Finite Group Databasebeta

← C22×C3×#3713 #13174418

Subgroups — C22×C3×#3713 #13174418

Order 1728.

1954 conjugacy classes of subgroups; only the 384 normal subgroups are shown.

normal subgroups only

This group has 1954 conjugacy classes of subgroups — too many to draw. Only its 384 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×C32×D12C212yesmaximal, complemented
2C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
3C2×C3×#3713C2112yesmaximal, direct product factor
4C22×C3×#213—32nomaximal
5C22×#3713C311yesmaximal, characteristic, direct product factor
6C23×C32×D3—22no—
7C22×C33×C4C2211yescharacteristic
8C2×C32×D12—26no—
9C2×C32×D12C22112yescomplemented
10C2×C3×C3⋊(C3⋊C8)C2216yescomplemented
11C3×#3713C22116yesdirect product factor
12C22×C32×D4—32no—
13C22×C3×D12—22no—
14C22×C3×D12D312yescomplemented
15C22×C3×D12C2×C312yescomplemented
16C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
17C2×C3×#213—324no—
18C22×C3×C3⋊C8—32no—
19C22×C3×C3⋊C8—61no—
20C2×#3713C2×C3112yesdirect product factor
21C23×C33D411yescharacteristic
22C22×C32×D3—212no—
23C22×C32×D3—48no—
24C2×C33×C4D413yes—
25C2×C33×C4C2313yes—
26C32×D12—212no—
27C32×D12C2318yescomplemented
28C3×C3⋊(C3⋊C8)C2314yescomplemented
29C22×C3×D8—91no—
30C22×#213—32no—
31C24×C32—62no—
32C23×C3×D3—24no—
33C23×C3×D3—22no—
34C23×C3×D3—22no—
35C2×C32×D4—312no—
36C2×C32×D4—66no—
37C22×C32×C4—21no—
38C22×C32×C4—22no—
39C22×C32×C4—41no—
40C22×C32×C4C2×D312yes—
41C22×C32×C4C22×C311yescharacteristic
42C2×C3×D12—212no—
43C2×C3×D12—212no—
44C2×C3×D12—26no—
45C2×C3×D12—26no—
46C2×C3×D12C2×D3112yescomplemented
47C2×C3×D12C22×C3112yescomplemented
48C2×C3⋊(C3⋊C8)C22×C316yescomplemented
49C3×#213—332no—
50C2×C3×C3⋊C8—312no—
51C2×C3×C3⋊C8—66no—
52#3713C22×C3116yesdirect product factor
53C22×C33D814yescomplemented
54C22×C33C2×D413yes—
55C2×C32×D3—28no—
56C2×C32×D3—424no—
57C33×C4C2×D413yes—
58C33×C4C2411yescharacteristic
59C22×C3×D4—32no—
60C22×C3×D4—32no—
61C22×C3×D4—62no—
62C22×D12C3×D312yescomplemented
63C2×C3×D8—912no—
64C22×C3×C8—91no—
65C2×#213—324no—
66C22×C3⋊C8—32no—
67C22×C3⋊C8—61no—
68C23×C32—21no—
69C23×C32—22no—
70C23×C32—41no—
71C23×C32—612no—
72C23×C32—128no—
73C23×C32#6712yes—
74C23×C32C3×D411yescharacteristic
75C22×C3×D3—224no—
76C22×C3×D3—212no—
77C22×C3×D3—212no—
78C22×C3×D3—416no—
79C22×C3×D3—48no—
80C22×C3×D3—48no—
81C32×D4—38no—
82C32×D4—612no—
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×C4#6716yes—
90C2×C32×C4C3×D413yes—
91C2×C32×C4C22×D316yes—
92C2×C32×C4C23×C313yes—
93C3×D12—224no—
94C3×D12—212no—
95C3×D12—212no—
96C3×D12—28no—
97C3×D12C22×D318yescomplemented
98C3×D12C23×C318yescomplemented
99C3⋊(C3⋊C8)C23×C314yescomplemented
100C3×C3⋊C8—38no—
101C3×C3⋊C8—64no—
102C22×D8—91noSylow
103C2×C33C2×D816yescomplemented
104C2×C33C22×D411yescharacteristic
105C32×D3—48no—
106C24×C3—64no—
107C24×C3—62no—
108C24×C3—62no—
109C23×D3—22no—
110C2×C3×D4—312no—
111C2×C3×D4—312no—
112C2×C3×D4—612no—
113C2×C3×D4—612no—
114C2×C3×D4—66no—
115C2×C3×D4—66no—
116C22×C3×C4—21no—
117C22×C3×C4—22no—
118C22×C3×C4—41no—
119C22×C3×C4D3211yescharacteristic
120C22×C3×C4C2×C3×D312yes—
121C2×D12—26no—
122C2×D12C2×C3×D3112yescomplemented
123C3×D8—916no—
124C2×C3×C8—96no—
125#213—332no—
126C2×C3⋊C8—312no—
127C2×C3⋊C8—66no—
128C22×C32—24no—
129C22×C32—28no—
130C22×C32—23no—
131C22×C32—26no—
132C22×C32—44no—
133C22×C32—43no—
134C22×C32—68no—
135C22×C32—1224no—
136C22×C32#21318yescomplemented
137C22×C32C3×D814yescomplemented
138C22×C32C2×#6716yes—
139C22×C32C2×C3×D413yes—
140C2×C3×D3—216no—
141C2×C3×D3—28no—
142C2×C3×D3—28no—
143C2×C3×D3—448no—
144C2×C3×D3—424no—
145C2×C3×D3—424no—
146C32×C4—21no—
147C32×C4—22no—
148C32×C4—23no—
149C32×C4—26no—
150C32×C4—43no—
151C32×C4—41no—
152C32×C4C2×#6716yes—
153C32×C4C2×C3×D413yes—
154C32×C4C23×D312yes—
155C32×C4C24×C311yescharacteristic
156C22×D4—32no—
157C2×D8—912no—
158C22×C8—91no—
159C33C22×D811yescharacteristic, complemented, Sylow
160C23×C3—22no—
161C23×C3—21no—
162C23×C3—41no—
163C23×C3—624no—
164C23×C3—612no—
165C23×C3—612no—
166C23×C3—1216no—
167C23×C3—128no—
168C23×C3—128no—
169C23×C3#62511yescharacteristic
170C23×C3C3×#6712yes—
171C22×D3—212no—
172C22×D3—48no—
173C3×D4—38no—
174C3×D4—38no—
175C3×D4—68no—
176C3×D4—624no—
177C3×D4—612no—
178C3×D4—612no—
179C2×C3×C4—26no—
180C2×C3×C4—23no—
181C2×C3×C4—26no—
182C2×C3×C4—23no—
183C2×C3×C4—43no—
184C2×C3×C4—43no—
185C2×C3×C4#62513yes—
186C2×C3×C4C3×#6716yes—
187C2×C3×C4C2×D3213yes—
188C2×C3×C4C22×C3×D316yes—
189D12—212no—
190D12C22×C3×D318yescomplemented
191C3×C8—94no—
192C3⋊C8—38no—
193C3⋊C8—64no—
194C2×C32—212no—
195C2×C32—21no—
196C2×C32—26no—
197C2×C32—22no—
198C2×C32—46no—
199C2×C32—41no—
200C2×C32—128no—
201C2×C32C2×#213112yescomplemented
202C2×C32C2×C3×D816yescomplemented
203C2×C32C22×#6712yes—
204C2×C32C22×C3×D411yescharacteristic
205C3×D3—416no—
206C3×D3—48no—
207C3×D3—48no—
208C24—62no—
209C2×D4—312no—
210C2×D4—66no—
211C22×C4C3×D3211yescharacteristic
212D8—916no—
213C2×C8—96no—
214C22×C3—26no—
215C22×C3—23no—
216C22×C3—28no—
217C22×C3—24no—
218C22×C3—43no—
219C22×C3—44no—
220C22×C3—616no—
221C22×C3—68no—
222C22×C3—68no—
223C22×C3—1248no—
224C22×C3—1224no—
225C22×C3—1224no—
226C22×C3#371314yesdirect product factor
227C22×C3C3×#21318yescomplemented
228C22×C3C2×#62513yes—
229C22×C3C2×C3×#6716yes—
230C2×D3—28no—
231C2×D3—424no—
232C3×C4—21no—
233C3×C4—22no—
234C3×C4—26no—
235C3×C4—23no—
236C3×C4—41no—
237C3×C4—43no—
238C3×C4C2×#62513yes—
239C3×C4C2×C3×#6716yes—
240C3×C4C22×D3211yescharacteristic
241C3×C4C23×C3×D312yes—
242C32—22no—
243C32—21no—
244C32—41no—
245C32C22×#21312yescomplemented
246C32C22×C3×D811yescharacteristic, complemented
247C23—612no—
248C23—128no—
249C23C3×#62511yescharacteristic
250D4—38no—
251D4—612no—
252C2×C4C3×#62513yes—
253C2×C4C2×C3×D3213yes—
254C8—94no—
255C2×C3—22no—
256C2×C3—21no—
257C2×C3—212no—
258C2×C3—26no—
259C2×C3—41no—
260C2×C3—46no—
261C2×C3—1216no—
262C2×C3—128no—
263C2×C3—128no—
264C2×C3C2×#371316yesdirect product factor
265C2×C3C2×C3×#213112yescomplemented
266C2×C3C22×#62511yescharacteristic
267C2×C3C22×C3×#6712yes—
268D3—48no—
269C22—68no—
270C22—1224no—
271C22C3×#371314yesdirect product factor
272C22C2×C3×#62513yes—
273C4C2×C3×#62513yes—
274C4C22×C3×D3211yescharacteristic
275C3—22no—
276C3—21no—
277C3—41no—
278C3C22×#371311yescharacteristic, direct product factor
279C3C22×C3×#21312yescomplemented
280C2—128no—
281C2C2×C3×#371316yesdirect product factor
282C2C22×C3×#62511yescharacteristic
subgroupG/H
centerC23×C3#625
commutator, cosocleC32×C4C24×C3
FrattiniC4C22×C3×D32
FittingC22×C33×C4C22
socleC23×C33D4
hypercenterC22×C3×C4D32
hypocenterC32C22×C3×D8
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C32×D12C2semidirect
2. C22×C3×C3⋊(C3⋊C8)C2semidirect
3. C2×C3×#3713C2direct (N × H)
5. C22×#3713C3direct (N × H)
9. C2×C32×D12C22semidirect
10. C2×C3×C3⋊(C3⋊C8)C22semidirect
11. C3×#3713C22direct (N × H)
14. C22×C3×D12D3semidirect
15. C22×C3×D12C2×C3semidirect
16. C22×C3⋊(C3⋊C8)C2×C3semidirect
20. C2×#3713C2×C3direct (N × H)
27. C32×D12C23semidirect
28. C3×C3⋊(C3⋊C8)C23semidirect
46. C2×C3×D12C2×D3semidirect
47. C2×C3×D12C22×C3semidirect
48. C2×C3⋊(C3⋊C8)C22×C3semidirect
52. #3713C22×C3direct (N × H)
53. C22×C33D8semidirect
62. C22×D12C3×D3semidirect
97. C3×D12C22×D3semidirect
98. C3×D12C23×C3semidirect
99. C3⋊(C3⋊C8)C23×C3semidirect
103. C2×C33C2×D8semidirect
122. C2×D12C2×C3×D3semidirect
136. C22×C32#213semidirect
137. C22×C32C3×D8semidirect
159. C33C22×D8semidirect
190. D12C22×C3×D3semidirect
201. C2×C32C2×#213semidirect
202. C2×C32C2×C3×D8semidirect
226. C22×C3#3713direct (N × H)
227. C22×C3C3×#213semidirect
245. C32C22×#213semidirect
246. C32C22×C3×D8semidirect
264. C2×C3C2×#3713direct (N × H)
265. C2×C3C2×C3×#213semidirect
271. C22C3×#3713direct (N × H)
278. C3C22×#3713direct (N × H)
279. C3C22×C3×#213semidirect
281. C2C2×C3×#3713direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×D122
C22×C3×C3⋊(C3⋊C8)2
C2×C3×#37132
C22×C3×#2133
C22×#37133