Finite Group Databasebeta

← (#126⋊C3)⋊Dic3 #11921289

Subgroups — (#126⋊C3)⋊Dic3 #11921289

Order 1152.

247 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 247 conjugacy classes of subgroups — too many to draw. Only its 43 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×C3×#126⋊C3C211yesmaximal, characteristic
2C3⋊(#383⋊C3)C211yesmaximal, characteristic, complemented
3C3⋊(#384⋊C3)C211yesmaximal, characteristic, complemented
4#83076C311yesmaximal, characteristic, complemented
5(#126⋊C3)⋊C4—31nomaximal
6C2×Dic3×A4—41nomaximal
7C3×#126⋊C3C412yescomplemented
8C2×C3×C42⋊C3C2211yescharacteristic
9C2×C3×#126C2×C311yescharacteristic
10C2×#126⋊C3—21no—
11C2×#126⋊C3D311yescharacteristic
12#4984—31no—
13#4980—31no—
14#4951—31no—
15#4950—31no—
16#4937C2×C311yescharacteristic, complemented
17#4935C2×C311yescharacteristic, complemented
18#384⋊C3—31no—
19#383⋊C3—31no—
20C22×C3×A4—41no—
21Dic3×A4—41no—
22Dic3×A4—41no—
23C3×C42⋊C3C2×C411yescharacteristic, complemented
24#2383—31noSylow
25C23×Dic3A411yescharacteristic
26C2×C4×A4—121no—
27C3×#126—32no—
28C3×#126C3×C412yescomplemented
29C2×C3×C4⋊C4—31no—
30C2×C3×C22⋊C4—31no—
31C2×C3×C42C22×C311yescharacteristic
32C2×#217—31no—
33C2×C4⋊Dic3—31no—
34C2×C4⋊Dic3—31no—
35C2×C4⋊Dic3—31no—
36C2×C4×Dic3—31no—
37#126⋊C3—22no—
38#126⋊C3Dic312yescomplemented
39C2×C42⋊C3—21no—
40C2×C42⋊C3C2×D311yescharacteristic
41C3⋊#95—31no—
42C3⋊#95—31no—
43C3⋊#95—31no—
44C3⋊#95—31no—
45C2×C3×A4—42no—
46C2×C3×A4—41no—
47C2×C3×Dic3—161no—
48C2×#126C3×D311yescharacteristic
49#389—91no—
50#388—91no—
51#387—91no—
52#385—91no—
53#384—31no—
54#383—31no—
55C24×C3C2×A411yescharacteristic
56C22×A4—41no—
57C22×A4—81no—
58C22×C3×C4—31no—
59C22×C3×C4—31no—
60C22×Dic3—31no—
61C22×Dic3—31no—
62C22×Dic3—121no—
63C22×Dic3C2×A411yescharacteristic
64C22×Dic3C2×A411yescharacteristic
65C4×A4—121no—
66C4×A4—121no—
67C3×C4⋊C4—32no—
68C3×C4⋊C4—32no—
69C3×C22⋊C4—32no—
70C3×C22⋊C4—32no—
71C3×C42—31no—
72C3×C42C2×C3×C411yescharacteristic, complemented
73#217—121no—
74C4⋊Dic3—121no—
75C4⋊Dic3—121no—
76C4⋊Dic3—121no—
77C4×Dic3—121no—
78C42⋊C3—21no—
79C42⋊C3C2×Dic311yescharacteristic, complemented
80C22×C32—161no—
81C3×A4—41no—
82C3×Dic3—161no—
83C3×Dic3—161no—
84C23×C4—31no—
85#126—32no—
86#126C3×Dic312yescomplemented
87C2×C4⋊C4—31no—
88C2×C4⋊C4—91no—
89C2×C4⋊C4—91no—
90C2×C4⋊C4—91no—
91C2×C22⋊C4—31no—
92C2×C22⋊C4—91no—
93C2×C42—91no—
94C2×C42C2×C3×D311yescharacteristic
95#95—91no—
96#95—91no—
97#95—91no—
98#95—91no—
99C23×C3—61no—
100C23×C3—61no—
101C23×C3C4×A412yes—
102C23×C3C22×A411yescharacteristic
103C2×A4—41no—
104C2×A4—42no—
105C2×A4—81no—
106C2×A4—82no—
107C2×C3×C4—31no—
108C2×C3×C4—31no—
109C2×C3×C4—32no—
110C2×C3×C4—61no—
111C2×C3×C4—61no—
112C2×C3×C4—61no—
113C2×C3×C4—61no—
114C2×C3×C4—481no—
115C2×Dic3—41no—
116C2×Dic3—61no—
117C2×Dic3—61no—
118C2×Dic3—61no—
119C2×Dic3—61no—
120C2×Dic3—61no—
121C2×Dic3—61no—
122C2×Dic3—62no—
123C2×Dic3—121no—
124C2×C32—162no—
125C2×C32—161no—
126C24D3×A411yescharacteristic
127C22×C4—31no—
128C22×C4—31no—
129C22×C4—31no—
130C22×C4—31no—
131C22×C4—91no—
132C22×C4—91no—
133C22×C4—361no—
134C4⋊C4—32no—
135C4⋊C4—32no—
136C4⋊C4—361no—
137C4⋊C4—361no—
138C4⋊C4—361no—
139C22⋊C4—32no—
140C22⋊C4—32no—
141C22⋊C4—361no—
142C42—31no—
143C42—361no—
144C42C2×C3×Dic311yescharacteristic, complemented
145C22×C3—31no—
146C22×C3—31no—
147C22×C3—41no—
148C22×C3—62no—
149C22×C3—121no—
150C22×C3—161no—
151C22×C3—321no—
152C22×C3C2×C4×A411yescharacteristic
153A4—41no—
154A4—81no—
155C3×C4—61no—
156C3×C4—61no—
157C3×C4—62no—
158C3×C4—481no—
159C3×C4—481no—
160Dic3—41no—
161Dic3—41no—
162Dic3—121no—
163Dic3—121no—
164C32—161noSylow
165C23—61no—
166C23—61no—
167C23Dic3×A412yes—
168C23C2×D3×A411yescharacteristic
169C2×C4—31no—
170C2×C4—31no—
171C2×C4—32no—
172C2×C4—61no—
173C2×C4—61no—
174C2×C4—61no—
175C2×C4—61no—
176C2×C4—121no—
177C2×C4—181no—
178C2×C4—181no—
179C2×C4—181no—
180C2×C4—181no—
181C2×C4—181no—
182C2×C4—181no—
183C2×C4—182no—
184C2×C4—361no—
185C2×C3—31no—
186C2×C3—31no—
187C2×C3—42no—
188C2×C3—162no—
189C2×C3—161no—
190C2×C3—322no—
191C2×C3—321no—
192C2×C3#560⋊C311yescharacteristic
193C22—31no—
194C22—31no—
195C22—41no—
196C22—62no—
197C22—121no—
198C22C2×Dic3×A411yescharacteristic
199C4—61no—
200C4—61no—
201C4—62no—
202C4—121no—
203C4—121no—
204C4—361no—
205C4—361no—
206C3—161no—
207C3—321no—
208C3(#126⋊C3)⋊C411yescharacteristic, complemented
209C2—31no—
210C2—31no—
211C2—42no—
212C2(#126⋊C3)⋊D311yescharacteristic
subgroupG/H
center, hypercenterC2(#126⋊C3)⋊D3
commutator, hypocenterC3×C42C2×C3×C4
FrattiniC23C2×D3×A4
FittingC2×C3×#126C2×C3
socleC23×C3C22×A4
prime radicalC2×C3×#126⋊C3C2
cosocleC2×C3×C42C22×C3
LCSG ▷ C3×C42
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. C3⋊(#383⋊C3)C2semidirect
3. C3⋊(#384⋊C3)C2semidirect
4. #83076C3semidirect
7. C3×#126⋊C3C4semidirect
16. #4937C2×C3semidirect
17. #4935C2×C3semidirect
23. C3×C42⋊C3C2×C4semidirect
28. C3×#126C3×C4semidirect
38. #126⋊C3Dic3semidirect
72. C3×C42C2×C3×C4semidirect
79. C42⋊C3C2×Dic3semidirect
86. #126C3×Dic3semidirect
144. C42C2×C3×Dic3semidirect
208. C3(#126⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#126⋊C32
C3⋊(#383⋊C3)2
C3⋊(#384⋊C3)2
#830763
(#126⋊C3)⋊C43
C2×Dic3×A44