Finite Group Databasebeta

← C3×C3⋊(#384⋊C3) #13157030

Subgroups — C3×C3⋊(#384⋊C3) #13157030

Order 1728.

284 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 48 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
1C2×C32×C42⋊C3C211yesmaximal, characteristic
2C3×#4935C311yesmaximal, characteristic, complemented
3C3×#384⋊C3—31nomaximal
4C3⋊(#384⋊C3)C313yesmaximal, direct product factor
5C3×Dic3×A4—41nomaximal
6C32×C42⋊C3C411yescharacteristic, complemented
7C2×C32×C42C2×C311yescharacteristic
8C2×C3×C4⋊Dic3—31no—
9C2×C3×C4⋊Dic3—31no—
10C2×C3×C42⋊C3—21no—
11C2×C3×C42⋊C3—23no—
12C2×C3×C42⋊C3D311yescharacteristic
13C2×C3×C42⋊C3C2×C313yes—
14C2×C32×A4—41no—
15C3×#384—31no—
16#4935C3211yescharacteristic, complemented
17#384⋊C3—33no—
18C22×C32×C4—31no—
19C22×C3×Dic3—31no—
20C22×C3×Dic3A411yescharacteristic
21C3×C4×A4—121no—
22Dic3×A4—43no—
23C32×C42—31no—
24C32×C42C3×C411yescharacteristic, complemented
25C3×C4⋊Dic3—62no—
26C3×C4⋊Dic3—62no—
27C3×C42⋊C3—21no—
28C3×C42⋊C3—23no—
29C3×C42⋊C3Dic311yescharacteristic, complemented
30C3×C42⋊C3C3×C413yescomplemented
31C32×A4—41no—
32C32×Dic3—161no—
33C2×C3×C4⋊C4—91no—
34C2×C3×C4⋊C4—91no—
35C2×C3×C42—21no—
36C2×C3×C42C3×D311yescharacteristic
37C2×C3×C42C2×C3211yescharacteristic
38C2×C4⋊Dic3—31no—
39C2×C4⋊Dic3—31no—
40C2×C42⋊C3—23no—
41C2×C42⋊C3C3×D313yes—
42C23×C32C2×A411yescharacteristic
43C2×C3×A4—41no—
44C2×C3×A4—43no—
45C2×C3×A4—83no—
46C2×C3×A4—81no—
47C2×C32×C4—32no—
48C2×C32×C4—32no—
49C2×C32×C4—31no—
50C2×C32×C4—31no—
51C2×C3×Dic3—61no—
52C2×C3×Dic3—61no—
53C2×C3×Dic3—62no—
54#384—31noSylow
55C2×C33—161no—
56C22×C3×C4—31no—
57C22×C3×C4—31no—
58C22×C3×C4—31no—
59C22×C3×C4—61no—
60C22×C3×C4—91no—
61C22×Dic3—31no—
62C22×Dic3C3×A411yescharacteristic
63C4×A4—123no—
64C3×C4⋊C4—182no—
65C3×C4⋊C4—182no—
66C3×C42—21no—
67C3×C42—31no—
68C3×C42—31no—
69C3×C42—61no—
70C3×C42C3×Dic311yescharacteristic, complemented
71C3×C42C32×C411yescharacteristic, complemented
72C4⋊Dic3—62no—
73C4⋊Dic3—62no—
74C42⋊C3—23no—
75C42⋊C3C3×Dic313yescomplemented
76C22×C32—31no—
77C22×C32—31no—
78C22×C32C4×A411yescharacteristic
79C3×A4—41no—
80C3×A4—43no—
81C3×A4—83no—
82C3×A4—81no—
83C32×C4—32no—
84C32×C4—32no—
85C32×C4—481no—
86C3×Dic3—41no—
87C3×Dic3—121no—
88C3×Dic3—163no—
89C2×C4⋊C4—91no—
90C2×C4⋊C4—91no—
91C2×C42C32×D311yescharacteristic
92C33—161noSylow
93C23×C3—21no—
94C23×C3D3×A411yescharacteristic
95C23×C3C2×C3×A411yescharacteristic
96C2×A4—43no—
97C2×A4—83no—
98C2×C3×C4—32no—
99C2×C3×C4—32no—
100C2×C3×C4—32no—
101C2×C3×C4—32no—
102C2×C3×C4—31no—
103C2×C3×C4—31no—
104C2×C3×C4—31no—
105C2×C3×C4—31no—
106C2×C3×C4—61no—
107C2×C3×C4—61no—
108C2×C3×C4—62no—
109C2×C3×C4—62no—
110C2×C3×C4—181no—
111C2×C3×C4—181no—
112C2×C3×C4—182no—
113C2×Dic3—61no—
114C2×Dic3—61no—
115C2×Dic3—62no—
116C2×C32—31no—
117C2×C32—31no—
118C2×C32—161no—
119C2×C32—163no—
120C2×C32—323no—
121C2×C32—321no—
122C2×C32#127⋊C311yescharacteristic
123C22×C4—31no—
124C22×C4—31no—
125C22×C4—91no—
126C4⋊C4—182no—
127C4⋊C4—182no—
128C42—31no—
129C42C32×Dic311yescharacteristic, complemented
130C22×C3—21no—
131C22×C3—31no—
132C22×C3—31no—
133C22×C3—31no—
134C22×C3—31no—
135C22×C3—61no—
136C22×C3—61no—
137C22×C3Dic3×A411yescharacteristic
138C22×C3C3×C4×A411yescharacteristic
139A4—43no—
140A4—83no—
141C3×C4—32no—
142C3×C4—32no—
143C3×C4—32no—
144C3×C4—32no—
145C3×C4—62no—
146C3×C4—62no—
147C3×C4—121no—
148C3×C4—361no—
149C3×C4—483no—
150Dic3—41no—
151Dic3—121no—
152C32—161no—
153C32—163no—
154C32—323no—
155C32—321no—
156C32#384⋊C311yescharacteristic, complemented
157C23C3×D3×A411yescharacteristic
158C2×C4—32no—
159C2×C4—32no—
160C2×C4—31no—
161C2×C4—31no—
162C2×C4—181no—
163C2×C4—181no—
164C2×C4—182no—
165C2×C3—21no—
166C2×C3—31no—
167C2×C3—31no—
168C2×C3—31no—
169C2×C3—31no—
170C2×C3—61no—
171C2×C3—61no—
172C2×C3—163no—
173C2×C3—323no—
174C2×C3C3⋊(#127⋊C3)11yescharacteristic
175C2×C3C3×#127⋊C311yescharacteristic
176C22—31no—
177C22—31no—
178C22C3×Dic3×A411yescharacteristic
179C4—32no—
180C4—32no—
181C4—121no—
182C4—361no—
183C3—21no—
184C3—163no—
185C3—323no—
186C3C3⋊(#384⋊C3)11yescharacteristic, direct product factor
187C3C3×#384⋊C311yescharacteristic, complemented
188C2—31no—
189C2—31no—
190C2C3×C3⋊(#127⋊C3)11yescharacteristic
subgroupG/H
center, hypercenterC2×C3C3⋊(#127⋊C3)
commutator, hypocenterC3×C42C32×C4
FrattiniC23C3×D3×A4
FittingC2×C32×C42C2×C3
socleC23×C32C2×A4
prime radicalC2×C32×C42⋊C3C2
cosocleC2×C3×C42C2×C32
LCSG ▷ C3×C42
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
2. C3×#4935C3semidirect
4. C3⋊(#384⋊C3)C3direct (N × H)
6. C32×C42⋊C3C4semidirect
16. #4935C32semidirect
24. C32×C42C3×C4semidirect
29. C3×C42⋊C3Dic3semidirect
30. C3×C42⋊C3C3×C4semidirect
70. C3×C42C3×Dic3semidirect
71. C3×C42C32×C4semidirect
75. C42⋊C3C3×Dic3semidirect
129. C42C32×Dic3semidirect
156. C32#384⋊C3semidirect
186. C3C3⋊(#384⋊C3)direct (N × H)
187. C3C3×#384⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C42⋊C32
C3×#49353
C3×#384⋊C33
C3⋊(#384⋊C3)3
C3×Dic3×A44