Finite Group Databasebeta

← C22×C3×#3710 #13174412

Subgroups — C22×C3×#3710 #13174412

Order 1728.

1794 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C22×C32×C3⋊C8C212yesmaximal, complemented
3C2×C3×#3710C2112yesmaximal, direct product factor
4C22×C3×D3×C8—32nomaximal
5C22×#3710C311yesmaximal, characteristic, direct product factor
6C23×C3×C3⋊D3C411yescharacteristic
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3C411yescharacteristic
9C2×C3×C4×C3⋊D3C416yes—
10C2×C3×C4×C3⋊D3C2216yes—
11C2×C32×C3⋊C8C22112yescomplemented
12C3×#3710C22116yesdirect product factor
13C22×C4×C3⋊D3C2×C311yescharacteristic
14C22×C3×C4×D3—32no—
15C22×C3×C4×D3—61no—
16C22×C32×C8—32no—
17C22×C3×C3⋊C8—22no—
18C22×C3×C3⋊C8D312yescomplemented
19C22×C3×C3⋊C8C2×C312yescomplemented
20C2×C3×D3×C8—324no—
21C2×#3710C2×C3112yesdirect product factor
22C23×C33C2×C411yescharacteristic
23C22×C3×C3⋊D3C818yescomplemented
24C22×C3×C3⋊D3C2×C416yes—
25C2×C33×C4C2×C413yes—
26C2×C33×C4C2313yes—
27C2×C3×C3⋊Dic3C2×C416yes—
28C3×C4×C3⋊D3C2×C4112yes—
29C3×C4×C3⋊D3C2314yes—
30C32×C3⋊C8C2318yescomplemented
31C23×C3×C8—91no—
32C22×D3×C8—32no—
33C23×C3⋊D3C3×C411yescharacteristic
34C23×C3×D3—32no—
35C23×C3×D3—61no—
36C22×C32×C4—21no—
37C22×C32×C4—22no—
38C22×C32×C4—41no—
39C22×C32×C4C2×D312yes—
40C22×C32×C4C22×C311yescharacteristic
41C22×C3⋊Dic3C3×C411yescharacteristic
42C2×C4×C3⋊D3C3×C416yes—
43C2×C4×C3⋊D3C22×C316yes—
44C22×C3×Dic3—32no—
45C22×C3×Dic3—61no—
46C2×C3×C4×D3—312no—
47C2×C3×C4×D3—312no—
48C2×C3×C4×D3—66no—
49C2×C3×C4×D3—66no—
50C2×C32×C8—312no—
51C2×C3×C3⋊C8—212no—
52C2×C3×C3⋊C8C2×D3112yescomplemented
53C2×C3×C3⋊C8C22×C3112yescomplemented
54C3×D3×C8—332no—
55#3710C22×C3116yesdirect product factor
56C22×C33C2×C814yescomplemented
57C22×C33C22×C413yes—
58C2×C3×C3⋊D3C2×C8124yescomplemented
59C2×C3×C3⋊D3C22×C414yes—
60C33×C4C22×C413yes—
61C33×C4C2411yescharacteristic
62C3×C3⋊Dic3C22×C414yes—
63C23×C3×C4—91no—
64C22×C4×D3—32no—
65C22×C4×D3—61no—
66C22×C3×C8—32no—
67C22×C3×C8—32no—
68C22×C3×C8—62no—
69C22×C3×C8—912no—
70C22×C3⋊C8C3×D312yescomplemented
71C2×D3×C8—324no—
72C23×C32—21no—
73C23×C32—22no—
74C23×C32—41no—
75C23×C32C4×D312yes—
76C23×C32C2×C3×C411yescharacteristic
77C22×C3⋊D3C3×C818yescomplemented
78C22×C3⋊D3C2×C3×C416yes—
79C22×C3×D3—316no—
80C22×C3×D3—312no—
81C22×C3×D3—68no—
82C22×C3×D3—66no—
83C2×C32×C4—23no—
84C2×C32×C4—26no—
85C2×C32×C4—23no—
86C2×C32×C4—26no—
87C2×C32×C4—43no—
88C2×C32×C4—43no—
89C2×C32×C4C4×D316yes—
90C2×C32×C4C2×C3×C413yes—
91C2×C32×C4C22×D316yes—
92C2×C32×C4C23×C313yes—
93C2×C3⋊Dic3C2×C3×C416yes—
94C4×C3⋊D3C2×C3×C4112yes—
95C4×C3⋊D3C23×C314yes—
96C2×C3×Dic3—312no—
97C2×C3×Dic3—66no—
98C3×C4×D3—324no—
99C3×C4×D3—38no—
100C3×C4×D3—612no—
101C3×C4×D3—64no—
102C32×C8—38no—
103C3×C3⋊C8—28no—
104C3×C3⋊C8C22×D318yescomplemented
105C3×C3⋊C8C23×C318yescomplemented
106C23×C8—91noSylow
107C2×C33C22×C816yescomplemented
108C2×C33C23×C411yescharacteristic
109C3×C3⋊D3C22×C818yescomplemented
110C24×C3—91no—
111C23×D3—32no—
112C23×D3—61no—
113C22×C3×C4—22no—
114C22×C3×C4—21no—
115C22×C3×C4—41no—
116C22×C3×C4—91no—
117C22×C3×C4—96no—
118C22×C3×C4—96no—
119C22×C3×C4D3211yescharacteristic
120C22×C3×C4C2×C3×D312yes—
121C22×Dic3—32no—
122C22×Dic3—61no—
123C2×C4×D3—312no—
124C2×C4×D3—312no—
125C2×C4×D3—66no—
126C2×C4×D3—66no—
127C2×C3×C8—312no—
128C2×C3×C8—312no—
129C2×C3×C8—612no—
130C2×C3×C8—916no—
131C2×C3⋊C8C2×C3×D3112yescomplemented
132D3×C8—332no—
133C22×C32—24no—
134C22×C32—28no—
135C22×C32—23no—
136C22×C32—26no—
137C22×C32—44no—
138C22×C32—43no—
139C22×C32D3×C818yescomplemented
140C22×C32C2×C3×C814yescomplemented
141C22×C32C2×C4×D316yes—
142C22×C32C22×C3×C413yes—
143C2×C3⋊D3C2×C3×C8124yescomplemented
144C2×C3⋊D3C22×C3×C414yes—
145C2×C3×D3—348no—
146C2×C3×D3—38no—
147C2×C3×D3—624no—
148C2×C3×D3—64no—
149C32×C4—21no—
150C32×C4—22no—
151C32×C4—23no—
152C32×C4—26no—
153C32×C4—43no—
154C32×C4—41no—
155C32×C4C2×C4×D316yes—
156C32×C4C22×C3×C413yes—
157C32×C4C23×D312yes—
158C32×C4C24×C311yescharacteristic
159C3⋊Dic3C22×C3×C414yes—
160C3×Dic3—38no—
161C3×Dic3—64no—
162C23×C4—91no—
163C22×C8—32no—
164C22×C8—912no—
165C33C23×C811yescharacteristic, complemented, Sylow
166C23×C3—22no—
167C23×C3—21no—
168C23×C3—41no—
169C23×C3—98no—
170C23×C3—96no—
171C23×C3#62411yescharacteristic
172C23×C3C3×C4×D312yes—
173C22×D3—316no—
174C22×D3—312no—
175C22×D3—68no—
176C22×D3—66no—
177C2×C3×C4—26no—
178C2×C3×C4—23no—
179C2×C3×C4—26no—
180C2×C3×C4—23no—
181C2×C3×C4—43no—
182C2×C3×C4—43no—
183C2×C3×C4—96no—
184C2×C3×C4—94no—
185C2×C3×C4—912no—
186C2×C3×C4#62413yes—
187C2×C3×C4C3×C4×D316yes—
188C2×C3×C4C2×D3213yes—
189C2×C3×C4C22×C3×D316yes—
190C2×Dic3—312no—
191C2×Dic3—66no—
192C4×D3—38no—
193C4×D3—324no—
194C4×D3—612no—
195C4×D3—64no—
196C3×C8—38no—
197C3×C8—38no—
198C3×C8—68no—
199C3⋊C8C22×C3×D318yescomplemented
200C2×C32—212no—
201C2×C32—21no—
202C2×C32—26no—
203C2×C32—22no—
204C2×C32—46no—
205C2×C32—41no—
206C2×C32C2×D3×C8112yescomplemented
207C2×C32C22×C3×C816yescomplemented
208C2×C32C22×C4×D312yes—
209C2×C32C23×C3×C411yescharacteristic
210C3⋊D3C22×C3×C818yescomplemented
211C3×D3—316no—
212C3×D3—68no—
213C24—91no—
214C22×C4—91no—
215C22×C4—96no—
216C22×C4—96no—
217C22×C4C3×D3211yescharacteristic
218C2×C8—312no—
219C2×C8—916no—
220C22×C3—23no—
221C22×C3—26no—
222C22×C3—24no—
223C22×C3—28no—
224C22×C3—43no—
225C22×C3—44no—
226C22×C3—94no—
227C22×C3—924no—
228C22×C3#371014yesdirect product factor
229C22×C3C3×D3×C818yescomplemented
230C22×C3C2×#62413yes—
231C22×C3C2×C3×C4×D316yes—
232C2×D3—348no—
233C2×D3—38no—
234C2×D3—64no—
235C2×D3—624no—
236C3×C4—22no—
237C3×C4—21no—
238C3×C4—23no—
239C3×C4—26no—
240C3×C4—41no—
241C3×C4—43no—
242C3×C4—94no—
243C3×C4C2×#62413yes—
244C3×C4C2×C3×C4×D316yes—
245C3×C4C22×D3211yescharacteristic
246C3×C4C23×C3×D312yes—
247Dic3—38no—
248Dic3—64no—
249C32—21no—
250C32—22no—
251C32—41no—
252C32C22×D3×C812yescomplemented
253C32C23×C3×C811yescharacteristic, complemented
254C23—96no—
255C23—98no—
256C23C3×#62411yescharacteristic
257C2×C4—96no—
258C2×C4—94no—
259C2×C4—912no—
260C2×C4C3×#62413yes—
261C2×C4C2×C3×D3213yes—
262C8—38no—
263C2×C3—21no—
264C2×C3—22no—
265C2×C3—26no—
266C2×C3—212no—
267C2×C3—41no—
268C2×C3—46no—
269C2×C3—98no—
270C2×C3C2×#371016yesdirect product factor
271C2×C3C2×C3×D3×C8112yescomplemented
272C2×C3C22×#62411yescharacteristic
273C2×C3C22×C3×C4×D312yes—
274D3—316no—
275D3—68no—
276C22—94no—
277C22—924no—
278C22C3×#371014yesdirect product factor
279C22C2×C3×#62413yes—
280C4—94no—
281C4C2×C3×#62413yes—
282C4C22×C3×D3211yescharacteristic
283C3—21no—
284C3—22no—
285C3—41no—
286C3C22×#371011yescharacteristic, direct product factor
287C3C22×C3×D3×C812yescomplemented
288C2—98no—
289C2C2×C3×#371016yesdirect product factor
290C2C22×C3×#62411yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4D32
commutator, hypocenterC32C23×C3×C8
FrattiniC4C22×C3×D32
FittingC22×C33×C4C22
socleC23×C33C2×C4
prime radicalC23×C3×C3⋊D3C4
cosocleC32×C4C24×C3
LCSG ▷ C32
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C22×C32×C3⋊C8C2semidirect
3. C2×C3×#3710C2direct (N × H)
5. C22×#3710C3direct (N × H)
11. C2×C32×C3⋊C8C22semidirect
12. C3×#3710C22direct (N × H)
18. C22×C3×C3⋊C8D3semidirect
19. C22×C3×C3⋊C8C2×C3semidirect
21. C2×#3710C2×C3direct (N × H)
23. C22×C3×C3⋊D3C8semidirect
30. C32×C3⋊C8C23semidirect
52. C2×C3×C3⋊C8C2×D3semidirect
53. C2×C3×C3⋊C8C22×C3semidirect
55. #3710C22×C3direct (N × H)
56. C22×C33C2×C8semidirect
58. C2×C3×C3⋊D3C2×C8semidirect
70. C22×C3⋊C8C3×D3semidirect
77. C22×C3⋊D3C3×C8semidirect
104. C3×C3⋊C8C22×D3semidirect
105. C3×C3⋊C8C23×C3semidirect
107. C2×C33C22×C8semidirect
109. C3×C3⋊D3C22×C8semidirect
131. C2×C3⋊C8C2×C3×D3semidirect
139. C22×C32D3×C8semidirect
140. C22×C32C2×C3×C8semidirect
143. C2×C3⋊D3C2×C3×C8semidirect
165. C33C23×C8semidirect
199. C3⋊C8C22×C3×D3semidirect
206. C2×C32C2×D3×C8semidirect
207. C2×C32C22×C3×C8semidirect
210. C3⋊D3C22×C3×C8semidirect
228. C22×C3#3710direct (N × H)
229. C22×C3C3×D3×C8semidirect
252. C32C22×D3×C8semidirect
253. C32C23×C3×C8semidirect
270. C2×C3C2×#3710direct (N × H)
271. C2×C3C2×C3×D3×C8semidirect
278. C22C3×#3710direct (N × H)
286. C3C22×#3710direct (N × H)
287. C3C22×C3×D3×C8semidirect
289. C2C2×C3×#3710direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C22×C32×C3⋊C82
C2×C3×#37102
C22×C3×D3×C83
C22×#37103