Finite Group Databasebeta

← C3×#10596191 #13179471

Subgroups — C3×#10596191 #13179471

Order 1728.

1944 conjugacy classes of subgroups; only the 342 normal subgroups are shown.

normal subgroups only

This group has 1944 conjugacy classes of subgroups — too many to draw. Only its 342 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
1C33×#127C211yesmaximal, characteristic, complemented
2C3×#64148C216yesmaximal, complemented
3C3×#64146C213yesmaximal, complemented
4C3×#64144C213yesmaximal, complemented
5C3×#64085C212yesmaximal, complemented
6#10596191C311yesmaximal, characteristic, direct product factor
7C3×#5720—34nomaximal
8C23×C3×C3⋊D3C2211yescharacteristic
9C2×C33×D4C2213yes—
10C2×C33×D4C2213yescomplemented
11C2×C3×#638C2216yes—
12C22×C3×C3⋊Dic3C2213yescomplemented
13C33×C42C2211yescharacteristic, complemented
14C3×#3757C2213yes—
15C3×#3757C2213yescomplemented
16C3×#3752C2216yes—
17C3×#3750C2216yescomplemented
18C32×#127—24no—
19C32×#127D314yescomplemented
20C32×#127C2×C311yescharacteristic, complemented
21#64148C2×C316yescomplemented
22#64146C2×C313yescomplemented
23#64144C2×C313yescomplemented
24#64085C2×C312yescomplemented
25C3×#939—324no—
26C3×#937—312no—
27C3×#935—312no—
28C3×#876—38no—
29C23×C33C2311yescharacteristic
30C23×C33C2313yes—
31C22×C3×C3⋊D3—43no—
32C22×C3×C3⋊D3C2312yes—
33C33×D4—43no—
34C33×D4—43no—
35C2×C33×C4C2313yes—
36C3×#638—46no—
37C2×C3×C3⋊Dic3—43no—
38C2×C3×C3⋊Dic3C2316yes—
39C3×#560—91no—
40#5720—34no—
41C23×C3⋊D3C22×C311yescharacteristic
42C23×C3×D3—34no—
43C2×C32×D4—212no—
44C2×C32×D4—212no—
45C2×C32×D4C2×D3112yes—
46C2×C32×D4C2×D3112yescomplemented
47C2×C32×D4C22×C313yes—
48C2×C32×D4C22×C313yescomplemented
49C2×#638C22×C316yes—
50C22×C3⋊Dic3C22×C313yescomplemented
51C2×C3×#67—324no—
52C22×C3×Dic3—312no—
53C32×C42—24no—
54C32×C42C2×D314yescomplemented
55C32×C42C22×C311yescharacteristic, complemented
56#3757C22×C313yes—
57#3757C22×C313yescomplemented
58#3752C22×C316yes—
59#3750C22×C316yescomplemented
60C3×#217—312no—
61C3×#217—312no—
62C3×(C2×D3)⋊C4—324no—
63C3×Dic3⋊C4—324no—
64C22×C33—23no—
65C22×C33—26no—
66C22×C33—23no—
67C22×C33C2411yescharacteristic
68C2×C3×C3⋊D3—26no—
69C2×C3×C3⋊D3—44no—
70C33×C4—23no—
71C3×C3⋊Dic3—26no—
72C3×#127—24no—
73C3×#127C3×D314yescomplemented
74C3×#127C3⋊D311yescharacteristic, complemented
75C3×#126—92no—
76C3×#123—93no—
77C3×#121—96no—
78C3×#120—93no—
79#939—324no—
80#937—312no—
81#935—312no—
82#876—38no—
83C23×C32—24no—
84C23×C32—212no—
85C23×C32C22×D314yes—
86C23×C32C22×D3112yes—
87C23×C32C23×C311yescharacteristic
88C23×C32C23×C313yes—
89C22×C3⋊D3—43no—
90C22×C3⋊D3C23×C312yes—
91C22×C3×D3—38no—
92C22×C3×D3—1212no—
93C32×D4—412no—
94C32×D4—412no—
95C32×D4—43no—
96C32×D4—43no—
97C32×D4—424no—
98C32×D4—424no—
99C2×C32×C4—212no—
100C2×C32×C4C22×D3112yes—
101C2×C32×C4C23×C313yes—
102#638—46no—
103C2×C3⋊Dic3—43no—
104C2×C3⋊Dic3C23×C316yes—
105C3×#67—1224no—
106C2×C3×Dic3—324no—
107C2×C3×Dic3—1212no—
108#560—91noSylow
109C2×C33—41no—
110C2×C33—43no—
111C2×C33H(5,𝔽2)13yes—
112C3×C3⋊D3—42no—
113C24×C3—91no—
114C23×D3—34no—
115C2×C3×D4—212no—
116C2×C3×D4—212no—
117C2×C3×D4—96no—
118C2×C3×D4C2×C3×D3112yes—
119C2×C3×D4C2×C3×D3112yescomplemented
120C2×C3×D4C2×C3⋊D313yes—
121C2×C3×D4C2×C3⋊D313yescomplemented
122C22×C3×C4—93no—
123C2×#67—324no—
124C22×Dic3—312no—
125C3×C4⋊C4—96no—
126C3×C22⋊C4—93no—
127C3×C22⋊C4—93no—
128C3×C22⋊C4—96no—
129C3×C42—24no—
130C3×C42C2×C3×D314yescomplemented
131C3×C42C2×C3⋊D311yescharacteristic, complemented
132#217—312no—
133#217—312no—
134(C2×D3)⋊C4—324no—
135Dic3⋊C4—324no—
136C22×C32—23no—
137C22×C32—26no—
138C22×C32—23no—
139C22×C32—24no—
140C22×C32—212no—
141C22×C32—224no—
142C22×C32—212no—
143C22×C32—412no—
144C22×C32—424no—
145C22×C32—412no—
146C22×C32C23×D314yes—
147C22×C32C24×C311yescharacteristic
148C2×C3⋊D3—26no—
149C2×C3⋊D3—44no—
150C2×C3×D3—624no—
151C2×C3×D3—1216no—
152C32×C4—224no—
153C32×C4—212no—
154C32×C4—23no—
155C3⋊Dic3—26no—
156C3×Dic3—624no—
157#127C3×C3⋊D311yescharacteristic, complemented
158#126—92no—
159#123—93no—
160#121—96no—
161#120—93no—
162C33#56011yescharacteristic, complemented, Sylow
163C23×C3—24no—
164C23×C3—212no—
165C23×C3—92no—
166C23×C3—363no—
167C23×C3C22×C3×D314yes—
168C23×C3C22×C3×D3112yes—
169C23×C3C22×C3⋊D311yescharacteristic
170C23×C3C22×C3⋊D313yes—
171C22×D3—38no—
172C22×D3—1212no—
173C3×D4—412no—
174C3×D4—412no—
175C3×D4—43no—
176C3×D4—43no—
177C3×D4—424no—
178C3×D4—424no—
179C3×D4—366no—
180C2×C3×C4—212no—
181C2×C3×C4—96no—
182C2×C3×C4—363no—
183C2×C3×C4C22×C3×D3112yes—
184C2×C3×C4C22×C3⋊D313yes—
185#67—1224no—
186C2×Dic3—324no—
187C2×Dic3—1212no—
188C2×C32—212no—
189C2×C32—41no—
190C2×C32—43no—
191C2×C32—44no—
192C2×C32—412no—
193C2×C32—84no—
194C2×C32—812no—
195C2×C32#1004112yes—
196C2×C32C3×H(5,𝔽2)13yes—
197C3⋊D3—42no—
198C3×D3—128no—
199C24—91no—
200C2×D4—96no—
201C2×D4C2×C3×C3⋊D313yes—
202C2×D4C2×C3×C3⋊D313yescomplemented
203C22×C4—93no—
204C4⋊C4—96no—
205C22⋊C4—93no—
206C22⋊C4—93no—
207C22⋊C4—96no—
208C42C2×C3×C3⋊D311yescharacteristic, complemented
209C22×C3—212no—
210C22×C3—224no—
211C22×C3—212no—
212C22×C3—23no—
213C22×C3—26no—
214C22×C3—23no—
215C22×C3—24no—
216C22×C3—412no—
217C22×C3—424no—
218C22×C3—412no—
219C22×C3—186no—
220C22×C3—364no—
221C22×C3C23×C3×D314yes—
222C22×C3C23×C3⋊D311yescharacteristic
223C2×D3—624no—
224C2×D3—1216no—
225C3×C4—224no—
226C3×C4—212no—
227C3×C4—23no—
228C3×C4—186no—
229Dic3—624no—
230C32—24no—
231C32#572014yescomplemented
232C32C3×#56011yescharacteristic, complemented
233C23—92no—
234C23—363no—
235C23C22×C3×C3⋊D311yescharacteristic
236C23C22×C3×C3⋊D313yes—
237D4—43no—
238D4—43no—
239D4—366no—
240C2×C4—96no—
241C2×C4—363no—
242C2×C4C22×C3×C3⋊D313yes—
243C2×C3—212no—
244C2×C3—41no—
245C2×C3—43no—
246C2×C3—44no—
247C2×C3—412no—
248C2×C3—84no—
249C2×C3—812no—
250C2×C3—362no—
251C2×C3C3×#1004112yes—
252C2×C3#6436113yes—
253D3—128no—
254C22—23no—
255C22—26no—
256C22—23no—
257C22—186no—
258C22—364no—
259C22C23×C3×C3⋊D311yescharacteristic
260C4—23no—
261C4—186no—
262C3—24no—
263C3C3×#572014yescomplemented
264C3#1059619111yescharacteristic, direct product factor
265C2—41no—
266C2—43no—
267C2—362no—
268C2C3×#6436113yes—
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×#127C2
socleC22×C33C24
hypercenterC3×#127C3⋊D3
hypocenterC32C3×#560
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×#127

Complements

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

NH ≅ G/Ntype
1. C33×#127C2semidirect
2. C3×#64148C2semidirect
3. C3×#64146C2semidirect
4. C3×#64144C2semidirect
5. C3×#64085C2semidirect
6. #10596191C3direct (N × H)
10. C2×C33×D4C22semidirect
12. C22×C3×C3⋊Dic3C22semidirect
13. C33×C42C22semidirect
15. C3×#3757C22semidirect
17. C3×#3750C22semidirect
19. C32×#127D3semidirect
20. C32×#127C2×C3semidirect
21. #64148C2×C3semidirect
22. #64146C2×C3semidirect
23. #64144C2×C3semidirect
24. #64085C2×C3semidirect
46. C2×C32×D4C2×D3semidirect
48. C2×C32×D4C22×C3semidirect
50. C22×C3⋊Dic3C22×C3semidirect
54. C32×C42C2×D3semidirect
55. C32×C42C22×C3semidirect
57. #3757C22×C3semidirect
59. #3750C22×C3semidirect
73. C3×#127C3×D3semidirect
74. C3×#127C3⋊D3semidirect
119. C2×C3×D4C2×C3×D3semidirect
121. C2×C3×D4C2×C3⋊D3semidirect
130. C3×C42C2×C3×D3semidirect
131. C3×C42C2×C3⋊D3semidirect
157. #127C3×C3⋊D3semidirect
162. C33#560semidirect
202. C2×D4C2×C3×C3⋊D3semidirect
208. C42C2×C3×C3⋊D3semidirect
231. C32#5720semidirect
232. C32C3×#560semidirect
263. C3C3×#5720semidirect
264. C3#10596191direct (N × H)

Maximal subgroups

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

subgroupindex
C33×#1272
C3×#641482
C3×#641462
C3×#641442
C3×#640852
#105961913
C3×#57203