Finite Group Databasebeta

← C3×#10588329 #13141272

Subgroups — C3×#10588329 #13141272

Order 1728.

658 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Dic9⋊C4C212yesmaximal
2C2×C3×C4×Dic9C212yesmaximal
3C2×C3×C4×Dic9C211yesmaximal, characteristic
4C3×C9×#95C211yesmaximal, characteristic
5C3×C9⋊#95C211yesmaximal, characteristic
6C3×#4649—31nomaximal
7#10588329C311yesmaximal, characteristic, direct product factor
8C22×C3×C4×C9C2211yescharacteristic
9C22×C3×C4×C9C2212yes—
10C22×C3×Dic9C2212yes—
11C22×C3×Dic9C2211yescharacteristic
12C22×C3×Dic9C2211yescharacteristic
13C3×Dic9⋊C4C418yescomplemented
14C3×C4×Dic9—24no—
15C3×C4×Dic9C414yescomplemented
16C2×C3×Dic3⋊C4—32no—
17C2×C3×C4×Dic3—31no—
18C2×C3×C4×Dic3—32no—
19C32×#95D311yescharacteristic
20C3×C3⋊#95—31no—
21C2×Dic9⋊C4C2×C312yes—
22C2×C4×Dic9C2×C312yes—
23C2×C4×Dic9C2×C311yescharacteristic
24C9×#95—21no—
25C9×#95C2×C311yescharacteristic
26C9⋊#95C2×C311yescharacteristic
27C23×C3×C9C2311yescharacteristic
28C2×C3×C4×C9—22no—
29C2×C3×C4×C9—22no—
30C2×C3×C4×C9—22no—
31C2×C3×C4×C9C2×C412yes—
32C2×C3×C4×C9C2×C414yes—
33C2×C3×Dic9—22no—
34C2×C3×Dic9—22no—
35C2×C3×Dic9C2×C414yes—
36C2×C3×Dic9C2×C412yes—
37C2×C3×Dic9C2×C414yes—
38C2×C3×Dic9C2×C412yes—
39C2×C3×Dic9D412yes—
40C2×C3×Dic9Q812yes—
41C3×C4×C4⋊C4—91no—
42#4649—31no—
43C22×C32×C4C2×D312yes—
44C22×C32×C4C2×D311yescharacteristic
45C22×C3×Dic3—31no—
46C22×C3×Dic3—31no—
47C22×C3×Dic3—32no—
48C3×Dic3⋊C4—38no—
49C3×C4×Dic3—34no—
50C3×C4×Dic3—64no—
51C22×C4×C9—21no—
52C22×C4×C9—22no—
53C22×C4×C9C22×C311yescharacteristic
54C22×C4×C9C22×C312yes—
55C22×Dic9C22×C312yes—
56C22×Dic9C22×C311yescharacteristic
57C22×Dic9C22×C311yescharacteristic
58Dic9⋊C4C3×C418yescomplemented
59C4×Dic9—24no—
60C4×Dic9C3×C414yescomplemented
61C22×C3×C9C22×C411yescharacteristic
62C22×C3×C9C22×C412yes—
63C22×C3×C9C2×D411yescharacteristic
64C22×C3×C9C2×Q811yescharacteristic
65C22×C3×C9#4112yes—
66C3×C4×C9—22no—
67C3×C4×C9—24no—
68C3×Dic9—24no—
69C3×Dic9C4214yescomplemented
70C3×Dic9C4⋊C414yescomplemented
71C2×C3×C4⋊C4—92no—
72C2×C3×C42—91no—
73C2×C3×C42—92no—
74C2×Dic3⋊C4—32no—
75C2×C4×Dic3—31no—
76C2×C4×Dic3—32no—
77C3×#95—21no—
78C3×#95—91no—
79C3×#95D911yescharacteristic
80C3×#95C3×D311yescharacteristic
81C3⋊#95—31no—
82C23×C32C22×D311yescharacteristic
83C2×C32×C4—22no—
84C2×C32×C4—22no—
85C2×C32×C4—22no—
86C2×C32×C4C4×D312yes—
87C2×C32×C4C4×D314yes—
88C2×C3×Dic3—32no—
89C2×C3×Dic3—32no—
90C2×C3×Dic3—34no—
91C2×C3×Dic3—32no—
92C2×C3×Dic3—32no—
93C2×C3×Dic3—34no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—62no—
96C23×C9—21no—
97C23×C9C23×C311yescharacteristic
98C2×C4×C9—22no—
99C2×C4×C9—22no—
100C2×C4×C9—22no—
101C2×C4×C9—22no—
102C2×C4×C9—24no—
103C2×C4×C9—42no—
104C2×C4×C9—42no—
105C2×C4×C9—42no—
106C2×C4×C9C2×C3×C412yes—
107C2×C4×C9C2×C3×C414yes—
108C2×Dic9—22no—
109C2×Dic9—22no—
110C2×Dic9C2×C3×C414yes—
111C2×Dic9C2×C3×C412yes—
112C2×Dic9C2×C3×C414yes—
113C2×Dic9C2×C3×C412yes—
114C2×Dic9C3×D412yes—
115C2×Dic9C3×Q812yes—
116C4×C4⋊C4—91noSylow
117C2×C3×C9C2×C4211yescharacteristic
118C2×C3×C9C2×C4⋊C411yescharacteristic
119C2×C3×C9#11711yescharacteristic
120C2×C3×C9C4×D412yes—
121C2×C3×C9C4×Q812yes—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4—91no—
125C22×C3×C4—91no—
126C22×C3×C4—92no—
127C22×C3×C4C2×D911yescharacteristic
128C22×C3×C4C2×D912yes—
129C22×C3×C4C2×C3×D311yescharacteristic
130C22×C3×C4C2×C3×D312yes—
131C22×Dic3—31no—
132C22×Dic3—31no—
133C22×Dic3—32no—
134C3×C4⋊C4—98no—
135C3×C42—94no—
136C3×C42—184no—
137Dic3⋊C4—38no—
138C4×Dic3—34no—
139C4×Dic3—64no—
140C22×C32C2×C4×D311yescharacteristic
141C22×C32C2×C4×D312yes—
142C22×C32D3×D411yescharacteristic
143C22×C32#23712yes—
144C22×C32D3×Q811yescharacteristic
145C32×C4—22no—
146C32×C4—24no—
147C3×Dic3—34no—
148C3×Dic3—34no—
149C3×Dic3—64no—
150C22×C9—21no—
151C22×C9—22no—
152C22×C9—21no—
153C22×C9—22no—
154C22×C9—21no—
155C22×C9C22×C3×C411yescharacteristic
156C22×C9C22×C3×C412yes—
157C22×C9C2×C3×D411yescharacteristic
158C22×C9C2×C3×Q811yescharacteristic
159C22×C9C3×#4112yes—
160C4×C9—22no—
161C4×C9—24no—
162C4×C9—42no—
163C4×C9—44no—
164Dic9—24no—
165Dic9C3×C4214yescomplemented
166Dic9C3×C4⋊C414yescomplemented
167C2×C4⋊C4—92no—
168C2×C42—91no—
169C2×C42—92no—
170#95—91no—
171#95C3×D911yescharacteristic
172C3×C9C4×C4⋊C411yescharacteristic, complemented, Sylow
173C23×C3—21no—
174C23×C3C22×D911yescharacteristic
175C23×C3C22×C3×D311yescharacteristic
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4—24no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—92no—
188C2×C3×C4—92no—
189C2×C3×C4—94no—
190C2×C3×C4—92no—
191C2×C3×C4—92no—
192C2×C3×C4—94no—
193C2×C3×C4—182no—
194C2×C3×C4—182no—
195C2×C3×C4C4×D912yes—
196C2×C3×C4C4×D914yes—
197C2×C3×C4C3×C4×D312yes—
198C2×C3×C4C3×C4×D314yes—
199C2×Dic3—32no—
200C2×Dic3—32no—
201C2×Dic3—34no—
202C2×Dic3—32no—
203C2×Dic3—32no—
204C2×Dic3—34no—
205C2×Dic3—62no—
206C2×Dic3—62no—
207C2×C32C42×D311yescharacteristic
208C2×C32#87711yescharacteristic
209C2×C32#67⋊C412yes—
210C2×C32#88712yes—
211C2×C32D3×C4⋊C411yescharacteristic
212C2×C9—21no—
213C2×C9—22no—
214C2×C9—22no—
215C2×C9—21no—
216C2×C9—21no—
217C2×C9C2×C3×C4211yescharacteristic
218C2×C9C2×C3×C4⋊C411yescharacteristic
219C2×C9C3×#11711yescharacteristic
220C2×C9C3×C4×D412yes—
221C2×C9C3×C4×Q812yes—
222C22×C4—91no—
223C22×C4—91no—
224C22×C4—92no—
225C22×C4C2×C3×D911yescharacteristic
226C22×C4C2×C3×D912yes—
227C4⋊C4—98no—
228C42—94no—
229C42—184no—
230C22×C3—21no—
231C22×C3—22no—
232C22×C3—21no—
233C22×C3—22no—
234C22×C3—21no—
235C22×C3C2×C4×D911yescharacteristic
236C22×C3C2×C4×D912yes—
237C22×C3D4×D911yescharacteristic
238C22×C3#369912yes—
239C22×C3Q8×D911yescharacteristic
240C22×C3C2×C3×C4×D311yescharacteristic
241C22×C3C2×C3×C4×D312yes—
242C22×C3C3×D3×D411yescharacteristic
243C22×C3C3×#23712yes—
244C22×C3C3×D3×Q811yescharacteristic
245C3×C4—22no—
246C3×C4—24no—
247C3×C4—22no—
248C3×C4—24no—
249C3×C4—42no—
250C3×C4—44no—
251C3×C4—94no—
252C3×C4—94no—
253C3×C4—184no—
254Dic3—34no—
255Dic3—34no—
256Dic3—64no—
257C32#464911yescharacteristic
258C9—21no—
259C9C3×C4×C4⋊C411yescharacteristic, complemented
260C23C22×C3×D911yescharacteristic
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—92no—
265C2×C4—92no—
266C2×C4—94no—
267C2×C4—92no—
268C2×C4—92no—
269C2×C4—94no—
270C2×C4—182no—
271C2×C4—182no—
272C2×C4C3×C4×D912yes—
273C2×C4C3×C4×D914yes—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—22no—
277C2×C3—22no—
278C2×C3—21no—
279C2×C3C42×D911yescharacteristic
280C2×C3#6343911yescharacteristic
281C2×C3#611⋊C412yes—
282C2×C3#6344912yes—
283C2×C3C4⋊C4×D911yescharacteristic
284C2×C3C3×C42×D311yescharacteristic
285C2×C3C3×#87711yescharacteristic
286C2×C3C3×#67⋊C412yes—
287C2×C3C3×#88712yes—
288C2×C3C3×D3×C4⋊C411yescharacteristic
289C22C2×C3×C4×D911yescharacteristic
290C22C2×C3×C4×D912yes—
291C22C3×D4×D911yescharacteristic
292C22C3×#369912yes—
293C22C3×Q8×D911yescharacteristic
294C4—22no—
295C4—24no—
296C4—94no—
297C4—94no—
298C4—184no—
299C3—21no—
300C3#1058832911yescharacteristic, direct product factor
301C3C3×#464911yescharacteristic
302C2C3×C42×D911yescharacteristic
303C2C3×#6343911yescharacteristic
304C2C3×#611⋊C412yes—
305C2C3×#6344912yes—
306C2C3×C4⋊C4×D911yescharacteristic
subgroupG/H
centerC23×C3C22×D9
commutatorC2×C9C2×C3×C42
FrattiniC23×C3C22×C3×D3
FittingC3×C9×#95C2
socle, prime radicalC23×C32C22×D3
cosocleC23×C9C23×C3
hypercenterC3×#95D9
hypocenterC9C3×C4×C4⋊C4
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C23×C3 ◁ C3×#95

Complements

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

NH ≅ G/Ntype
7. #10588329C3direct (N × H)
13. C3×Dic9⋊C4C4semidirect
15. C3×C4×Dic9C4semidirect
58. Dic9⋊C4C3×C4semidirect
60. C4×Dic9C3×C4semidirect
69. C3×Dic9C42semidirect
70. C3×Dic9C4⋊C4semidirect
165. Dic9C3×C42semidirect
166. Dic9C3×C4⋊C4semidirect
172. C3×C9C4×C4⋊C4semidirect
259. C9C3×C4×C4⋊C4semidirect
300. C3#10588329direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Dic9⋊C42
C2×C3×C4×Dic92
C3×C9×#952
C3×C9⋊#952
C3×#46493
#105883293