Finite Group Databasebeta

← C3×C3⋊(#396⋊C3) #13157028

Subgroups — C3×C3⋊(#396⋊C3) #13157028

Order 1728.

303 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 40 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, complemented
2C3×#4676C311yesmaximal, characteristic, complemented
3C3×#396⋊C3—31nomaximal
4C3⋊(#396⋊C3)C313yesmaximal, direct product factor
5C2×C3×D3×A4—41nomaximal
6C2×C3×(C2×D3)⋊C4—31no—
7C2×C3×Dic3⋊C4—31no—
8C32×#95C2×C311yescharacteristic, complemented
9C3×#95⋊C3—21no—
10C3×#95⋊C3—23no—
11C3×#95⋊C3D311yescharacteristic, complemented
12C3×#95⋊C3C2×C313yescomplemented
13C2×C32×A4—41no—
14C3×D3×A4—42no—
15C3×#396—31no—
16#4676C3211yescharacteristic, complemented
17#396⋊C3—33no—
18C23×C3×D3A411yescharacteristic
19C22×C3×A4—121no—
20C2×D3×A4—43no—
21C22×C32×C4—31no—
22C22×C3×Dic3—31no—
23C3×(C2×D3)⋊C4—62no—
24C3×Dic3⋊C4—62no—
25C2×C32×D3—161no—
26C32×A4—41no—
27C2×C3×C4⋊C4—91no—
28C2×C3×C22⋊C4—91no—
29C2×(C2×D3)⋊C4—31no—
30C2×Dic3⋊C4—31no—
31C3×#95—21no—
32C3×#95C3×D311yescharacteristic, complemented
33C3×#95C2×C3211yescharacteristic, complemented
34#95⋊C3—23no—
35#95⋊C3C3×D313yescomplemented
36C23×C32C2×A411yescharacteristic
37C22×C3×D3—61no—
38C22×C3×D3—61no—
39C22×C3×D3SL(2,𝔽3)12yes—
40C2×C3×A4—41no—
41C2×C3×A4—43no—
42C2×C3×A4—83no—
43C2×C3×A4—81no—
44C2×C3×A4—122no—
45D3×A4—46no—
46C2×C32×C4—61no—
47C2×C32×C4—61no—
48C2×C32×C4—61no—
49C2×C3×Dic3—32no—
50C2×C3×Dic3—61no—
51C2×C3×Dic3—61no—
52#396—31noSylow
53C2×C33—161no—
54C32×D3—162no—
55C24×C3—31no—
56C23×D3C3×A411yescharacteristic
57C22×A4—123no—
58C22×C3×C4—31no—
59C22×C3×C4—31no—
60C22×C3×C4—61no—
61C22×C3×C4—91no—
62C22×Dic3—31no—
63C3×C4⋊C4—182no—
64C3×C22⋊C4—182no—
65(C2×D3)⋊C4—62no—
66Dic3⋊C4—62no—
67C22×C32—31no—
68C22×C32—31no—
69C22×C32—481no—
70C22×C32C2×SL(2,𝔽3)11yescharacteristic
71C2×C3×D3—41no—
72C2×C3×D3—62no—
73C2×C3×D3—121no—
74C2×C3×D3—163no—
75C3×A4—41no—
76C3×A4—43no—
77C3×A4—83no—
78C3×A4—81no—
79C32×C4—121no—
80C3×Dic3—62no—
81C2×C4⋊C4—91no—
82C2×C22⋊C4—91no—
83#95C32×D311yescharacteristic, complemented
84C33—161noSylow
85C23×C3—21no—
86C23×C3—32no—
87C23×C3—181no—
88C23×C3—181no—
89C23×C3D3×A411yescharacteristic
90C23×C3C2×C3×A411yescharacteristic
91C22×D3—61no—
92C22×D3—61no—
93C22×D3C3×SL(2,𝔽3)12yes—
94C2×A4—43no—
95C2×A4—83no—
96C2×A4—126no—
97C2×C3×C4—61no—
98C2×C3×C4—61no—
99C2×C3×C4—61no—
100C2×C3×C4—61no—
101C2×C3×C4—61no—
102C2×C3×C4—61no—
103C2×C3×C4—62no—
104C2×C3×C4—92no—
105C2×C3×C4—121no—
106C2×C3×C4—121no—
107C2×C3×C4—181no—
108C2×C3×C4—181no—
109C2×Dic3—32no—
110C2×Dic3—61no—
111C2×Dic3—61no—
112C2×C32—31no—
113C2×C32—31no—
114C2×C32—161no—
115C2×C32—163no—
116C2×C32—323no—
117C2×C32—321no—
118C2×C32—482no—
119C2×C32#126⋊C311yescharacteristic
120C3×D3—42no—
121C3×D3—166no—
122C24—31no—
123C22×C4—31no—
124C22×C4—91no—
125C4⋊C4—182no—
126C22⋊C4—182no—
127C22×C3—21no—
128C22×C3—31no—
129C22×C3—31no—
130C22×C3—31no—
131C22×C3—31no—
132C22×C3—61no—
133C22×C3—61no—
134C22×C3—121no—
135C22×C3—182no—
136C22×C3—361no—
137C22×C3—483no—
138C22×C3D3×SL(2,𝔽3)11yescharacteristic
139C22×C3C2×C3×SL(2,𝔽3)11yescharacteristic
140C2×D3—41no—
141C2×D3—62no—
142C2×D3—121no—
143A4—43no—
144A4—83no—
145C3×C4—121no—
146C3×C4—121no—
147C3×C4—122no—
148C3×C4—182no—
149Dic3—62no—
150C32—161no—
151C32—163no—
152C32—323no—
153C32—321no—
154C32#396⋊C311yescharacteristic, complemented
155C23—32no—
156C23—181no—
157C23—181no—
158C23C3×D3×A411yescharacteristic
159C2×C4—61no—
160C2×C4—61no—
161C2×C4—61no—
162C2×C4—92no—
163C2×C4—181no—
164C2×C4—181no—
165C2×C3—21no—
166C2×C3—31no—
167C2×C3—31no—
168C2×C3—31no—
169C2×C3—31no—
170C2×C3—61no—
171C2×C3—61no—
172C2×C3—122no—
173C2×C3—163no—
174C2×C3—323no—
175C2×C3—486no—
176C2×C3C3⋊(#126⋊C3)11yescharacteristic
177C2×C3C3×#126⋊C311yescharacteristic
178D3—42no—
179C22—31no—
180C22—31no—
181C22—121no—
182C22—182no—
183C22—361no—
184C22C3×D3×SL(2,𝔽3)11yescharacteristic
185C4—121no—
186C4—182no—
187C3—21no—
188C3—163no—
189C3—323no—
190C3C3⋊(#396⋊C3)11yescharacteristic, direct product factor
191C3C3×#396⋊C311yescharacteristic, complemented
192C2—31no—
193C2—31no—
194C2—122no—
195C2C3×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
LCSG ▷ C3×#95
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C32×#95⋊C3C2semidirect
2. C3×#4676C3semidirect
4. C3⋊(#396⋊C3)C3direct (N × H)
8. C32×#95C2×C3semidirect
11. C3×#95⋊C3D3semidirect
12. C3×#95⋊C3C2×C3semidirect
16. #4676C32semidirect
32. C3×#95C3×D3semidirect
33. C3×#95C2×C32semidirect
35. #95⋊C3C3×D3semidirect
83. #95C32×D3semidirect
154. C32#396⋊C3semidirect
190. C3C3⋊(#396⋊C3)direct (N × H)
191. C3C3×#396⋊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×#46763
C3×#396⋊C33
C3⋊(#396⋊C3)3
C2×C3×D3×A44