Finite Group Databasebeta

← C3×C4×#638 #11723883

Subgroups — C3×C4×#638 #11723883

Order 864.

780 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 194 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×C33×C4C211yesmaximal, characteristic, complemented
2C2×C3×#638C211yesmaximal, characteristic
3C2×C3×C4×C3⋊D3C211yesmaximal, characteristic, complemented
4C3×#3757C211yesmaximal, characteristic, complemented
5C3×#3752C211yesmaximal, characteristic, complemented
6C3×#3750C211yesmaximal, characteristic, complemented
7C3×C4×C3⋊Dic3C211yesmaximal, characteristic, complemented
8C4×#638C311yesmaximal, characteristic, direct product factor
9C3×C4×#67—34nomaximal
10C23×C33C2211yescharacteristic
11C22×C3×C3⋊D3C2211yescharacteristic
12C2×C33×C4—22no—
13C2×C33×C4C2211yescharacteristic
14C2×C33×C4C2211yescharacteristic
15C3×#638C414yesdirect product factor
16C2×C3×C3⋊Dic3C2211yescharacteristic
17C2×C3×C3⋊Dic3C2211yescharacteristic
18C2×C3×C3⋊Dic3C2211yescharacteristic
19C3×C4×C3⋊D3—22no—
20C22×C32×C4—24no—
21C22×C32×C4D314yescomplemented
22C22×C32×C4C2×C311yescharacteristic, complemented
23C2×#638C2×C311yescharacteristic
24C2×C4×C3⋊D3C2×C311yescharacteristic, complemented
25C2×C3×#67—34no—
26C2×C3×C4×D3—34no—
27#3757C2×C311yescharacteristic, complemented
28#3752C2×C311yescharacteristic, complemented
29#3750C2×C311yescharacteristic, complemented
30C4×C3⋊Dic3C2×C311yescharacteristic, complemented
31C3×#217—34no—
32C3×(C2×D3)⋊C4—34no—
33C3×Dic3⋊C4—34no—
34C3×C4×Dic3—34no—
35C22×C33—21no—
36C22×C33—21no—
37C22×C33C2×C412yescomplemented
38C22×C33C2311yescharacteristic
39C2×C3×C3⋊D3—21no—
40C2×C3×C3⋊D3—21no—
41C2×C3×C3⋊D3C2×C412yescomplemented
42C33×C4—21no—
43C33×C4D412yescomplemented
44C3×C3⋊Dic3—21no—
45C3×C3⋊Dic3—21no—
46C3×C3⋊Dic3C2×C412yescomplemented
47C3×C4×D4—91no—
48C4×#67—34no—
49C23×C32—24no—
50C23×C32C2×D314yes—
51C23×C32C22×C311yescharacteristic
52C22×C3⋊D3C22×C311yescharacteristic
53C22×C3×D3—34no—
54C2×C32×C4—216no—
55C2×C32×C4—28no—
56C2×C32×C4—22no—
57C2×C32×C4—24no—
58C2×C32×C4—24no—
59C2×C32×C4C2×D314yes—
60C2×C32×C4C2×D314yes—
61C2×C32×C4C22×C311yescharacteristic
62C2×C32×C4C22×C311yescharacteristic
63#638C3×C414yesdirect product factor
64C2×C3⋊Dic3C22×C311yescharacteristic
65C2×C3⋊Dic3C22×C311yescharacteristic
66C2×C3⋊Dic3C22×C311yescharacteristic
67C4×C3⋊D3—22no—
68C3×#67—316no—
69C2×C3×Dic3—34no—
70C2×C3×Dic3—34no—
71C2×C3×Dic3—34no—
72C3×C4×D3—68no—
73C2×C33—22no—
74C2×C33C22×C411yescharacteristic
75C2×C33C2×D411yescharacteristic
76C2×C33#4111yescharacteristic
77C3×C3⋊D3—22no—
78C2×C3×D4—91no—
79C22×C3×C4—24no—
80C22×C3×C4—91no—
81C22×C3×C4C3×D314yescomplemented
82C22×C3×C4C3⋊D311yescharacteristic, complemented
83C2×#67—34no—
84C2×C4×D3—34no—
85C3×C4⋊C4—91no—
86C3×C22⋊C4—91no—
87C3×C22⋊C4—91no—
88C3×C42—91no—
89#217—34no—
90(C2×D3)⋊C4—34no—
91Dic3⋊C4—34no—
92C4×Dic3—34no—
93C22×C32—28no—
94C22×C32—28no—
95C22×C32—24no—
96C22×C32—24no—
97C22×C32—21no—
98C22×C32—21no—
99C22×C32—28no—
100C22×C32—24no—
101C22×C32C4×D318yescomplemented
102C22×C32C2×C3×C412yescomplemented
103C22×C32C22×D314yes—
104C22×C32C23×C311yescharacteristic
105C2×C3⋊D3—21no—
106C2×C3⋊D3—21no—
107C2×C3⋊D3C2×C3×C412yescomplemented
108C2×C3×D3—38no—
109C2×C3×D3—64no—
110C2×C3×D3—64no—
111C32×C4—28no—
112C32×C4—24no—
113C32×C4—21no—
114C32×C4—28no—
115C32×C4#6718yescomplemented
116C32×C4C3×D412yescomplemented
117C3⋊Dic3—21no—
118C3⋊Dic3—21no—
119C3⋊Dic3C2×C3×C412yescomplemented
120C3×Dic3—38no—
121C3×Dic3—64no—
122C3×Dic3—64no—
123C4×D4—91noSylow
124C33C4×D411yescharacteristic, complemented, Sylow
125C23×C3—24no—
126C23×C3—91no—
127C23×C3C2×C3×D314yes—
128C23×C3C2×C3⋊D311yescharacteristic
129C22×D3—34no—
130C3×D4—94no—
131C2×C3×C4—216no—
132C2×C3×C4—28no—
133C2×C3×C4—22no—
134C2×C3×C4—24no—
135C2×C3×C4—24no—
136C2×C3×C4—91no—
137C2×C3×C4—91no—
138C2×C3×C4—91no—
139C2×C3×C4—182no—
140C2×C3×C4C2×C3×D314yes—
141C2×C3×C4C2×C3×D314yes—
142C2×C3×C4C2×C3⋊D311yescharacteristic
143C2×C3×C4C2×C3⋊D311yescharacteristic
144#67—316no—
145C2×Dic3—34no—
146C2×Dic3—34no—
147C2×Dic3—34no—
148C4×D3—68no—
149C2×C32—216no—
150C2×C32—28no—
151C2×C32—22no—
152C2×C32—24no—
153C2×C32—24no—
154C2×C32—24no—
155C2×C32C2×C4×D314yes—
156C2×C32#23514yes—
157C2×C32C2×#6714yes—
158C2×C32C22×C3×C411yescharacteristic
159C2×C32C2×C3×D411yescharacteristic
160C2×C32C3×#4111yescharacteristic
161C3⋊D3—22no—
162C3×D3—68no—
163C2×D4—91no—
164C22×C4—91no—
165C22×C4C3×C3⋊D311yescharacteristic, complemented
166C4⋊C4—91no—
167C22⋊C4—91no—
168C22⋊C4—91no—
169C42—91no—
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—92no—
179C22×C3—181no—
180C22×C3—181no—
181C22×C3C3×C4×D318yescomplemented
182C22×C3C4×C3⋊D312yescomplemented
183C22×C3C22×C3×D314yes—
184C22×C3C22×C3⋊D311yescharacteristic
185C2×D3—38no—
186C2×D3—64no—
187C2×D3—64no—
188C3×C4—28no—
189C3×C4—24no—
190C3×C4—21no—
191C3×C4—28no—
192C3×C4—92no—
193C3×C4—181no—
194C3×C4—181no—
195C3×C4C3×#6718yescomplemented
196C3×C4#63812yesdirect product factor
197Dic3—38no—
198Dic3—64no—
199Dic3—64no—
200C32—24no—
201C32C4×#6714yescomplemented
202C32C3×C4×D411yescharacteristic, complemented
203C23—91no—
204C23C2×C3×C3⋊D311yescharacteristic
205D4—94no—
206C2×C4—22no—
207C2×C4—91no—
208C2×C4—91no—
209C2×C4—91no—
210C2×C4—182no—
211C2×C4C2×C3×C3⋊D311yescharacteristic
212C2×C4C2×C3×C3⋊D311yescharacteristic
213C2×C3—216no—
214C2×C3—28no—
215C2×C3—22no—
216C2×C3—24no—
217C2×C3—24no—
218C2×C3—24no—
219C2×C3—182no—
220C2×C3C2×C3×C4×D314yes—
221C2×C3C3×#23514yes—
222C2×C3C2×C3×#6714yes—
223C2×C3C2×C4×C3⋊D311yescharacteristic
224C2×C3#382811yescharacteristic
225C2×C3C2×#63811yescharacteristic
226D3—68no—
227C22—21no—
228C22—21no—
229C22—92no—
230C22—181no—
231C22—181no—
232C22C3×C4×C3⋊D312yescomplemented
233C22C22×C3×C3⋊D311yescharacteristic
234C4—21no—
235C4—92no—
236C4—181no—
237C4—181no—
238C4C3×#63812yesdirect product factor
239C3—24no—
240C3C3×C4×#6714yescomplemented
241C3C4×#63811yescharacteristic, direct product factor
242C2—22no—
243C2—182no—
244C2C2×C3×C4×C3⋊D311yescharacteristic
245C2C3×#382811yescharacteristic
246C2C2×C3×#63811yescharacteristic
subgroupG/H
centerC2×C3×C4C2×C3⋊D3
commutatorC2×C32C22×C3×C4
FrattiniC22C22×C3×C3⋊D3
FittingC22×C33×C4C2
socleC22×C33C23
prime radicalC2×C3×#638C2
cosocleC22×C32C23×C3
hypercenterC22×C3×C4C3⋊D3
hypocenterC32C3×C4×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C33×C4C2semidirect
3. C2×C3×C4×C3⋊D3C2semidirect
4. C3×#3757C2semidirect
5. C3×#3752C2semidirect
6. C3×#3750C2semidirect
7. C3×C4×C3⋊Dic3C2semidirect
8. C4×#638C3direct (N × H)
15. C3×#638C4direct (N × H)
21. C22×C32×C4D3semidirect
22. C22×C32×C4C2×C3semidirect
24. C2×C4×C3⋊D3C2×C3semidirect
27. #3757C2×C3semidirect
28. #3752C2×C3semidirect
29. #3750C2×C3semidirect
30. C4×C3⋊Dic3C2×C3semidirect
37. C22×C33C2×C4semidirect
41. C2×C3×C3⋊D3C2×C4semidirect
43. C33×C4D4semidirect
46. C3×C3⋊Dic3C2×C4semidirect
63. #638C3×C4direct (N × H)
81. C22×C3×C4C3×D3semidirect
82. C22×C3×C4C3⋊D3semidirect
101. C22×C32C4×D3semidirect
102. C22×C32C2×C3×C4semidirect
107. C2×C3⋊D3C2×C3×C4semidirect
115. C32×C4#67semidirect
116. C32×C4C3×D4semidirect
119. C3⋊Dic3C2×C3×C4semidirect
124. C33C4×D4semidirect
165. C22×C4C3×C3⋊D3semidirect
181. C22×C3C3×C4×D3semidirect
182. C22×C3C4×C3⋊D3semidirect
195. C3×C4C3×#67semidirect
196. C3×C4#638direct (N × H)
201. C32C4×#67semidirect
202. C32C3×C4×D4semidirect
232. C22C3×C4×C3⋊D3semidirect
238. C4C3×#638direct (N × H)
240. C3C3×C4×#67semidirect
241. C3C4×#638direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C33×C42
C2×C3×#6382
C2×C3×C4×C3⋊D32
C3×#37572
C3×#37522
C3×#37502
C3×C4×C3⋊Dic32
C4×#6383
C3×C4×#673