Finite Group Databasebeta

← C3×#10596782 #13181843

Subgroups — C3×#10596782 #13181843

Order 1728.

1058 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 54 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#10596782C319yesmaximal, direct product factor
4C3×A42—43nomaximal
5C2×C3×C22⋊A4—33no—
6C26×C3C3211yescharacteristic, complemented
7C22⋊(C22⋊A4)C3213yescomplemented
8C22×C22⋊A4C3219yescomplemented
9C3×C22⋊A4—43no—
10C3×C22⋊A4—123no—
11C3×C22⋊A4—123no—
12C3×C22⋊A4A413yescomplemented
13C22×C3×A4—46no—
14C22×C3×A4—123no—
15A42—427no—
16C32×A4—163no—
17C25×C3—33no—
18C25×C3—93no—
19C25×C3—93no—
20C2×C22⋊A4—39no—
21C2×C3×A4—126no—
22C2×C3×A4—129no—
23C26C3311yescharacteristic, complemented, Sylow
24C24×C3—33no—
25C24×C3—33no—
26C24×C3—33no—
27C24×C3—93no—
28C24×C3—96no—
29C24×C3—99no—
30C24×C3—99no—
31C24×C3—99no—
32C24×C3—93no—
33C24×C3—93no—
34C24×C3—918no—
35C24×C3—99no—
36C24×C3C3×A413yescomplemented
37C22⋊A4—49no—
38C22⋊A4—129no—
39C22⋊A4—129no—
40C22⋊A4C3×A419yescomplemented
41C22×A4—418no—
42C22×A4—129no—
43C22×C32—163no—
44C3×A4—46no—
45C3×A4—123no—
46C3×A4—1627no—
47C3×A4—163no—
48C3×A4—483no—
49C3×A4—483no—
50C25—33no—
51C25—93no—
52C25—93no—
53C33—641noSylow
54C23×C3—39no—
55C23×C3—36no—
56C23×C3—93no—
57C23×C3—93no—
58C23×C3—99no—
59C23×C3—96no—
60C23×C3—99no—
61C23×C3—99no—
62C23×C3—99no—
63C23×C3—93no—
64C23×C3—918no—
65C23×C3—99no—
66C23×C3—99no—
67C23×C3—936no—
68C23×C3—96no—
69C23×C3—918no—
70C23×C3—93no—
71C2×A4—1218no—
72C2×A4—1227no—
73C2×C32—483no—
74C24—33no—
75C24—33no—
76C24—33no—
77C24—93no—
78C24—96no—
79C24—99no—
80C24—99no—
81C24—99no—
82C24—93no—
83C24—93no—
84C24—918no—
85C24—99no—
86C24C32×A413yescomplemented
87C22×C3—33no—
88C22×C3—33no—
89C22×C3—33no—
90C22×C3—93no—
91C22×C3—96no—
92C22×C3—99no—
93C22×C3—99no—
94C22×C3—99no—
95C22×C3—93no—
96C22×C3—918no—
97C22×C3—99no—
98C22×C3—93no—
99C22×C3—169no—
100C22×C3A4213yescomplemented
101A4—418no—
102A4—129no—
103A4—169no—
104A4—489no—
105A4—489no—
106C32—163no—
107C32—649no—
108C32—641no—
109C23—39no—
110C23—36no—
111C23—93no—
112C23—93no—
113C23—99no—
114C23—96no—
115C23—99no—
116C23—99no—
117C23—99no—
118C23—93no—
119C23—918no—
120C23—99no—
121C23—99no—
122C23—936no—
123C23—96no—
124C23—918no—
125C23—93no—
126C2×C3—33no—
127C2×C3—93no—
128C2×C3—93no—
129C2×C3—489no—
130C22—33no—
131C22—33no—
132C22—33no—
133C22—93no—
134C22—96no—
135C22—99no—
136C22—99no—
137C22—99no—
138C22—93no—
139C22—918no—
140C22—99no—
141C22—93no—
142C22C3×A4213yescomplemented
143C3—169no—
144C3—643no—
145C3#1059678211yescharacteristic, direct product factor
146C2—33no—
147C2—93no—
148C2—93no—
subgroupG/H
center, hypercenterC3#10596782
commutator, cosocle, hypocenterC26C33
FrattiniC1G
Fitting, socleC26×C3C32
LCSG ▷ 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
3. #10596782C3direct (N × H)
6. C26×C3C32semidirect
7. C22⋊(C22⋊A4)C32semidirect
8. C22×C22⋊A4C32semidirect
12. C3×C22⋊A4A4semidirect
23. C26C33semidirect
36. C24×C3C3×A4semidirect
40. C22⋊A4C3×A4semidirect
86. C24C32×A4semidirect
100. C22×C3A42semidirect
142. C22C3×A42semidirect
145. C3#10596782direct (N × H)

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
#105967823
C3×A424