Finite Group Databasebeta

← C3×#63912 #11723237

Subgroups — C3×#63912 #11723237

Order 864.

514 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 96 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×C4×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×C3×#626C212yesmaximal, complemented
3C2×C3×#625C211yesmaximal, characteristic, complemented
4C32×(C2×D3)⋊C4C212yesmaximal, complemented
5C3×#3724C211yesmaximal, characteristic, complemented
6C3×#884—32nomaximal
7#63912C311yesmaximal, characteristic, direct product factor
8C22×C3×C3⋊D3C2211yescharacteristic
9C22×C32×D3C2212yes—
10C2×C33×C4C2211yescharacteristic
11C2×C3×C3⋊Dic3C2211yescharacteristic
12C3×C4×C3⋊D3—22no—
13C2×C32×Dic3C2212yes—
14C3×#626—24no—
15C3×#625—22no—
16C2×C4×C3⋊D3C2×C311yescharacteristic, complemented
17C2×C3×#67—32no—
18C2×C3×#67—32no—
19C2×C3×D12—32no—
20C2×C3×C4×D3—32no—
21C2×C3×C4×D3—61no—
22C2×#626C2×C312yescomplemented
23C2×#625C2×C311yescharacteristic, complemented
24C32×C22⋊C4—32no—
25C3×(C2×D3)⋊C4—22no—
26C3×(C2×D3)⋊C4D312yescomplemented
27C3×(C2×D3)⋊C4C2×C312yescomplemented
28C3×Dic3⋊C4—32no—
29#3724C2×C311yescharacteristic, complemented
30C22×C33C2311yescharacteristic
31C2×C3×C3⋊D3—21no—
32C2×C3×C3⋊D3—21no—
33C2×C3×C3⋊D3D412yescomplemented
34C2×C32×D3—22no—
35C2×C32×D3—22no—
36C2×C32×D3—22no—
37C33×C4—21no—
38C3×C3⋊Dic3D412yescomplemented
39C32×Dic3—22no—
40C3×#121—91no—
41#884—32no—
42C23×C32—32no—
43C22×C3⋊D3C22×C311yescharacteristic
44C22×C3×D3—22no—
45C22×C3×D3—32no—
46C22×C3×D3—61no—
47C22×C3×D3C2×D312yes—
48C22×C3×D3C22×C312yes—
49C2×C32×C4—21no—
50C2×C32×C4—22no—
51C2×C32×C4—32no—
52C2×C32×C4—41no—
53C2×C32×C4C2×D312yes—
54C2×C32×C4C22×C311yescharacteristic
55C2×C3⋊Dic3C22×C311yescharacteristic
56C4×C3⋊D3—22no—
57C3×#67—64no—
58C3×#67—64no—
59C2×C3×Dic3—22no—
60C2×C3×Dic3—32no—
61C2×C3×Dic3—61no—
62C2×C3×Dic3C2×D312yes—
63C2×C3×Dic3C22×C312yes—
64C3×D12—64no—
65C3×C4×D3—64no—
66C3×C4×D3—64no—
67#626—24no—
68#625—22no—
69C2×C33C2×D411yescharacteristic
70C2×C33C2×D411yescharacteristic
71C2×C33#4111yescharacteristic
72C3×C3⋊D3—22no—
73C32×D3—42no—
74C2×C3×D4—92no—
75C2×C3×D4—91no—
76C22×C3×C4—91no—
77C2×#67—32no—
78C2×#67—32no—
79C2×D12—32no—
80C2×C4×D3—32no—
81C2×C4×D3—61no—
82C3×C4⋊C4—91no—
83C3×C22⋊C4—32no—
84C3×C22⋊C4—32no—
85C3×C22⋊C4—62no—
86(C2×D3)⋊C4C3×D312yescomplemented
87Dic3⋊C4—32no—
88C22×C32—21no—
89C22×C32—22no—
90C22×C32—41no—
91C22×C32—62no—
92C22×C32—62no—
93C22×C32—62no—
94C22×C32C22×D312yes—
95C22×C32C23×C311yescharacteristic
96C2×C3⋊D3—21no—
97C2×C3⋊D3—21no—
98C2×C3⋊D3C3×D412yescomplemented
99C2×C3×D3—24no—
100C2×C3×D3—22no—
101C2×C3×D3—22no—
102C2×C3×D3—22no—
103C2×C3×D3—22no—
104C2×C3×D3—22no—
105C2×C3×D3—22no—
106C2×C3×D3—34no—
107C2×C3×D3—42no—
108C2×C3×D3—42no—
109C2×C3×D3—62no—
110C2×C3×D3—62no—
111C2×C3×D3—62no—
112C2×C3×D3—62no—
113C2×C3×D3—62no—
114C32×C4—21no—
115C32×C4—22no—
116C32×C4—22no—
117C32×C4—42no—
118C32×C4—42no—
119C32×C4—62no—
120C3⋊Dic3C3×D412yescomplemented
121C3×Dic3—22no—
122C3×Dic3—22no—
123C3×Dic3—34no—
124C3×Dic3—42no—
125C3×Dic3—62no—
126#121—91noSylow
127C33#12111yescharacteristic, complemented, Sylow
128C23×C3—32no—
129C23×C3—32no—
130C23×C3—62no—
131C23×C3—91no—
132C22×D3—32no—
133C22×D3—61no—
134C22×D3C2×C3×D312yes—
135C3×D4—184no—
136C3×D4—182no—
137C2×C3×C4—21no—
138C2×C3×C4—22no—
139C2×C3×C4—32no—
140C2×C3×C4—32no—
141C2×C3×C4—41no—
142C2×C3×C4—62no—
143C2×C3×C4—91no—
144C2×C3×C4—182no—
145C2×C3×C4D3211yescharacteristic
146C2×C3×C4C2×C3×D312yes—
147#67—64no—
148#67—64no—
149C2×Dic3—32no—
150C2×Dic3—61no—
151C2×Dic3C2×C3×D312yes—
152D12—64no—
153C4×D3—64no—
154C4×D3—64no—
155C2×C32—21no—
156C2×C32—21no—
157C2×C32—21no—
158C2×C32—22no—
159C2×C32—22no—
160C2×C32—22no—
161C2×C32—41no—
162C2×C32—41no—
163C2×C32—41no—
164C2×C32—122no—
165C2×C32#23512yes—
166C2×C32D3×D412yes—
167C2×C32D3×D412yes—
168C2×C32C2×C3×D411yescharacteristic
169C2×C32C2×C3×D411yescharacteristic
170C2×C32C3×#4111yescharacteristic
171C3⋊D3—22no—
172C3×D3—44no—
173C3×D3—42no—
174C3×D3—42no—
175C3×D3—64no—
176C3×D3—64no—
177C2×D4—92no—
178C2×D4—91no—
179C22×C4—91no—
180C4⋊C4—91no—
181C22⋊C4—32no—
182C22×C3—21no—
183C22×C3—22no—
184C22×C3—41no—
185C22×C3—64no—
186C22×C3—62no—
187C22×C3—62no—
188C22×C3—62no—
189C22×C3—62no—
190C22×C3—62no—
191C22×C3—62no—
192C22×C3—92no—
193C22×C3—122no—
194C22×C3—122no—
195C22×C3—181no—
196C22×C3—181no—
197C22×C3C2×D3211yescharacteristic
198C22×C3C22×C3×D312yes—
199C2×D3—22no—
200C2×D3—22no—
201C2×D3—22no—
202C2×D3—34no—
203C2×D3—62no—
204C2×D3—62no—
205C2×D3—62no—
206C2×D3—62no—
207C2×D3—62no—
208C3×C4—22no—
209C3×C4—21no—
210C3×C4—22no—
211C3×C4—42no—
212C3×C4—42no—
213C3×C4—62no—
214C3×C4—62no—
215C3×C4—92no—
216C3×C4—122no—
217Dic3—22no—
218Dic3—34no—
219Dic3—62no—
220C32—22no—
221C32—21no—
222C32—41no—
223C32#88412yescomplemented
224C32C3×#12111yescharacteristic, complemented
225C23—32no—
226C23—91no—
227D4—184no—
228D4—182no—
229C2×C4—32no—
230C2×C4—91no—
231C2×C4—182no—
232C2×C4C3×D3211yescharacteristic
233C2×C3—21no—
234C2×C3—22no—
235C2×C3—22no—
236C2×C3—21no—
237C2×C3—21no—
238C2×C3—22no—
239C2×C3—41no—
240C2×C3—41no—
241C2×C3—41no—
242C2×C3—124no—
243C2×C3—122no—
244C2×C3—122no—
245C2×C3—182no—
246C2×C3#379811yescharacteristic
247C2×C3#380211yescharacteristic
248C2×C3#381111yescharacteristic
249C2×C3C3×#23512yes—
250C2×C3C3×D3×D412yes—
251C2×C3C3×D3×D412yes—
252D3—42no—
253D3—64no—
254D3—64no—
255C22—62no—
256C22—62no—
257C22—62no—
258C22—92no—
259C22—181no—
260C22—181no—
261C22C2×C3×D3211yescharacteristic
262C4—21no—
263C4—62no—
264C4—92no—
265C3—21no—
266C3—22no—
267C3—41no—
268C3#6391211yescharacteristic, direct product factor
269C3C3×#88412yescomplemented
270C2—122no—
271C2—182no—
272C2C3×#379811yescharacteristic
273C2C3×#380211yescharacteristic
274C2C3×#381111yescharacteristic
subgroupG/H
centerC22×C3C2×D32
commutator, cosocleC22×C32C23×C3
FrattiniC22C2×C3×D32
FittingC2×C33×C4C22
socleC22×C33C23
hypercenterC2×C3×C4D32
hypocenterC32C3×#121
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C3⋊D3C2semidirect
2. C2×C3×#626C2semidirect
3. C2×C3×#625C2semidirect
4. C32×(C2×D3)⋊C4C2semidirect
5. C3×#3724C2semidirect
7. #63912C3direct (N × H)
16. C2×C4×C3⋊D3C2×C3semidirect
22. C2×#626C2×C3semidirect
23. C2×#625C2×C3semidirect
26. C3×(C2×D3)⋊C4D3semidirect
27. C3×(C2×D3)⋊C4C2×C3semidirect
29. #3724C2×C3semidirect
33. C2×C3×C3⋊D3D4semidirect
38. C3×C3⋊Dic3D4semidirect
86. (C2×D3)⋊C4C3×D3semidirect
98. C2×C3⋊D3C3×D4semidirect
120. C3⋊Dic3C3×D4semidirect
127. C33#121semidirect
223. C32#884semidirect
224. C32C3×#121semidirect
268. C3#63912direct (N × H)
269. C3C3×#884semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊D32
C2×C3×#6262
C2×C3×#6252
C32×(C2×D3)⋊C42
C3×#37242
C3×#8843
#639123