no—
119C23—96no—
120C23—918no—
121C23—93no—
122C2×C3—33no—
123C2×C3—93no—
124C2×C3—93no—
125C2×C3—489no—
126C22—33no—
127C22—33no—
128C22—33no—
129C22—93no—
130C22—96no—
131C22—99no—
132C22—99no—
133C22—99no—
134C22—93no—
135C22—918no—
136C22—99no—
137C22—93no—
138C22(C3×A4)⋊A413yescomplemented
139C3—483no—
140C3—1921no—
141C3#1059678211yescharacteristic
142C2—33no—
143C2—93no—
144C2—93no—
subgroupG/H
center, Frattini, hypercenterC3#10596782
commutator, Fitting, socle, cosocleC26×C3C32
hypocenterC26H(3,𝔽3)
LCSG ▷ C26×C3 ▷ C26
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. C3×C22⋊(C22⋊A4)C3semidirect
2. C22×C3×C22⋊A4C3semidirect
11. C3×C22⋊A4A4semidirect
21. C26H(3,𝔽3)semidirect
83. C24C22⋊H(3,𝔽3)semidirect
138. C22(C3×A4)⋊A4semidirect

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×C3×C22⋊A43
(C3×A4)⋊A44