Finite Group Databasebeta

← C3×#63982 #11723307

Subgroups — C3×#63982 #11723307

Order 864.

450 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 108 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⋊Dic3C211yesmaximal, characteristic
2C2×C3×#627C211yesmaximal, characteristic, complemented
3C32×#217C212yesmaximal
4C3×#3724C211yesmaximal, characteristic, complemented
5C3×Dic3⋊Dic3C212yesmaximal, complemented
6C3×#878—32nomaximal
7#63982C311yesmaximal, characteristic, direct product factor
8C23×C33C2211yescharacteristic
9C2×C3×C3⋊Dic3—22no—
10C2×C3×C3⋊Dic3C2211yescharacteristic
11C2×C3×C3⋊Dic3C2211yescharacteristic
12C2×C32×Dic3C2212yes—
13C2×C32×Dic3C2212yes—
14C3×#627—22no—
15C22×C3⋊Dic3C2×C311yescharacteristic
16C22×C3×Dic3—32no—
17C22×C3×Dic3—61no—
18C2×C3×Dic6—32no—
19C2×#627C2×C311yescharacteristic, complemented
20C32×C22⋊C4—32no—
21C3×#217—22no—
22C3×#217D312yes—
23C3×#217C2×C312yes—
24C3×C4⋊Dic3—32no—
25C3×Dic3⋊C4—32no—
26C3×Dic3⋊C4—32no—
27#3724C2×C311yescharacteristic, complemented
28Dic3⋊Dic3C2×C312yescomplemented
29C22×C33—21no—
30C22×C33—21no—
31C22×C33Q812yescomplemented
32C22×C33C2311yescharacteristic
33C3×C3⋊Dic3—21no—
34C3×C3⋊Dic3D412yes—
35C32×Dic3—22no—
36C32×Dic3—22no—
37C3×C22⋊Q8—91no—
38#878—32no—
39C23×C32—21no—
40C23×C32—22no—
41C23×C32—41no—
42C23×C32C2×D312yes—
43C23×C32C22×C311yescharacteristic
44C2×C32×C4—32no—
45C2×C32×C4—32no—
46C2×C3⋊Dic3—22no—
47C2×C3⋊Dic3C22×C311yescharacteristic
48C2×C3⋊Dic3C22×C311yescharacteristic
49C2×C3×Dic3—22no—
50C2×C3×Dic3—22no—
51C2×C3×Dic3—32no—
52C2×C3×Dic3—32no—
53C2×C3×Dic3—61no—
54C2×C3×Dic3—61no—
55C2×C3×Dic3—64no—
56C2×C3×Dic3—64no—
57C2×C3×Dic3C2×D312yes—
58C2×C3×Dic3C2×D312yes—
59C2×C3×Dic3C22×C312yes—
60C2×C3×Dic3C22×C312yes—
61C3×Dic6—64no—
62#627—22no—
63C2×C33—22no—
64C2×C33C2×D411yescharacteristic
65C2×C33C2×Q811yescharacteristic
66C2×C33#4111yescharacteristic
67C2×C3×Q8—91no—
68C22×C3×C4—91no—
69C22×Dic3—32no—
70C22×Dic3—61no—
71C2×Dic6—32no—
72C3×C4⋊C4—91no—
73C3×C4⋊C4—92no—
74C3×C22⋊C4—32no—
75C3×C22⋊C4—32no—
76C3×C22⋊C4—62no—
77#217C3×D312yes—
78C4⋊Dic3—32no—
79Dic3⋊C4—32no—
80Dic3⋊C4—32no—
81C22×C32—21no—
82C22×C32—21no—
83C22×C32—22no—
84C22×C32—22no—
85C22×C32—22no—
86C22×C32—22no—
87C22×C32—22no—
88C22×C32—24no—
89C22×C32—21no—
90C22×C32—22no—
91C22×C32—42no—
92C22×C32—41no—
93C22×C32—42no—
94C22×C32—42no—
95C22×C32—42no—
96C22×C32—42no—
97C22×C32Dic614yescomplemented
98C22×C32C3×Q812yescomplemented
99C22×C32C22×D312yes—
100C22×C32C23×C311yescharacteristic
101C32×C4—62no—
102C32×C4—62no—
103C3⋊Dic3—21no—
104C3⋊Dic3C3×D412yes—
105C3×Dic3—22no—
106C3×Dic3—22no—
107C3×Dic3—22no—
108C3×Dic3—22no—
109C3×Dic3—34no—
110C3×Dic3—42no—
111C3×Dic3—42no—
112C3×Dic3—62no—
113C3×Dic3—62no—
114C3×Dic3—62no—
115C22⋊Q8—91noSylow
116C33C22⋊Q811yescharacteristic, complemented, Sylow
117C23×C3—21no—
118C23×C3—22no—
119C23×C3—41no—
120C23×C3D3211yescharacteristic
121C23×C3C2×C3×D312yes—
122C3×Q8—182no—
123C2×C3×C4—32no—
124C2×C3×C4—32no—
125C2×C3×C4—32no—
126C2×C3×C4—32no—
127C2×C3×C4—62no—
128C2×C3×C4—62no—
129C2×C3×C4—91no—
130C2×C3×C4—91no—
131C2×C3×C4—182no—
132C2×Dic3—32no—
133C2×Dic3—32no—
134C2×Dic3—61no—
135C2×Dic3—61no—
136C2×Dic3—64no—
137C2×Dic3—64no—
138C2×Dic3C2×C3×D312yes—
139C2×Dic3C2×C3×D312yes—
140Dic6—64no—
141C2×C32—22no—
142C2×C32—24no—
143C2×C32—24no—
144C2×C32—21no—
145C2×C32—21no—
146C2×C32—21no—
147C2×C32—22no—
148C2×C32—22no—
149C2×C32—22no—
150C2×C32—41no—
151C2×C32—41no—
152C2×C32—41no—
153C2×C32—44no—
154C2×C32—44no—
155C2×C32C2×Dic612yes—
156C2×C32D3×D412yes—
157C2×C32#23712yes—
158C2×C32C2×C3×D411yescharacteristic
159C2×C32C2×C3×Q811yescharacteristic
160C2×C32C3×#4111yescharacteristic
161C2×Q8—91no—
162C22×C4—91no—
163C4⋊C4—91no—
164C4⋊C4—92no—
165C22⋊C4—32no—
166C22×C3—22no—
167C22×C3—22no—
168C22×C3—21no—
169C22×C3—21no—
170C22×C3—22no—
171C22×C3—22no—
172C22×C3—22no—
173C22×C3—24no—
174C22×C3—21no—
175C22×C3—22no—
176C22×C3—42no—
177C22×C3—42no—
178C22×C3—42no—
179C22×C3—42no—
180C22×C3—42no—
181C22×C3—41no—
182C22×C3#62712yescomplemented
183C22×C3C3×Dic614yescomplemented
184C22×C3C2×D3211yescharacteristic
185C22×C3C22×C3×D312yes—
186C3×C4—62no—
187C3×C4—62no—
188C3×C4—62no—
189C3×C4—62no—
190C3×C4—92no—
191C3×C4—122no—
192C3×C4—122no—
193C3×C4—181no—
194Dic3—22no—
195Dic3—22no—
196Dic3—34no—
197Dic3—62no—
198Dic3—62no—
199Dic3—62no—
200C32—22no—
201C32—21no—
202C32—41no—
203C32#87812yescomplemented
204C32C3×C22⋊Q811yescharacteristic, complemented
205C23C3×D3211yescharacteristic
206Q8—182no—
207C2×C4—32no—
208C2×C4—32no—
209C2×C4—91no—
210C2×C4—91no—
211C2×C4—182no—
212C2×C3—24no—
213C2×C3—24no—
214C2×C3—22no—
215C2×C3—22no—
216C2×C3—21no—
217C2×C3—22no—
218C2×C3—21no—
219C2×C3—21no—
220C2×C3—22no—
221C2×C3—44no—
222C2×C3—44no—
223C2×C3—41no—
224C2×C3—41no—
225C2×C3—41no—
226C2×C3#380511yescharacteristic
227C2×C3C2×#62711yescharacteristic
228C2×C3#381111yescharacteristic
229C2×C3C2×C3×Dic612yes—
230C2×C3C3×D3×D412yes—
231C2×C3C3×#23712yes—
232C22—21no—
233C22—21no—
234C22C3×#62712yescomplemented
235C22C2×C3×D3211yescharacteristic
236C4—62no—
237C4—62no—
238C4—92no—
239C4—181no—
240C3—21no—
241C3—22no—
242C3—41no—
243C3#6398211yescharacteristic, direct product factor
244C3C3×#87812yescomplemented
245C2—22no—
246C2C3×#380511yescharacteristic
247C2C2×C3×#62711yescharacteristic
248C2C3×#381111yescharacteristic
subgroupG/H
centerC22×C3C2×D32
commutator, cosocleC22×C32C23×C3
FrattiniC22C2×C3×D32
Fitting, prime radicalC23×C33C22
socleC22×C33C23
hypercenterC23×C3D32
hypocenterC32C3×C22⋊Q8
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
4. C3×#3724C2semidirect
5. C3×Dic3⋊Dic3C2semidirect
7. #63982C3direct (N × H)
19. C2×#627C2×C3semidirect
27. #3724C2×C3semidirect
28. Dic3⋊Dic3C2×C3semidirect
31. C22×C33Q8semidirect
97. C22×C32Dic6semidirect
98. C22×C32C3×Q8semidirect
116. C33C22⋊Q8semidirect
182. C22×C3#627semidirect
183. C22×C3C3×Dic6semidirect
203. C32#878semidirect
204. C32C3×C22⋊Q8semidirect
234. C22C3×#627semidirect
243. C3#63982direct (N × H)
244. C3C3×#878semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊Dic32
C2×C3×#6272
C32×#2172
C3×#37242
C3×Dic3⋊Dic32
C3×#8783
#639823