Finite Group Databasebeta

← C3×C4×Dic9⋊C4 #13142381

Subgroups — C3×C4×Dic9⋊C4 #13142381

Order 1728.

690 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×C42×C9C211yesmaximal, characteristic
2C2×C3×Dic9⋊C4C212yesmaximal
3C2×C3×C4×Dic9C211yesmaximal, characteristic
4C2×C3×C4×Dic9C211yesmaximal, characteristic
5C3×C9⋊#95C212yesmaximal
6C3×C4×Dic3⋊C4—31nomaximal
7C4×Dic9⋊C4C311yesmaximal, characteristic, direct product factor
8C22×C3×C4×C9C2212yes—
9C22×C3×C4×C9C2211yescharacteristic
10C22×C3×Dic9C2211yescharacteristic
11C22×C3×Dic9C2212yes—
12C22×C3×Dic9C2211yescharacteristic
13C3×C42×C9—22no—
14C3×Dic9⋊C4C418yesdirect product factor
15C3×C4×Dic9—22no—
16C3×C4×Dic9C414yescomplemented
17C2×C32×C42D311yescharacteristic
18C2×C3×Dic3⋊C4—32no—
19C2×C3×C4×Dic3—31no—
20C2×C3×C4×Dic3—31no—
21C3×C3⋊#95—32no—
22C2×C42×C9—21no—
23C2×C42×C9C2×C311yescharacteristic
24C2×Dic9⋊C4C2×C312yes—
25C2×C4×Dic9C2×C311yescharacteristic
26C2×C4×Dic9C2×C311yescharacteristic
27C9⋊#95C2×C312yes—
28C23×C3×C9C2311yescharacteristic
29C2×C3×C4×C9—22no—
30C2×C3×C4×C9—22no—
31C2×C3×C4×C9C2×C414yes—
32C2×C3×C4×C9C2×C412yes—
33C2×C3×C4×C9D412yes—
34C2×C3×C4×C9Q812yes—
35C2×C3×Dic9—22no—
36C2×C3×Dic9—22no—
37C2×C3×Dic9—22no—
38C2×C3×Dic9C2×C412yes—
39C2×C3×Dic9C2×C414yes—
40C2×C3×Dic9C2×C412yes—
41C2×C3×Dic9C2×C414yes—
42C3×C4×C4⋊C4—91no—
43C4×Dic3⋊C4—31no—
44C22×C32×C4C2×D312yes—
45C22×C32×C4C2×D311yescharacteristic
46C22×C3×Dic3—31no—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—31no—
49C32×C42—22no—
50C3×Dic3⋊C4—38no—
51C3×C4×Dic3—34no—
52C3×C4×Dic3—62no—
53C22×C4×C9—22no—
54C22×C4×C9—21no—
55C22×C4×C9C22×C312yes—
56C22×C4×C9C22×C311yescharacteristic
57C22×Dic9C22×C311yescharacteristic
58C22×Dic9C22×C312yes—
59C22×Dic9C22×C311yescharacteristic
60C42×C9—24no—
61C42×C9—22no—
62Dic9⋊C4C3×C418yesdirect product factor
63C4×Dic9—22no—
64C4×Dic9C3×C414yescomplemented
65C22×C3×C9C22×C411yescharacteristic
66C22×C3×C9C22×C412yes—
67C22×C3×C9C2×D411yescharacteristic
68C22×C3×C9C2×Q811yescharacteristic
69C22×C3×C9#4112yes—
70C3×C4×C9—24no—
71C3×C4×C9C4⋊C414yescomplemented
72C3×Dic9—22no—
73C3×Dic9—24no—
74C3×Dic9C4214yescomplemented
75C2×C3×C4⋊C4—92no—
76C2×C3×C42—21no—
77C2×C3×C42—91no—
78C2×C3×C42—91no—
79C2×C3×C42D911yescharacteristic
80C2×C3×C42C3×D311yescharacteristic
81C2×Dic3⋊C4—32no—
82C2×C4×Dic3—31no—
83C2×C4×Dic3—31no—
84C3×#95—92no—
85C3⋊#95—32no—
86C23×C32C22×D311yescharacteristic
87C2×C32×C4—22no—
88C2×C32×C4—22no—
89C2×C32×C4Dic612yes—
90C2×C32×C4C4×D314yes—
91C2×C32×C4C4×D312yes—
92C2×C32×C4#6712yes—
93C2×C3×Dic3—32no—
94C2×C3×Dic3—34no—
95C2×C3×Dic3—32no—
96C2×C3×Dic3—34no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C23×C9—21no—
101C23×C9C23×C311yescharacteristic
102C2×C4×C9—24no—
103C2×C4×C9—24no—
104C2×C4×C9—22no—
105C2×C4×C9—22no—
106C2×C4×C9—22no—
107C2×C4×C9—24no—
108C2×C4×C9—22no—
109C2×C4×C9—22no—
110C2×C4×C9C2×C3×C414yes—
111C2×C4×C9C2×C3×C412yes—
112C2×C4×C9C3×D412yes—
113C2×C4×C9C3×Q812yes—
114C2×Dic9—22no—
115C2×Dic9—22no—
116C2×Dic9—22no—
117C2×Dic9C2×C3×C412yes—
118C2×Dic9C2×C3×C414yes—
119C2×Dic9C2×C3×C412yes—
120C2×Dic9C2×C3×C414yes—
121C4×C4⋊C4—91noSylow
122C2×C3×C9C2×C4211yescharacteristic
123C2×C3×C9C2×C4⋊C411yescharacteristic
124C2×C3×C9#11711yescharacteristic
125C2×C3×C9C4×D412yes—
126C2×C3×C9C4×Q812yes—
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4—91no—
130C22×C3×C4—91no—
131C22×C3×C4—92no—
132C22×C3×C4C2×D912yes—
133C22×C3×C4C2×D911yescharacteristic
134C22×C3×C4C2×C3×D312yes—
135C22×C3×C4C2×C3×D311yescharacteristic
136C22×Dic3—31no—
137C22×Dic3—31no—
138C22×Dic3—32no—
139C3×C4⋊C4—98no—
140C3×C42—24no—
141C3×C42—22no—
142C3×C42—22no—
143C3×C42—94no—
144C3×C42—182no—
145Dic3⋊C4—38no—
146C4×Dic3—34no—
147C4×Dic3—62no—
148C22×C32C2×Dic611yescharacteristic
149C22×C32C2×C4×D312yes—
150C22×C32C2×C4×D311yescharacteristic
151C22×C32#23512yes—
152C22×C32C2×#6711yescharacteristic
153C32×C4—24no—
154C32×C4Dic3⋊C414yes—
155C3×Dic3—34no—
156C3×Dic3—62no—
157C3×Dic3—64no—
158C22×C9—21no—
159C22×C9—21no—
160C22×C9—22no—
161C22×C9—22no—
162C22×C9—21no—
163C22×C9C22×C3×C411yescharacteristic
164C22×C9C22×C3×C412yes—
165C22×C9C2×C3×D411yescharacteristic
166C22×C9C2×C3×Q811yescharacteristic
167C22×C9C3×#4112yes—
168C4×C9—28no—
169C4×C9—24no—
170C4×C9—24no—
171C4×C9C3×C4⋊C414yescomplemented
172Dic9—22no—
173Dic9—24no—
174Dic9C3×C4214yescomplemented
175C2×C4⋊C4—92no—
176C2×C42—91no—
177C2×C42—91no—
178C2×C42C3×D911yescharacteristic
179#95—92no—
180C3×C9C4×C4⋊C411yescharacteristic, complemented, Sylow
181C23×C3—21no—
182C23×C3C22×D911yescharacteristic
183C23×C3C22×C3×D311yescharacteristic
184C2×C3×C4—24no—
185C2×C3×C4—24no—
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—22no—
194C2×C3×C4—92no—
195C2×C3×C4—94no—
196C2×C3×C4—92no—
197C2×C3×C4—94no—
198C2×C3×C4—182no—
199C2×C3×C4—182no—
200C2×C3×C4—182no—
201C2×C3×C4Dic1812yes—
202C2×C3×C4C4×D914yes—
203C2×C3×C4C4×D912yes—
204C2×C3×C4#61112yes—
205C2×C3×C4C3×Dic612yes—
206C2×C3×C4C3×C4×D314yes—
207C2×C3×C4C3×C4×D312yes—
208C2×C3×C4C3×#6712yes—
209C2×Dic3—32no—
210C2×Dic3—34no—
211C2×Dic3—32no—
212C2×Dic3—34no—
213C2×Dic3—62no—
214C2×Dic3—62no—
215C2×Dic3—62no—
216C2×C32C4×Dic612yes—
217C2×C32C42×D311yescharacteristic
218C2×C32#87211yescharacteristic
219C2×C32C2×Dic3⋊C411yescharacteristic
220C2×C32C4×#6712yes—
221C2×C9—21no—
222C2×C9—22no—
223C2×C9—22no—
224C2×C9—21no—
225C2×C9—21no—
226C2×C9C2×C3×C4211yescharacteristic
227C2×C9C2×C3×C4⋊C411yescharacteristic
228C2×C9C3×#11711yescharacteristic
229C2×C9C3×C4×D412yes—
230C2×C9C3×C4×Q812yes—
231C22×C4—91no—
232C22×C4—91no—
233C22×C4—92no—
234C22×C4C2×C3×D912yes—
235C22×C4C2×C3×D911yescharacteristic
236C4⋊C4—98no—
237C42—22no—
238C42—94no—
239C42—182no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—22no—
243C22×C3—22no—
244C22×C3—21no—
245C22×C3C2×Dic1811yescharacteristic
246C22×C3C2×C4×D912yes—
247C22×C3C2×C4×D911yescharacteristic
248C22×C3#369712yes—
249C22×C3C2×#61111yescharacteristic
250C22×C3C2×C3×Dic611yescharacteristic
251C22×C3C2×C3×C4×D311yescharacteristic
252C22×C3C2×C3×C4×D312yes—
253C22×C3C3×#23512yes—
254C22×C3C2×C3×#6711yescharacteristic
255C3×C4—28no—
256C3×C4—24no—
257C3×C4—24no—
258C3×C4—24no—
259C3×C4—94no—
260C3×C4—184no—
261C3×C4—182no—
262C3×C4Dic9⋊C414yesdirect product factor
263C3×C4C3×Dic3⋊C414yes—
264Dic3—34no—
265Dic3—64no—
266Dic3—62no—
267C32C4×Dic3⋊C411yescharacteristic
268C9—21no—
269C9C3×C4×C4⋊C411yescharacteristic, complemented
270C23C22×C3×D911yescharacteristic
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—92no—
274C2×C4—94no—
275C2×C4—92no—
276C2×C4—94no—
277C2×C4—182no—
278C2×C4—182no—
279C2×C4—182no—
280C2×C4C3×Dic1812yes—
281C2×C4C3×C4×D914yes—
282C2×C4C3×C4×D912yes—
283C2×C4C3×#61112yes—
284C2×C3—21no—
285C2×C3—22no—
286C2×C3—22no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3C4×Dic1812yes—
290C2×C3C42×D911yescharacteristic
291C2×C3#6343411yescharacteristic
292C2×C3C2×Dic9⋊C411yescharacteristic
293C2×C3C4×#61112yes—
294C2×C3C3×C4×Dic612yes—
295C2×C3C3×C42×D311yescharacteristic
296C2×C3C3×#87211yescharacteristic
297C2×C3C2×C3×Dic3⋊C411yescharacteristic
298C2×C3C3×C4×#6712yes—
299C22C2×C3×Dic1811yescharacteristic
300C22C2×C3×C4×D911yescharacteristic
301C22C2×C3×C4×D912yes—
302C22C3×#369712yes—
303C22C2×C3×#61111yescharacteristic
304C4—24no—
305C4—94no—
306C4—184no—
307C4—182no—
308C4C3×Dic9⋊C414yesdirect product factor
309C3—21no—
310C3C4×Dic9⋊C411yescharacteristic, direct product factor
311C3C3×C4×Dic3⋊C411yescharacteristic
312C2C3×C4×Dic1812yes—
313C2C3×C42×D911yescharacteristic
314C2C3×#6343411yescharacteristic
315C2C2×C3×Dic9⋊C411yescharacteristic
316C2C3×C4×#61112yes—
subgroupG/H
centerC22×C3×C4C2×D9
commutatorC2×C9C2×C3×C42
FrattiniC23×C3C22×C3×D3
FittingC2×C3×C42×C9C2
socle, prime radicalC23×C32C22×D3
cosocleC23×C9C23×C3
hypercenterC2×C3×C42D9
hypocenterC9C3×C4×C4⋊C4
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×C42

Complements

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

NH ≅ G/Ntype
7. C4×Dic9⋊C4C3direct (N × H)
14. C3×Dic9⋊C4C4direct (N × H)
16. C3×C4×Dic9C4semidirect
62. Dic9⋊C4C3×C4direct (N × H)
64. C4×Dic9C3×C4semidirect
71. C3×C4×C9C4⋊C4semidirect
74. C3×Dic9C42semidirect
171. C4×C9C3×C4⋊C4semidirect
174. Dic9C3×C42semidirect
180. C3×C9C4×C4⋊C4semidirect
262. C3×C4Dic9⋊C4direct (N × H)
269. C9C3×C4×C4⋊C4semidirect
308. C4C3×Dic9⋊C4direct (N × H)
310. C3C4×Dic9⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C42×C92
C2×C3×Dic9⋊C42
C2×C3×C4×Dic92
C3×C9⋊#952
C3×C4×Dic3⋊C43
C4×Dic9⋊C43