Finite Group Databasebeta

← C2×C3×#63515 #13153761

Subgroups — C2×C3×#63515 #13153761

Order 1728.

2734 conjugacy classes of subgroups; only the 426 normal subgroups are shown.

normal subgroups only

This group has 2734 conjugacy classes of subgroups — too many to draw. Only its 426 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×C3×C9C211yesmaximal, characteristic, complemented
2C22×C3×#611C213yesmaximal, complemented
3C3×#63515C218yesmaximal, direct product factor
4C2×C3×#3676C213yesmaximal, complemented
5C2×C3×#953—31nomaximal
6C2×#63515C311yesmaximal, characteristic, direct product factor
7C24×C3×C9—212no—
8C24×C3×C9C2213yes—
9C24×C3×C9C2214yescomplemented
10C23×C3×D9C2211yescharacteristic, complemented
11C2×C3×#611—212no—
12C2×C3×#611C22112yescomplemented
13C22×C3×Dic9C2213yescomplemented
14C3×#3676C22112yescomplemented
15C25×C32D311yescharacteristic
16C22×C3×#67—33no—
17C25×C9—21no—
18C25×C9C2×C311yescharacteristic, complemented
19C3×#953—38no—
20C2×C3×#217—33no—
21C22×#611C2×C313yescomplemented
22#63515C2×C318yesdirect product factor
23C2×#3676C2×C313yescomplemented
24C23×C3×C9—232no—
25C23×C3×C9—212no—
26C23×C3×C9—224no—
27C23×C3×C9D4112yescomplemented
28C23×C3×C9C2316yes—
29C23×C3×C9C2311yescharacteristic
30C22×C3×D9—26no—
31C22×C3×D9C2312yescomplemented
32C3×#611—224no—
33C2×C3×Dic9—26no—
34C2×C3×Dic9C2316yescomplemented
35C2×C3×#120—91no—
36C2×#953—31no—
37C24×C32—212no—
38C24×C32C2×D314yes—
39C24×C32C2×D313yes—
40C23×C3×D3—31no—
41C2×C3×#67—312no—
42C2×C3×#67—612no—
43C22×C3×Dic3—33no—
44C24×C9—224no—
45C24×C9—23no—
46C24×C9—24no—
47C24×C9—212no—
48C24×C9C22×C313yes—
49C24×C9C22×C314yescomplemented
50C23×D9C22×C311yescharacteristic, complemented
51C3×#217—312no—
52C2×#611—212no—
53C2×#611C22×C3112yescomplemented
54C22×Dic9C22×C313yescomplemented
55#3676C22×C3112yescomplemented
56C22×C3×C9—232no—
57C22×C3×C9—224no—
58C22×C3×C9—212no—
59C22×C3×C9C2×D416yes—
60C22×C3×C9C2×D4112yescomplemented
61C22×C3×C9C2411yescharacteristic
62C2×C3×D9—26no—
63C2×C3×D9—44no—
64C3×Dic9—26no—
65C25×C3—21no—
66C25×C3D911yescharacteristic, complemented
67C25×C3C3×D311yescharacteristic
68C22×C3×D4—93no—
69C22×#67—33no—
70C3×#120—98no—
71C2×C3×C22⋊C4—93no—
72#953—38no—
73C2×#217—33no—
74C23×C32—232no—
75C23×C32—212no—
76C23×C32—224no—
77C23×C32#67112yes—
78C23×C32C22×D316yes—
79C23×C32C22×D311yescharacteristic
80C22×C3×D3—32no—
81C22×C3×D3—66no—
82C3×#67—624no—
83C2×C3×Dic3—36no—
84C2×C3×Dic3—66no—
85C23×C9—264no—
86C23×C9—232no—
87C23×C9—224no—
88C23×C9—248no—
89C23×C9—212no—
90C23×C9—224no—
91C23×C9—212no—
92C23×C9—26no—
93C23×C9—21no—
94C23×C9C3×D4112yescomplemented
95C23×C9C23×C316yes—
96C23×C9C23×C311yescharacteristic
97C22×D9—26no—
98C22×D9C23×C312yescomplemented
99#611—224no—
100C2×Dic9—26no—
101C2×Dic9C23×C316yescomplemented
102C2×#120—91noSylow
103C2×C3×C9—212no—
104C2×C3×C9#12014yescomplemented
105C2×C3×C9C22×D413yes—
106C3×D9—42no—
107C24×C3—224no—
108C24×C3—212no—
109C24×C3—212no—
110C24×C3—23no—
111C24×C3—24no—
112C24×C3—91no—
113C24×C3C2×D913yes—
114C24×C3C2×D914yescomplemented
115C24×C3C2×C3×D313yes—
116C24×C3C2×C3×D314yes—
117C23×D3—31no—
118C2×C3×D4—912no—
119C2×C3×D4—1812no—
120C22×C3×C4—93no—
121C2×#67—312no—
122C2×#67—612no—
123C22×Dic3—33no—
124C3×C22⋊C4—912no—
125#217—312no—
126C22×C32—232no—
127C22×C32—224no—
128C22×C32—212no—
129C22×C32C2×#6716yes—
130C22×C32C2×#67112yes—
131C22×C32C23×D311yescharacteristic
132C2×C3×D3—66no—
133C2×C3×D3—124no—
134C3×Dic3—66no—
135C22×C9—264no—
136C22×C9—232no—
137C22×C9—248no—
138C22×C9—224no—
139C22×C9—224no—
140C22×C9—212no—
141C22×C9—21no—
142C22×C9—26no—
143C22×C9—212no—
144C22×C9C2×C3×D416yes—
145C22×C9C2×C3×D4112yescomplemented
146C22×C9C24×C311yescharacteristic
147C2×D9—26no—
148C2×D9—44no—
149Dic9—26no—
150C25C3×D911yescharacteristic, complemented
151C22×D4—93no—
152#120—98no—
153C2×C22⋊C4—93no—
154C3×C9C2×#12011yescharacteristic, complemented, Sylow
155C23×C3—232no—
156C23×C3—264no—
157C23×C3—232no—
158C23×C3—224no—
159C23×C3—248no—
160C23×C3—212no—
161C23×C3—224no—
162C23×C3—212no—
163C23×C3—224no—
164C23×C3—212no—
165C23×C3—26no—
166C23×C3—21no—
167C23×C3—92no—
168C23×C3—186no—
169C23×C3#611112yescomplemented
170C23×C3C22×D916yes—
171C23×C3C22×D911yescharacteristic
172C23×C3C3×#67112yes—
173C23×C3C22×C3×D316yes—
174C23×C3C22×C3×D311yescharacteristic
175C22×D3—32no—
176C22×D3—66no—
177C3×D4—1824no—
178C2×C3×C4—96no—
179C2×C3×C4—186no—
180#67—624no—
181C2×Dic3—36no—
182C2×Dic3—66no—
183C2×C32—212no—
184C2×C32#95314yes—
185C2×C32C22×#6713yes—
186C3×D3—122no—
187C2×C9—224no—
188C2×C9—212no—
189C2×C9—24no—
190C2×C9—23no—
191C2×C9C3×#12014yescomplemented
192C2×C9C22×C3×D413yes—
193D9—42no—
194C24—212no—
195C24—91no—
196C24C2×C3×D913yes—
197C24C2×C3×D914yescomplemented
198C2×D4—912no—
199C2×D4—1812no—
200C22×C4—93no—
201C22⋊C4—912no—
202C22×C3—232no—
203C22×C3—264no—
204C22×C3—232no—
205C22×C3—224no—
206C22×C3—248no—
207C22×C3—224no—
208C22×C3—224no—
209C22×C3—212no—
210C22×C3—212no—
211C22×C3—21no—
212C22×C3—26no—
213C22×C3—212no—
214C22×C3—186no—
215C22×C3—364no—
216C22×C3C2×#61116yes—
217C22×C3C2×#611112yescomplemented
218C22×C3C23×D911yescharacteristic
219C22×C3C2×C3×#6716yes—
220C22×C3C2×C3×#67112yes—
221C22×C3C23×C3×D311yescharacteristic
222C2×D3—66no—
223C2×D3—124no—
224C3×C4—186no—
225Dic3—66no—
226C32C2×#95311yescharacteristic
227C9—21no—
228C9C2×C3×#12011yescharacteristic, complemented
229C23—232no—
230C23—212no—
231C23—224no—
232C23—92no—
233C23—186no—
234C23C3×#611112yescomplemented
235C23C22×C3×D916yes—
236C23C22×C3×D911yescharacteristic
237D4—1824no—
238C2×C4—96no—
239C2×C4—186no—
240C2×C3—224no—
241C2×C3—212no—
242C2×C3—212no—
243C2×C3—23no—
244C2×C3—24no—
245C2×C3—362no—
246C2×C3#6351514yesdirect product factor
247C2×C3C22×#61113yes—
248C2×C3C3×#95314yes—
249C2×C3C22×C3×#6713yes—
250D3—122no—
251C22—232no—
252C22—212no—
253C22—224no—
254C22—186no—
255C22—364no—
256C22C2×C3×#61116yes—
257C22C2×C3×#611112yescomplemented
258C22C23×C3×D911yescharacteristic
259C4—186no—
260C3—21no—
261C3C2×#6351511yescharacteristic, direct product factor
262C3C2×C3×#95311yescharacteristic
263C2—212no—
264C2—362no—
265C2C3×#6351514yesdirect product factor
266C2C22×C3×#61113yes—
subgroupG/H
centerC23×C3C22×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC25×C3×C9C2
socleC23×C32C22×D3
hypercenterC25×C3D9
hypocenterC9C2×C3×#120
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23×C3 ◁ C25×C3

Complements

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

NH ≅ G/Ntype
1. C25×C3×C9C2semidirect
2. C22×C3×#611C2semidirect
3. C3×#63515C2direct (N × H)
4. C2×C3×#3676C2semidirect
6. C2×#63515C3direct (N × H)
9. C24×C3×C9C22semidirect
10. C23×C3×D9C22semidirect
12. C2×C3×#611C22semidirect
13. C22×C3×Dic9C22semidirect
14. C3×#3676C22semidirect
18. C25×C9C2×C3semidirect
21. C22×#611C2×C3semidirect
22. #63515C2×C3direct (N × H)
23. C2×#3676C2×C3semidirect
27. C23×C3×C9D4semidirect
31. C22×C3×D9C23semidirect
34. C2×C3×Dic9C23semidirect
49. C24×C9C22×C3semidirect
50. C23×D9C22×C3semidirect
53. C2×#611C22×C3semidirect
54. C22×Dic9C22×C3semidirect
55. #3676C22×C3semidirect
60. C22×C3×C9C2×D4semidirect
66. C25×C3D9semidirect
94. C23×C9C3×D4semidirect
98. C22×D9C23×C3semidirect
101. C2×Dic9C23×C3semidirect
104. C2×C3×C9#120semidirect
114. C24×C3C2×D9semidirect
145. C22×C9C2×C3×D4semidirect
150. C25C3×D9semidirect
154. C3×C9C2×#120semidirect
169. C23×C3#611semidirect
191. C2×C9C3×#120semidirect
197. C24C2×C3×D9semidirect
217. C22×C3C2×#611semidirect
228. C9C2×C3×#120semidirect
234. C23C3×#611semidirect
246. C2×C3#63515direct (N × H)
257. C22C2×C3×#611semidirect
261. C3C2×#63515direct (N × H)
265. C2C3×#63515direct (N × H)

Maximal subgroups

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

subgroupindex
C25×C3×C92
C22×C3×#6112
C3×#635152
C2×C3×#36762
C2×C3×#9533
C2×#635153