Finite Group Databasebeta

← C2×C3×#64162 #13179591

Subgroups — C2×C3×#64162 #13179591

Order 1728.

6964 conjugacy classes of subgroups; only the 750 normal subgroups are shown.

normal subgroups only

This group has 6964 conjugacy classes of subgroups — too many to draw. Only its 750 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
1C25×C33C211yesmaximal, characteristic, complemented
2C22×C3×#638C213yesmaximal, complemented
3C3×#64162C218yesmaximal, direct product factor
4C2×C3×#3757C213yesmaximal, complemented
5C2×#64162C311yesmaximal, characteristic, direct product factor
6C2×C3×#953—34nomaximal
7C24×C33—212no—
8C24×C33C2213yes—
9C24×C33C2214yescomplemented
10C23×C3×C3⋊D3C2211yescharacteristic, complemented
11C2×C3×#638—212no—
12C2×C3×#638C22112yescomplemented
13C22×C3×C3⋊Dic3C2213yescomplemented
14C3×#3757C22112yescomplemented
15C25×C32—24no—
16C25×C32D314yescomplemented
17C25×C32C2×C311yescharacteristic, complemented
18C22×#638C2×C313yescomplemented
19C22×C3×#67—312no—
20#64162C2×C318yesdirect product factor
21C2×#3757C2×C313yescomplemented
22C3×#953—332no—
23C2×C3×#217—312no—
24C23×C33—232no—
25C23×C33—212no—
26C23×C33—224no—
27C23×C33D4112yescomplemented
28C23×C33C2316yes—
29C23×C33C2311yescharacteristic
30C22×C3×C3⋊D3—26no—
31C22×C3×C3⋊D3C2312yescomplemented
32C3×#638—224no—
33C2×C3×C3⋊Dic3—26no—
34C2×C3×C3⋊Dic3C2316yescomplemented
35C2×C3×#120—91no—
36C2×#953—34no—
37C24×C32—296no—
38C24×C32—212no—
39C24×C32—216no—
40C24×C32—248no—
41C24×C32—212no—
42C24×C32C2×D3112yes—
43C24×C32C2×D3116yescomplemented
44C24×C32C22×C313yes—
45C24×C32C22×C314yescomplemented
46C23×C3⋊D3C22×C311yescharacteristic, complemented
47C23×C3×D3—34no—
48C2×#638—212no—
49C2×#638C22×C3112yescomplemented
50C22×C3⋊Dic3C22×C313yescomplemented
51C2×C3×#67—348no—
52C2×C3×#67—648no—
53C22×C3×Dic3—312no—
54#3757C22×C3112yescomplemented
55C3×#217—348no—
56C22×C33—232no—
57C22×C33—224no—
58C22×C33—212no—
59C22×C33C2×D416yes—
60C22×C33C2×D4112yescomplemented
61C22×C33C2411yescharacteristic
62C2×C3×C3⋊D3—26no—
63C2×C3×C3⋊D3—44no—
64C3×C3⋊Dic3—26no—
65C25×C3—24no—
66C25×C3C3×D314yescomplemented
67C25×C3C3⋊D311yescharacteristic, complemented
68C22×C3×D4—93no—
69C22×#67—312no—
70C3×#120—98no—
71C2×C3×C22⋊C4—93no—
72#953—332no—
73C2×#217—312no—
74C23×C32—2256no—
75C23×C32—2128no—
76C23×C32—232no—
77C23×C32—296no—
78C23×C32—2192no—
79C23×C32—248no—
80C23×C32—296no—
81C23×C32—212no—
82C23×C32—224no—
83C23×C32—248no—
84C23×C32—224no—
85C23×C32—24no—
86C23×C32#67148yescomplemented
87C23×C32C3×D4112yescomplemented
88C23×C32C22×D3124yes—
89C23×C32C22×D314yes—
90C23×C32C23×C316yes—
91C23×C32C23×C311yescharacteristic
92C22×C3⋊D3—26no—
93C22×C3⋊D3C23×C312yescomplemented
94C22×C3×D3—38no—
95C22×C3×D3—624no—
96#638—224no—
97C2×C3⋊Dic3—26no—
98C2×C3⋊Dic3C23×C316yescomplemented
99C3×#67—696no—
100C2×C3×Dic3—324no—
101C2×C3×Dic3—624no—
102C2×#120—91noSylow
103C2×C33—212no—
104C2×C33#12014yescomplemented
105C2×C33C22×D413yes—
106C3×C3⋊D3—42no—
107C24×C3—296no—
108C24×C3—248no—
109C24×C3—212no—
110C24×C3—212no—
111C24×C3—216no—
112C24×C3—91no—
113C24×C3C2×C3×D3112yes—
114C24×C3C2×C3×D3116yescomplemented
115C24×C3C2×C3⋊D313yes—
116C24×C3C2×C3⋊D314yescomplemented
117C23×D3—34no—
118C2×C3×D4—912no—
119C2×C3×D4—1812no—
120C22×C3×C4—93no—
121C2×#67—348no—
122C2×#67—648no—
123C22×Dic3—312no—
124C3×C22⋊C4—912no—
125#217—348no—
126C22×C32—2256no—
127C22×C32—2128no—
128C22×C32—232no—
129C22×C32—2192no—
130C22×C32—296no—
131C22×C32—296no—
132C22×C32—224no—
133C22×C32—248no—
134C22×C32—212no—
135C22×C32—24no—
136C22×C32—224no—
137C22×C32—248no—
138C22×C32C2×#67124yes—
139C22×C32C2×#67148yescomplemented
140C22×C32C2×C3×D416yes—
141C22×C32C2×C3×D4112yescomplemented
142C22×C32C23×D314yes—
143C22×C32C24×C311yescharacteristic
144C2×C3⋊D3—26no—
145C2×C3⋊D3—44no—
146C2×C3×D3—624no—
147C2×C3×D3—1216no—
148C3⋊Dic3—26no—
149C3×Dic3—624no—
150C25C3×C3⋊D311yescharacteristic, complemented
151C22×D4—93no—
152#120—98no—
153C2×C22⋊C4—93no—
154C33C2×#12011yescharacteristic, complemented, Sylow
155C23×C3—2256no—
156C23×C3—2128no—
157C23×C3—232no—
158C23×C3—296no—
159C23×C3—2192no—
160C23×C3—248no—
161C23×C3—296no—
162C23×C3—212no—
163C23×C3—224no—
164C23×C3—248no—
165C23×C3—224no—
166C23×C3—24no—
167C23×C3—92no—
168C23×C3—186no—
169C23×C3C3×#67148yescomplemented
170C23×C3#638112yescomplemented
171C23×C3C22×C3×D3124yes—
172C23×C3C22×C3×D314yes—
173C23×C3C22×C3⋊D316yes—
174C23×C3C22×C3⋊D311yescharacteristic
175C22×D3—38no—
176C22×D3—624no—
177C3×D4—1824no—
178C2×C3×C4—96no—
179C2×C3×C4—186no—
180#67—696no—
181C2×Dic3—324no—
182C2×Dic3—624no—
183C2×C32—296no—
184C2×C32—248no—
185C2×C32—212no—
186C2×C32—216no—
187C2×C32—212no—
188C2×C32#953116yescomplemented
189C2×C32C3×#12014yescomplemented
190C2×C32C22×#67112yes—
191C2×C32C22×C3×D413yes—
192C3⋊D3—42no—
193C3×D3—128no—
194C24—212no—
195C24—91no—
196C24C2×C3×C3⋊D313yes—
197C24C2×C3×C3⋊D314yescomplemented
198C2×D4—912no—
199C2×D4—1812no—
200C22×C4—93no—
201C22⋊C4—912no—
202C22×C3—2256no—
203C22×C3—2128no—
204C22×C3—232no—
205C22×C3—296no—
206C22×C3—2192no—
207C22×C3—296no—
208C22×C3—248no—
209C22×C3—212no—
210C22×C3—224no—
211C22×C3—24no—
212C22×C3—224no—
213C22×C3—248no—
214C22×C3—186no—
215C22×C3—364no—
216C22×C3C2×C3×#67124yes—
217C22×C3C2×C3×#67148yescomplemented
218C22×C3C2×#63816yes—
219C22×C3C2×#638112yescomplemented
220C22×C3C23×C3×D314yes—
221C22×C3C23×C3⋊D311yescharacteristic
222C2×D3—624no—
223C2×D3—1216no—
224C3×C4—186no—
225Dic3—624no—
226C32—24no—
227C32C2×#95314yescomplemented
228C32C2×C3×#12011yescharacteristic, complemented
229C23—232no—
230C23—212no—
231C23—224no—
232C23—92no—
233C23—186no—
234C23C3×#638112yescomplemented
235C23C22×C3×C3⋊D316yes—
236C23C22×C3×C3⋊D311yescharacteristic
237D4—1824no—
238C2×C4—96no—
239C2×C4—186no—
240C2×C3—296no—
241C2×C3—248no—
242C2×C3—212no—
243C2×C3—212no—
244C2×C3—216no—
245C2×C3—362no—
246C2×C3C3×#953116yescomplemented
247C2×C3#6416214yesdirect product factor
248C2×C3C22×C3×#67112yes—
249C2×C3C22×#63813yes—
250D3—128no—
251C22—232no—
252C22—212no—
253C22—224no—
254C22—186no—
255C22—364no—
256C22C2×C3×#63816yes—
257C22C2×C3×#638112yescomplemented
258C22C23×C3×C3⋊D311yescharacteristic
259C4—186no—
260C3—24no—
261C3C2×C3×#95314yescomplemented
262C3C2×#6416211yescharacteristic, direct product factor
263C2—212no—
264C2—362no—
265C2C3×#6416214yesdirect product factor
266C2C22×C3×#63813yes—
subgroupG/H
centerC23×C3C22×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC25×C33C2
socleC23×C33C23
hypercenterC25×C3C3⋊D3
hypocenterC32C2×C3×#120
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C25×C3

Complements

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

NH ≅ G/Ntype
1. C25×C33C2semidirect
2. C22×C3×#638C2semidirect
3. C3×#64162C2direct (N × H)
4. C2×C3×#3757C2semidirect
5. C2×#64162C3direct (N × H)
9. C24×C33C22semidirect
10. C23×C3×C3⋊D3C22semidirect
12. C2×C3×#638C22semidirect
13. C22×C3×C3⋊Dic3C22semidirect
14. C3×#3757C22semidirect
16. C25×C32D3semidirect
17. C25×C32C2×C3semidirect
18. C22×#638C2×C3semidirect
20. #64162C2×C3direct (N × H)
21. C2×#3757C2×C3semidirect
27. C23×C33D4semidirect
31. C22×C3×C3⋊D3C23semidirect
34. C2×C3×C3⋊Dic3C23semidirect
43. C24×C32C2×D3semidirect
45. C24×C32C22×C3semidirect
46. C23×C3⋊D3C22×C3semidirect
49. C2×#638C22×C3semidirect
50. C22×C3⋊Dic3C22×C3semidirect
54. #3757C22×C3semidirect
60. C22×C33C2×D4semidirect
66. C25×C3C3×D3semidirect
67. C25×C3C3⋊D3semidirect
86. C23×C32#67semidirect
87. C23×C32C3×D4semidirect
93. C22×C3⋊D3C23×C3semidirect
98. C2×C3⋊Dic3C23×C3semidirect
104. C2×C33#120semidirect
114. C24×C3C2×C3×D3semidirect
116. C24×C3C2×C3⋊D3semidirect
139. C22×C32C2×#67semidirect
141. C22×C32C2×C3×D4semidirect
150. C25C3×C3⋊D3semidirect
154. C33C2×#120semidirect
169. C23×C3C3×#67semidirect
170. C23×C3#638semidirect
188. C2×C32#953semidirect
189. C2×C32C3×#120semidirect
197. C24C2×C3×C3⋊D3semidirect
217. C22×C3C2×C3×#67semidirect
219. C22×C3C2×#638semidirect
227. C32C2×#953semidirect
228. C32C2×C3×#120semidirect
234. C23C3×#638semidirect
246. C2×C3C3×#953semidirect
247. C2×C3#64162direct (N × H)
257. C22C2×C3×#638semidirect
261. C3C2×C3×#953semidirect
262. C3C2×#64162direct (N × H)
265. C2C3×#64162direct (N × H)

Maximal subgroups

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

subgroupindex
C25×C332
C22×C3×#6382
C3×#641622
C2×C3×#37572
C2×#641623
C2×C3×#9533