Finite Group Databasebeta

← #67⋊(C3⋊Dic3) #11723625

Subgroups — #67⋊(C3⋊Dic3) #11723625

Order 864.

780 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 178 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
1C22×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C2×C32×#67C211yesmaximal, characteristic
3C2×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C3×#3757C211yesmaximal, characteristic, complemented
5Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
6(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
7Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
8D4×C3⋊Dic3—31nomaximal
9#67⋊Dic3—34nomaximal
10C23×C33C2211yescharacteristic
11C22×C32×D3C2211yescharacteristic
12C2×C3⋊(C3⋊Dic3)—22no—
13C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊Dic3C2211yescharacteristic
16C2×C3×C3⋊Dic3C2211yescharacteristic
17C32×#67C414yescomplemented
18C2×C32×Dic3C2211yescharacteristic
19D3×C3⋊Dic3—22no—
20C2×C32×D4—31no—
21C22×C3⋊Dic3—31no—
22C22×C3⋊Dic3—34no—
23C22×C3⋊Dic3—31no—
24C22×C3⋊Dic3—64no—
25C2×C3×#67D314yes—
26C2×D3×Dic3—34no—
27#3757—31no—
28#3757D311yescharacteristic, complemented
29C4⋊(C3⋊Dic3)—31no—
30C4×C3⋊Dic3—31no—
31C3×#217—34no—
32Dic3⋊Dic3—34no—
33(C2×D3)⋊Dic3—34no—
34Dic32—34no—
35C22×C33—21no—
36C22×C33—21no—
37C22×C33C2×C412yescomplemented
38C22×C33C2311yescharacteristic
39C2×C32×D3—21no—
40C2×C32×D3—21no—
41C2×C32×D3C2×C412yescomplemented
42C3⋊(C3⋊Dic3)—21no—
43C3⋊(C3⋊Dic3)D412yescomplemented
44C3×C3⋊Dic3—21no—
45C3×C3⋊Dic3—21no—
46C32×Dic3C2×C412yescomplemented
47D4×Dic3—94no—
48#67⋊C4—91no—
49C23×C32—24no—
50C23×C32—31no—
51C23×C32C2×D314yes—
52C23×C32C2×D311yescharacteristic
53C22×C3×D3C2×D314yes—
54C32×D4—34no—
55C2×C32×C4—31no—
56C2×C3⋊Dic3—31no—
57C2×C3⋊Dic3—31no—
58C2×C3⋊Dic3—34no—
59C2×C3⋊Dic3—34no—
60C2×C3⋊Dic3—64no—
61C2×C3⋊Dic3—64no—
62C2×C3⋊Dic3—62no—
63C2×C3⋊Dic3—62no—
64C2×C3⋊Dic3—68no—
65C2×C3⋊Dic3—616no—
66C2×C3⋊Dic3C2×D311yescharacteristic
67C2×C3⋊Dic3C2×D311yescharacteristic
68C3×#67Dic3116yescomplemented
69C2×C3×Dic3—34no—
70C2×C3×Dic3—34no—
71C2×C3×Dic3C2×D314yes—
72D3×Dic3—68no—
73C2×C33—22no—
74C2×C33C22×C411yescharacteristic
75C2×C33C2×D411yescharacteristic
76C2×C33#4111yescharacteristic
77C32×D3—22no—
78C2×C3×D4—34no—
79C2×#67C3⋊D311yescharacteristic
80C22×Dic3—94no—
81C22×Dic3—94no—
82C22×Dic3—91no—
83C22×Dic3—184no—
84C2×C4×D3—91no—
85C3×C22⋊C4—91no—
86#217—34no—
87#217—94no—
88(C2×D3)⋊C4—91no—
89C4⋊Dic3—94no—
90Dic3⋊C4—91no—
91C4×Dic3—94no—
92C4×Dic3—91no—
93C22×C32—28no—
94C22×C32—28no—
95C22×C32—24no—
96C22×C32—21no—
97C22×C32—24no—
98C22×C32—21no—
99C22×C32—28no—
100C22×C32—24no—
101C22×C32—32no—
102C22×C32—61no—
103C22×C32—61no—
104C22×C32C4×D312yescomplemented
105C22×C32C2×Dic318yescomplemented
106C22×C32C22×D314yes—
107C22×C32C22×D311yescharacteristic
108C2×C3×D3—24no—
109C2×C3×D3—24no—
110C2×C3×D3C2×Dic318yescomplemented
111C32×C4—32no—
112C3⋊Dic3—21no—
113C3⋊Dic3—21no—
114C3⋊Dic3—32no—
115C3⋊Dic3—38no—
116C3⋊Dic3—68no—
117C3⋊Dic3—68no—
118C3⋊Dic3—64no—
119C3⋊Dic3—61no—
120C3×Dic3—64no—
121C3×Dic3—64no—
122C3×Dic3C2×Dic318yescomplemented
123C4×D4—271noSylow
124C33C4×D411yescharacteristic, complemented, Sylow
125C23×C3—24no—
126C23×C3—34no—
127C23×C3D3214yes—
128C23×C3C2×C3⋊D311yescharacteristic
129C22×D3C2×C3⋊D311yescharacteristic
130C3×D4—316no—
131C2×C3×C4—34no—
132C2×C3×C4—91no—
133C2×C3×C4—91no—
134#67C3⋊Dic314yescomplemented
135C2×Dic3—34no—
136C2×Dic3—34no—
137C2×Dic3—94no—
138C2×Dic3—91no—
139C2×Dic3—94no—
140C2×Dic3—91no—
141C2×Dic3—184no—
142C2×Dic3—184no—
143C2×Dic3—188no—
144C2×Dic3—188no—
145C2×Dic3—182no—
146C2×Dic3—1816no—
147C2×Dic3C2×C3⋊D311yescharacteristic
148C4×D3—182no—
149C2×C32—216no—
150C2×C32—22no—
151C2×C32—28no—
152C2×C32—24no—
153C2×C32—24no—
154C2×C32—24no—
155C2×C32—62no—
156C2×C32C2×C4×D311yescharacteristic
157C2×C32D3×D414yes—
158C2×C32D3×D411yescharacteristic
159C2×C32#23714yes—
160C2×C32#23711yescharacteristic
161C2×C32C22×Dic314yes—
162C3×D3—28no—
163C2×D4—31no—
164C22×C4—271no—
165C22×C4—271no—
166C4⋊C4—271no—
167C22⋊C4—91no—
168C22⋊C4—271no—
169C42—271no—
170C22×C3—28no—
171C22×C3—28no—
172C22×C3—24no—
173C22×C3—24no—
174C22×C3—21no—
175C22×C3—21no—
176C22×C3—28no—
177C22×C3—24no—
178C22×C3—38no—
179C22×C3—64no—
180C22×C3—64no—
181C22×C3D3×Dic318yescomplemented
182C22×C3C2×C3⋊Dic312yescomplemented
183C22×C3C2×D3214yes—
184C22×C3C22×C3⋊D311yescharacteristic
185C2×D3—21no—
186C2×D3—21no—
187C2×D3C2×C3⋊Dic312yescomplemented
188C3×C4—38no—
189C3×C4—181no—
190C3×C4—181no—
191Dic3—64no—
192Dic3—64no—
193Dic3—98no—
194Dic3—92no—
195Dic3—188no—
196Dic3—188no—
197Dic3—184no—
198Dic3—181no—
199Dic3C2×C3⋊Dic312yescomplemented
200C32—24no—
201C32#67⋊C411yescharacteristic, complemented
202C32D4×Dic314yescomplemented
203C23—31no—
204C23D3×C3⋊D311yescharacteristic
205D4—34no—
206C2×C4—31no—
207C2×C4—91no—
208C2×C4—91no—
209C2×C4—271no—
210C2×C4—271no—
211C2×C4—542no—
212C2×C4—542no—
213C2×C3—216no—
214C2×C3—22no—
215C2×C3—28no—
216C2×C3—24no—
217C2×C3—24no—
218C2×C3—24no—
219C2×C3—68no—
220C2×C3C2×D3×Dic314yes—
221C2×C3#380514yes—
222C2×C3#381114yes—
223C2×C3D4×C3⋊D311yescharacteristic
224C2×C3#383011yescharacteristic
225C2×C3C22×C3⋊Dic311yescharacteristic
226D3—22no—
227C22—21no—
228C22—21no—
229C22—32no—
230C22—61no—
231C22—61no—
232C22D3×C3⋊Dic312yescomplemented
233C22C2×D3×C3⋊D311yescharacteristic
234C4—32no—
235C4—181no—
236C4—181no—
237C4—272no—
238C4—541no—
239C3—24no—
240C3#67⋊Dic314yescomplemented
241C3D4×C3⋊Dic311yescharacteristic, complemented
242C2—22no—
243C2—62no—
244C2C2×D3×C3⋊Dic311yescharacteristic
245C2#8902911yescharacteristic
246C2#8903911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutatorC2×C33C22×C4
FittingC23×C33C22
socle, cosocleC22×C33C23
prime radicalC2×C32×#67C2
hypercenterC23D3×C3⋊D3
hypocenterC33C4×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C3⋊(C3⋊Dic3)C2semidirect
3. C2×D3×C3⋊Dic3C2semidirect
4. C3×#3757C2semidirect
5. Dic3⋊(C3⋊Dic3)C2semidirect
6. (C2×D3)⋊(C3⋊Dic3)C2semidirect
7. Dic3×C3⋊Dic3C2semidirect
17. C32×#67C4semidirect
28. #3757D3semidirect
37. C22×C33C2×C4semidirect
41. C2×C32×D3C2×C4semidirect
43. C3⋊(C3⋊Dic3)D4semidirect
46. C32×Dic3C2×C4semidirect
68. C3×#67Dic3semidirect
104. C22×C32C4×D3semidirect
105. C22×C32C2×Dic3semidirect
110. C2×C3×D3C2×Dic3semidirect
122. C3×Dic3C2×Dic3semidirect
124. C33C4×D4semidirect
134. #67C3⋊Dic3semidirect
181. C22×C3D3×Dic3semidirect
182. C22×C3C2×C3⋊Dic3semidirect
187. C2×D3C2×C3⋊Dic3semidirect
199. Dic3C2×C3⋊Dic3semidirect
201. C32#67⋊C4semidirect
202. C32D4×Dic3semidirect
232. C22D3×C3⋊Dic3semidirect
240. C3#67⋊Dic3semidirect
241. C3D4×C3⋊Dic3semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊(C3⋊Dic3)2
C2×C32×#672
C2×D3×C3⋊Dic32
C3×#37572
Dic3⋊(C3⋊Dic3)2
(C2×D3)⋊(C3⋊Dic3)2
Dic3×C3⋊Dic32
D4×C3⋊Dic33
#67⋊Dic33