Finite Group Databasebeta

← C3×#10588341 #13141284

Subgroups — C3×#10588341 #13141284

Order 1728.

526 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×C4⋊Dic9C211yesmaximal, characteristic
2C2×C3×C4⋊Dic9C212yesmaximal
3C2×C3×Dic9⋊C4C211yesmaximal, characteristic
4C3×C9×#95C211yesmaximal, characteristic
5C3×C9⋊#95C212yesmaximal
6C3×#4661—31nomaximal
7#10588341C311yesmaximal, characteristic, direct product factor
8C22×C3×C4×C9C2211yescharacteristic
9C22×C3×C4×C9C2212yes—
10C22×C3×Dic9C2211yescharacteristic
11C22×C3×Dic9C2212yes—
12C22×C3×Dic9C2211yescharacteristic
13C3×C4⋊Dic9—22no—
14C3×C4⋊Dic9—24no—
15C3×Dic9⋊C4—22no—
16C2×C3×C4⋊Dic3—32no—
17C2×C3×C4⋊Dic3—31no—
18C2×C3×Dic3⋊C4—31no—
19C32×#95D311yescharacteristic
20C3×C3⋊#95—32no—
21C2×C4⋊Dic9C2×C311yescharacteristic
22C2×C4⋊Dic9C2×C312yes—
23C2×Dic9⋊C4C2×C311yescharacteristic
24C9×#95—21no—
25C9×#95C2×C311yescharacteristic
26C9⋊#95C2×C312yes—
27C23×C3×C9C2311yescharacteristic
28C2×C3×C4×C9—22no—
29C2×C3×C4×C9—22no—
30C2×C3×C4×C9—22no—
31C2×C3×C4×C9D412yes—
32C2×C3×C4×C9Q814yes—
33C2×C3×Dic9—21no—
34C2×C3×Dic9—21no—
35C2×C3×Dic9—21no—
36C2×C3×Dic9—22no—
37C2×C3×Dic9—22no—
38C2×C3×Dic9—22no—
39C2×C3×Dic9—22no—
40C2×C3×Dic9D412yes—
41C3×#398—91no—
42#4661—31no—
43C22×C32×C4C2×D312yes—
44C22×C32×C4C2×D311yescharacteristic
45C22×C3×Dic3—31no—
46C22×C3×Dic3—31no—
47C22×C3×Dic3—32no—
48C3×C4⋊Dic3—64no—
49C3×C4⋊Dic3—62no—
50C3×Dic3⋊C4—62no—
51C22×C4×C9—22no—
52C22×C4×C9—21no—
53C22×C4×C9C22×C311yescharacteristic
54C22×C4×C9C22×C312yes—
55C22×Dic9C22×C311yescharacteristic
56C22×Dic9C22×C312yes—
57C22×Dic9C22×C311yescharacteristic
58C4⋊Dic9—22no—
59C4⋊Dic9—24no—
60Dic9⋊C4—22no—
61C22×C3×C9C2×D411yescharacteristic
62C22×C3×C9C2×D411yescharacteristic
63C22×C3×C9C2×Q812yes—
64C22×C3×C9#4111yescharacteristic
65C22×C3×C9#4112yes—
66C3×C4×C9—24no—
67C3×C4×C9—22no—
68C3×Dic9—22no—
69C3×Dic9—42no—
70C3×Dic9—41no—
71C2×C3×C4⋊C4—91no—
72C2×C3×C4⋊C4—92no—
73C2×C3×C4⋊C4—91no—
74C2×C4⋊Dic3—32no—
75C2×C4⋊Dic3—31no—
76C2×Dic3⋊C4—31no—
77C3×#95—21no—
78C3×#95—92no—
79C3×#95D911yescharacteristic
80C3×#95C3×D311yescharacteristic
81C3⋊#95—32no—
82C23×C32C22×D311yescharacteristic
83C2×C32×C4—22no—
84C2×C32×C4—22no—
85C2×C32×C4—22no—
86C2×C32×C4Dic614yes—
87C2×C32×C4D1212yes—
88C2×C3×Dic3—32no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
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—24no—
102C2×C4×C9—22no—
103C2×C4×C9—42no—
104C2×C4×C9—42no—
105C2×C4×C9—42no—
106C2×C4×C9C3×D412yes—
107C2×C4×C9C3×Q814yes—
108C2×Dic9—21no—
109C2×Dic9—21no—
110C2×Dic9—21no—
111C2×Dic9—22no—
112C2×Dic9—22no—
113C2×Dic9—22no—
114C2×Dic9—22no—
115C2×Dic9C3×D412yes—
116#398—91noSylow
117C2×C3×C9#12111yescharacteristic
118C2×C3×C9C22⋊Q812yes—
119C2×C3×C9#12311yescharacteristic
120C2×C3×C9#12512yes—
121C2×C3×C9C4⋊Q811yescharacteristic
122C22×C3×C4—22no—
123C22×C3×C4—21no—
124C22×C3×C4—91no—
125C22×C3×C4—91no—
126C22×C3×C4—92no—
127C22×C3×C4C2×D912yes—
128C22×C3×C4C2×D911yescharacteristic
129C22×C3×C4C2×C3×D312yes—
130C22×C3×C4C2×C3×D311yescharacteristic
131C22×Dic3—31no—
132C22×Dic3—31no—
133C22×Dic3—32no—
134C3×C4⋊C4—184no—
135C3×C4⋊C4—182no—
136C3×C4⋊C4—182no—
137C4⋊Dic3—64no—
138C4⋊Dic3—62no—
139Dic3⋊C4—62no—
140C22×C32C2×Dic612yes—
141C22×C32C2×D1211yescharacteristic
142C22×C32D3×D411yescharacteristic
143C22×C32#23712yes—
144C22×C32#23911yescharacteristic
145C32×C4—24no—
146C32×C4—22no—
147C3×Dic3—62no—
148C3×Dic3—121no—
149C3×Dic3—122no—
150C22×C9—22no—
151C22×C9—21no—
152C22×C9—21no—
153C22×C9—22no—
154C22×C9—21no—
155C22×C9C2×C3×D411yescharacteristic
156C22×C9C2×C3×D411yescharacteristic
157C22×C9C2×C3×Q812yes—
158C22×C9C3×#4112yes—
159C22×C9C3×#4111yescharacteristic
160C4×C9—24no—
161C4×C9—22no—
162C4×C9—44no—
163C4×C9—42no—
164Dic9—22no—
165Dic9—42no—
166Dic9—41no—
167C2×C4⋊C4—91no—
168C2×C4⋊C4—92no—
169C2×C4⋊C4—91no—
170#95—92no—
171#95C3×D911yescharacteristic
172C3×C9#39811yescharacteristic, 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—24no—
183C2×C3×C4—22no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—92no—
188C2×C3×C4—181no—
189C2×C3×C4—181no—
190C2×C3×C4—181no—
191C2×C3×C4—182no—
192C2×C3×C4—182no—
193C2×C3×C4—182no—
194C2×C3×C4—182no—
195C2×C3×C4Dic1814yes—
196C2×C3×C4D3612yes—
197C2×C3×C4C3×Dic614yes—
198C2×C3×C4C3×D1212yes—
199C2×Dic3—32no—
200C2×Dic3—61no—
201C2×Dic3—61no—
202C2×Dic3—61no—
203C2×Dic3—62no—
204C2×Dic3—62no—
205C2×Dic3—62no—
206C2×Dic3—62no—
207C2×C32#86911yescharacteristic
208C2×C32#87812yes—
209C2×C32#88611yescharacteristic
210C2×C32#89012yes—
211C2×C32#89511yescharacteristic
212C2×C9—22no—
213C2×C9—21no—
214C2×C9—22no—
215C2×C9—21no—
216C2×C9—21no—
217C2×C9C3×#12111yescharacteristic
218C2×C9C3×C22⋊Q812yes—
219C2×C9C3×#12311yescharacteristic
220C2×C9C3×#12512yes—
221C2×C9C3×C4⋊Q811yescharacteristic
222C22×C4—91no—
223C22×C4—91no—
224C22×C4—92no—
225C22×C4C2×C3×D912yes—
226C22×C4C2×C3×D911yescharacteristic
227C4⋊C4—184no—
228C4⋊C4—182no—
229C4⋊C4—182no—
230C22×C3—22no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—22no—
234C22×C3—21no—
235C22×C3C2×Dic1812yes—
236C22×C3C2×D3611yescharacteristic
237C22×C3D4×D911yescharacteristic
238C22×C3#369912yes—
239C22×C3#370111yescharacteristic
240C22×C3C2×C3×Dic612yes—
241C22×C3C2×C3×D1211yescharacteristic
242C22×C3C3×D3×D411yescharacteristic
243C22×C3C3×#23712yes—
244C22×C3C3×#23911yescharacteristic
245C3×C4—24no—
246C3×C4—22no—
247C3×C4—24no—
248C3×C4—22no—
249C3×C4—44no—
250C3×C4—42no—
251C3×C4—182no—
252C3×C4—361no—
253C3×C4—362no—
254Dic3—62no—
255Dic3—121no—
256Dic3—122no—
257C32#466111yescharacteristic
258C9—21no—
259C9C3×#39811yescharacteristic, complemented
260C23C22×C3×D911yescharacteristic
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—92no—
265C2×C4—181no—
266C2×C4—181no—
267C2×C4—181no—
268C2×C4—182no—
269C2×C4—182no—
270C2×C4—182no—
271C2×C4—182no—
272C2×C4C3×Dic1814yes—
273C2×C4C3×D3612yes—
274C2×C3—21no—
275C2×C3—22no—
276C2×C3—21no—
277C2×C3—22no—
278C2×C3—21no—
279C2×C3#6343111yescharacteristic
280C2×C3#6344012yes—
281C2×C3#6344811yescharacteristic
282C2×C3#6345212yes—
283C2×C3#6345711yescharacteristic
284C2×C3C3×#86911yescharacteristic
285C2×C3C3×#87812yes—
286C2×C3C3×#88611yescharacteristic
287C2×C3C3×#89012yes—
288C2×C3C3×#89511yescharacteristic
289C22C2×C3×Dic1812yes—
290C22C2×C3×D3611yescharacteristic
291C22C3×D4×D911yescharacteristic
292C22C3×#369912yes—
293C22C3×#370111yescharacteristic
294C4—24no—
295C4—22no—
296C4—182no—
297C4—361no—
298C4—362no—
299C3—21no—
300C3#1058834111yescharacteristic, direct product factor
301C3C3×#466111yescharacteristic
302C2C3×#6343111yescharacteristic
303C2C3×#6344012yes—
304C2C3×#6344811yescharacteristic
305C2C3×#6345212yes—
306C2C3×#6345711yescharacteristic
subgroupG/H
centerC23×C3C22×D9
commutator, cosocleC23×C9C23×C3
FrattiniC23×C3C22×C3×D3
FittingC3×C9×#95C2
socle, prime radicalC23×C32C22×D3
hypercenterC3×#95D9
hypocenterC9C3×#398
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23×C3 ◁ C3×#95

Complements

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

NH ≅ G/Ntype
7. #10588341C3direct (N × H)
172. C3×C9#398semidirect
259. C9C3×#398semidirect
300. C3#10588341direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊Dic92
C2×C3×Dic9⋊C42
C3×C9×#952
C3×C9⋊#952
C3×#46613
#105883413