Finite Group Databasebeta

← C3×#10588661 #13142563

Subgroups — C3×#10588661 #13142563

Order 1728.

542 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×C4⋊C4C211yesmaximal, characteristic
2C2×C3×C4⋊Dic9C211yesmaximal, characteristic
3C2×C3×Dic9⋊C4C212yesmaximal
4C3×C9⋊#95C212yesmaximal
5C3×C9⋊#95C211yesmaximal, characteristic
6C3×#4981—31nomaximal
7#10588661C311yesmaximal, characteristic, direct product factor
8C22×C3×C4×C9C2211yescharacteristic
9C22×C3×C4×C9C2211yescharacteristic
10C22×C3×C4×C9C2211yescharacteristic
11C22×C3×Dic9C2212yes—
12C22×C3×Dic9C2212yes—
13C3×C9×C4⋊C4—22no—
14C3×C4⋊Dic9—22no—
15C3×Dic9⋊C4—24no—
16C2×C32×C4⋊C4D311yescharacteristic
17C2×C3×C4⋊Dic3—31no—
18C2×C3×Dic3⋊C4—32no—
19C3×C3⋊#95—31no—
20C3×C3⋊#95—32no—
21C2×C9×C4⋊C4—21no—
22C2×C9×C4⋊C4C2×C311yescharacteristic
23C2×C4⋊Dic9C2×C311yescharacteristic
24C2×Dic9⋊C4C2×C312yes—
25C9⋊#95C2×C312yes—
26C9⋊#95C2×C311yescharacteristic
27C23×C3×C9C2311yescharacteristic
28C2×C3×C4×C9—21no—
29C2×C3×C4×C9—21no—
30C2×C3×C4×C9—22no—
31C2×C3×C4×C9—21no—
32C2×C3×C4×C9—22no—
33C2×C3×C4×C9D412yes—
34C2×C3×C4×C9D412yes—
35C2×C3×Dic9—22no—
36C2×C3×Dic9—22no—
37C2×C3×Dic9—22no—
38C2×C3×Dic9—22no—
39C2×C3×Dic9—22no—
40C2×C3×Dic9Q814yes—
41C3×#398—91no—
42#4981—31no—
43C22×C32×C4C2×D311yescharacteristic
44C22×C32×C4C2×D311yescharacteristic
45C22×C32×C4C2×D311yescharacteristic
46C22×C3×Dic3—32no—
47C22×C3×Dic3—32no—
48C32×C4⋊C4—22no—
49C3×C4⋊Dic3—62no—
50C3×Dic3⋊C4—64no—
51C22×C4×C9—21no—
52C22×C4×C9—21no—
53C22×C4×C9—21no—
54C22×C4×C9C22×C311yescharacteristic
55C22×C4×C9C22×C311yescharacteristic
56C22×C4×C9C22×C311yescharacteristic
57C22×Dic9C22×C312yes—
58C22×Dic9C22×C312yes—
59C9×C4⋊C4—24no—
60C9×C4⋊C4—22no—
61C4⋊Dic9—22no—
62Dic9⋊C4—24no—
63C22×C3×C9C2×D411yescharacteristic
64C22×C3×C9C2×D411yescharacteristic
65C22×C3×C9C2×Q812yes—
66C22×C3×C9#4111yescharacteristic
67C22×C3×C9#4112yes—
68C3×C4×C9—22no—
69C3×C4×C9—22no—
70C3×C4×C9—41no—
71C3×Dic9—24no—
72C3×Dic9—42no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—92no—
75C2×C3×C4⋊C4—91no—
76C2×C3×C4⋊C4D911yescharacteristic
77C2×C3×C4⋊C4C3×D311yescharacteristic
78C2×C4⋊Dic3—31no—
79C2×Dic3⋊C4—32no—
80C3×#95—91no—
81C3×#95—92no—
82C3⋊#95—31no—
83C3⋊#95—32no—
84C23×C32C22×D311yescharacteristic
85C2×C32×C4—21no—
86C2×C32×C4—21no—
87C2×C32×C4—21no—
88C2×C32×C4—22no—
89C2×C32×C4—22no—
90C2×C32×C4D1212yes—
91C2×C32×C4#6712yes—
92C2×C3×Dic3—34no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—62no—
98C23×C9—21no—
99C23×C9C23×C311yescharacteristic
100C2×C4×C9—22no—
101C2×C4×C9—22no—
102C2×C4×C9—21no—
103C2×C4×C9—21no—
104C2×C4×C9—22no—
105C2×C4×C9—22no—
106C2×C4×C9—21no—
107C2×C4×C9—22no—
108C2×C4×C9—22no—
109C2×C4×C9—22no—
110C2×C4×C9—42no—
111C2×C4×C9—42no—
112C2×C4×C9C3×D412yes—
113C2×C4×C9C3×D412yes—
114C2×Dic9—22no—
115C2×Dic9—22no—
116C2×Dic9—22no—
117C2×Dic9—22no—
118C2×Dic9—22no—
119C2×Dic9C3×Q814yes—
120#398—91noSylow
121C2×C3×C9#12111yescharacteristic
122C2×C3×C9C22⋊Q812yes—
123C2×C3×C9#12311yescharacteristic
124C2×C3×C9#12512yes—
125C2×C3×C9C4⋊Q811yescharacteristic
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—92no—
130C22×C3×C4—92no—
131C22×C3×C4C2×D911yescharacteristic
132C22×C3×C4C2×D911yescharacteristic
133C22×C3×C4C2×D911yescharacteristic
134C22×C3×C4C2×C3×D311yescharacteristic
135C22×C3×C4C2×C3×D311yescharacteristic
136C22×C3×C4C2×C3×D311yescharacteristic
137C22×Dic3—32no—
138C22×Dic3—32no—
139C3×C4⋊C4—24no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—184no—
143C3×C4⋊C4—182no—
144C4⋊Dic3—62no—
145Dic3⋊C4—64no—
146C22×C32C2×D1211yescharacteristic
147C22×C32#23511yescharacteristic
148C22×C32#23712yes—
149C22×C32D3×Q812yes—
150C22×C32C2×#6711yescharacteristic
151C32×C4—22no—
152C32×C4—22no—
153C32×C4—41no—
154C3×Dic3—64no—
155C3×Dic3—122no—
156C22×C9—21no—
157C22×C9—21no—
158C22×C9—22no—
159C22×C9—22no—
160C22×C9—21no—
161C22×C9C2×C3×D411yescharacteristic
162C22×C9C2×C3×D411yescharacteristic
163C22×C9C2×C3×Q812yes—
164C22×C9C3×#4111yescharacteristic
165C22×C9C3×#4112yes—
166C4×C9—24no—
167C4×C9—22no—
168C4×C9—22no—
169C4×C9—42no—
170C4×C9—42no—
171C4×C9—41no—
172Dic9—24no—
173Dic9—42no—
174C2×C4⋊C4—92no—
175C2×C4⋊C4—91no—
176C2×C4⋊C4C3×D911yescharacteristic
177#95—91no—
178#95—92no—
179C3×C9#39811yescharacteristic, complemented, Sylow
180C23×C3—21no—
181C23×C3C22×D911yescharacteristic
182C23×C3C22×C3×D311yescharacteristic
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—21no—
186C2×C3×C4—22no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—22no—
190C2×C3×C4—21no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—21no—
194C2×C3×C4—22no—
195C2×C3×C4—21no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—94no—
201C2×C3×C4—182no—
202C2×C3×C4—182no—
203C2×C3×C4—182no—
204C2×C3×C4—182no—
205C2×C3×C4—182no—
206C2×C3×C4D3612yes—
207C2×C3×C4#61112yes—
208C2×C3×C4C3×D1212yes—
209C2×C3×C4C3×#6712yes—
210C2×Dic3—34no—
211C2×Dic3—62no—
212C2×Dic3—62no—
213C2×Dic3—62no—
214C2×Dic3—62no—
215C2×Dic3—62no—
216C2×C32#88912yes—
217C2×C32#89712yes—
218C2×C32#93011yescharacteristic
219C2×C32#93511yescharacteristic
220C2×C32#94411yescharacteristic
221C2×C9—22no—
222C2×C9—22no—
223C2×C9—21no—
224C2×C9—21no—
225C2×C9—21no—
226C2×C9C3×#12111yescharacteristic
227C2×C9C3×C22⋊Q812yes—
228C2×C9C3×#12311yescharacteristic
229C2×C9C3×#12512yes—
230C2×C9C3×C4⋊Q811yescharacteristic
231C22×C4—92no—
232C22×C4—92no—
233C22×C4C2×C3×D911yescharacteristic
234C22×C4C2×C3×D911yescharacteristic
235C22×C4C2×C3×D911yescharacteristic
236C4⋊C4—22no—
237C4⋊C4—184no—
238C4⋊C4—182no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C3—21no—
244C22×C3C2×D3611yescharacteristic
245C22×C3#369711yescharacteristic
246C22×C3#369912yes—
247C22×C3Q8×D912yes—
248C22×C3C2×#61111yescharacteristic
249C22×C3C2×C3×D1211yescharacteristic
250C22×C3C3×#23511yescharacteristic
251C22×C3C3×#23712yes—
252C22×C3C3×D3×Q812yes—
253C22×C3C2×C3×#6711yescharacteristic
254C3×C4—24no—
255C3×C4—22no—
256C3×C4—22no—
257C3×C4—22no—
258C3×C4—22no—
259C3×C4—42no—
260C3×C4—42no—
261C3×C4—41no—
262C3×C4—41no—
263C3×C4—184no—
264C3×C4—362no—
265Dic3—64no—
266Dic3—122no—
267C32#498111yescharacteristic
268C9—21no—
269C9C3×#39811yescharacteristic, complemented
270C23C22×C3×D911yescharacteristic
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—94no—
277C2×C4—182no—
278C2×C4—182no—
279C2×C4—182no—
280C2×C4—182no—
281C2×C4—182no—
282C2×C4C3×D3612yes—
283C2×C4C3×#61112yes—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—22no—
288C2×C3—22no—
289C2×C3#6345112yes—
290C2×C3#6345912yes—
291C2×C3#6349211yescharacteristic
292C2×C3#6349711yescharacteristic
293C2×C3#6350611yescharacteristic
294C2×C3C3×#88912yes—
295C2×C3C3×#89712yes—
296C2×C3C3×#93011yescharacteristic
297C2×C3C3×#93511yescharacteristic
298C2×C3C3×#94411yescharacteristic
299C22C2×C3×D3611yescharacteristic
300C22C3×#369711yescharacteristic
301C22C3×#369912yes—
302C22C3×Q8×D912yes—
303C22C2×C3×#61111yescharacteristic
304C4—22no—
305C4—22no—
306C4—41no—
307C4—184no—
308C4—362no—
309C3—21no—
310C3#1058866111yescharacteristic, direct product factor
311C3C3×#498111yescharacteristic
312C2C3×#6345112yes—
313C2C3×#6345912yes—
314C2C3×#6349211yescharacteristic
315C2C3×#6349711yescharacteristic
316C2C3×#6350611yescharacteristic
subgroupG/H
centerC23×C3C22×D9
commutator, cosocleC23×C9C23×C3
FrattiniC23×C3C22×C3×D3
FittingC2×C3×C9×C4⋊C4C2
socle, prime radicalC23×C32C22×D3
hypercenterC2×C3×C4⋊C4D9
hypocenterC9C3×#398
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
7. #10588661C3direct (N × H)
179. C3×C9#398semidirect
269. C9C3×#398semidirect
310. C3#10588661direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C9×C4⋊C42
C2×C3×C4⋊Dic92
C2×C3×Dic9⋊C42
C3×C9⋊#952
C3×#49813
#105886613