Finite Group Databasebeta

← #13154266

Subgroups — #13154266

Order 1728.

1260 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 251 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×#127C211yesmaximal, characteristic, complemented
2#11721839C216yesmaximal, complemented
3#11721837C213yesmaximal, complemented
4#11721835C213yesmaximal, complemented
5#11721713C212yesmaximal, complemented
6#10596191—31nomaximal
7#10589822—33nomaximal
8C23×C3⋊D9C2211yescharacteristic
9C2×C3×D4×C9C2213yes—
10C2×C3×D4×C9C2213yescomplemented
11C2×#6233C2216yes—
12C22×C3⋊Dic9C2213yescomplemented
13C3×C42×C9C2211yescharacteristic, complemented
14#88551C2213yes—
15#88551C2213yescomplemented
16#88536C2216yes—
17#88534C2216yescomplemented
18C32×#127D311yescharacteristic
19#64148—36no—
20#64146—33no—
21#64144—33no—
22#64085—32no—
23C9×#127D313yescomplemented
24#63501—318no—
25#63499—39no—
26#63497—39no—
27#63438—36no—
28C23×C3×C9C2311yescharacteristic
29C23×C3×C9C2313yes—
30C22×C3⋊D9—43no—
31C22×C3⋊D9C2312yes—
32C3×D4×C9—43no—
33C3×D4×C9—43no—
34C2×C3×C4×C9C2313yes—
35#6233—46no—
36C2×C3⋊Dic9—43no—
37C2×C3⋊Dic9C2316yes—
38#5720—93no—
39#5720—91no—
40C23×C3⋊D3—31no—
41C2×C32×D4C2×D313yes—
42C2×C32×D4C2×D313yes—
43C2×#638—36no—
44C22×C3⋊Dic3—33no—
45C23×D9—33no—
46C32×C42C2×D311yescharacteristic
47#3757—33no—
48#3757—33no—
49#3752—36no—
50#3750—36no—
51C2×D4×C9C2×D319yes—
52C2×D4×C9C2×D319yescomplemented
53C2×#611—318no—
54C22×Dic9—39no—
55C42×C9C2×D313yescomplemented
56#3676—39no—
57#3676—39no—
58(C2×D9)⋊C4—318no—
59Dic9⋊C4—318no—
60C22×C3×C9—23no—
61C22×C3×C9—26no—
62C22×C3×C9—23no—
63C22×C3×C9C2411yescharacteristic
64C2×C3⋊D9—26no—
65C2×C3⋊D9—44no—
66C3×C4×C9—23no—
67C3⋊Dic9—26no—
68C3×#127D913yescomplemented
69C3×#127C3⋊D311yescharacteristic
70#939—918no—
71#939—96no—
72#937—99no—
73#937—93no—
74#935—99no—
75#935—93no—
76#876—96no—
77#876—92no—
78C23×C32C22×D311yescharacteristic
79C23×C32C22×D313yes—
80C22×C3⋊D3—32no—
81C22×C3⋊D3—123no—
82C32×D4—43no—
83C32×D4—43no—
84C2×C32×C4C22×D313yes—
85#638—126no—
86C2×C3⋊Dic3—36no—
87C2×C3⋊Dic3—123no—
88C23×C9C22×D313yes—
89C23×C9C22×D319yes—
90C22×D9—36no—
91C22×D9—129no—
92D4×C9—49no—
93D4×C9—49no—
94C2×C4×C9C22×D319yes—
95#611—1218no—
96C2×Dic9—318no—
97C2×Dic9—129no—
98#560—271noSylow
99C2×C3×C9—41no—
100C2×C3×C9—43no—
101C2×C3×C9H(5,𝔽2)13yes—
102C3⋊D9—42no—
103C23×D3—93no—
104C23×D3—91no—
105C2×C3×D4C2×D919yes—
106C2×C3×D4C2×D919yescomplemented
107C2×C3×D4C2×C3⋊D313yes—
108C2×C3×D4C2×C3⋊D313yes—
109C2×#67—918no—
110C2×#67—96no—
111C22×Dic3—99no—
112C22×Dic3—93no—
113C3×C42C2×D913yescomplemented
114C3×C42C2×C3⋊D311yescharacteristic
115#217—99no—
116#217—99no—
117#217—93no—
118#217—93no—
119(C2×D3)⋊C4—918no—
120(C2×D3)⋊C4—96no—
121Dic3⋊C4—918no—
122Dic3⋊C4—96no—
123C22×C32—23no—
124C22×C32—26no—
125C22×C32—23no—
126C22×C32C23×D311yescharacteristic
127C2×C3⋊D3—66no—
128C2×C3⋊D3—124no—
129C32×C4—23no—
130C3⋊Dic3—66no—
131C22×C9—29no—
132C22×C9—218no—
133C22×C9—29no—
134C22×C9C23×D313yes—
135C2×D9—618no—
136C2×D9—1212no—
137C4×C9—29no—
138Dic9—618no—
139#127C3⋊D911yescharacteristic, complemented
140#126—272no—
141#123—273no—
142#121—276no—
143#120—273no—
144C3×C9#56011yescharacteristic, complemented, Sylow
145C23×C3C22×D913yes—
146C23×C3C22×D919yes—
147C23×C3C22×C3⋊D311yescharacteristic
148C23×C3C22×C3⋊D313yes—
149C22×D3—92no—
150C22×D3—96no—
151C22×D3—363no—
152C22×D3—369no—
153C3×D4—43no—
154C3×D4—43no—
155C3×D4—49no—
156C3×D4—49no—
157C2×C3×C4C22×D919yes—
158C2×C3×C4C22×C3⋊D313yes—
159#67—366no—
160#67—3618no—
161C2×Dic3—96no—
162C2×Dic3—918no—
163C2×Dic3—363no—
164C2×Dic3—369no—
165C2×C32—41no—
166C2×C32—43no—
167C2×C32#100413yes—
168C3⋊D3—122no—
169C2×C9—43no—
170C2×C9—49no—
171C2×C9#100419yes—
172D9—126no—
173C24—271no—
174C2×D4—276no—
175C2×D4C2×C3⋊D913yes—
176C2×D4C2×C3⋊D913yescomplemented
177C22×C4—273no—
178C4⋊C4—276no—
179C22⋊C4—273no—
180C22⋊C4—273no—
181C22⋊C4—276no—
182C42C2×C3⋊D911yescharacteristic, complemented
183C22×C3—23no—
184C22×C3—26no—
185C22×C3—23no—
186C22×C3—29no—
187C22×C3—218no—
188C22×C3—29no—
189C22×C3C23×D913yes—
190C22×C3C23×C3⋊D311yescharacteristic
191C2×D3—186no—
192C2×D3—1818no—
193C2×D3—364no—
194C2×D3—3612no—
195C3×C4—23no—
196C3×C4—29no—
197Dic3—186no—
198Dic3—1818no—
199C32#572011yescharacteristic
200C9#572013yescomplemented
201C23—272no—
202C23—1083no—
203C23C22×C3⋊D911yescharacteristic
204C23C22×C3⋊D913yes—
205D4—43no—
206D4—43no—
207D4—1086no—
208C2×C4—276no—
209C2×C4—1083no—
210C2×C4C22×C3⋊D913yes—
211C2×C3—41no—
212C2×C3—43no—
213C2×C3—43no—
214C2×C3—49no—
215C2×C3#6371019yes—
216C2×C3#6436113yes—
217D3—362no—
218D3—366no—
219C22—23no—
220C22—26no—
221C22—23no—
222C22—546no—
223C22—1084no—
224C22C23×C3⋊D911yescharacteristic
225C4—23no—
226C4—546no—
227C3#1058982213yescomplemented
228C3#1059619111yescharacteristic
229C2—41no—
230C2—43no—
231C2—1082no—
232C2#1172266113yes—
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C9×#127C2
socleC22×C32C23×D3
hypercenter#127C3⋊D9
hypocenterC3×C9#560
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C3×C9×#127C2semidirect
2. #11721839C2semidirect
3. #11721837C2semidirect
4. #11721835C2semidirect
5. #11721713C2semidirect
10. C2×C3×D4×C9C22semidirect
12. C22×C3⋊Dic9C22semidirect
13. C3×C42×C9C22semidirect
15. #88551C22semidirect
17. #88534C22semidirect
23. C9×#127D3semidirect
52. C2×D4×C9C2×D3semidirect
55. C42×C9C2×D3semidirect
68. C3×#127D9semidirect
106. C2×C3×D4C2×D9semidirect
113. C3×C42C2×D9semidirect
139. #127C3⋊D9semidirect
144. C3×C9#560semidirect
176. C2×D4C2×C3⋊D9semidirect
182. C42C2×C3⋊D9semidirect
200. C9#5720semidirect
227. C3#10589822semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1272
#117218392
#117218372
#117218352
#117217132
#105961913
#105898223