Finite Group Databasebeta

← C2×C4⋊C4×AGL(1,ℤ/9) #13152453

Subgroups — C2×C4⋊C4×AGL(1,ℤ/9) #13152453

Order 1728.

1300 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×AGL(1,ℤ/9)C212yesmaximal
2C22×C4×AGL(1,ℤ/9)C211yesmaximal, characteristic
3C2×C4⋊C4×3−1+2C211yesmaximal, characteristic, complemented
4C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
5C2×(3−1+2⋊C4)⋊C4C212yesmaximal, complemented
6C4⋊C4×AGL(1,ℤ/9)C218yesmaximal, direct product factor
7C2×C3×D3×C4⋊C4—31nomaximal
8C2×C4⋊C4×D9C311yesmaximal, 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)—28no—
15C2×C4×AGL(1,ℤ/9)C418yescomplemented
16C2×C4×AGL(1,ℤ/9)C2218yes—
17C2×C4×AGL(1,ℤ/9)C2214yes—
18C4⋊C4×3−1+2C2214yescomplemented
19C4⋊(3−1+2⋊C4)C2214yescomplemented
20(3−1+2⋊C4)⋊C4C2218yescomplemented
21C22×C3×C4×D3—31no—
22C22×C3×C4×D3—32no—
23C2×C32×C4⋊C4D311yescharacteristic
24C2×C3×C4⋊Dic3—31no—
25C2×C3×Dic3⋊C4—32no—
26C3×D3×C4⋊C4—38no—
27C22×C4×D9C2×C312yes—
28C22×C4×D9C2×C311yescharacteristic
29C2×C9×C4⋊C4—21no—
30C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
31C2×C4⋊Dic9C2×C311yescharacteristic, complemented
32C2×Dic9⋊C4C2×C312yescomplemented
33C4⋊C4×D9C2×C318yescomplemented
34C23×3−1+2C2311yescharacteristic
35C22×AGL(1,ℤ/9)C2×C414yes—
36C22×AGL(1,ℤ/9)D414yes—
37C22×AGL(1,ℤ/9)Q814yes—
38C22×AGL(1,ℤ/9)C2312yes—
39C2×C4×3−1+2—24no—
40C2×C4×3−1+2C2×C414yescomplemented
41C2×C4×3−1+2C2314yes—
42C2×C4×3−1+2C2312yes—
43C2×3−1+2⋊C4—24no—
44C2×3−1+2⋊C4C2×C414yescomplemented
45C2×3−1+2⋊C4C2314yes—
46C2×3−1+2⋊C4C2312yes—
47C4×AGL(1,ℤ/9)—216no—
48C4×AGL(1,ℤ/9)C2×C4116yescomplemented
49C22×C3×C4⋊C4—91no—
50C2×D3×C4⋊C4—31no—
51C23×C3×D3—31no—
52C22×C32×C4C2×D312yes—
53C22×C32×C4C2×D311yescharacteristic
54C22×C3×Dic3—31no—
55C22×C3×Dic3—32no—
56C2×C3×C4×D3—38no—
57C2×C3×C4×D3—34no—
58C2×C3×C4×D3—38no—
59C2×C3×C4×D3—68no—
60C23×D9C22×C311yescharacteristic
61C32×C4⋊C4C2×D314yes—
62C3×C4⋊Dic3—34no—
63C3×Dic3⋊C4—38no—
64C22×C4×C9—21no—
65C22×C4×C9—22no—
66C22×C4×C9C22×C312yes—
67C22×C4×C9C22×C311yescharacteristic
68C22×Dic9C22×C312yes—
69C22×Dic9C22×C311yescharacteristic
70C2×C4×D9—28no—
71C2×C4×D9C3×C418yescomplemented
72C2×C4×D9C22×C318yes—
73C2×C4×D9C22×C314yes—
74C9×C4⋊C4—24no—
75C9×C4⋊C4C22×C314yescomplemented
76C4⋊Dic9C22×C314yescomplemented
77Dic9⋊C4C22×C318yescomplemented
78C22×3−1+2C22×C412yes—
79C22×3−1+2C2×D412yes—
80C22×3−1+2C2×Q812yes—
81C22×3−1+2C2411yescharacteristic
82C2×AGL(1,ℤ/9)C4⋊C4116yesdirect product factor
83C2×AGL(1,ℤ/9)C22×C414yes—
84C2×AGL(1,ℤ/9)C2×D414yes—
85C2×AGL(1,ℤ/9)C2×Q814yes—
86C4×3−1+2—24no—
87C4×3−1+2C22×C414yescomplemented
883−1+2⋊C4—24no—
893−1+2⋊C4C22×C414yescomplemented
90C23×C3×C4—91no—
91C23×C3×C4—92no—
92C22×C4×D3—31no—
93C22×C4×D3—32no—
94C2×C3×C4⋊C4—31no—
95C2×C3×C4⋊C4—98no—
96C2×C3×C4⋊C4—92no—
97C2×C3×C4⋊C4—91no—
98C2×C3×C4⋊C4C3×D311yescharacteristic
99C2×C4⋊Dic3—31no—
100C2×Dic3⋊C4—32no—
101D3×C4⋊C4—38no—
102C23×C32C22×D311yescharacteristic
103C22×C3×D3—34no—
104C22×C3×D3—34no—
105C22×C3×D3—34no—
106C22×C3×D3—32no—
107C2×C32×C4—24no—
108C2×C32×C4C4×D314yes—
109C2×C32×C4C22×D314yes—
110C2×C32×C4C22×D312yes—
111C2×C3×Dic3—34no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—34no—
114C2×C3×Dic3—64no—
115C3×C4×D3—316no—
116C3×C4×D3—616no—
117C23×C9—21no—
118C23×C9C23×C311yescharacteristic
119C22×D9C2×C3×C414yes—
120C22×D9C3×D414yes—
121C22×D9C3×Q814yes—
122C22×D9C23×C312yes—
123C2×C4×C9—24no—
124C2×C4×C9—24no—
125C2×C4×C9—22no—
126C2×C4×C9—24no—
127C2×C4×C9—44no—
128C2×C4×C9C2×C3×C414yescomplemented
129C2×C4×C9C23×C314yes—
130C2×C4×C9C23×C312yes—
131C2×Dic9—24no—
132C2×Dic9C2×C3×C414yescomplemented
133C2×Dic9C23×C314yes—
134C2×Dic9C23×C312yes—
135C4×D9—216no—
136C4×D9C2×C3×C4116yescomplemented
137C22×C4⋊C4—91noSylow
138C2×3−1+2C2×C4⋊C414yescomplemented
139C2×3−1+2C23×C411yescharacteristic
140C2×3−1+2C22×D411yescharacteristic
141C2×3−1+2C22×Q811yescharacteristic
142AGL(1,ℤ/9)C2×C4⋊C418yesdirect product factor
143C24×C3—91no—
144C23×D3—31no—
145C22×C3×C4—31no—
146C22×C3×C4—32no—
147C22×C3×C4—94no—
148C22×C3×C4—98no—
149C22×C3×C4—98no—
150C22×C3×C4—91no—
151C22×C3×C4—92no—
152C22×C3×C4—188no—
153C22×C3×C4C2×C3×D312yes—
154C22×C3×C4C2×C3×D311yescharacteristic
155C22×Dic3—31no—
156C22×Dic3—32no—
157C2×C4×D3—38no—
158C2×C4×D3—34no—
159C2×C4×D3—38no—
160C2×C4×D3—68no—
161C3×C4⋊C4—34no—
162C3×C4⋊C4—98no—
163C3×C4⋊C4—94no—
164C3×C4⋊C4C2×C3×D314yes—
165C4⋊Dic3—34no—
166Dic3⋊C4—38no—
167C22×C32C2×C4×D312yes—
168C22×C32D3×D412yes—
169C22×C32D3×Q812yes—
170C22×C32C23×D311yescharacteristic
171C2×C3×D3—316no—
172C2×C3×D3—34no—
173C2×C3×D3—34no—
174C2×C3×D3—34no—
175C32×C4—24no—
176C32×C4C2×C4×D314yes—
177C3×Dic3—34no—
178C3×Dic3—64no—
179C22×C9—21no—
180C22×C9—22no—
181C22×C9—22no—
182C22×C9—22no—
183C22×C9C22×C3×C412yes—
184C22×C9C2×C3×D412yes—
185C22×C9C2×C3×Q812yes—
186C22×C9C24×C311yescharacteristic
187C2×D9C3×C4⋊C4116yescomplemented
188C2×D9C22×C3×C414yes—
189C2×D9C2×C3×D414yes—
190C2×D9C2×C3×Q814yes—
191C4×C9—24no—
192C4×C9—24no—
193C4×C9—44no—
194C4×C9C22×C3×C414yescomplemented
195Dic9—24no—
196Dic9C22×C3×C414yescomplemented
197C23×C4—91no—
198C23×C4—92no—
199C2×C4⋊C4—92no—
200C2×C4⋊C4—91no—
201C2×C4⋊C4—98no—
202C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic, direct product factor
2033−1+2C22×C4⋊C411yescharacteristic, complemented, Sylow
204C23×C3—31no—
205C23×C3—92no—
206C23×C3—94no—
207C23×C3—94no—
208C23×C3—94no—
209C23×C3C22×C3×D311yescharacteristic
210C22×D3—32no—
211C22×D3—34no—
212C22×D3—34no—
213C22×D3—34no—
214C2×C3×C4—24no—
215C2×C3×C4—34no—
216C2×C3×C4—32no—
217C2×C3×C4—34no—
218C2×C3×C4—64no—
219C2×C3×C4—916no—
220C2×C3×C4—92no—
221C2×C3×C4—94no—
222C2×C3×C4—94no—
223C2×C3×C4—1816no—
224C2×C3×C4—184no—
225C2×C3×C4C3×C4×D314yes—
226C2×C3×C4C22×C3×D314yes—
227C2×C3×C4C22×C3×D312yes—
228C2×Dic3—32no—
229C2×Dic3—34no—
230C2×Dic3—34no—
231C2×Dic3—64no—
232C4×D3—316no—
233C4×D3—616no—
234C2×C32D3×C4⋊C414yes—
235C2×C32C22×C4×D311yescharacteristic
236C2×C32C2×D3×D411yescharacteristic
237C2×C32C2×D3×Q811yescharacteristic
238C3×D3—38no—
239C2×C9—24no—
240C2×C9—21no—
241C2×C9—21no—
242C2×C9—21no—
243C2×C9C2×C3×C4⋊C414yescomplemented
244C2×C9C23×C3×C411yescharacteristic
245C2×C9C22×C3×D411yescharacteristic
246C2×C9C22×C3×Q811yescharacteristic
247D9C2×C3×C4⋊C418yescomplemented
248C24—91no—
249C22×C4—91no—
250C22×C4—98no—
251C22×C4—94no—
252C22×C4—92no—
253C22×C4—98no—
254C22×C4—188no—
255C22×C4C2×AGL(1,ℤ/9)12yes—
256C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
257C4⋊C4—98no—
258C4⋊C4—94no—
259C4⋊C4C2×AGL(1,ℤ/9)14yesdirect product factor
260C22×C3—31no—
261C22×C3—32no—
262C22×C3—32no—
263C22×C3—32no—
264C22×C3—94no—
265C22×C3—94no—
266C22×C3—916no—
267C22×C3—94no—
268C22×C3C2×C3×C4×D312yes—
269C22×C3C3×D3×D412yes—
270C22×C3C3×D3×Q812yes—
271C22×C3C23×C3×D311yescharacteristic
272C2×D3—316no—
273C2×D3—34no—
274C2×D3—34no—
275C2×D3—34no—
276C3×C4—24no—
277C3×C4—34no—
278C3×C4—64no—
279C3×C4—94no—
280C3×C4—184no—
281C3×C4C2×C3×C4×D314yes—
282Dic3—34no—
283Dic3—64no—
284C32C2×D3×C4⋊C411yescharacteristic
285C9—21no—
286C9C22×C3×C4⋊C411yescharacteristic, complemented
287C23—92no—
288C23—94no—
289C23—94no—
290C23—94no—
291C23C22×AGL(1,ℤ/9)11yescharacteristic
292C2×C4—24no—
293C2×C4—92no—
294C2×C4—94no—
295C2×C4—916no—
296C2×C4—94no—
297C2×C4—1816no—
298C2×C4—184no—
299C2×C4C4×AGL(1,ℤ/9)14yescomplemented
300C2×C4C22×AGL(1,ℤ/9)14yes—
301C2×C4C22×AGL(1,ℤ/9)12yes—
302C2×C3—34no—
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—31no—
306C2×C3—98no—
307C2×C3C3×D3×C4⋊C414yes—
308C2×C3C22×C3×C4×D311yescharacteristic
309C2×C3C2×C3×D3×D411yescharacteristic
310C2×C3C2×C3×D3×Q811yescharacteristic
311D3—38no—
312C22—916no—
313C22—94no—
314C22—94no—
315C22—94no—
316C22C2×C4×AGL(1,ℤ/9)12yes—
317C22D4×AGL(1,ℤ/9)12yes—
318C22Q8×AGL(1,ℤ/9)12yes—
319C22C23×AGL(1,ℤ/9)11yescharacteristic
320C4—24no—
321C4—94no—
322C4—184no—
323C4C2×C4×AGL(1,ℤ/9)14yescomplemented
324C3—31no—
325C3C2×C3×D3×C4⋊C411yescharacteristic
326C2—98no—
327C2C4⋊C4×AGL(1,ℤ/9)14yesdirect product factor
328C2C22×C4×AGL(1,ℤ/9)11yescharacteristic
329C2C2×D4×AGL(1,ℤ/9)11yescharacteristic
330C2C2×Q8×AGL(1,ℤ/9)11yescharacteristic
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)
hypocenterC9C22×C3×C4⋊C4
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊C4×3−1+2C2semidirect
4. C2×C4⋊(3−1+2⋊C4)C2semidirect
5. C2×(3−1+2⋊C4)⋊C4C2semidirect
6. C4⋊C4×AGL(1,ℤ/9)C2direct (N × H)
8. C2×C4⋊C4×D9C3semidirect
15. C2×C4×AGL(1,ℤ/9)C4semidirect
18. C4⋊C4×3−1+2C22semidirect
19. C4⋊(3−1+2⋊C4)C22semidirect
20. (3−1+2⋊C4)⋊C4C22semidirect
30. C2×C9×C4⋊C4C2×C3semidirect
31. C2×C4⋊Dic9C2×C3semidirect
32. C2×Dic9⋊C4C2×C3semidirect
33. C4⋊C4×D9C2×C3semidirect
40. C2×C4×3−1+2C2×C4semidirect
44. C2×3−1+2⋊C4C2×C4semidirect
48. C4×AGL(1,ℤ/9)C2×C4semidirect
71. C2×C4×D9C3×C4semidirect
75. C9×C4⋊C4C22×C3semidirect
76. C4⋊Dic9C22×C3semidirect
77. Dic9⋊C4C22×C3semidirect
82. C2×AGL(1,ℤ/9)C4⋊C4direct (N × H)
87. C4×3−1+2C22×C4semidirect
89. 3−1+2⋊C4C22×C4semidirect
128. C2×C4×C9C2×C3×C4semidirect
132. C2×Dic9C2×C3×C4semidirect
136. C4×D9C2×C3×C4semidirect
138. C2×3−1+2C2×C4⋊C4semidirect
142. AGL(1,ℤ/9)C2×C4⋊C4direct (N × H)
187. C2×D9C3×C4⋊C4semidirect
194. C4×C9C22×C3×C4semidirect
196. Dic9C22×C3×C4semidirect
202. C2×C4⋊C4AGL(1,ℤ/9)direct (N × H)
203. 3−1+2C22×C4⋊C4semidirect
243. C2×C9C2×C3×C4⋊C4semidirect
247. D9C2×C3×C4⋊C4semidirect
259. C4⋊C4C2×AGL(1,ℤ/9)direct (N × H)
286. C9C22×C3×C4⋊C4semidirect
299. C2×C4C4×AGL(1,ℤ/9)semidirect
323. C4C2×C4×AGL(1,ℤ/9)semidirect
327. C2C4⋊C4×AGL(1,ℤ/9)direct (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×C4⋊(3−1+2⋊C4)2
C2×(3−1+2⋊C4)⋊C42
C4⋊C4×AGL(1,ℤ/9)2
C2×C3×D3×C4⋊C43
C2×C4⋊C4×D93