Finite Group Databasebeta

← C2×#63454⋊C3 #13152454

Subgroups — C2×#63454⋊C3 #13152454

Order 1728.

1036 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×C4×AGL(1,ℤ/9)C211yesmaximal, characteristic
2C2×C4⋊C4×3−1+2C211yesmaximal, characteristic, complemented
3C2×(C2×AGL(1,ℤ/9))⋊C4C212yesmaximal
4C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
5C2×C4×3−1+2⋊C4C212yesmaximal, complemented
6#63454⋊C3C218yesmaximal, direct product factor
7C2×C3×#892—31nomaximal
8C2×#63454C311yesmaximal, characteristic, complemented
9C23×AGL(1,ℤ/9)C2211yescharacteristic
10C22×C4×3−1+2C2212yes—
11C22×C4×3−1+2C2211yescharacteristic
12C22×3−1+2⋊C4C2212yes—
13C22×3−1+2⋊C4C2211yescharacteristic
14C2×C4×AGL(1,ℤ/9)C418yescomplemented
15C2×C4×AGL(1,ℤ/9)C2214yes—
16C4⋊C4×3−1+2C2214yescomplemented
17(C2×AGL(1,ℤ/9))⋊C4C2218yes—
18C4⋊(3−1+2⋊C4)C2214yescomplemented
19C4×3−1+2⋊C4C2218yescomplemented
20C22×C3×C4×D3—31no—
21C2×C32×C4⋊C4D311yescharacteristic
22C2×C3×(C2×D3)⋊C4—32no—
23C2×C3×C4⋊Dic3—31no—
24C2×C3×C4×Dic3—32no—
25C3×#892—38no—
26C22×C4×D9C2×C311yescharacteristic
27C2×C9×C4⋊C4—21no—
28C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
29C2×(C2×D9)⋊C4C2×C312yes—
30C2×C4⋊Dic9C2×C311yescharacteristic, complemented
31C2×C4×Dic9C2×C312yescomplemented
32#63454C2×C318yescomplemented
33C23×3−1+2C2311yescharacteristic
34C22×AGL(1,ℤ/9)—22no—
35C22×AGL(1,ℤ/9)—22no—
36C22×AGL(1,ℤ/9)C2×C414yes—
37C22×AGL(1,ℤ/9)C2312yes—
38C2×C4×3−1+2—24no—
39C2×C4×3−1+2C2×C414yes—
40C2×C4×3−1+2C2314yes—
41C2×C4×3−1+2C2312yes—
42C2×3−1+2⋊C4—24no—
43C2×3−1+2⋊C4C2×C414yes—
44C2×3−1+2⋊C4C2314yes—
45C2×3−1+2⋊C4C2312yes—
46C4×AGL(1,ℤ/9)C2×C4116yescomplemented
47C2×C3×#117—91no—
48C2×#892—31no—
49C23×C3×D3—31no—
50C22×C32×C4C2×D312yes—
51C22×C32×C4C2×D311yescharacteristic
52C22×C3×Dic3—31no—
53C22×C3×Dic3—32no—
54C2×C3×C4×D3—38no—
55C2×C3×C4×D3—34no—
56C23×D9C22×C311yescharacteristic
57C32×C4⋊C4C2×D314yes—
58C3×(C2×D3)⋊C4—38no—
59C3×C4⋊Dic3—34no—
60C3×C4×Dic3—38no—
61C22×C4×C9—21no—
62C22×C4×C9—22no—
63C22×C4×C9C22×C312yes—
64C22×C4×C9C22×C311yescharacteristic
65C22×Dic9C22×C312yes—
66C22×Dic9C22×C311yescharacteristic
67C2×C4×D9C3×C418yescomplemented
68C2×C4×D9C22×C314yes—
69C9×C4⋊C4—24no—
70C9×C4⋊C4C22×C314yescomplemented
71(C2×D9)⋊C4C22×C318yes—
72C4⋊Dic9C22×C314yescomplemented
73C4×Dic9C22×C318yescomplemented
74C22×3−1+2C22×C412yes—
75C22×3−1+2#4112yes—
76C22×3−1+2#4112yes—
77C22×3−1+2C2411yescharacteristic
78C2×AGL(1,ℤ/9)—28no—
79C2×AGL(1,ℤ/9)—22no—
80C2×AGL(1,ℤ/9)—22no—
81C2×AGL(1,ℤ/9)C22×C414yes—
82C4×3−1+2—24no—
83C4×3−1+2C22×C414yes—
843−1+2⋊C4—24no—
853−1+2⋊C4C22×C414yes—
86C23×C3×C4—91no—
87C22×C4×D3—31no—
88C3×#117—98no—
89C2×C3×C4⋊C4—31no—
90C2×C3×C4⋊C4—91no—
91C2×C3×C4⋊C4C3×D311yescharacteristic
92C2×C3×C22⋊C4—92no—
93C2×C3×C42—92no—
94C2×(C2×D3)⋊C4—32no—
95C2×C4⋊Dic3—31no—
96C2×C4×Dic3—32no—
97#892—38no—
98C23×C32C22×D311yescharacteristic
99C22×C3×D3—34no—
100C22×C3×D3—32no—
101C22×C3×D3—62no—
102C22×C3×D3—62no—
103C2×C32×C4—24no—
104C2×C32×C4C4×D314yes—
105C2×C32×C4C22×D314yes—
106C2×C32×C4C22×D312yes—
107C2×C3×Dic3—34no—
108C2×C3×Dic3—32no—
109C2×C3×Dic3—34no—
110C2×C3×Dic3—64no—
111C3×C4×D3—316no—
112C23×C9—21no—
113C23×C9C23×C311yescharacteristic
114C22×D9—22no—
115C22×D9—22no—
116C22×D9C2×C3×C414yes—
117C22×D9C23×C312yes—
118C2×C4×C9—24no—
119C2×C4×C9—24no—
120C2×C4×C9—22no—
121C2×C4×C9—24no—
122C2×C4×C9—44no—
123C2×C4×C9C2×C3×C414yes—
124C2×C4×C9C23×C314yes—
125C2×C4×C9C23×C312yes—
126C2×Dic9—24no—
127C2×Dic9C2×C3×C414yes—
128C2×Dic9C23×C314yes—
129C2×Dic9C23×C312yes—
130C4×D9C2×C3×C4116yescomplemented
131C2×#117—91noSylow
132C2×3−1+2#11714yescomplemented
133C2×3−1+2C23×C411yescharacteristic
134C2×3−1+2C2×#4111yescharacteristic
135C2×3−1+2C2×#4111yescharacteristic
136AGL(1,ℤ/9)—24no—
137C24×C3—91no—
138C23×D3—31no—
139C22×C3×C4—31no—
140C22×C3×C4—32no—
141C22×C3×C4—94no—
142C22×C3×C4—98no—
143C22×C3×C4—91no—
144C22×C3×C4—92no—
145C22×C3×C4C2×C3×D312yes—
146C22×C3×C4C2×C3×D311yescharacteristic
147C22×Dic3—31no—
148C22×Dic3—32no—
149C2×C4×D3—38no—
150C2×C4×D3—34no—
151C3×C4⋊C4—34no—
152C3×C4⋊C4—94no—
153C3×C4⋊C4C2×C3×D314yes—
154C3×C22⋊C4—98no—
155C3×C42—98no—
156(C2×D3)⋊C4—38no—
157C4⋊Dic3—34no—
158C4×Dic3—38no—
159C22×C32C2×C4×D312yes—
160C22×C32#23712yes—
161C22×C32#23912yes—
162C22×C32C23×D311yescharacteristic
163C2×C3×D3—34no—
164C2×C3×D3—68no—
165C2×C3×D3—62no—
166C2×C3×D3—62no—
167C32×C4—24no—
168C32×C4C2×C4×D314yes—
169C3×Dic3—34no—
170C3×Dic3—64no—
171C22×C9—21no—
172C22×C9—22no—
173C22×C9—22no—
174C22×C9—22no—
175C22×C9C22×C3×C412yes—
176C22×C9C3×#4112yes—
177C22×C9C3×#4112yes—
178C22×C9C24×C311yescharacteristic
179C2×D9—28no—
180C2×D9—22no—
181C2×D9—22no—
182C2×D9C22×C3×C414yes—
183C4×C9—24no—
184C4×C9—24no—
185C4×C9—44no—
186C4×C9C22×C3×C414yes—
187Dic9—24no—
188Dic9C22×C3×C414yes—
189C23×C4—91no—
190#117—98no—
191C2×C4⋊C4—91no—
192C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
193C2×C22⋊C4—92no—
194C2×C42—92no—
1953−1+2C2×#11711yescharacteristic, complemented, Sylow
196C23×C3—31no—
197C23×C3—92no—
198C23×C3—94no—
199C23×C3—182no—
200C23×C3—182no—
201C23×C3C22×C3×D311yescharacteristic
202C22×D3—32no—
203C22×D3—34no—
204C22×D3—62no—
205C22×D3—62no—
206C2×C3×C4—24no—
207C2×C3×C4—34no—
208C2×C3×C4—32no—
209C2×C3×C4—34no—
210C2×C3×C4—64no—
211C2×C3×C4—916no—
212C2×C3×C4—92no—
213C2×C3×C4—94no—
214C2×C3×C4—94no—
215C2×C3×C4—184no—
216C2×C3×C4C3×C4×D314yes—
217C2×C3×C4C22×C3×D314yes—
218C2×C3×C4C22×C3×D312yes—
219C2×Dic3—32no—
220C2×Dic3—34no—
221C2×Dic3—34no—
222C2×Dic3—64no—
223C4×D3—316no—
224C2×C32#89214yes—
225C2×C32C22×C4×D311yescharacteristic
226C2×C32C2×#23711yescharacteristic
227C2×C32C2×#23911yescharacteristic
228C3×D3—64no—
229C2×C9—24no—
230C2×C9—21no—
231C2×C9—21no—
232C2×C9—21no—
233C2×C9C3×#11714yescomplemented
234C2×C9C23×C3×C411yescharacteristic
235C2×C9C2×C3×#4111yescharacteristic
236C2×C9C2×C3×#4111yescharacteristic
237D9—24no—
238C24—91no—
239C22×C4—91no—
240C22×C4—98no—
241C22×C4—94no—
242C22×C4—92no—
243C22×C4C2×AGL(1,ℤ/9)12yes—
244C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
245C4⋊C4—94no—
246C4⋊C4C2×AGL(1,ℤ/9)14yescomplemented
247C22⋊C4—98no—
248C42—98no—
249C22×C3—31no—
250C22×C3—32no—
251C22×C3—32no—
252C22×C3—32no—
253C22×C3—94no—
254C22×C3—188no—
255C22×C3—182no—
256C22×C3—182no—
257C22×C3C2×C3×C4×D312yes—
258C22×C3C3×#23712yes—
259C22×C3C3×#23912yes—
260C22×C3C23×C3×D311yescharacteristic
261C2×D3—34no—
262C2×D3—68no—
263C2×D3—62no—
264C2×D3—62no—
265C3×C4—24no—
266C3×C4—34no—
267C3×C4—64no—
268C3×C4—94no—
269C3×C4—184no—
270C3×C4C2×C3×C4×D314yes—
271Dic3—34no—
272Dic3—64no—
273C32C2×#89211yescharacteristic
274C9—21no—
275C9C2×C3×#11711yescharacteristic, complemented
276C23—92no—
277C23—94no—
278C23—182no—
279C23—182no—
280C23C22×AGL(1,ℤ/9)11yescharacteristic
281C2×C4—24no—
282C2×C4—92no—
283C2×C4—94no—
284C2×C4—916no—
285C2×C4—94no—
286C2×C4—184no—
287C2×C4C4×AGL(1,ℤ/9)14yes—
288C2×C4C22×AGL(1,ℤ/9)14yes—
289C2×C4C22×AGL(1,ℤ/9)12yes—
290C2×C3—34no—
291C2×C3—31no—
292C2×C3—31no—
293C2×C3—31no—
294C2×C3—184no—
295C2×C3C3×#89214yes—
296C2×C3C22×C3×C4×D311yescharacteristic
297C2×C3C2×C3×#23711yescharacteristic
298C2×C3C2×C3×#23911yescharacteristic
299D3—64no—
300C22—94no—
301C22—188no—
302C22—182no—
303C22—182no—
304C22C2×C4×AGL(1,ℤ/9)12yes—
305C22#3699⋊C312yes—
306C22#8872412yes—
307C22C23×AGL(1,ℤ/9)11yescharacteristic
308C4—24no—
309C4—94no—
310C4—184no—
311C4C2×C4×AGL(1,ℤ/9)14yes—
312C3—31no—
313C3C2×C3×#89211yescharacteristic
314C2—184no—
315C2#63454⋊C314yesdirect product factor
316C2C22×C4×AGL(1,ℤ/9)11yescharacteristic
317C2C2×#3699⋊C311yescharacteristic
318C2C2×#8872411yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutatorC2×C9C23×C3×C4
FrattiniC22×C3C23×C3×D3
FittingC2×C4⋊C4×3−1+2C2
socleC23×C3C22×C3×D3
prime radicalC23×AGL(1,ℤ/9)C22
cosocleC22×C9C24×C3
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C2×C3×#117
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×3−1+2C2semidirect
4. C2×C4⋊(3−1+2⋊C4)C2semidirect
5. C2×C4×3−1+2⋊C4C2semidirect
6. #63454⋊C3C2direct (N × H)
8. C2×#63454C3semidirect
14. C2×C4×AGL(1,ℤ/9)C4semidirect
16. C4⋊C4×3−1+2C22semidirect
18. C4⋊(3−1+2⋊C4)C22semidirect
19. C4×3−1+2⋊C4C22semidirect
28. C2×C9×C4⋊C4C2×C3semidirect
30. C2×C4⋊Dic9C2×C3semidirect
31. C2×C4×Dic9C2×C3semidirect
32. #63454C2×C3semidirect
46. C4×AGL(1,ℤ/9)C2×C4semidirect
67. C2×C4×D9C3×C4semidirect
70. C9×C4⋊C4C22×C3semidirect
72. C4⋊Dic9C22×C3semidirect
73. C4×Dic9C22×C3semidirect
130. C4×D9C2×C3×C4semidirect
132. C2×3−1+2#117semidirect
192. C2×C4⋊C4AGL(1,ℤ/9)semidirect
195. 3−1+2C2×#117semidirect
233. C2×C9C3×#117semidirect
246. C4⋊C4C2×AGL(1,ℤ/9)semidirect
275. C9C2×C3×#117semidirect
315. C2#63454⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×AGL(1,ℤ/9)2
C2×C4⋊C4×3−1+22
C2×(C2×AGL(1,ℤ/9))⋊C42
C2×C4⋊(3−1+2⋊C4)2
C2×C4×3−1+2⋊C42
#63454⋊C32
C2×C3×#8923
C2×#634543