Finite Group Databasebeta

← C3×#10589800 #13153159

Subgroups — C3×#10589800 #13153159

Order 1728.

632 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#125C211yesmaximal, characteristic, complemented
2C3×#63460C214yesmaximal
3C3×#63459C212yesmaximal
4C3×#63452C214yesmaximal, complemented
5C3×#63449C212yesmaximal, complemented
6C3×#63434C211yesmaximal, characteristic
7C3×#63431C211yesmaximal, characteristic, complemented
8C3×#5698—31nomaximal
9#10589800C311yesmaximal, characteristic, direct product factor
10C2×C3×C4×D9C2211yescharacteristic
11C2×C3×Dic18C2212yes—
12C3×C9×C4⋊C4C2212yes—
13C3×C9×C4⋊C4C2214yes—
14C3×C42×C9C2211yescharacteristic
15C3×(C2×D9)⋊C4C2212yes—
16C3×(C2×D9)⋊C4C2214yes—
17C3×C4⋊Dic9C2214yes—
18C3×C4⋊Dic9C2214yes—
19C3×Dic9⋊C4C2212yes—
20C3×Dic9⋊C4C2214yes—
21C3×C4×Dic9C2211yescharacteristic
22C3×C4×Dic9C2214yes—
23C32×#125D311yescharacteristic
24C3×#898—34no—
25C3×#897—32no—
26C3×#890—34no—
27C3×#887—32no—
28C3×#872—31no—
29C3×#869—31no—
30C9×#125—21no—
31C9×#125C2×C311yescharacteristic, complemented
32#63460C2×C314yes—
33#63459C2×C312yes—
34#63452C2×C314yescomplemented
35#63449C2×C312yescomplemented
36#63434C2×C311yescharacteristic
37#63431C2×C311yescharacteristic, complemented
38C22×C3×D9C2311yescharacteristic
39C2×C3×C4×C9C2314yes—
40C2×C3×C4×C9C2312yes—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3×Dic9C2314yes—
43C2×C3×Dic9C2311yescharacteristic
44C2×C3×Dic9C2312yes—
45C3×C4×D9—22no—
46C3×Dic18—24no—
47C3×#541—91no—
48#5698—31no—
49C2×C3×C4×D3—31no—
50C2×C3×Dic6—32no—
51C32×C4⋊C4C2×D312yes—
52C32×C4⋊C4C2×D314yes—
53C32×C42C2×D311yescharacteristic
54C3×(C2×D3)⋊C4—34no—
55C3×(C2×D3)⋊C4—32no—
56C3×C4⋊Dic3—34no—
57C3×C4⋊Dic3—34no—
58C3×Dic3⋊C4—34no—
59C3×Dic3⋊C4—32no—
60C3×C4×Dic3—31no—
61C3×C4×Dic3—34no—
62C2×C4×D9C22×C311yescharacteristic
63C2×Dic18C22×C312yes—
64C9×C4⋊C4—22no—
65C9×C4⋊C4—24no—
66C9×C4⋊C4C22×C312yes—
67C9×C4⋊C4C22×C314yes—
68C42×C9—21no—
69C42×C9C22×C311yescharacteristic
70(C2×D9)⋊C4C22×C312yes—
71(C2×D9)⋊C4C22×C314yes—
72C4⋊Dic9C22×C314yes—
73C4⋊Dic9C22×C314yes—
74Dic9⋊C4C22×C312yes—
75Dic9⋊C4C22×C314yes—
76C4×Dic9C22×C311yescharacteristic
77C4×Dic9C22×C314yes—
78C22×C3×C9C2411yescharacteristic
79C2×C3×D9—21no—
80C2×C3×D9—22no—
81C3×C4×C9—24no—
82C3×C4×C9—22no—
83C3×C4×C9#4112yes—
84C3×Dic9—24no—
85C3×Dic9—21no—
86C3×Dic9—22no—
87C3×C4⋊Q8—91no—
88C3×#126—94no—
89C3×#125—21no—
90C3×#125—94no—
91C3×#125D911yescharacteristic, complemented
92C3×#125C3×D311yescharacteristic
93C3×C22⋊Q8—92no—
94C3×C4×Q8—92no—
95C3×#117—91no—
96#898—34no—
97#897—32no—
98#890—34no—
99#887—32no—
100#872—31no—
101#869—31no—
102C22×C3×D3—31no—
103C2×C32×C4C22×D314yes—
104C2×C32×C4C22×D312yes—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3×Dic3—32no—
107C2×C3×Dic3—34no—
108C2×C3×Dic3—31no—
109C3×C4×D3—62no—
110C3×Dic6—64no—
111C22×D9C23×C311yescharacteristic
112C2×C4×C9—21no—
113C2×C4×C9—22no—
114C2×C4×C9—24no—
115C2×C4×C9C23×C314yes—
116C2×C4×C9C23×C312yes—
117C2×C4×C9C23×C311yescharacteristic
118C2×Dic9C23×C312yes—
119C2×Dic9C23×C314yes—
120C2×Dic9C23×C311yescharacteristic
121C4×D9—22no—
122Dic18—24no—
123#541—91noSylow
124C2×C3×C9C2×#4111yescharacteristic
125C2×C3×C92−1+412yes—
126C3×D9—41no—
127C2×C3×Q8—92no—
128C22×C3×C4—91no—
129C2×C4×D3—31no—
130C2×Dic6—32no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—24no—
133C3×C4⋊C4—92no—
134C3×C4⋊C4—94no—
135C3×C4⋊C4—94no—
136C3×C4⋊C4—94no—
137C3×C4⋊C4C2×D912yes—
138C3×C4⋊C4C2×D914yes—
139C3×C4⋊C4C2×C3×D312yes—
140C3×C4⋊C4C2×C3×D314yes—
141C3×C22⋊C4—92no—
142C3×C22⋊C4—94no—
143C3×C42—21no—
144C3×C42—91no—
145C3×C42—94no—
146C3×C42C2×D911yescharacteristic
147C3×C42C2×C3×D311yescharacteristic
148(C2×D3)⋊C4—32no—
149(C2×D3)⋊C4—34no—
150C4⋊Dic3—34no—
151C4⋊Dic3—34no—
152Dic3⋊C4—32no—
153Dic3⋊C4—34no—
154C4×Dic3—31no—
155C4×Dic3—34no—
156C22×C32C23×D311yescharacteristic
157C2×C3×D3—61no—
158C2×C3×D3—62no—
159C32×C4—24no—
160C32×C4—22no—
161C32×C4#23912yes—
162C3×Dic3—64no—
163C3×Dic3—61no—
164C3×Dic3—62no—
165C22×C9—21no—
166C22×C9C24×C311yescharacteristic
167C2×D9—21no—
168C2×D9—22no—
169C4×C9—24no—
170C4×C9—22no—
171C4×C9—22no—
172C4×C9—42no—
173C4×C9—44no—
174C4×C9C3×#4112yes—
175Dic9—24no—
176Dic9—21no—
177Dic9—22no—
178C4⋊Q8—91no—
179#126—94no—
180#125—94no—
181#125C3×D911yescharacteristic, complemented
182C22⋊Q8—92no—
183C4×Q8—92no—
184#117—91no—
185C3×C9#54111yescharacteristic, complemented, Sylow
186C23×C3—91no—
187C22×D3—31no—
188C3×Q8—184no—
189C2×C3×C4—21no—
190C2×C3×C4—22no—
191C2×C3×C4—24no—
192C2×C3×C4—92no—
193C2×C3×C4—94no—
194C2×C3×C4—91no—
195C2×C3×C4—182no—
196C2×C3×C4C22×D914yes—
197C2×C3×C4C22×D912yes—
198C2×C3×C4C22×D911yescharacteristic
199C2×C3×C4C22×C3×D314yes—
200C2×C3×C4C22×C3×D312yes—
201C2×C3×C4C22×C3×D311yescharacteristic
202C2×Dic3—32no—
203C2×Dic3—34no—
204C2×Dic3—31no—
205C4×D3—62no—
206Dic6—64no—
207C2×C32C2×#23911yescharacteristic
208C2×C32#101012yes—
209C3×D3—121no—
210C2×C9—21no—
211C2×C9—22no—
212C2×C9C2×C3×#4111yescharacteristic
213C2×C9C3×2−1+412yes—
214D9—41no—
215C2×Q8—92no—
216C22×C4—91no—
217C4⋊C4—92no—
218C4⋊C4—94no—
219C4⋊C4—94no—
220C4⋊C4—94no—
221C4⋊C4C2×C3×D912yes—
222C4⋊C4C2×C3×D914yes—
223C22⋊C4—92no—
224C22⋊C4—94no—
225C42—91no—
226C42—94no—
227C42C2×C3×D911yescharacteristic
228C22×C3—21no—
229C22×C3—181no—
230C22×C3—182no—
231C22×C3C23×D911yescharacteristic
232C22×C3C23×C3×D311yescharacteristic
233C2×D3—61no—
234C2×D3—62no—
235C3×C4—24no—
236C3×C4—22no—
237C3×C4—24no—
238C3×C4—22no—
239C3×C4—22no—
240C3×C4—42no—
241C3×C4—44no—
242C3×C4—181no—
243C3×C4—184no—
244C3×C4—182no—
245C3×C4#370112yes—
246C3×C4C3×#23912yes—
247Dic3—64no—
248Dic3—61no—
249Dic3—62no—
250C32#569811yescharacteristic
251C9—21no—
252C9C3×#54111yescharacteristic, complemented
253C23—91no—
254Q8—184no—
255C2×C4—92no—
256C2×C4—94no—
257C2×C4—91no—
258C2×C4—182no—
259C2×C4C22×C3×D914yes—
260C2×C4C22×C3×D912yes—
261C2×C4C22×C3×D911yescharacteristic
262C2×C3—21no—
263C2×C3—22no—
264C2×C3—361no—
265C2×C3C2×#370111yescharacteristic
266C2×C3#6371612yes—
267C2×C3C2×C3×#23911yescharacteristic
268C2×C3C3×#101012yes—
269D3—121no—
270C22—181no—
271C22—182no—
272C22C23×C3×D911yescharacteristic
273C4—24no—
274C4—22no—
275C4—181no—
276C4—184no—
277C4—182no—
278C4C3×#370112yes—
279C3—21no—
280C3#1058980011yescharacteristic, direct product factor
281C3C3×#569811yescharacteristic
282C2—361no—
283C2C2×C3×#370111yescharacteristic
284C2C3×#6371612yes—
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#125C2
socleC22×C32C23×D3
prime radicalC22×C3×D9C23
hypercenterC3×#125D9
hypocenterC9C3×#541
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
1. C3×C9×#125C2semidirect
4. C3×#63452C2semidirect
5. C3×#63449C2semidirect
7. C3×#63431C2semidirect
9. #10589800C3direct (N × H)
31. C9×#125C2×C3semidirect
34. #63452C2×C3semidirect
35. #63449C2×C3semidirect
37. #63431C2×C3semidirect
91. C3×#125D9semidirect
181. #125C3×D9semidirect
185. C3×C9#541semidirect
252. C9C3×#541semidirect
280. C3#10589800direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×#1252
C3×#634602
C3×#634592
C3×#634522
C3×#634492
C3×#634342
C3×#634312
C3×#56983
#105898003