Finite Group Databasebeta

← C3×C4×#3676 #13143513

Subgroups — C3×C4×#3676 #13143513

Order 1728.

1010 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4×C9C211yesmaximal, characteristic
2C2×C3×#3676C212yesmaximal
3C2×C3×C4×Dic9C212yesmaximal, complemented
4C3×C9⋊#95C212yesmaximal, complemented
5C3×C4×#217—31nomaximal
6C4×#3676C311yesmaximal, characteristic, direct product factor
7C24×C3×C9C2211yescharacteristic
8C22×C3×C4×C9—24no—
9C22×C3×C4×C9C414yescomplemented
10C22×C3×C4×C9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3×Dic9C2214yes—
13C3×#3676C418yesdirect product factor
14C3×C4×Dic9—24no—
15C23×C32×C4D311yescharacteristic
16C2×C3×#217—32no—
17C2×C3×C4×Dic3—32no—
18C23×C4×C9—21no—
19C23×C4×C9C2×C311yescharacteristic
20C3×C3⋊#95—32no—
21C2×#3676C2×C312yes—
22C2×C4×Dic9C2×C312yescomplemented
23C9⋊#95C2×C312yescomplemented
24C23×C3×C9—22no—
25C23×C3×C9—22no—
26C23×C3×C9C2×C414yes—
27C23×C3×C9C2×C412yes—
28C23×C3×C9C2311yescharacteristic
29C2×C3×C4×C9—22no—
30C2×C3×C4×C9—28no—
31C2×C3×C4×C9C2×C412yes—
32C2×C3×C4×C9C2×C412yes—
33C2×C3×C4×C9D414yes—
34C2×C3×Dic9—24no—
35C2×C3×Dic9—24no—
36C2×C3×Dic9C2×C418yescomplemented
37C3×C4×C22⋊C4—91no—
38C4×#217—31no—
39C24×C32C2×D311yescharacteristic
40C22×C32×C4—24no—
41C22×C32×C4Dic314yes—
42C22×C32×C4C2×D311yescharacteristic
43C22×C32×C4C2×D311yescharacteristic
44C22×C3×Dic3—34no—
45C24×C9—21no—
46C24×C9C22×C311yescharacteristic
47C3×#217—38no—
48C3×C4×Dic3—64no—
49C22×C4×C9—28no—
50C22×C4×C9—24no—
51C22×C4×C9—21no—
52C22×C4×C9—21no—
53C22×C4×C9—24no—
54C22×C4×C9C3×C414yescomplemented
55C22×C4×C9C22×C311yescharacteristic
56C22×C4×C9C22×C311yescharacteristic
57C22×Dic9C22×C314yes—
58#3676C3×C418yesdirect product factor
59C4×Dic9—24no—
60C22×C3×C9—22no—
61C22×C3×C9—22no—
62C22×C3×C9—28no—
63C22×C3×C9C4214yescomplemented
64C22×C3×C9C22×C411yescharacteristic
65C22×C3×C9C22×C412yes—
66C22×C3×C9C2×D412yes—
67C22×C3×C9#4112yes—
68C3×C4×C9—22no—
69C3×C4×C9C22⋊C414yescomplemented
70C3×Dic9—28no—
71C23×C3×C4—21no—
72C23×C3×C4D911yescharacteristic
73C23×C3×C4C3×D311yescharacteristic
74C2×C3×C22⋊C4—92no—
75C2×C3×C42—92no—
76C2×#217—32no—
77C2×C4×Dic3—32no—
78C3×#95—92no—
79C3⋊#95—32no—
80C23×C32—22no—
81C23×C32—22no—
82C23×C32C4×D314yes—
83C23×C32C2×Dic312yes—
84C23×C32C22×D311yescharacteristic
85C2×C32×C4—22no—
86C2×C32×C4—28no—
87C2×C32×C4C2×Dic312yes—
88C2×C32×C4C2×Dic312yes—
89C2×C32×C4#6714yes—
90C2×C3×Dic3—38no—
91C2×C3×Dic3—64no—
92C2×C3×Dic3—64no—
93C23×C9—24no—
94C23×C9—24no—
95C23×C9—22no—
96C23×C9—22no—
97C23×C9—24no—
98C23×C9—22no—
99C23×C9—21no—
100C23×C9C2×C3×C414yes—
101C23×C9C2×C3×C412yes—
102C23×C9C23×C311yescharacteristic
103C2×C4×C9—24no—
104C2×C4×C9—216no—
105C2×C4×C9—22no—
106C2×C4×C9—28no—
107C2×C4×C9—22no—
108C2×C4×C9—22no—
109C2×C4×C9—24no—
110C2×C4×C9C2×C3×C412yes—
111C2×C4×C9C2×C3×C412yes—
112C2×C4×C9C3×D414yes—
113C2×Dic9—24no—
114C2×Dic9—24no—
115C2×Dic9C2×C3×C418yescomplemented
116C4×C22⋊C4—91noSylow
117C2×C3×C9—24no—
118C2×C3×C9C2×C4211yescharacteristic
119C2×C3×C9C2×C22⋊C411yescharacteristic
120C2×C3×C9#11711yescharacteristic
121C2×C3×C9C4×D414yes—
122C24×C3—21no—
123C24×C3C2×D911yescharacteristic
124C24×C3C2×C3×D311yescharacteristic
125C22×C3×C4—28no—
126C22×C3×C4—24no—
127C22×C3×C4—24no—
128C22×C3×C4—24no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—94no—
132C22×C3×C4Dic914yescomplemented
133C22×C3×C4C2×D911yescharacteristic
134C22×C3×C4C2×D911yescharacteristic
135C22×C3×C4C3×Dic314yes—
136C22×C3×C4C2×C3×D311yescharacteristic
137C22×C3×C4C2×C3×D311yescharacteristic
138C22×Dic3—34no—
139C3×C22⋊C4—98no—
140C3×C42—184no—
141#217—38no—
142C4×Dic3—64no—
143C22×C32—22no—
144C22×C32—28no—
145C22×C32—22no—
146C22×C32C4×Dic314yes—
147C22×C32C2×C4×D312yes—
148C22×C32#23512yes—
149C22×C32C22×Dic311yescharacteristic
150C22×C32C2×#6712yes—
151C32×C4—22no—
152C32×C4#21714yes—
153C3×Dic3—68no—
154C22×C9—24no—
155C22×C9—24no—
156C22×C9—216no—
157C22×C9—22no—
158C22×C9—22no—
159C22×C9—28no—
160C22×C9—24no—
161C22×C9—22no—
162C22×C9—22no—
163C22×C9—21no—
164C22×C9—22no—
165C22×C9C3×C4214yescomplemented
166C22×C9C22×C3×C412yes—
167C22×C9C22×C3×C411yescharacteristic
168C22×C9C2×C3×D412yes—
169C22×C9C3×#4112yes—
170C4×C9—24no—
171C4×C9—22no—
172C4×C9—24no—
173C4×C9C3×C22⋊C414yescomplemented
174Dic9—28no—
175C23×C4C3×D911yescharacteristic
176C2×C22⋊C4—92no—
177C2×C42—92no—
178#95—92no—
179C3×C9C4×C22⋊C411yescharacteristic, complemented, Sylow
180C23×C3—24no—
181C23×C3—24no—
182C23×C3—22no—
183C23×C3—22no—
184C23×C3—22no—
185C23×C3—22no—
186C23×C3—24no—
187C23×C3—22no—
188C23×C3—21no—
189C23×C3C4×D914yes—
190C23×C3C2×Dic912yes—
191C23×C3C22×D911yescharacteristic
192C23×C3C3×C4×D314yes—
193C23×C3C2×C3×Dic312yes—
194C23×C3C22×C3×D311yescharacteristic
195C2×C3×C4—24no—
196C2×C3×C4—216no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—28no—
200C2×C3×C4—28no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—24no—
204C2×C3×C4—98no—
205C2×C3×C4—184no—
206C2×C3×C4—184no—
207C2×C3×C4C2×Dic912yes—
208C2×C3×C4C2×Dic912yes—
209C2×C3×C4#61114yes—
210C2×C3×C4C2×C3×Dic312yes—
211C2×C3×C4C2×C3×Dic312yes—
212C2×C3×C4C3×#6714yes—
213C2×Dic3—38no—
214C2×Dic3—64no—
215C2×Dic3—64no—
216C2×C32—24no—
217C2×C32C2×C4×Dic311yescharacteristic
218C2×C32#92611yescharacteristic
219C2×C32C4×#6714yes—
220C2×C32C2×#21711yescharacteristic
221C2×C9—28no—
222C2×C9—24no—
223C2×C9—21no—
224C2×C9—21no—
225C2×C9—24no—
226C2×C9—21no—
227C2×C9C2×C3×C4211yescharacteristic
228C2×C9C2×C3×C22⋊C411yescharacteristic
229C2×C9C3×#11711yescharacteristic
230C2×C9C3×C4×D414yes—
231C24C2×C3×D911yescharacteristic
232C22×C4—24no—
233C22×C4—94no—
234C22×C4C3×Dic914yescomplemented
235C22×C4C2×C3×D911yescharacteristic
236C22×C4C2×C3×D911yescharacteristic
237C22⋊C4—98no—
238C42—184no—
239C22×C3—24no—
240C22×C3—24no—
241C22×C3—216no—
242C22×C3—22no—
243C22×C3—28no—
244C22×C3—22no—
245C22×C3—22no—
246C22×C3—28no—
247C22×C3—22no—
248C22×C3—22no—
249C22×C3—22no—
250C22×C3—21no—
251C22×C3—22no—
252C22×C3—24no—
253C22×C3C4×Dic914yescomplemented
254C22×C3C2×C4×D912yes—
255C22×C3#369712yes—
256C22×C3C22×Dic911yescharacteristic
257C22×C3C2×#61112yes—
258C22×C3C3×C4×Dic314yes—
259C22×C3C2×C3×C4×D312yes—
260C22×C3C3×#23512yes—
261C22×C3C22×C3×Dic311yescharacteristic
262C22×C3C2×C3×#6712yes—
263C3×C4—24no—
264C3×C4—22no—
265C3×C4—22no—
266C3×C4—24no—
267C3×C4—188no—
268C3×C4#367614yesdirect product factor
269C3×C4C3×#21714yes—
270Dic3—68no—
271C32C4×#21711yescharacteristic
272C9—21no—
273C9C3×C4×C22⋊C411yescharacteristic, complemented
274C23—22no—
275C23—22no—
276C23C3×C4×D914yes—
277C23C2×C3×Dic912yes—
278C23C22×C3×D911yescharacteristic
279C2×C4—22no—
280C2×C4—28no—
281C2×C4—98no—
282C2×C4—184no—
283C2×C4—184no—
284C2×C4C2×C3×Dic912yes—
285C2×C4C2×C3×Dic912yes—
286C2×C4C3×#61114yes—
287C2×C3—28no—
288C2×C3—24no—
289C2×C3—24no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—24no—
293C2×C3—21no—
294C2×C3C2×C4×Dic911yescharacteristic
295C2×C3#6348811yescharacteristic
296C2×C3C4×#61114yes—
297C2×C3C2×#367611yescharacteristic
298C2×C3C2×C3×C4×Dic311yescharacteristic
299C2×C3C3×#92611yescharacteristic
300C2×C3C3×C4×#6714yes—
301C2×C3C2×C3×#21711yescharacteristic
302C22—22no—
303C22—28no—
304C22—22no—
305C22C3×C4×Dic914yescomplemented
306C22C2×C3×C4×D912yes—
307C22C3×#369712yes—
308C22C22×C3×Dic911yescharacteristic
309C22C2×C3×#61112yes—
310C4—22no—
311C4—188no—
312C4C3×#367614yesdirect product factor
313C3—21no—
314C3C4×#367611yescharacteristic, direct product factor
315C3C3×C4×#21711yescharacteristic
316C2—24no—
317C2C2×C3×C4×Dic911yescharacteristic
318C2C3×#6348811yescharacteristic
319C2C3×C4×#61114yes—
320C2C2×C3×#367611yescharacteristic
subgroupG/H
centerC22×C3×C4C2×D9
commutatorC2×C9C2×C3×C42
FrattiniC23×C3C22×C3×D3
FittingC23×C3×C4×C9C2
socleC23×C32C22×D3
prime radicalC24×C32C2×D3
cosocleC23×C9C23×C3
hypercenterC23×C3×C4D9
hypocenterC9C3×C4×C22⋊C4
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3×C4 ◁ C23×C3×C4

Complements

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

NH ≅ G/Ntype
3. C2×C3×C4×Dic9C2semidirect
4. C3×C9⋊#95C2semidirect
6. C4×#3676C3direct (N × H)
9. C22×C3×C4×C9C4semidirect
13. C3×#3676C4direct (N × H)
22. C2×C4×Dic9C2×C3semidirect
23. C9⋊#95C2×C3semidirect
36. C2×C3×Dic9C2×C4semidirect
54. C22×C4×C9C3×C4semidirect
58. #3676C3×C4direct (N × H)
63. C22×C3×C9C42semidirect
69. C3×C4×C9C22⋊C4semidirect
115. C2×Dic9C2×C3×C4semidirect
132. C22×C3×C4Dic9semidirect
165. C22×C9C3×C42semidirect
173. C4×C9C3×C22⋊C4semidirect
179. C3×C9C4×C22⋊C4semidirect
234. C22×C4C3×Dic9semidirect
253. C22×C3C4×Dic9semidirect
268. C3×C4#3676direct (N × H)
273. C9C3×C4×C22⋊C4semidirect
305. C22C3×C4×Dic9semidirect
312. C4C3×#3676direct (N × H)
314. C3C4×#3676direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4×C92
C2×C3×#36762
C2×C3×C4×Dic92
C3×C9⋊#952
C3×C4×#2173
C4×#36763