Finite Group Databasebeta

← C3×C3⋊(#401⋊C3) #13157025

Subgroups — C3×C3⋊(#401⋊C3) #13157025

Order 1728.

215 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 215 conjugacy classes of subgroups — too many to draw. Only its 36 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
1C32×#95⋊C3C211yesmaximal, characteristic
2C3×C3⋊#401C311yesmaximal, characteristic, complemented
3C3×#401⋊C3—31nomaximal
4C3⋊(#401⋊C3)C313yesmaximal, direct product factor
5C3×Dic3×A4—41nomaximal
6C32×#95C2×C311yescharacteristic
7C3×C3⋊#95—31no—
8C3×C3⋊#95—31no—
9C3×#95⋊C3—21no—
10C3×#95⋊C3—23no—
11C3×#95⋊C3D311yescharacteristic
12C3×#95⋊C3C2×C313yes—
13C2×C32×A4—41no—
14C3×#401—31no—
15C3⋊#401C3211yescharacteristic, complemented
16#401⋊C3—33no—
17C22×C32×C4—31no—
18C22×C3×Dic3—31no—
19C22×C3×Dic3A411yescharacteristic
20C3×C4×A4—121no—
21Dic3×A4—43no—
22C32×A4—41no—
23C32×Dic3—161no—
24C3×#95—21no—
25C3×#95—91no—
26C3×#95—91no—
27C3×#95C3×D311yescharacteristic
28C3×#95C2×C3211yescharacteristic
29C3⋊#95—31no—
30C3⋊#95—31no—
31#95⋊C3—23no—
32#95⋊C3C3×D313yes—
33C23×C32C2×A411yescharacteristic
34C2×C3×A4—41no—
35C2×C3×A4—43no—
36C2×C3×A4—83no—
37C2×C3×A4—81no—
38C2×C32×C4—61no—
39C2×C32×C4—62no—
40C2×C3×Dic3—61no—
41C2×C3×Dic3—61no—
42C2×C3×Dic3—61no—
43C2×C3×Dic3—61no—
44#401—31noSylow
45C2×C33—161no—
46C22×C3×C4—31no—
47C22×C3×C4—31no—
48C22×C3×C4—31no—
49C22×C3×C4—61no—
50C22×C3×C4—91no—
51C22×Dic3—31no—
52C22×Dic3C3×A411yescharacteristic
53C4×A4—123no—
54C22×C32—31no—
55C22×C32—31no—
56C22×C32#41⋊C311yescharacteristic
57C3×A4—41no—
58C3×A4—43no—
59C3×A4—83no—
60C3×A4—81no—
61C32×C4—121no—
62C32×C4—481no—
63C3×Dic3—41no—
64C3×Dic3—121no—
65C3×Dic3—163no—
66#95—91no—
67#95—91no—
68#95C32×D311yescharacteristic
69C33—161noSylow
70C23×C3—21no—
71C23×C3D3×A411yescharacteristic
72C23×C3C2×C3×A411yescharacteristic
73C2×A4—43no—
74C2×A4—83no—
75C2×C3×C4—61no—
76C2×C3×C4—61no—
77C2×C3×C4—62no—
78C2×C3×C4—62no—
79C2×C3×C4—62no—
80C2×C3×C4—122no—
81C2×C3×C4—181no—
82C2×C3×C4—181no—
83C2×C3×C4—181no—
84C2×C3×C4—181no—
85C2×Dic3—61no—
86C2×Dic3—61no—
87C2×Dic3—61no—
88C2×Dic3—61no—
89C2×C32—31no—
90C2×C32—31no—
91C2×C32—161no—
92C2×C32—163no—
93C2×C32—323no—
94C2×C32—321no—
95C2×C32#126⋊C311yescharacteristic
96C22×C4—31no—
97C22×C4—31no—
98C22×C4—91no—
99C22×C3—21no—
100C22×C3—31no—
101C22×C3—31no—
102C22×C3—31no—
103C22×C3—31no—
104C22×C3—61no—
105C22×C3—61no—
106C22×C3C3⋊(#41⋊C3)11yescharacteristic
107C22×C3C3×#41⋊C311yescharacteristic
108A4—43no—
109A4—83no—
110C3×C4—121no—
111C3×C4—121no—
112C3×C4—122no—
113C3×C4—121no—
114C3×C4—361no—
115C3×C4—483no—
116Dic3—41no—
117Dic3—121no—
118C32—161no—
119C32—163no—
120C32—323no—
121C32—321no—
122C32#401⋊C311yescharacteristic, complemented
123C23C3×D3×A411yescharacteristic
124C2×C4—61no—
125C2×C4—62no—
126C2×C4—181no—
127C2×C4—181no—
128C2×C4—181no—
129C2×C4—181no—
130C2×C3—21no—
131C2×C3—31no—
132C2×C3—31no—
133C2×C3—31no—
134C2×C3—31no—
135C2×C3—61no—
136C2×C3—61no—
137C2×C3—163no—
138C2×C3—323no—
139C2×C3C3⋊(#126⋊C3)11yescharacteristic
140C2×C3C3×#126⋊C311yescharacteristic
141C22—31no—
142C22—31no—
143C22C3×C3⋊(#41⋊C3)11yescharacteristic
144C4—121no—
145C4—121no—
146C4—361no—
147C3—21no—
148C3—163no—
149C3—323no—
150C3C3⋊(#401⋊C3)11yescharacteristic, direct product factor
151C3C3×#401⋊C311yescharacteristic, complemented
152C2—31no—
153C2—31no—
154C2C3×C3⋊(#126⋊C3)11yescharacteristic
subgroupG/H
center, hypercenterC2×C3C3⋊(#126⋊C3)
commutator, cosocle, hypocenterC3×#95C2×C32
FrattiniC23C3×D3×A4
FittingC32×#95C2×C3
socleC23×C32C2×A4
prime radicalC32×#95⋊C3C2
LCSG ▷ C3×#95
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
2. C3×C3⋊#401C3semidirect
4. C3⋊(#401⋊C3)C3direct (N × H)
15. C3⋊#401C32semidirect
122. C32#401⋊C3semidirect
150. C3C3⋊(#401⋊C3)direct (N × H)
151. C3C3×#401⋊C3semidirect

Maximal subgroups

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

subgroupindex
C32×#95⋊C32
C3×C3⋊#4013
C3×#401⋊C33
C3⋊(#401⋊C3)3
C3×Dic3×A44