Finite Group Databasebeta

← C3×#63953 #11723278

Subgroups — C3×#63953 #11723278

Order 864.

458 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 92 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
2C2×C3×#627C211yesmaximal, characteristic, complemented
3C32×#217C212yesmaximal, complemented
4C3×#3722C212yesmaximal, complemented
5C3×Dic32C211yesmaximal, characteristic, complemented
6C3×#885—32nomaximal
7#63953C311yesmaximal, characteristic, direct product factor
8C23×C33C2211yescharacteristic
9C22×C3×C3⋊D3C2211yescharacteristic
10C3×#638—22no—
11C2×C3×C3⋊Dic3C2211yescharacteristic
12C2×C32×Dic3C2212yes—
13C2×C32×Dic3C2212yes—
14C3×#627—22no—
15C2×#638C2×C311yescharacteristic
16C2×C3×#67—32no—
17C2×C3×#67—61no—
18C2×C3×Dic6—32no—
19C2×#627C2×C311yescharacteristic, complemented
20C32×C22⋊C4—32no—
21C3×#217—22no—
22C3×#217D312yescomplemented
23C3×#217C2×C312yescomplemented
24C3×(C2×D3)⋊C4—32no—
25C3×(C2×D3)⋊C4—32no—
26C3×C4×Dic3—32no—
27#3722C2×C312yescomplemented
28Dic32C2×C311yescharacteristic, complemented
29C22×C33—21no—
30C22×C33—22no—
31C22×C33C2311yescharacteristic
32C2×C3×C3⋊D3—21no—
33C2×C3×C3⋊D3—22no—
34C3×C3⋊Dic3D412yes—
35C32×Dic3—22no—
36C32×Dic3—22no—
37C3×#124—91no—
38#885—32no—
39C23×C32—21no—
40C23×C32—22no—
41C23×C32—41no—
42C23×C32C2×D312yes—
43C23×C32C22×C311yescharacteristic
44C22×C3⋊D3C22×C311yescharacteristic
45C22×C3×D3—32no—
46C22×C3×D3—61no—
47C2×C32×C4—32no—
48C2×C32×C4—32no—
49#638—22no—
50C2×C3⋊Dic3C22×C311yescharacteristic
51C3×#67—64no—
52C3×#67—64no—
53C2×C3×Dic3—22no—
54C2×C3×Dic3—22no—
55C2×C3×Dic3—32no—
56C2×C3×Dic3—61no—
57C2×C3×Dic3C2×D312yes—
58C2×C3×Dic3C2×D312yes—
59C2×C3×Dic3C22×C312yes—
60C2×C3×Dic3C22×C312yes—
61C3×Dic6—64no—
62#627—22no—
63C2×C33—41no—
64C2×C33C2×D411yescharacteristic
65C2×C33#4112yes—
66C3×C3⋊D3—41no—
67C2×C3×Q8—91no—
68C2×C3×D4—91no—
69C2×#67—32no—
70C2×#67—61no—
71C2×Dic6—32no—
72C3×C22⋊C4—32no—
73C3×C22⋊C4—32no—
74C3×C22⋊C4—62no—
75C3×C22⋊C4—92no—
76C3×C42—91no—
77#217C3×D312yescomplemented
78(C2×D3)⋊C4—32no—
79(C2×D3)⋊C4—32no—
80C4×Dic3—32no—
81C22×C32—22no—
82C22×C32—21no—
83C22×C32—22no—
84C22×C32—24no—
85C22×C32—22no—
86C22×C32—22no—
87C22×C32—22no—
88C22×C32—21no—
89C22×C32—22no—
90C22×C32—41no—
91C22×C32—42no—
92C22×C32—42no—
93C22×C32—42no—
94C22×C32—42no—
95C22×C32—44no—
96C22×C32C22×D312yes—
97C22×C32C23×C311yescharacteristic
98C2×C3⋊D3—21no—
99C2×C3⋊D3—22no—
100C2×C3×D3—62no—
101C2×C3×D3—62no—
102C2×C3×D3—62no—
103C2×C3×D3—62no—
104C2×C3×D3—122no—
105C32×C4—62no—
106C32×C4—62no—
107C3⋊Dic3C3×D412yes—
108C3×Dic3—22no—
109C3×Dic3—22no—
110C3×Dic3—22no—
111C3×Dic3—22no—
112C3×Dic3—34no—
113C3×Dic3—42no—
114C3×Dic3—42no—
115C3×Dic3—62no—
116#124—91noSylow
117C33#12411yescharacteristic, complemented, Sylow
118C23×C3—22no—
119C23×C3—21no—
120C23×C3—41no—
121C23×C3—91no—
122C23×C3D3211yescharacteristic
123C23×C3C2×C3×D312yes—
124C22×D3—32no—
125C22×D3—61no—
126C3×Q8—182no—
127C3×D4—182no—
128C2×C3×C4—32no—
129C2×C3×C4—32no—
130C2×C3×C4—32no—
131C2×C3×C4—32no—
132C2×C3×C4—62no—
133C2×C3×C4—62no—
134C2×C3×C4—91no—
135#67—64no—
136#67—64no—
137C2×Dic3—32no—
138C2×Dic3—61no—
139C2×Dic3C2×C3×D312yes—
140C2×Dic3C2×C3×D312yes—
141Dic6—64no—
142C2×C32—22no—
143C2×C32—21no—
144C2×C32—22no—
145C2×C32—22no—
146C2×C32—22no—
147C2×C32—41no—
148C2×C32—42no—
149C2×C32—44no—
150C2×C32—42no—
151C2×C32—44no—
152C2×C32—42no—
153C2×C32—41no—
154C2×C32#23512yes—
155C2×C32D3×D412yes—
156C2×C32#23712yes—
157C2×C32C2×C3×D411yescharacteristic
158C2×C32C3×#4112yes—
159C3⋊D3—41no—
160C3×D3—122no—
161C3×D3—122no—
162C2×Q8—91no—
163C2×D4—91no—
164C22⋊C4—32no—
165C22⋊C4—92no—
166C42—91no—
167C22×C3—22no—
168C22×C3—24no—
169C22×C3—22no—
170C22×C3—21no—
171C22×C3—22no—
172C22×C3—22no—
173C22×C3—22no—
174C22×C3—21no—
175C22×C3—22no—
176C22×C3—42no—
177C22×C3—42no—
178C22×C3—42no—
179C22×C3—44no—
180C22×C3—42no—
181C22×C3—41no—
182C22×C3—182no—
183C22×C3—181no—
184C22×C3C2×D3211yescharacteristic
185C22×C3C22×C3×D312yes—
186C2×D3—62no—
187C2×D3—62no—
188C2×D3—62no—
189C2×D3—62no—
190C2×D3—122no—
191C3×C4—62no—
192C3×C4—62no—
193C3×C4—62no—
194C3×C4—62no—
195C3×C4—92no—
196C3×C4—122no—
197C3×C4—122no—
198Dic3—22no—
199Dic3—22no—
200Dic3—34no—
201Dic3—62no—
202C32—22no—
203C32—21no—
204C32—41no—
205C32#88512yescomplemented
206C32C3×#12411yescharacteristic, complemented
207C23—91no—
208C23C3×D3211yescharacteristic
209Q8—182no—
210D4—182no—
211C2×C4—32no—
212C2×C4—32no—
213C2×C4—91no—
214C2×C3—22no—
215C2×C3—22no—
216C2×C3—22no—
217C2×C3—21no—
218C2×C3—22no—
219C2×C3—42no—
220C2×C3—44no—
221C2×C3—44no—
222C2×C3—42no—
223C2×C3—41no—
224C2×C3—42no—
225C2×C3—41no—
226C2×C3—361no—
227C2×C3#380412yes—
228C2×C3#381111yescharacteristic
229C2×C3C3×#23512yes—
230C2×C3C3×D3×D412yes—
231C2×C3C3×#23712yes—
232D3—122no—
233D3—122no—
234C22—22no—
235C22—21no—
236C22—182no—
237C22—181no—
238C22C2×C3×D3211yescharacteristic
239C4—62no—
240C4—62no—
241C4—92no—
242C3—21no—
243C3—22no—
244C3—41no—
245C3#6395311yescharacteristic, direct product factor
246C3C3×#88512yescomplemented
247C2—41no—
248C2—361no—
249C2C3×#380412yes—
250C2C3×#381111yescharacteristic
subgroupG/H
centerC22×C3C2×D32
commutator, cosocleC22×C32C23×C3
FrattiniC22C2×C3×D32
FittingC23×C33C22
socleC22×C33C23
prime radicalC2×C3×#638C2
hypercenterC23×C3D32
hypocenterC32C3×#124
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×#627C2semidirect
3. C32×#217C2semidirect
4. C3×#3722C2semidirect
5. C3×Dic32C2semidirect
7. #63953C3direct (N × H)
19. C2×#627C2×C3semidirect
22. C3×#217D3semidirect
23. C3×#217C2×C3semidirect
27. #3722C2×C3semidirect
28. Dic32C2×C3semidirect
77. #217C3×D3semidirect
117. C33#124semidirect
205. C32#885semidirect
206. C32C3×#124semidirect
245. C3#63953direct (N × H)
246. C3C3×#885semidirect

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×#6272
C32×#2172
C3×#37222
C3×Dic322
C3×#8853
#639533