Finite Group Databasebeta

← #126×C3⋊D9 #13154238

Subgroups — #126×C3⋊D9 #13154238

Order 1728.

1230 conjugacy classes of subgroups; only the 261 normal subgroups are shown.

normal subgroups only

This group has 1230 conjugacy classes of subgroups — too many to draw. Only its 261 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
1C3×C9×#126C211yesmaximal, characteristic, complemented
2#11721754C213yesmaximal, complemented
3C4⋊C4×C3⋊D9C213yesmaximal, complemented
4C22⋊C4×C3⋊D9C213yesmaximal
5#11721725C213yesmaximal, complemented
6#11721713C211yesmaximal, characteristic, complemented
7C42×C3⋊D9C211yesmaximal, characteristic, complemented
8C3⋊D3×#126—31nomaximal
9D9×#126—33nomaximal
10C23×C3⋊D9C2211yescharacteristic
11C2×C4×C3⋊D9C2213yes—
12C2×C4×C3⋊D9C2213yes—
13C3×C9×C4⋊C4C2213yescomplemented
14C3×C9×C22⋊C4C2213yes—
15C3×C42×C9C2211yescharacteristic, complemented
16#88551C2213yes—
17#88536C2213yes—
18#88536C2213yes—
19C4⋊(C3⋊Dic9)C2213yescomplemented
20#88534C2213yescomplemented
21#88534C2213yescomplemented
22C4×C3⋊Dic9C2213yescomplemented
23C32×#126D311yescharacteristic
24#64107—33no—
25C4⋊C4×C3⋊D3—33no—
26C22⋊C4×C3⋊D3—33no—
27#64088—33no—
28#64085—31no—
29C42×C3⋊D3—31no—
30C9×#126D313yescomplemented
31#63460—39no—
32C4⋊C4×D9—39no—
33C22⋊C4×D9—39no—
34#63441—39no—
35#63438—33no—
36C42×D9—33no—
37C23×C3×C9C2311yescharacteristic
38C22×C3⋊D9—26no—
39C22×C3⋊D9C2311yescharacteristic
40C22×C3⋊D9C2311yescharacteristic
41C2×C3×C4×C9C2313yes—
42C2×C3×C4×C9C2313yes—
43C2×C3⋊Dic9C2313yes—
44C2×C3⋊Dic9C2313yes—
45C4×C3⋊D9—26no—
46C4×C3⋊D9—26no—
47D3×#126—93no—
48D3×#126—91no—
49C23×C3⋊D3—31no—
50C2×C4×C3⋊D3—33no—
51C2×C4×C3⋊D3—33no—
52C23×D9—33no—
53C32×C4⋊C4C2×D313yes—
54C32×C22⋊C4C2×D313yes—
55C32×C42C2×D311yescharacteristic
56#3757—33no—
57#3752—33no—
58#3752—33no—
59C4⋊(C3⋊Dic3)—33no—
60#3750—33no—
61#3750—33no—
62C4×C3⋊Dic3—33no—
63C2×C4×D9—39no—
64C2×C4×D9—39no—
65C9×C4⋊C4C2×D319yescomplemented
66C9×C22⋊C4C2×D319yes—
67C42×C9C2×D313yescomplemented
68#3676—39no—
69(C2×D9)⋊C4—39no—
70(C2×D9)⋊C4—39no—
71C4⋊Dic9—39no—
72Dic9⋊C4—39no—
73Dic9⋊C4—39no—
74C4×Dic9—39no—
75C22×C3×C9—23no—
76C22×C3×C9C2411yescharacteristic
77C2×C3⋊D9—23no—
78C2×C3⋊D9—44no—
79C2×C3⋊D9#4116yes—
80C3×C4×C9—23no—
81C3×C4×C9—23no—
82C3⋊Dic9—23no—
83C3⋊Dic9—23no—
84C3×#126D913yescomplemented
85C3×#126C3⋊D311yescharacteristic
86#898—99no—
87#898—93no—
88D3×C4⋊C4—99no—
89D3×C4⋊C4—93no—
90D3×C22⋊C4—99no—
91D3×C22⋊C4—93no—
92#879—99no—
93#879—93no—
94#876—93no—
95#876—91no—
96C42×D3—93no—
97C42×D3—91no—
98C23×C32C22×D311yescharacteristic
99C22×C3⋊D3—31no—
100C22×C3⋊D3—31no—
101C22×C3⋊D3—66no—
102C2×C32×C4C22×D313yes—
103C2×C32×C4C22×D313yes—
104C2×C3⋊Dic3—33no—
105C2×C3⋊Dic3—33no—
106C4×C3⋊D3—66no—
107C4×C3⋊D3—66no—
108C23×C9C22×D313yes—
109C22×D9—33no—
110C22×D9—33no—
111C22×D9—618no—
112C2×C4×C9C22×D319yes—
113C2×C4×C9C22×D319yes—
114C2×Dic9—39no—
115C2×Dic9—39no—
116C4×D9—618no—
117C4×D9—618no—
118C2×#126—271noSylow
119C2×C3×C9—41no—
120C2×C3×C9C2×#4113yes—
121C3⋊D9—41no—
122C3⋊D9#12614yesdirect product factor
123C23×D3—93no—
124C23×D3—91no—
125C2×C4×D3—99no—
126C2×C4×D3—99no—
127C2×C4×D3—93no—
128C2×C4×D3—93no—
129C3×C4⋊C4C2×D919yescomplemented
130C3×C4⋊C4C2×C3⋊D313yes—
131C3×C22⋊C4C2×D919yes—
132C3×C22⋊C4C2×C3⋊D313yes—
133C3×C42C2×D913yescomplemented
134C3×C42C2×C3⋊D311yescharacteristic
135#217—99no—
136#217—93no—
137(C2×D3)⋊C4—99no—
138(C2×D3)⋊C4—99no—
139(C2×D3)⋊C4—93no—
140(C2×D3)⋊C4—93no—
141C4⋊Dic3—99no—
142C4⋊Dic3—93no—
143Dic3⋊C4—99no—
144Dic3⋊C4—99no—
145Dic3⋊C4—93no—
146Dic3⋊C4—93no—
147C4×Dic3—99no—
148C4×Dic3—93no—
149C22×C32—23no—
150C22×C32C23×D311yescharacteristic
151C2×C3⋊D3—36no—
152C2×C3⋊D3—63no—
153C2×C3⋊D3—124no—
154C32×C4—23no—
155C32×C4—23no—
156C3⋊Dic3—63no—
157C3⋊Dic3—63no—
158C22×C9—29no—
159C22×C9C23×D313yes—
160C2×D9—318no—
161C2×D9—69no—
162C2×D9—1212no—
163C4×C9—29no—
164C4×C9—29no—
165Dic9—69no—
166Dic9—69no—
167#126—273no—
168#126—273no—
169#126—271no—
170#126C3⋊D911yescharacteristic, direct product factor
171C2×C4⋊C4—273no—
172C2×C22⋊C4—273no—
173C2×C42—271no—
174C3×C9C2×#12611yescharacteristic, complemented, Sylow
175C23×C3C22×D913yes—
176C23×C3C22×C3⋊D311yescharacteristic
177C22×D3—91no—
178C22×D3—93no—
179C22×D3—91no—
180C22×D3—93no—
181C22×D3—186no—
182C22×D3—1818no—
183C2×C3×C4C22×D919yes—
184C2×C3×C4C22×D919yes—
185C2×C3×C4C22×C3⋊D313yes—
186C2×C3×C4C22×C3⋊D313yes—
187C2×Dic3—93no—
188C2×Dic3—93no—
189C2×Dic3—99no—
190C2×Dic3—99no—
191C4×D3—186no—
192C4×D3—186no—
193C4×D3—1818no—
194C4×D3—1818no—
195C2×C32—41no—
196C2×C32D3×#4113yes—
197C3⋊D3—34no—
198C3⋊D3—121no—
199C2×C9—43no—
200C2×C9D3×#4119yes—
201D9—312no—
202D9—123no—
203C24—271no—
204C22×C4—273no—
205C22×C4—273no—
206C4⋊C4—273no—
207C4⋊C4—273no—
208C4⋊C4—273no—
209C4⋊C4C2×C3⋊D913yescomplemented
210C22⋊C4—273no—
211C22⋊C4—273no—
212C22⋊C4—273no—
213C22⋊C4C2×C3⋊D913yes—
214C42—273no—
215C42C2×C3⋊D911yescharacteristic, complemented
216C22×C3—23no—
217C22×C3—29no—
218C22×C3C23×D913yes—
219C22×C3C23×C3⋊D311yescharacteristic
220C2×D3—96no—
221C2×D3—918no—
222C2×D3—183no—
223C2×D3—189no—
224C2×D3—364no—
225C2×D3—3612no—
226C3×C4—23no—
227C3×C4—23no—
228C3×C4—29no—
229C3×C4—29no—
230Dic3—183no—
231Dic3—183no—
232Dic3—189no—
233Dic3—189no—
234C32D3×#12611yescharacteristic
235C9D3×#12613yescomplemented
236C23—271no—
237C23—271no—
238C23—546no—
239C23C22×C3⋊D911yescharacteristic
240C2×C4—273no—
241C2×C4—273no—
242C2×C4—546no—
243C2×C4—546no—
244C2×C4C22×C3⋊D913yes—
245C2×C4C22×C3⋊D913yes—
246C2×C3—41no—
247C2×C3—43no—
248C2×C3#41×D919yes—
249C2×C3#41×C3⋊D313yes—
250D3—94no—
251D3—912no—
252D3—361no—
253D3—363no—
254C22—23no—
255C22—276no—
256C22—543no—
257C22—1084no—
258C22C23×C3⋊D911yescharacteristic
259C4—23no—
260C4—23no—
261C4—543no—
262C4—543no—
263C3D9×#12613yescomplemented
264C3C3⋊D3×#12611yescharacteristic
265C2—41no—
266C2—274no—
267C2—1081no—
268C2#41×C3⋊D913yes—
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C9×#126C2
socleC22×C32C23×D3
prime radicalC23×C3⋊D9C22
hypercenter#126C3⋊D9
hypocenterC3×C9C2×#126
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. C3×C9×#126C2semidirect
2. #11721754C2semidirect
3. C4⋊C4×C3⋊D9C2semidirect
5. #11721725C2semidirect
6. #11721713C2semidirect
7. C42×C3⋊D9C2semidirect
13. C3×C9×C4⋊C4C22semidirect
15. C3×C42×C9C22semidirect
19. C4⋊(C3⋊Dic9)C22semidirect
20. #88534C22semidirect
21. #88534C22semidirect
22. C4×C3⋊Dic9C22semidirect
30. C9×#126D3semidirect
65. C9×C4⋊C4C2×D3semidirect
67. C42×C9C2×D3semidirect
84. C3×#126D9semidirect
122. C3⋊D9#126direct (N × H)
129. C3×C4⋊C4C2×D9semidirect
133. C3×C42C2×D9semidirect
170. #126C3⋊D9direct (N × H)
174. C3×C9C2×#126semidirect
209. C4⋊C4C2×C3⋊D9semidirect
215. C42C2×C3⋊D9semidirect
235. C9D3×#126semidirect
263. C3D9×#126semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1262
#117217542
C4⋊C4×C3⋊D92
C22⋊C4×C3⋊D92
#117217252
#117217132
C42×C3⋊D92
C3⋊D3×#1263
D9×#1263