Finite Group Databasebeta

← C4×#6292 #11723511

Subgroups — C4×#6292 #11723511

Order 864.

780 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 150 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×C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×C3×C4×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×#6292C211yesmaximal, characteristic
4C32×C4×Dic3C211yesmaximal, characteristic, complemented
5Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
6#88804C211yesmaximal, characteristic, complemented
7#88803C211yesmaximal, characteristic, complemented
8C4×C4⋊(C3⋊D3)—31nomaximal
9C4×#626—34nomaximal
10C22×C3⋊(C3⋊D3)C2211yescharacteristic
11C22×C3×C3⋊D3C2211yescharacteristic
12C2×C33×C4C2211yescharacteristic
13C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C4×C3⋊(C3⋊D3)—22no—
15C2×C3×C3⋊Dic3C2211yescharacteristic
16C3×C4×C3⋊D3—22no—
17C2×C32×Dic3C2211yescharacteristic
18C2×C32×Dic3C2211yescharacteristic
19#6292C414yesdirect product factor
20C2×C4⋊(C3⋊D3)—31no—
21C2×C4×C3⋊D3—31no—
22C2×C4×C3⋊D3—34no—
23C2×C4×C3⋊D3—64no—
24C2×C4×C3⋊D3D311yescharacteristic, complemented
25C2×C3×C4×D3—34no—
26C2×#626—34no—
27C32×C42—31no—
28#3752—31no—
29#3752—31no—
30C4⋊(C3⋊Dic3)—31no—
31C3×C4×Dic3D314yescomplemented
32Dic3⋊Dic3—34no—
33#3722—34no—
34(C2×D3)⋊Dic3—34no—
35C22×C33C2311yescharacteristic
36C2×C3⋊(C3⋊D3)—21no—
37C2×C3⋊(C3⋊D3)—21no—
38C2×C3⋊(C3⋊D3)C2×C412yescomplemented
39C2×C3×C3⋊D3—21no—
40C2×C3×C3⋊D3—21no—
41C2×C3×C3⋊D3C2×C412yescomplemented
42C33×C4D412yescomplemented
43C3⋊(C3⋊Dic3)—21no—
44C3×C3⋊Dic3—21no—
45C32×Dic3—21no—
46C32×Dic3C2×C412yescomplemented
47C4×#67—91no—
48C4×D12—94no—
49C22×C3⋊D3—31no—
50C22×C3⋊D3—34no—
51C22×C3⋊D3—64no—
52C22×C3⋊D3C2×D311yescharacteristic
53C22×C3×D3—34no—
54C2×C32×C4—24no—
55C2×C32×C4—31no—
56C2×C32×C4—31no—
57C2×C32×C4C2×D314yes—
58C2×C32×C4C2×D311yescharacteristic
59C2×C3⋊Dic3—31no—
60C2×C3⋊Dic3—34no—
61C2×C3⋊Dic3—64no—
62C2×C3⋊Dic3C2×D311yescharacteristic
63C4⋊(C3⋊D3)—34no—
64C4×C3⋊D3—22no—
65C4×C3⋊D3—616no—
66C4×C3⋊D3—68no—
67C4×C3⋊D3—62no—
68C2×C3×Dic3—34no—
69C2×C3×Dic3C2×D314yes—
70C2×C3×Dic3C2×D314yes—
71C3×C4×D3—68no—
72#626—316no—
73C2×C33C22×C411yescharacteristic
74C2×C33C2×D411yescharacteristic
75C2×C33#4111yescharacteristic
76C3⋊(C3⋊D3)—22no—
77C3×C3⋊D3—22no—
78C22×C3×C4—91no—
79C2×#67—91no—
80C2×D12—94no—
81C2×C4×D3—34no—
82C2×C4×D3—94no—
83C2×C4×D3—91no—
84C2×C4×D3—184no—
85C3×C42—34no—
86#217—91no—
87(C2×D3)⋊C4—94no—
88(C2×D3)⋊C4—94no—
89(C2×D3)⋊C4—91no—
90C4⋊Dic3—94no—
91Dic3⋊C4—91no—
92C4×Dic3C3⋊D311yescharacteristic, complemented
93C22×C32—24no—
94C22×C32C22×D314yes—
95C22×C32C22×D311yescharacteristic
96C2×C3⋊D3—21no—
97C2×C3⋊D3—21no—
98C2×C3⋊D3—32no—
99C2×C3⋊D3—38no—
100C2×C3⋊D3—68no—
101C2×C3⋊D3—68no—
102C2×C3⋊D3—68no—
103C2×C3⋊D3—64no—
104C2×C3⋊D3—64no—
105C2×C3⋊D3—61no—
106C2×C3⋊D3—61no—
107C2×C3⋊D3C4×D312yescomplemented
108C2×C3×D3—38no—
109C2×C3×D3—64no—
110C2×C3×D3—64no—
111C32×C4—28no—
112C32×C4—32no—
113C32×C4—61no—
114C32×C4D1218yescomplemented
115C32×C4#6712yescomplemented
116C3⋊Dic3—21no—
117C3⋊Dic3—68no—
118C3⋊Dic3—64no—
119C3⋊Dic3—61no—
120C3×Dic3—24no—
121C3×Dic3—64no—
122C3×Dic3C4×D318yescomplemented
123C4×D4—271noSylow
124C33C4×D411yescharacteristic, complemented, Sylow
125C23×C3—91no—
126C22×D3—34no—
127C22×D3—94no—
128C22×D3—91no—
129C22×D3—184no—
130C2×C3×C4—24no—
131C2×C3×C4—34no—
132C2×C3×C4—34no—
133C2×C3×C4—91no—
134C2×C3×C4—182no—
135C2×C3×C4D3214yes—
136C2×C3×C4C2×C3⋊D311yescharacteristic
137#67—94no—
138C2×Dic3—34no—
139C2×Dic3—94no—
140C2×Dic3—91no—
141C2×Dic3—184no—
142C2×Dic3C2×C3⋊D311yescharacteristic
143C2×Dic3C2×C3⋊D311yescharacteristic
144D12—916no—
145C4×D3—68no—
146C4×D3—1816no—
147C4×D3—188no—
148C4×D3—182no—
149C2×C32—24no—
150C2×C32—24no—
151C2×C32—24no—
152C2×C32C2×C4×D314yes—
153C2×C32C2×C4×D311yescharacteristic
154C2×C32C2×D1214yes—
155C2×C32#23514yes—
156C2×C32#23511yescharacteristic
157C2×C32C2×#6711yescharacteristic
158C3⋊D3—22no—
159C3⋊D3—616no—
160C3⋊D3—68no—
161C3⋊D3—62no—
162C3×D3—68no—
163C2×D4—271no—
164C22×C4—91no—
165C22×C4—271no—
166C4⋊C4—271no—
167C22⋊C4—271no—
168C22⋊C4—271no—
169C42—31no—
170C22×C3—24no—
171C22×C3—92no—
172C22×C3—181no—
173C22×C3—181no—
174C22×C3C2×D3214yes—
175C22×C3C22×C3⋊D311yescharacteristic
176C2×D3—38no—
177C2×D3—64no—
178C2×D3—64no—
179C2×D3—98no—
180C2×D3—92no—
181C2×D3—188no—
182C2×D3—188no—
183C2×D3—188no—
184C2×D3—184no—
185C2×D3—184no—
186C2×D3—181no—
187C2×D3—181no—
188C3×C4—28no—
189C3×C4—38no—
190C3×C4—64no—
191C3×C4—181no—
192C3×C4#62618yescomplemented
193C3×C4C4⋊(C3⋊D3)12yescomplemented
194Dic3—21no—
195Dic3—64no—
196Dic3—188no—
197Dic3—184no—
198Dic3—181no—
199Dic3C4×C3⋊D312yescomplemented
200C32—24no—
201C32C4×D1214yescomplemented
202C32C4×#6711yescharacteristic, complemented
203C23—91no—
204C23—271no—
205D4—274no—
206C2×C4—31no—
207C2×C4—31no—
208C2×C4—91no—
209C2×C4—182no—
210C2×C4—271no—
211C2×C4—542no—
212C2×C4D3×C3⋊D311yescharacteristic
213C2×C3—24no—
214C2×C3—24no—
215C2×C3—24no—
216C2×C3—182no—
217C2×C3#379814yes—
218C2×C3C4×D3214yes—
219C2×C3C2×#62614yes—
220C2×C3C2×C4×C3⋊D311yescharacteristic
221C2×C3C2×C4⋊(C3⋊D3)11yescharacteristic
222C2×C3#382811yescharacteristic
223D3—68no—
224D3—1816no—
225D3—182no—
226D3—188no—
227C22—92no—
228C22—181no—
229C22—181no—
230C22—272no—
231C22—541no—
232C22—541no—
233C22C2×D3×C3⋊D311yescharacteristic
234C4—32no—
235C4—61no—
236C4—181no—
237C4—541no—
238C4#629212yesdirect product factor
239C3—24no—
240C3C4×#62614yescomplemented
241C3C4×C4⋊(C3⋊D3)11yescharacteristic, complemented
242C2—182no—
243C2—542no—
244C2#8902011yescharacteristic
245C2C4×D3×C3⋊D311yescharacteristic
246C2C2×#629211yescharacteristic
subgroupG/H
center, hypercenterC2×C4D3×C3⋊D3
commutatorC2×C33C22×C4
FrattiniC22C2×D3×C3⋊D3
FittingC2×C33×C4C22
socle, cosocleC22×C33C23
prime radicalC2×#6292C2
hypocenterC33C4×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C3⋊(C3⋊D3)C2semidirect
2. C2×C3×C4×C3⋊D3C2semidirect
4. C32×C4×Dic3C2semidirect
5. Dic3⋊(C3⋊Dic3)C2semidirect
6. #88804C2semidirect
7. #88803C2semidirect
19. #6292C4direct (N × H)
24. C2×C4×C3⋊D3D3semidirect
31. C3×C4×Dic3D3semidirect
38. C2×C3⋊(C3⋊D3)C2×C4semidirect
41. C2×C3×C3⋊D3C2×C4semidirect
42. C33×C4D4semidirect
46. C32×Dic3C2×C4semidirect
92. C4×Dic3C3⋊D3semidirect
107. C2×C3⋊D3C4×D3semidirect
114. C32×C4D12semidirect
115. C32×C4#67semidirect
122. C3×Dic3C4×D3semidirect
124. C33C4×D4semidirect
192. C3×C4#626semidirect
193. C3×C4C4⋊(C3⋊D3)semidirect
199. Dic3C4×C3⋊D3semidirect
201. C32C4×D12semidirect
202. C32C4×#67semidirect
238. C4#6292direct (N × H)
240. C3C4×#626semidirect
241. C3C4×C4⋊(C3⋊D3)semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊(C3⋊D3)2
C2×C3×C4×C3⋊D32
C2×#62922
C32×C4×Dic32
Dic3⋊(C3⋊Dic3)2
#888042
#888032
C4×C4⋊(C3⋊D3)3
C4×#6263