Finite Group Databasebeta

← C3×#10588336 #13141278

Subgroups — C3×#10588336 #13141278

Order 1728.

622 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×Dic18C211yesmaximal, characteristic
2C2×C3×C4⋊Dic9C212yesmaximal
3C2×C3×Dic9⋊C4C211yesmaximal, characteristic
4C3×C9×#95C211yesmaximal, characteristic
5C3×C9⋊#95C212yesmaximal
6C3×#4656—31nomaximal
7#10588336C311yesmaximal, characteristic, direct product factor
8C22×C3×C4×C9C2211yescharacteristic
9C22×C3×C4×C9C2212yes—
10C22×C3×Dic9C2211yescharacteristic
11C22×C3×Dic9C2212yes—
12C22×C3×Dic9C2211yescharacteristic
13C2×C3×Dic18—22no—
14C2×C3×Dic18—24no—
15C3×C4⋊Dic9—24no—
16C3×Dic9⋊C4—22no—
17C22×C3×Dic6—31no—
18C2×C3×C4⋊Dic3—32no—
19C2×C3×Dic3⋊C4—31no—
20C22×Dic18C2×C311yescharacteristic
21C32×#95D311yescharacteristic
22C3×C3⋊#95—32no—
23C2×C4⋊Dic9C2×C312yes—
24C2×Dic9⋊C4C2×C311yescharacteristic
25C9×#95—21no—
26C9×#95C2×C311yescharacteristic
27C9⋊#95C2×C312yes—
28C23×C3×C9C2311yescharacteristic
29C2×C3×C4×C9—22no—
30C2×C3×C4×C9—22no—
31C2×C3×C4×C9—22no—
32C2×C3×C4×C9D412yes—
33C2×C3×C4×C9Q814yescomplemented
34C2×C3×Dic9—22no—
35C2×C3×Dic9—22no—
36C2×C3×Dic9—21no—
37C2×C3×Dic9—21no—
38C2×C3×Dic9—21no—
39C2×C3×Dic9—22no—
40C2×C3×Dic9D414yes—
41C2×C3×Dic9Q812yescomplemented
42C3×Dic18—28no—
43C3×#395—91no—
44#4656—31no—
45C22×C32×C4C2×D312yes—
46C22×C32×C4C2×D311yescharacteristic
47C22×C3×Dic3—31no—
48C22×C3×Dic3—32no—
49C22×C3×Dic3—31no—
50C2×C3×Dic6—64no—
51C2×C3×Dic6—62no—
52C3×C4⋊Dic3—64no—
53C3×Dic3⋊C4—62no—
54C22×C4×C9—22no—
55C22×C4×C9—21no—
56C22×C4×C9C22×C311yescharacteristic
57C22×C4×C9C22×C312yes—
58C22×Dic9C22×C311yescharacteristic
59C22×Dic9C22×C312yes—
60C22×Dic9C22×C311yescharacteristic
61C2×Dic18—22no—
62C2×Dic18—24no—
63C4⋊Dic9—24no—
64Dic9⋊C4—22no—
65C22×C3×C9C2×D411yescharacteristic
66C22×C3×C9C2×D412yes—
67C22×C3×C9C2×Q812yes—
68C22×C3×C9C2×Q811yescharacteristic
69C22×C3×C9#4111yescharacteristic
70C3×C4×C9—22no—
71C3×C4×C9—24no—
72C3×Dic9—24no—
73C3×Dic9—22no—
74C3×Dic9—41no—
75C22×C3×Q8—91no—
76C22×Dic6—31no—
77C2×C3×C4⋊C4—91no—
78C2×C3×C4⋊C4—92no—
79C2×C4⋊Dic3—32no—
80C2×Dic3⋊C4—31no—
81C3×#95—21no—
82C3×#95—92no—
83C3×#95D911yescharacteristic
84C3×#95C3×D311yescharacteristic
85C3⋊#95—32no—
86C23×C32C22×D311yescharacteristic
87C2×C32×C4—22no—
88C2×C32×C4—22no—
89C2×C32×C4—22no—
90C2×C32×C4Dic614yes—
91C2×C32×C4D1212yes—
92C2×C3×Dic3—34no—
93C2×C3×Dic3—32no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—62no—
100C3×Dic6—68no—
101C23×C9—21no—
102C23×C9C23×C311yescharacteristic
103C2×C4×C9—22no—
104C2×C4×C9—22no—
105C2×C4×C9—22no—
106C2×C4×C9—24no—
107C2×C4×C9—22no—
108C2×C4×C9—42no—
109C2×C4×C9—42no—
110C2×C4×C9—42no—
111C2×C4×C9C3×D412yes—
112C2×C4×C9C3×Q814yescomplemented
113C2×Dic9—21no—
114C2×Dic9—22no—
115C2×Dic9—22no—
116C2×Dic9—21no—
117C2×Dic9—21no—
118C2×Dic9—22no—
119C2×Dic9C3×D414yes—
120C2×Dic9C3×Q812yescomplemented
121Dic18—28no—
122#395—91noSylow
123C2×C3×C9#12011yescharacteristic
124C2×C3×C9C22⋊Q812yes—
125C2×C3×C9C22⋊Q811yescharacteristic
126C2×C3×C9C4⋊Q811yescharacteristic
127C2×C3×C9C4⋊Q812yes—
128C2×C3×Q8—184no—
129C2×C3×Q8—182no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—91no—
133C22×C3×C4—92no—
134C22×C3×C4—91no—
135C22×C3×C4C2×D912yes—
136C22×C3×C4C2×D911yescharacteristic
137C22×C3×C4C2×C3×D312yes—
138C22×C3×C4C2×C3×D311yescharacteristic
139C22×Dic3—31no—
140C22×Dic3—32no—
141C22×Dic3—31no—
142C2×Dic6—64no—
143C2×Dic6—62no—
144C3×C4⋊C4—184no—
145C3×C4⋊C4—182no—
146C4⋊Dic3—64no—
147Dic3⋊C4—62no—
148C22×C32C2×Dic612yes—
149C22×C32C2×D1211yescharacteristic
150C22×C32D3×D412yes—
151C22×C32#23711yescharacteristic
152C22×C32D3×Q811yescharacteristic
153C32×C4—24no—
154C32×C4—22no—
155C3×Dic3—64no—
156C3×Dic3—62no—
157C3×Dic3—121no—
158C22×C9—22no—
159C22×C9—21no—
160C22×C9—22no—
161C22×C9—21no—
162C22×C9—21no—
163C22×C9C2×C3×D411yescharacteristic
164C22×C9C2×C3×D412yes—
165C22×C9C2×C3×Q812yes—
166C22×C9C2×C3×Q811yescharacteristic
167C22×C9C3×#4111yescharacteristic
168C4×C9—22no—
169C4×C9—24no—
170C4×C9—44no—
171C4×C9—42no—
172Dic9—24no—
173Dic9—22no—
174Dic9—41no—
175C22×Q8—91no—
176C2×C4⋊C4—91no—
177C2×C4⋊C4—92no—
178#95—92no—
179#95C3×D911yescharacteristic
180C3×C9#39511yescharacteristic, complemented, Sylow
181C23×C3—21no—
182C23×C3C22×D911yescharacteristic
183C23×C3C22×C3×D311yescharacteristic
184C3×Q8—188no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—24no—
192C2×C3×C4—22no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—94no—
197C2×C3×C4—92no—
198C2×C3×C4—181no—
199C2×C3×C4—182no—
200C2×C3×C4—182no—
201C2×C3×C4—181no—
202C2×C3×C4—181no—
203C2×C3×C4—182no—
204C2×C3×C4Dic1814yescomplemented
205C2×C3×C4D3612yes—
206C2×C3×C4C3×Dic614yes—
207C2×C3×C4C3×D1212yes—
208C2×Dic3—34no—
209C2×Dic3—32no—
210C2×Dic3—61no—
211C2×Dic3—62no—
212C2×Dic3—62no—
213C2×Dic3—61no—
214C2×Dic3—61no—
215C2×Dic3—62no—
216Dic6—68no—
217C2×C32#86911yescharacteristic
218C2×C32#87812yes—
219C2×C32#88211yescharacteristic
220C2×C32#88812yes—
221C2×C32#89711yescharacteristic
222C2×C9—22no—
223C2×C9—21no—
224C2×C9—22no—
225C2×C9—21no—
226C2×C9—21no—
227C2×C9C3×#12011yescharacteristic
228C2×C9C3×C22⋊Q812yes—
229C2×C9C3×C22⋊Q811yescharacteristic
230C2×C9C3×C4⋊Q812yes—
231C2×C9C3×C4⋊Q811yescharacteristic
232C2×Q8—184no—
233C2×Q8—182no—
234C22×C4—91no—
235C22×C4—92no—
236C22×C4—91no—
237C22×C4C2×C3×D912yes—
238C22×C4C2×C3×D911yescharacteristic
239C4⋊C4—184no—
240C4⋊C4—182no—
241C22×C3—22no—
242C22×C3—21no—
243C22×C3—22no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3C2×Dic1812yes—
247C22×C3C2×D3611yescharacteristic
248C22×C3D4×D912yes—
249C22×C3#369911yescharacteristic
250C22×C3Q8×D911yescharacteristic
251C22×C3C2×C3×Dic612yes—
252C22×C3C2×C3×D1211yescharacteristic
253C22×C3C3×D3×D412yes—
254C22×C3C3×#23711yescharacteristic
255C22×C3C3×D3×Q811yescharacteristic
256C3×C4—24no—
257C3×C4—22no—
258C3×C4—24no—
259C3×C4—22no—
260C3×C4—44no—
261C3×C4—42no—
262C3×C4—184no—
263C3×C4—182no—
264C3×C4—361no—
265Dic3—64no—
266Dic3—62no—
267Dic3—121no—
268C32#465611yescharacteristic
269C9—21no—
270C9C3×#39511yescharacteristic, complemented
271C23C22×C3×D911yescharacteristic
272Q8—188no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—94no—
277C2×C4—92no—
278C2×C4—181no—
279C2×C4—182no—
280C2×C4—182no—
281C2×C4—181no—
282C2×C4—181no—
283C2×C4—182no—
284C2×C4C3×Dic1814yescomplemented
285C2×C4C3×D3612yes—
286C2×C3—21no—
287C2×C3—22no—
288C2×C3—21no—
289C2×C3—22no—
290C2×C3—21no—
291C2×C3#6343111yescharacteristic
292C2×C3#6344012yes—
293C2×C3#6344411yescharacteristic
294C2×C3#6345012yes—
295C2×C3#6345911yescharacteristic
296C2×C3C3×#86911yescharacteristic
297C2×C3C3×#87812yes—
298C2×C3C3×#88211yescharacteristic
299C2×C3C3×#88812yes—
300C2×C3C3×#89711yescharacteristic
301C22C2×C3×Dic1812yes—
302C22C2×C3×D3611yescharacteristic
303C22C3×D4×D912yes—
304C22C3×#369911yescharacteristic
305C22C3×Q8×D911yescharacteristic
306C4—24no—
307C4—22no—
308C4—184no—
309C4—182no—
310C4—361no—
311C3—21no—
312C3#1058833611yescharacteristic, direct product factor
313C3C3×#465611yescharacteristic
314C2C3×#6343111yescharacteristic
315C2C3×#6344012yes—
316C2C3×#6344411yescharacteristic
317C2C3×#6345012yes—
318C2C3×#6345911yescharacteristic
subgroupG/H
centerC23×C3C22×D9
commutator, cosocleC23×C9C23×C3
FrattiniC23×C3C22×C3×D3
FittingC3×C9×#95C2
socle, prime radicalC23×C32C22×D3
hypercenterC3×#95D9
hypocenterC9C3×#395
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23×C3 ◁ C3×#95

Complements

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

NH ≅ G/Ntype
7. #10588336C3direct (N × H)
33. C2×C3×C4×C9Q8semidirect
41. C2×C3×Dic9Q8semidirect
112. C2×C4×C9C3×Q8semidirect
120. C2×Dic9C3×Q8semidirect
180. C3×C9#395semidirect
204. C2×C3×C4Dic18semidirect
270. C9C3×#395semidirect
284. C2×C4C3×Dic18semidirect
312. C3#10588336direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Dic182
C2×C3×C4⋊Dic92
C2×C3×Dic9⋊C42
C3×C9×#952
C3×C9⋊#952
C3×#46563
#105883363