Finite Group Databasebeta

← C3×#10592564 #13162904

Subgroups — C3×#10592564 #13162904

Order 1728.

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

normal subgroups only

This group has 924 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
1C2×C3×#3745C211yesmaximal, characteristic, complemented
2C2×C3×#3743C211yesmaximal, characteristic, complemented
3C3×#64104C212yesmaximal, complemented
4C33×#106C211yesmaximal, characteristic, complemented
5C3×#63636C212yesmaximal, complemented
6#10592564C311yesmaximal, characteristic, direct product factor
7C3×#4865—34nomaximal
8C2×C3×C4⋊(C3⋊D3)—22no—
9C2×C3×C4⋊(C3⋊D3)C2212yes—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C33×C8C2211yescharacteristic
12C33×C4⋊C4C2212yes—
13C3×#3752—22no—
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×#3745—22no—
16C3×#3743—22no—
17C2×#3745C2×C311yescharacteristic, complemented
18C2×#3743C2×C311yescharacteristic, complemented
19#64104C2×C312yescomplemented
20C2×C3×D24—34no—
21C2×C3×ΓL(1,𝔽25)—34no—
22C3×#895—38no—
23C32×#106—24no—
24C32×#106D314yescomplemented
25C32×#106C2×C311yescharacteristic, complemented
26#63636C2×C312yescomplemented
27C3×#809—38no—
28C22×C3×C3⋊D3—22no—
29C22×C3×C3⋊D3D411yescharacteristic
30C2×C33×C4—22no—
31C2×C33×C4C2311yescharacteristic
32C2×C3×C3⋊Dic3D411yescharacteristic
33C3×C4⋊(C3⋊D3)—22no—
34C3×C4⋊(C3⋊D3)—22no—
35C3×C4⋊(C3⋊D3)—44no—
36C3×C4×C3⋊D3—21no—
37C3×C4×C3⋊D3—21no—
38C33×C8D412yescomplemented
39C3×C3⋊(C3⋊C8)—21no—
40C3×#469—91no—
41#4865—34no—
42C2×C4⋊(C3⋊D3)—22no—
43C2×C4⋊(C3⋊D3)C22×C312yes—
44C2×C4×C3⋊D3C22×C311yescharacteristic
45C2×C3×D12—38no—
46C2×C3×D12—68no—
47C2×C3×C4×D3—34no—
48C2×C32×C8—24no—
49C2×C32×C8C2×D314yes—
50C2×C32×C8C22×C311yescharacteristic
51C32×C4⋊C4—28no—
52C32×C4⋊C4C2×D318yes—
53C32×C4⋊C4C22×C312yes—
54#3752—22no—
55C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
56#3745—22no—
57#3743—22no—
58C3×(C2×D3)⋊C4—68no—
59C2×C3×C3⋊C8—34no—
60C3×D24—68no—
61C3×ΓL(1,𝔽25)—68no—
62C22×C33C2×D411yescharacteristic
63C2×C3×C3⋊D3—21no—
64C2×C3×C3⋊D3—22no—
65C2×C3×C3⋊D3—42no—
66C2×C3×C3⋊D3—42no—
67C2×C3×C3⋊D3—42no—
68C33×C4—42no—
69C33×C4C2×D411yescharacteristic
70C33×C4#4111yescharacteristic
71C3×C3⋊Dic3—21no—
72C2×C3×D8—91no—
73C2×C3×M16—91no—
74C3×#121—92no—
75C2×D24—34no—
76C2×ΓL(1,𝔽25)—34no—
77#895—38no—
78C3×#106—24no—
79C3×#106C3×D314yescomplemented
80C3×#106C3⋊D311yescharacteristic, complemented
81C3×#102—92no—
82#809—38no—
83C22×C3⋊D3—22no—
84C22×C3⋊D3C3×D411yescharacteristic
85C22×C3×D3—34no—
86C22×C3×D3—68no—
87C2×C32×C4—22no—
88C2×C32×C4—24no—
89C2×C32×C4—28no—
90C2×C32×C4—48no—
91C2×C32×C4C22×D314yes—
92C2×C32×C4C23×C311yescharacteristic
93C2×C3⋊Dic3C3×D411yescharacteristic
94C4⋊(C3⋊D3)—22no—
95C4⋊(C3⋊D3)—22no—
96C4⋊(C3⋊D3)—44no—
97C4×C3⋊D3—21no—
98C4×C3⋊D3—21no—
99C2×C3×Dic3—34no—
100C3×D12—68no—
101C3×D12—68no—
102C3×D12—1216no—
103C3×C4×D3—64no—
104C3×C4×D3—64no—
105C32×C8—28no—
106C32×C8D1218yescomplemented
107C32×C8C3×D412yescomplemented
108C3⋊(C3⋊C8)—21no—
109C3×C3⋊C8—64no—
110#469—91noSylow
111C2×C33#12111yescharacteristic
112C2×C33AGL(1,ℤ/8)12yes—
113C3×C3⋊D3—41no—
114C3×C3⋊D3—82no—
115C2×C3×D4—92no—
116C2×C3×D4—182no—
117C22×C3×C4—91no—
118C2×D12—38no—
119C2×D12—68no—
120C2×C4×D3—34no—
121C3×D8—182no—
122C3×M16—182no—
123C2×C3×C8—24no—
124C2×C3×C8—91no—
125C2×C3×C8C2×C3×D314yes—
126C2×C3×C8C2×C3⋊D311yescharacteristic
127C3×C4⋊C4—28no—
128C3×C4⋊C4C2×C3×D318yes—
129C3×C4⋊C4C2×C3⋊D312yes—
130C3×C22⋊C4—182no—
131(C2×D3)⋊C4—68no—
132C2×C3⋊C8—34no—
133D24—68no—
134ΓL(1,𝔽25)—68no—
135C22×C32—24no—
136C22×C32D3×D414yes—
137C22×C32C2×C3×D411yescharacteristic
138C2×C3⋊D3—21no—
139C2×C3⋊D3—22no—
140C2×C3⋊D3—42no—
141C2×C3⋊D3—42no—
142C2×C3⋊D3—42no—
143C2×C3×D3—64no—
144C2×C3×D3—68no—
145C2×C3×D3—128no—
146C2×C3×D3—128no—
147C2×C3×D3—128no—
148C32×C4—24no—
149C32×C4—24no—
150C32×C4—42no—
151C32×C4—48no—
152C32×C4—88no—
153C32×C4C2×D1214yes—
154C32×C4#23914yes—
155C32×C4C2×C3×D411yescharacteristic
156C32×C4C3×#4111yescharacteristic
157C3⋊Dic3—21no—
158C3×Dic3—64no—
159C2×D8—91no—
160C2×M16—91no—
161#121—92no—
162#106C3×C3⋊D311yescharacteristic, complemented
163#102—92no—
164C33#46911yescharacteristic, complemented, Sylow
165C23×C3—91no—
166C23×C3—182no—
167C22×D3—34no—
168C22×D3—68no—
169C3×D4—182no—
170C3×D4—182no—
171C3×D4—364no—
172C2×C3×C4—28no—
173C2×C3×C4—22no—
174C2×C3×C4—24no—
175C2×C3×C4—48no—
176C2×C3×C4—91no—
177C2×C3×C4—181no—
178C2×C3×C4—181no—
179C2×C3×C4C22×C3×D314yes—
180C2×C3×C4C22×C3⋊D311yescharacteristic
181C2×Dic3—34no—
182D12—68no—
183D12—68no—
184D12—1216no—
185C4×D3—64no—
186C4×D3—64no—
187C3×C8—28no—
188C3×C8—181no—
189C3×C8C3×D1218yescomplemented
190C3×C8C4⋊(C3⋊D3)12yescomplemented
191C3⋊C8—64no—
192C2×C32—24no—
193C2×C32—28no—
194C2×C32#89514yes—
195C2×C32#91418yes—
196C2×C32C3×#12111yescharacteristic
197C2×C32C3×AGL(1,ℤ/8)12yes—
198C3⋊D3—41no—
199C3⋊D3—82no—
200C3×D3—124no—
201C3×D3—248no—
202C2×D4—92no—
203C2×D4—182no—
204C22×C4—91no—
205D8—182no—
206M16—182no—
207C2×C8—91no—
208C2×C8C2×C3×C3⋊D311yescharacteristic
209C4⋊C4C2×C3×C3⋊D312yes—
210C22⋊C4—182no—
211C22×C3—24no—
212C22×C3—181no—
213C22×C3—182no—
214C22×C3—362no—
215C22×C3—362no—
216C22×C3—362no—
217C22×C3C3×D3×D414yes—
218C22×C3D4×C3⋊D311yescharacteristic
219C2×D3—64no—
220C2×D3—68no—
221C2×D3—128no—
222C2×D3—128no—
223C2×D3—128no—
224C3×C4—24no—
225C3×C4—24no—
226C3×C4—48no—
227C3×C4—42no—
228C3×C4—88no—
229C3×C4—181no—
230C3×C4C2×C3×D1214yes—
231C3×C4C3×#23914yes—
232C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
233C3×C4#383211yescharacteristic
234Dic3—64no—
235C32—24no—
236C32#486514yescomplemented
237C32C3×#46911yescharacteristic, complemented
238C23—91no—
239C23—182no—
240D4—182no—
241D4—182no—
242D4—364no—
243C2×C4—22no—
244C2×C4—91no—
245C2×C4—181no—
246C2×C4—181no—
247C2×C4C22×C3×C3⋊D311yescharacteristic
248C8—181no—
249C8C3×C4⋊(C3⋊D3)12yescomplemented
250C2×C3—24no—
251C2×C3—28no—
252C2×C3—361no—
253C2×C3—722no—
254C2×C3C3×#89514yes—
255C2×C3C3×#91418yes—
256C2×C3#6410411yescharacteristic
257C2×C3#6412312yes—
258D3—124no—
259D3—248no—
260C22—181no—
261C22—182no—
262C22—362no—
263C22—362no—
264C22—362no—
265C22C3×D4×C3⋊D311yescharacteristic
266C4—42no—
267C4—181no—
268C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
269C4C3×#383211yescharacteristic
270C3—24no—
271C3C3×#486514yescomplemented
272C3#1059256411yescharacteristic, direct product factor
273C2—361no—
274C2—722no—
275C2C3×#6410411yescharacteristic
276C2C3×#6412312yes—
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#106C2
socleC22×C33C2×D4
hypercenterC3×#106C3⋊D3
hypocenterC32C3×#469
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#106

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3745C2semidirect
2. C2×C3×#3743C2semidirect
3. C3×#64104C2semidirect
4. C33×#106C2semidirect
5. C3×#63636C2semidirect
6. #10592564C3direct (N × H)
17. C2×#3745C2×C3semidirect
18. C2×#3743C2×C3semidirect
19. #64104C2×C3semidirect
24. C32×#106D3semidirect
25. C32×#106C2×C3semidirect
26. #63636C2×C3semidirect
38. C33×C8D4semidirect
79. C3×#106C3×D3semidirect
80. C3×#106C3⋊D3semidirect
106. C32×C8D12semidirect
107. C32×C8C3×D4semidirect
162. #106C3×C3⋊D3semidirect
164. C33#469semidirect
189. C3×C8C3×D12semidirect
190. C3×C8C4⋊(C3⋊D3)semidirect
236. C32#4865semidirect
237. C32C3×#469semidirect
249. C8C3×C4⋊(C3⋊D3)semidirect
271. C3C3×#4865semidirect
272. C3#10592564direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37452
C2×C3×#37432
C3×#641042
C33×#1062
C3×#636362
#105925643
C3×#48653