Finite Group Databasebeta

← C3×#64093 #11723839

Subgroups — C3×#64093 #11723839

Order 864.

700 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 138 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×#638C211yesmaximal, characteristic, complemented
2C2×C3×#638C211yesmaximal, characteristic, complemented
3C2×C3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C2×C3×C4×C3⋊D3C211yesmaximal, characteristic, complemented
5C33×C22⋊C4C211yesmaximal, characteristic, complemented
6C3×#3752C211yesmaximal, characteristic, complemented
7C3×#3750C211yesmaximal, characteristic, complemented
8#64093C311yesmaximal, characteristic, direct product factor
9C3×#884—34nomaximal
10C23×C33C2211yescharacteristic
11C22×C3×C3⋊D3C2211yescharacteristic
12C22×C3×C3⋊D3C2211yescharacteristic
13C2×C33×C4C2211yescharacteristic
14C2×C33×C4C2211yescharacteristic
15C3×#638—22no—
16C3×#638—22no—
17C2×C3×C3⋊Dic3C2211yescharacteristic
18C2×C3×C3⋊Dic3C2211yescharacteristic
19C3×C4⋊(C3⋊D3)—22no—
20C3×C4×C3⋊D3—22no—
21C2×#638C2×C311yescharacteristic, complemented
22C2×#638C2×C311yescharacteristic, complemented
23C2×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
24C2×C4×C3⋊D3C2×C311yescharacteristic, complemented
25C2×C3×#67—34no—
26C2×C3×#67—34no—
27C2×C3×D12—34no—
28C2×C3×C4×D3—34no—
29C32×C22⋊C4—24no—
30C32×C22⋊C4D314yescomplemented
31C32×C22⋊C4C2×C311yescharacteristic, complemented
32#3752C2×C311yescharacteristic, complemented
33#3750C2×C311yescharacteristic, complemented
34C3×(C2×D3)⋊C4—34no—
35C3×Dic3⋊C4—34no—
36C22×C33—21no—
37C22×C33—21no—
38C22×C33—21no—
39C22×C33C2311yescharacteristic
40C2×C3×C3⋊D3—21no—
41C2×C3×C3⋊D3—21no—
42C2×C3×C3⋊D3—21no—
43C2×C3×C3⋊D3—21no—
44C2×C3×C3⋊D3—21no—
45C2×C3×C3⋊D3D412yescomplemented
46C33×C4—21no—
47C33×C4—21no—
48C3×C3⋊Dic3—21no—
49C3×C3⋊Dic3D412yescomplemented
50C3×#121—91no—
51#884—34no—
52C23×C32—24no—
53C23×C32C2×D314yes—
54C23×C32C22×C311yescharacteristic
55C22×C3⋊D3C22×C311yescharacteristic
56C22×C3⋊D3C22×C311yescharacteristic
57C22×C3×D3—34no—
58C22×C3×D3—34no—
59C2×C32×C4—24no—
60C2×C32×C4—24no—
61C2×C32×C4C2×D314yes—
62C2×C32×C4C2×D314yes—
63C2×C32×C4C22×C311yescharacteristic
64C2×C32×C4C22×C311yescharacteristic
65#638—22no—
66#638—22no—
67C2×C3⋊Dic3C22×C311yescharacteristic
68C2×C3⋊Dic3C22×C311yescharacteristic
69C4⋊(C3⋊D3)—22no—
70C4×C3⋊D3—22no—
71C3×#67—68no—
72C3×#67—68no—
73C2×C3×Dic3—34no—
74C2×C3×Dic3—34no—
75C3×D12—68no—
76C3×C4×D3—68no—
77C2×C33—41no—
78C2×C33C2×D411yescharacteristic
79C2×C33C2×D411yescharacteristic
80C2×C33#4111yescharacteristic
81C3×C3⋊D3—22no—
82C3×C3⋊D3—41no—
83C2×C3×D4—91no—
84C2×C3×D4—91no—
85C2×C3×D4—91no—
86C22×C3×C4—91no—
87C2×#67—34no—
88C2×#67—34no—
89C2×D12—34no—
90C2×C4×D3—34no—
91C3×C4⋊C4—91no—
92C3×C22⋊C4—24no—
93C3×C22⋊C4—91no—
94C3×C22⋊C4C3×D314yescomplemented
95C3×C22⋊C4C3⋊D311yescharacteristic, complemented
96(C2×D3)⋊C4—34no—
97Dic3⋊C4—34no—
98C22×C32—28no—
99C22×C32—24no—
100C22×C32—21no—
101C22×C32—21no—
102C22×C32—21no—
103C22×C32—24no—
104C22×C32—24no—
105C22×C32—24no—
106C22×C32—44no—
107C22×C32—44no—
108C22×C32C22×D314yes—
109C22×C32C23×C311yescharacteristic
110C2×C3⋊D3—21no—
111C2×C3⋊D3—21no—
112C2×C3⋊D3—21no—
113C2×C3⋊D3—21no—
114C2×C3⋊D3—21no—
115C2×C3⋊D3C3×D412yescomplemented
116C2×C3×D3—38no—
117C2×C3×D3—64no—
118C2×C3×D3—64no—
119C2×C3×D3—64no—
120C2×C3×D3—64no—
121C2×C3×D3—64no—
122C32×C4—21no—
123C32×C4—21no—
124C32×C4—24no—
125C32×C4—24no—
126C32×C4—44no—
127C32×C4—44no—
128C3⋊Dic3—21no—
129C3⋊Dic3C3×D412yescomplemented
130C3×Dic3—38no—
131C3×Dic3—64no—
132#121—91noSylow
133C33#12111yescharacteristic, complemented, Sylow
134C23×C3—24no—
135C23×C3—91no—
136C23×C3—91no—
137C23×C3C2×C3×D314yes—
138C23×C3C2×C3⋊D311yescharacteristic
139C22×D3—34no—
140C22×D3—34no—
141C3×D4—182no—
142C3×D4—182no—
143C3×D4—182no—
144C2×C3×C4—24no—
145C2×C3×C4—24no—
146C2×C3×C4—91no—
147C2×C3×C4—91no—
148C2×C3×C4—182no—
149C2×C3×C4C2×C3×D314yes—
150C2×C3×C4C2×C3×D314yes—
151C2×C3×C4C2×C3⋊D311yescharacteristic
152C2×C3×C4C2×C3⋊D311yescharacteristic
153#67—68no—
154#67—68no—
155C2×Dic3—34no—
156C2×Dic3—34no—
157D12—68no—
158C4×D3—68no—
159C2×C32—24no—
160C2×C32—24no—
161C2×C32—24no—
162C2×C32—48no—
163C2×C32—44no—
164C2×C32—41no—
165C2×C32#23514yes—
166C2×C32D3×D414yes—
167C2×C32D3×D414yes—
168C2×C32C2×C3×D411yescharacteristic
169C2×C32C2×C3×D411yescharacteristic
170C2×C32C3×#4111yescharacteristic
171C3⋊D3—22no—
172C3⋊D3—41no—
173C3×D3—68no—
174C3×D3—124no—
175C2×D4—91no—
176C2×D4—91no—
177C2×D4—91no—
178C22×C4—91no—
179C4⋊C4—91no—
180C22⋊C4—91no—
181C22⋊C4C3×C3⋊D311yescharacteristic, complemented
182C22×C3—28no—
183C22×C3—24no—
184C22×C3—24no—
185C22×C3—24no—
186C22×C3—21no—
187C22×C3—21no—
188C22×C3—21no—
189C22×C3—24no—
190C22×C3—44no—
191C22×C3—44no—
192C22×C3—92no—
193C22×C3—181no—
194C22×C3—181no—
195C22×C3—181no—
196C22×C3—181no—
197C22×C3—181no—
198C22×C3C22×C3×D314yes—
199C22×C3C22×C3⋊D311yescharacteristic
200C2×D3—38no—
201C2×D3—64no—
202C2×D3—64no—
203C2×D3—64no—
204C2×D3—64no—
205C2×D3—64no—
206C3×C4—24no—
207C3×C4—24no—
208C3×C4—21no—
209C3×C4—21no—
210C3×C4—44no—
211C3×C4—44no—
212C3×C4—92no—
213C3×C4—181no—
214Dic3—38no—
215Dic3—64no—
216C32—24no—
217C32#88414yescomplemented
218C32C3×#12111yescharacteristic, complemented
219C23—91no—
220C23—91no—
221C23C2×C3×C3⋊D311yescharacteristic
222D4—182no—
223D4—182no—
224D4—182no—
225C2×C4—91no—
226C2×C4—91no—
227C2×C4—182no—
228C2×C4C2×C3×C3⋊D311yescharacteristic
229C2×C4C2×C3×C3⋊D311yescharacteristic
230C2×C3—24no—
231C2×C3—24no—
232C2×C3—24no—
233C2×C3—48no—
234C2×C3—44no—
235C2×C3—41no—
236C2×C3—182no—
237C2×C3—361no—
238C2×C3C3×#23514yes—
239C2×C3C3×D3×D414yes—
240C2×C3C3×D3×D414yes—
241C2×C3#382811yescharacteristic
242C2×C3D4×C3⋊D311yescharacteristic
243C2×C3D4×C3⋊D311yescharacteristic
244D3—68no—
245D3—124no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—92no—
250C22—181no—
251C22—181no—
252C22—181no—
253C22—181no—
254C22—181no—
255C22C22×C3×C3⋊D311yescharacteristic
256C4—21no—
257C4—21no—
258C4—92no—
259C4—181no—
260C3—24no—
261C3C3×#88414yescomplemented
262C3#6409311yescharacteristic, direct product factor
263C2—41no—
264C2—182no—
265C2—361no—
266C2C3×#382811yescharacteristic
267C2C3×D4×C3⋊D311yescharacteristic
268C2C3×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC22×C3C22×C3⋊D3
commutator, cosocleC22×C32C23×C3
FrattiniC22C22×C3×C3⋊D3
FittingC33×C22⋊C4C2
socleC22×C33C23
hypercenterC3×C22⋊C4C3⋊D3
hypocenterC32C3×#121
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#638C2semidirect
2. C2×C3×#638C2semidirect
3. C2×C3×C4⋊(C3⋊D3)C2semidirect
4. C2×C3×C4×C3⋊D3C2semidirect
5. C33×C22⋊C4C2semidirect
6. C3×#3752C2semidirect
7. C3×#3750C2semidirect
8. #64093C3direct (N × H)
21. C2×#638C2×C3semidirect
22. C2×#638C2×C3semidirect
23. C2×C4⋊(C3⋊D3)C2×C3semidirect
24. C2×C4×C3⋊D3C2×C3semidirect
30. C32×C22⋊C4D3semidirect
31. C32×C22⋊C4C2×C3semidirect
32. #3752C2×C3semidirect
33. #3750C2×C3semidirect
45. C2×C3×C3⋊D3D4semidirect
49. C3×C3⋊Dic3D4semidirect
94. C3×C22⋊C4C3×D3semidirect
95. C3×C22⋊C4C3⋊D3semidirect
115. C2×C3⋊D3C3×D4semidirect
129. C3⋊Dic3C3×D4semidirect
133. C33#121semidirect
181. C22⋊C4C3×C3⋊D3semidirect
217. C32#884semidirect
218. C32C3×#121semidirect
261. C3C3×#884semidirect
262. C3#64093direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#6382
C2×C3×C4⋊(C3⋊D3)2
C2×C3×C4×C3⋊D32
C33×C22⋊C42
C3×#37522
C3×#37502
#640933
C3×#8843