Finite Group Databasebeta

← C3×#10588663 #13142558

Subgroups — C3×#10588663 #13142558

Order 1728.

542 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×C4⋊C4C211yesmaximal, characteristic
2C2×C3×Dic9⋊C4C211yesmaximal, characteristic
3C2×C3×Dic9⋊C4C212yesmaximal
4C3×C9⋊#95C212yesmaximal
5C3×C9⋊#95C211yesmaximal, characteristic
6C3×#4983—31nomaximal
7#10588663C311yesmaximal, characteristic, direct product factor
8C22×C3×C4×C9C2212yes—
9C22×C3×C4×C9C2211yescharacteristic
10C22×C3×Dic9C2211yescharacteristic
11C22×C3×Dic9C2211yescharacteristic
12C22×C3×Dic9C2212yes—
13C3×C9×C4⋊C4—22no—
14C3×Dic9⋊C4—22no—
15C3×Dic9⋊C4—24no—
16C2×C32×C4⋊C4D311yescharacteristic
17C2×C3×Dic3⋊C4—32no—
18C2×C3×Dic3⋊C4—31no—
19C3×C3⋊#95—31no—
20C3×C3⋊#95—32no—
21C2×C9×C4⋊C4—21no—
22C2×C9×C4⋊C4C2×C311yescharacteristic
23C2×Dic9⋊C4C2×C311yescharacteristic
24C2×Dic9⋊C4C2×C312yes—
25C9⋊#95C2×C312yes—
26C9⋊#95C2×C311yescharacteristic
27C23×C3×C9C2311yescharacteristic
28C2×C3×C4×C9—22no—
29C2×C3×C4×C9—22no—
30C2×C3×C4×C9—22no—
31C2×C3×C4×C9—22no—
32C2×C3×C4×C9D412yes—
33C2×C3×Dic9—21no—
34C2×C3×Dic9—22no—
35C2×C3×Dic9—21no—
36C2×C3×Dic9—22no—
37C2×C3×Dic9—21no—
38C2×C3×Dic9—22no—
39C2×C3×Dic9D412yes—
40C2×C3×Dic9Q814yes—
41C3×#398—91no—
42#4983—31no—
43C22×C32×C4C2×D312yes—
44C22×C32×C4C2×D311yescharacteristic
45C22×C3×Dic3—31no—
46C22×C3×Dic3—32no—
47C22×C3×Dic3—31no—
48C32×C4⋊C4—22no—
49C3×Dic3⋊C4—64no—
50C3×Dic3⋊C4—62no—
51C22×C4×C9—21no—
52C22×C4×C9—22no—
53C22×C4×C9C22×C312yes—
54C22×C4×C9C22×C311yescharacteristic
55C22×Dic9C22×C311yescharacteristic
56C22×Dic9C22×C311yescharacteristic
57C22×Dic9C22×C312yes—
58C9×C4⋊C4—24no—
59C9×C4⋊C4—22no—
60Dic9⋊C4—22no—
61Dic9⋊C4—24no—
62C22×C3×C9C2×D411yescharacteristic
63C22×C3×C9C2×D411yescharacteristic
64C22×C3×C9C2×Q812yes—
65C22×C3×C9#4111yescharacteristic
66C22×C3×C9#4112yes—
67C3×C4×C9—22no—
68C3×C4×C9—42no—
69C3×Dic9—22no—
70C3×Dic9—24no—
71C3×Dic9—41no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—92no—
74C2×C3×C4⋊C4—91no—
75C2×C3×C4⋊C4D911yescharacteristic
76C2×C3×C4⋊C4C3×D311yescharacteristic
77C2×Dic3⋊C4—32no—
78C2×Dic3⋊C4—31no—
79C3×#95—91no—
80C3×#95—92no—
81C3⋊#95—31no—
82C3⋊#95—32no—
83C23×C32C22×D311yescharacteristic
84C2×C32×C4—22no—
85C2×C32×C4—22no—
86C2×C32×C4—22no—
87C2×C32×C4—22no—
88C2×C32×C4#6712yes—
89C2×C3×Dic3—32no—
90C2×C3×Dic3—34no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—61no—
97C23×C9—21no—
98C23×C9C23×C311yescharacteristic
99C2×C4×C9—24no—
100C2×C4×C9—22no—
101C2×C4×C9—24no—
102C2×C4×C9—22no—
103C2×C4×C9—22no—
104C2×C4×C9—22no—
105C2×C4×C9—22no—
106C2×C4×C9—42no—
107C2×C4×C9—42no—
108C2×C4×C9C3×D412yes—
109C2×Dic9—22no—
110C2×Dic9—21no—
111C2×Dic9—22no—
112C2×Dic9—21no—
113C2×Dic9—21no—
114C2×Dic9—22no—
115C2×Dic9C3×D412yes—
116C2×Dic9C3×Q814yes—
117#398—91noSylow
118C2×C3×C9#12111yescharacteristic
119C2×C3×C9C22⋊Q812yes—
120C2×C3×C9#12311yescharacteristic
121C2×C3×C9#12512yes—
122C2×C3×C9C4⋊Q811yescharacteristic
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—92no—
126C22×C3×C4—91no—
127C22×C3×C4—91no—
128C22×C3×C4C2×D912yes—
129C22×C3×C4C2×D911yescharacteristic
130C22×C3×C4C2×C3×D312yes—
131C22×C3×C4C2×C3×D311yescharacteristic
132C22×Dic3—32no—
133C22×Dic3—31no—
134C22×Dic3—31no—
135C3×C4⋊C4—24no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—184no—
139C3×C4⋊C4—182no—
140Dic3⋊C4—64no—
141Dic3⋊C4—62no—
142C22×C32#23512yes—
143C22×C32D3×D411yescharacteristic
144C22×C32#23711yescharacteristic
145C22×C32D3×Q812yes—
146C22×C32C2×#6711yescharacteristic
147C32×C4—22no—
148C32×C4—42no—
149C3×Dic3—64no—
150C3×Dic3—62no—
151C3×Dic3—121no—
152C22×C9—22no—
153C22×C9—21no—
154C22×C9—21no—
155C22×C9—22no—
156C22×C9—21no—
157C22×C9C2×C3×D411yescharacteristic
158C22×C9C2×C3×D411yescharacteristic
159C22×C9C2×C3×Q812yes—
160C22×C9C3×#4112yes—
161C22×C9C3×#4111yescharacteristic
162C4×C9—24no—
163C4×C9—22no—
164C4×C9—44no—
165C4×C9—42no—
166Dic9—22no—
167Dic9—24no—
168Dic9—41no—
169C2×C4⋊C4—92no—
170C2×C4⋊C4—91no—
171C2×C4⋊C4C3×D911yescharacteristic
172#95—91no—
173#95—92no—
174C3×C9#39811yescharacteristic, complemented, Sylow
175C23×C3—21no—
176C23×C3C22×D911yescharacteristic
177C23×C3C22×C3×D311yescharacteristic
178C2×C3×C4—24no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—24no—
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—92no—
192C2×C3×C4—94no—
193C2×C3×C4—182no—
194C2×C3×C4—182no—
195C2×C3×C4—181no—
196C2×C3×C4—181no—
197C2×C3×C4—182no—
198C2×C3×C4—181no—
199C2×C3×C4#61112yes—
200C2×C3×C4C3×#6712yes—
201C2×Dic3—32no—
202C2×Dic3—34no—
203C2×Dic3—62no—
204C2×Dic3—62no—
205C2×Dic3—61no—
206C2×Dic3—61no—
207C2×Dic3—62no—
208C2×Dic3—61no—
209C2×C32#88912yes—
210C2×C32#89612yes—
211C2×C32#92911yescharacteristic
212C2×C32#93811yescharacteristic
213C2×C32#94411yescharacteristic
214C2×C9—22no—
215C2×C9—22no—
216C2×C9—21no—
217C2×C9—21no—
218C2×C9—21no—
219C2×C9C3×#12111yescharacteristic
220C2×C9C3×C22⋊Q812yes—
221C2×C9C3×#12311yescharacteristic
222C2×C9C3×#12512yes—
223C2×C9C3×C4⋊Q811yescharacteristic
224C22×C4—92no—
225C22×C4—91no—
226C22×C4—91no—
227C22×C4C2×C3×D912yes—
228C22×C4C2×C3×D911yescharacteristic
229C4⋊C4—22no—
230C4⋊C4—184no—
231C4⋊C4—182no—
232C22×C3—22no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—22no—
236C22×C3—21no—
237C22×C3#369712yes—
238C22×C3D4×D911yescharacteristic
239C22×C3#369911yescharacteristic
240C22×C3Q8×D912yes—
241C22×C3C2×#61111yescharacteristic
242C22×C3C3×#23512yes—
243C22×C3C3×D3×D411yescharacteristic
244C22×C3C3×#23711yescharacteristic
245C22×C3C3×D3×Q812yes—
246C22×C3C2×C3×#6711yescharacteristic
247C3×C4—24no—
248C3×C4—22no—
249C3×C4—22no—
250C3×C4—44no—
251C3×C4—42no—
252C3×C4—42no—
253C3×C4—182no—
254C3×C4—184no—
255C3×C4—361no—
256Dic3—64no—
257Dic3—62no—
258Dic3—121no—
259C32#498311yescharacteristic
260C9—21no—
261C9C3×#39811yescharacteristic, complemented
262C23C22×C3×D911yescharacteristic
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—92no—
268C2×C4—94no—
269C2×C4—182no—
270C2×C4—182no—
271C2×C4—181no—
272C2×C4—181no—
273C2×C4—182no—
274C2×C4—181no—
275C2×C4C3×#61112yes—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—22no—
280C2×C3—22no—
281C2×C3#6345112yes—
282C2×C3#6345812yes—
283C2×C3#6349111yescharacteristic
284C2×C3#6350011yescharacteristic
285C2×C3#6350611yescharacteristic
286C2×C3C3×#88912yes—
287C2×C3C3×#89612yes—
288C2×C3C3×#92911yescharacteristic
289C2×C3C3×#93811yescharacteristic
290C2×C3C3×#94411yescharacteristic
291C22C3×#369712yes—
292C22C3×D4×D911yescharacteristic
293C22C3×#369911yescharacteristic
294C22C3×Q8×D912yes—
295C22C2×C3×#61111yescharacteristic
296C4—22no—
297C4—42no—
298C4—182no—
299C4—184no—
300C4—361no—
301C3—21no—
302C3#1058866311yescharacteristic, direct product factor
303C3C3×#498311yescharacteristic
304C2C3×#6345112yes—
305C2C3×#6345812yes—
306C2C3×#6349111yescharacteristic
307C2C3×#6350011yescharacteristic
308C2C3×#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. #10588663C3direct (N × H)
174. C3×C9#398semidirect
261. C9C3×#398semidirect
302. C3#10588663direct (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×Dic9⋊C42
C3×C9⋊#952
C3×#49833
#105886633