Finite Group Databasebeta

← C3×C3⋊(#383⋊C3) #13157031

Subgroups — C3×C3⋊(#383⋊C3) #13157031

Order 1728.

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

normal subgroups only

This group has 252 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×#4937C311yesmaximal, characteristic, complemented
3C3×#383⋊C3—31nomaximal
4C3⋊(#383⋊C3)C313yesmaximal, direct product factor
5C3×Dic3×A4—41nomaximal
6C32×C42⋊C3C411yescharacteristic, complemented
7C2×C32×C42C2×C311yescharacteristic
8C2×C3×C42⋊C3—21no—
9C2×C3×C42⋊C3—23no—
10C2×C3×C42⋊C3D311yescharacteristic
11C2×C3×C42⋊C3C2×C313yes—
12C3×C3⋊#95—31no—
13C3×C3⋊#95—31no—
14C2×C32×A4—41no—
15C3×#383—31no—
16#4937C3211yescharacteristic, complemented
17#383⋊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×C42⋊C3—21no—
26C3×C42⋊C3—23no—
27C3×C42⋊C3Dic311yescharacteristic, complemented
28C3×C42⋊C3C3×C413yescomplemented
29C32×A4—41no—
30C32×Dic3—161no—
31C2×C3×C42—21no—
32C2×C3×C42C3×D311yescharacteristic
33C2×C3×C42C2×C3211yescharacteristic
34C2×C42⋊C3—23no—
35C2×C42⋊C3C3×D313yes—
36C3×#95—91no—
37C3×#95—91no—
38C3⋊#95—31no—
39C3⋊#95—31no—
40C23×C32C2×A411yescharacteristic
41C2×C3×A4—41no—
42C2×C3×A4—43no—
43C2×C3×A4—83no—
44C2×C3×A4—81no—
45C2×C32×C4—31no—
46C2×C32×C4—31no—
47C2×C32×C4—61no—
48C2×C32×C4—61no—
49C2×C3×Dic3—61no—
50C2×C3×Dic3—62no—
51C2×C3×Dic3—61no—
52#383—31noSylow
53C2×C33—161no—
54C22×C3×C4—31no—
55C22×C3×C4—31no—
56C22×C3×C4—31no—
57C22×C3×C4—61no—
58C22×C3×C4—91no—
59C22×Dic3—31no—
60C22×Dic3C3×A411yescharacteristic
61C4×A4—123no—
62C3×C42—21no—
63C3×C42—31no—
64C3×C42—31no—
65C3×C42—61no—
66C3×C42C3×Dic311yescharacteristic, complemented
67C3×C42C32×C411yescharacteristic, complemented
68C42⋊C3—23no—
69C42⋊C3C3×Dic313yescomplemented
70C22×C32—31no—
71C22×C32—31no—
72C22×C32C4×A411yescharacteristic
73C3×A4—41no—
74C3×A4—43no—
75C3×A4—83no—
76C3×A4—81no—
77C32×C4—61no—
78C32×C4—61no—
79C32×C4—481no—
80C3×Dic3—41no—
81C3×Dic3—121no—
82C3×Dic3—163no—
83C2×C42C32×D311yescharacteristic
84#95—91no—
85#95—91no—
86C33—161noSylow
87C23×C3—21no—
88C23×C3D3×A411yescharacteristic
89C23×C3C2×C3×A411yescharacteristic
90C2×A4—43no—
91C2×A4—83no—
92C2×C3×C4—31no—
93C2×C3×C4—31no—
94C2×C3×C4—31no—
95C2×C3×C4—31no—
96C2×C3×C4—61no—
97C2×C3×C4—61no—
98C2×C3×C4—61no—
99C2×C3×C4—61no—
100C2×C3×C4—61no—
101C2×C3×C4—61no—
102C2×C3×C4—62no—
103C2×C3×C4—62no—
104C2×C3×C4—181no—
105C2×C3×C4—182no—
106C2×C3×C4—181no—
107C2×Dic3—61no—
108C2×Dic3—62no—
109C2×Dic3—61no—
110C2×C32—31no—
111C2×C32—31no—
112C2×C32—161no—
113C2×C32—163no—
114C2×C32—323no—
115C2×C32—321no—
116C2×C32#126⋊C311yescharacteristic
117C22×C4—31no—
118C22×C4—31no—
119C22×C4—91no—
120C42—31no—
121C42C32×Dic311yescharacteristic, complemented
122C22×C3—21no—
123C22×C3—31no—
124C22×C3—31no—
125C22×C3—31no—
126C22×C3—31no—
127C22×C3—61no—
128C22×C3—61no—
129C22×C3Dic3×A411yescharacteristic
130C22×C3C3×C4×A411yescharacteristic
131A4—43no—
132A4—83no—
133C3×C4—61no—
134C3×C4—61no—
135C3×C4—61no—
136C3×C4—61no—
137C3×C4—62no—
138C3×C4—62no—
139C3×C4—121no—
140C3×C4—361no—
141C3×C4—483no—
142Dic3—41no—
143Dic3—121no—
144C32—161no—
145C32—163no—
146C32—323no—
147C32—321no—
148C32#383⋊C311yescharacteristic, complemented
149C23C3×D3×A411yescharacteristic
150C2×C4—31no—
151C2×C4—31no—
152C2×C4—61no—
153C2×C4—61no—
154C2×C4—181no—
155C2×C4—182no—
156C2×C4—181no—
157C2×C3—21no—
158C2×C3—31no—
159C2×C3—31no—
160C2×C3—31no—
161C2×C3—31no—
162C2×C3—61no—
163C2×C3—61no—
164C2×C3—163no—
165C2×C3—323no—
166C2×C3C3⋊(#126⋊C3)11yescharacteristic
167C2×C3C3×#126⋊C311yescharacteristic
168C22—31no—
169C22—31no—
170C22C3×Dic3×A411yescharacteristic
171C4—61no—
172C4—61no—
173C4—121no—
174C4—361no—
175C3—21no—
176C3—163no—
177C3—323no—
178C3C3⋊(#383⋊C3)11yescharacteristic, direct product factor
179C3C3×#383⋊C311yescharacteristic, complemented
180C2—31no—
181C2—31no—
182C2C3×C3⋊(#126⋊C3)11yescharacteristic
subgroupG/H
center, hypercenterC2×C3C3⋊(#126⋊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×#4937C3semidirect
4. C3⋊(#383⋊C3)C3direct (N × H)
6. C32×C42⋊C3C4semidirect
16. #4937C32semidirect
24. C32×C42C3×C4semidirect
27. C3×C42⋊C3Dic3semidirect
28. C3×C42⋊C3C3×C4semidirect
66. C3×C42C3×Dic3semidirect
67. C3×C42C32×C4semidirect
69. C42⋊C3C3×Dic3semidirect
121. C42C32×Dic3semidirect
148. C32#383⋊C3semidirect
178. C3C3⋊(#383⋊C3)direct (N × H)
179. C3C3×#383⋊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×#49373
C3×#383⋊C33
C3⋊(#383⋊C3)3
C3×Dic3×A44