Finite Group Databasebeta

← C4⋊C4×C3⋊AGL(1,ℤ/9) #72254656

Subgroups — C4⋊C4×C3⋊AGL(1,ℤ/9) #72254656

Order 2592.

1037 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C3⋊AGL(1,ℤ/9)C212yesmaximal
2C2×C4×C3⋊AGL(1,ℤ/9)C211yesmaximal, characteristic
3C3×C4⋊C4×3−1+2C211yesmaximal, characteristic, complemented
4C4⋊(C3⋊(3−1+2⋊C4))C211yesmaximal, characteristic, complemented
5#13047947C212yesmaximal, complemented
6C3×C4⋊C4×C3⋊D3—31nomaximal
7C4⋊C4×C3⋊D9C311yesmaximal, characteristic, complemented
8C4⋊C4×AGL(1,ℤ/9)—33nomaximal
9C22×C3⋊AGL(1,ℤ/9)C2211yescharacteristic
10C2×C3×C4×3−1+2C2212yes—
11C2×C3×C4×3−1+2C2211yescharacteristic
12C2×C3⋊(3−1+2⋊C4)C2212yes—
13C2×C3⋊(3−1+2⋊C4)C2211yescharacteristic
14C4×C3⋊AGL(1,ℤ/9)—24no—
15C4×C3⋊AGL(1,ℤ/9)C414yescomplemented
16C2×C3×C4×C3⋊D3—31no—
17C2×C3×C4×C3⋊D3—32no—
18C33×C4⋊C4D311yescharacteristic
19C3×C4⋊(C3⋊Dic3)—31no—
20C3×#3750—32no—
21C2×C4×C3⋊D9C2×C312yes—
22C2×C4×C3⋊D9C2×C311yescharacteristic
23C2×C4×AGL(1,ℤ/9)—33no—
24C2×C4×AGL(1,ℤ/9)—36no—
25C4⋊C4×3−1+2—23no—
26C4⋊C4×3−1+2D313yescomplemented
27C3×C9×C4⋊C4—21no—
28C3×C9×C4⋊C4C2×C311yescharacteristic, complemented
29C4⋊(C3⋊Dic9)C2×C311yescharacteristic, complemented
30#88534C2×C312yescomplemented
31C4⋊(3−1+2⋊C4)—33no—
32(3−1+2⋊C4)⋊C4—36no—
33C22×C3×3−1+2C2311yescharacteristic
34C2×C3⋊AGL(1,ℤ/9)C2×C412yes—
35C2×C3⋊AGL(1,ℤ/9)D412yes—
36C2×C3⋊AGL(1,ℤ/9)Q812yes—
37C3×C4×3−1+2—22no—
38C3×C4×3−1+2C2×C412yescomplemented
39C3⋊(3−1+2⋊C4)—22no—
40C3⋊(3−1+2⋊C4)C2×C412yescomplemented
41C4⋊C4×C3⋊D3—31no—
42C3×D3×C4⋊C4—92no—
43C3×D3×C4⋊C4—91no—
44C3×D3×C4⋊C4—91no—
45C4⋊C4×D9—33no—
46C22×C3×C3⋊D3—31no—
47C2×C33×C4C2×D312yes—
48C2×C33×C4C2×D311yescharacteristic
49C2×C3×C3⋊Dic3—31no—
50C2×C3×C3⋊Dic3—32no—
51C3×C4×C3⋊D3—34no—
52C3×C4×C3⋊D3—64no—
53C22×C3⋊D9C22×C311yescharacteristic
54C22×AGL(1,ℤ/9)—33no—
55C2×C4×3−1+2—23no—
56C2×C4×3−1+2—26no—
57C2×C4×3−1+2C2×D316yes—
58C2×C4×3−1+2C2×D313yes—
59C2×C3×C4×C9—21no—
60C2×C3×C4×C9—22no—
61C2×C3×C4×C9C22×C312yes—
62C2×C3×C4×C9C22×C311yescharacteristic
63C2×C3⋊Dic9C22×C312yes—
64C2×C3⋊Dic9C22×C311yescharacteristic
65C4×C3⋊D9—24no—
66C4×C3⋊D9C3×C414yescomplemented
67C2×3−1+2⋊C4—33no—
68C2×3−1+2⋊C4—36no—
69C4×AGL(1,ℤ/9)—312no—
70C4×AGL(1,ℤ/9)—612no—
71C2×C3×3−1+2C22×C411yescharacteristic
72C2×C3×3−1+2C2×D411yescharacteristic
73C2×C3×3−1+2C2×Q811yescharacteristic
74C3⋊AGL(1,ℤ/9)C4⋊C414yesdirect product factor
75C2×C4×C3⋊D3—31no—
76C2×C4×C3⋊D3—32no—
77C2×C3×C4×D3—92no—
78C2×C3×C4×D3—91no—
79C2×C3×C4×D3—91no—
80C2×C3×C4×D3—94no—
81C2×C3×C4×D3—92no—
82C2×C3×C4×D3—92no—
83C32×C4⋊C4—21no—
84C32×C4⋊C4—32no—
85C32×C4⋊C4—61no—
86C32×C4⋊C4C3×D311yescharacteristic
87C32×C4⋊C4C3⋊D311yescharacteristic
88C4⋊(C3⋊Dic3)—31no—
89#3750—32no—
90C3×C4⋊Dic3—92no—
91C3×C4⋊Dic3—91no—
92C3×C4⋊Dic3—91no—
93C3×Dic3⋊C4—94no—
94C3×Dic3⋊C4—92no—
95C3×Dic3⋊C4—92no—
96C2×C4×D9—33no—
97C2×C4×D9—36no—
98C9×C4⋊C4—23no—
99C9×C4⋊C4C3×D313yescomplemented
100C4⋊Dic9—33no—
101Dic9⋊C4—36no—
102C22×C33C22×D311yescharacteristic
103C2×C3×C3⋊D3—32no—
104C2×C3×C3⋊D3—32no—
105C2×C3×C3⋊D3—32no—
106C33×C4—22no—
107C33×C4C4×D312yes—
108C3×C3⋊Dic3—32no—
109C3×C3⋊Dic3—62no—
110C22×3−1+2—23no—
111C22×3−1+2C22×D313yes—
112C22×C3×C9—21no—
113C22×C3×C9C23×C311yescharacteristic
114C2×C3⋊D9C2×C3×C412yes—
115C2×C3⋊D9C3×D412yes—
116C2×C3⋊D9C3×Q812yes—
117C2×AGL(1,ℤ/9)—36no—
118C2×AGL(1,ℤ/9)—36no—
119C2×AGL(1,ℤ/9)—36no—
120C4×3−1+2—26no—
121C4×3−1+2—26no—
122C4×3−1+2—46no—
123C4×3−1+2C4×D316yescomplemented
124C3×C4×C9—22no—
125C3×C4×C9—22no—
126C3×C4×C9—42no—
127C3×C4×C9C2×C3×C412yescomplemented
128C3⋊Dic9—22no—
129C3⋊Dic9C2×C3×C412yescomplemented
1303−1+2⋊C4—36no—
1313−1+2⋊C4—66no—
132C2×C3×C4⋊C4—271no—
133D3×C4⋊C4—92no—
134D3×C4⋊C4—91no—
135D3×C4⋊C4—91no—
136C3×3−1+2C2×C4⋊C411yescharacteristic, complemented, Sylow
137C22×C3⋊D3—31no—
138C22×C3×D3—92no—
139C22×C3×D3—91no—
140C22×C3×D3—91no—
141C2×C32×C4—21no—
142C2×C32×C4—22no—
143C2×C32×C4—32no—
144C2×C32×C4—34no—
145C2×C32×C4—61no—
146C2×C32×C4—62no—
147C2×C32×C4C2×C3×D312yes—
148C2×C32×C4C2×C3×D311yescharacteristic
149C2×C32×C4C2×C3⋊D312yes—
150C2×C32×C4C2×C3⋊D311yescharacteristic
151C2×C3⋊Dic3—31no—
152C2×C3⋊Dic3—32no—
153C4×C3⋊D3—34no—
154C4×C3⋊D3—64no—
155C2×C3×Dic3—92no—
156C2×C3×Dic3—91no—
157C2×C3×Dic3—91no—
158C2×C3×Dic3—94no—
159C2×C3×Dic3—92no—
160C2×C3×Dic3—92no—
161C3×C4×D3—98no—
162C3×C4×D3—94no—
163C3×C4×D3—94no—
164C3×C4×D3—184no—
165C3×C4×D3—188no—
166C3×C4×D3—184no—
167C22×D9—33no—
168C2×C4×C9—23no—
169C2×C4×C9—26no—
170C2×C4×C9C2×C3×D316yes—
171C2×C4×C9C2×C3×D313yes—
172C2×Dic9—33no—
173C2×Dic9—36no—
174C4×D9—312no—
175C4×D9—612no—
176C2×C33C2×C4×D311yescharacteristic
177C2×C33D3×D411yescharacteristic
178C2×C33D3×Q811yescharacteristic
179C3×C3⋊D3—34no—
180C2×3−1+2—23no—
181C2×3−1+2—23no—
182C2×3−1+2—23no—
183C2×3−1+2C2×C4×D313yes—
184C2×3−1+2D3×D413yes—
185C2×3−1+2D3×Q813yes—
186C2×C3×C9—21no—
187C2×C3×C9—21no—
188C2×C3×C9—21no—
189C2×C3×C9C22×C3×C411yescharacteristic
190C2×C3×C9C2×C3×D411yescharacteristic
191C2×C3×C9C2×C3×Q811yescharacteristic
192C3⋊D9C3×C4⋊C414yescomplemented
193AGL(1,ℤ/9)—312no—
194C22×C3×C4—271no—
195C22×C3×C4—272no—
196C2×C4×D3—92no—
197C2×C4×D3—91no—
198C2×C4×D3—91no—
199C2×C4×D3—94no—
200C2×C4×D3—92no—
201C2×C4×D3—92no—
202C3×C4⋊C4—61no—
203C3×C4⋊C4—61no—
204C3×C4⋊C4—272no—
205C3×C4⋊C4—271no—
206C3×C4⋊C4AGL(1,ℤ/9)12yescomplemented
207C3×C4⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
208C3×C4⋊C4C3×C3⋊D311yescharacteristic
209C4⋊Dic3—92no—
210C4⋊Dic3—91no—
211C4⋊Dic3—91no—
212Dic3⋊C4—94no—
213Dic3⋊C4—92no—
214Dic3⋊C4—92no—
215C22×C32—21no—
216C22×C32—32no—
217C22×C32—61no—
218C22×C32C22×C3×D311yescharacteristic
219C22×C32C22×C3⋊D311yescharacteristic
220C2×C3⋊D3—32no—
221C2×C3⋊D3—32no—
222C2×C3⋊D3—32no—
223C2×C3×D3—94no—
224C2×C3×D3—94no—
225C2×C3×D3—94no—
226C2×C3×D3—92no—
227C2×C3×D3—92no—
228C2×C3×D3—92no—
229C2×C3×D3—92no—
230C2×C3×D3—92no—
231C2×C3×D3—92no—
232C32×C4—22no—
233C32×C4—22no—
234C32×C4—22no—
235C32×C4—34no—
236C32×C4—42no—
237C32×C4—64no—
238C32×C4—62no—
239C32×C4—122no—
240C32×C4C3×C4×D312yes—
241C32×C4C4×C3⋊D312yes—
242C3⋊Dic3—32no—
243C3⋊Dic3—62no—
244C3×Dic3—94no—
245C3×Dic3—92no—
246C3×Dic3—92no—
247C3×Dic3—184no—
248C3×Dic3—182no—
249C3×Dic3—182no—
250C22×C9—23no—
251C22×C9C22×C3×D313yes—
252C2×D9—36no—
253C2×D9—36no—
254C2×D9—36no—
255C4×C9—26no—
256C4×C9—26no—
257C4×C9—46no—
258C4×C9C3×C4×D316yescomplemented
259Dic9—36no—
260Dic9—66no—
261C2×C4⋊C4—271noSylow
262C33D3×C4⋊C411yescharacteristic
2633−1+2—23no—
2643−1+2D3×C4⋊C413yescomplemented
265C3×C9—21no—
266C3×C9C2×C3×C4⋊C411yescharacteristic, complemented
267C23×C3—271no—
268C22×D3—92no—
269C22×D3—91no—
270C22×D3—91no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—62no—
274C2×C3×C4—62no—
275C2×C3×C4—274no—
276C2×C3×C4—271no—
277C2×C3×C4—272no—
278C2×C3×C4—544no—
279C2×C3×C4C2×AGL(1,ℤ/9)14yes—
280C2×C3×C4C2×AGL(1,ℤ/9)12yes—
281C2×C3×C4C2×AGL(1,ℤ/9)12yes—
282C2×C3×C4C2×AGL(1,ℤ/9)11yescharacteristic
283C2×C3×C4C2×C3×C3⋊D312yes—
284C2×C3×C4C2×C3×C3⋊D311yescharacteristic
285C2×Dic3—92no—
286C2×Dic3—91no—
287C2×Dic3—91no—
288C2×Dic3—94no—
289C2×Dic3—92no—
290C2×Dic3—92no—
291C4×D3—98no—
292C4×D3—94no—
293C4×D3—94no—
294C4×D3—184no—
295C4×D3—188no—
296C4×D3—184no—
297C2×C32—21no—
298C2×C32—21no—
299C2×C32—21no—
300C2×C32—32no—
301C2×C32—32no—
302C2×C32—32no—
303C2×C32—61no—
304C2×C32—61no—
305C2×C32—61no—
306C2×C32C2×C3×C4×D311yescharacteristic
307C2×C32C3×D3×D411yescharacteristic
308C2×C32C3×D3×Q811yescharacteristic
309C2×C32C2×C4×C3⋊D311yescharacteristic
310C2×C32D4×C3⋊D311yescharacteristic
311C2×C32Q8×C3⋊D311yescharacteristic
312C3⋊D3—34no—
313C3×D3—98no—
314C3×D3—94no—
315C3×D3—94no—
316C2×C9—23no—
317C2×C9—23no—
318C2×C9—23no—
319C2×C9C2×C3×C4×D313yes—
320C2×C9C3×D3×D413yes—
321C2×C9C3×D3×Q813yes—
322D9—312no—
323C22×C4—271no—
324C22×C4—272no—
325C4⋊C4—272no—
326C4⋊C4—271no—
327C4⋊C4C3⋊AGL(1,ℤ/9)11yescharacteristic, direct product factor
328C22×C3—61no—
329C22×C3—61no—
330C22×C3—272no—
331C22×C3—272no—
332C22×C3—272no—
333C22×C3C22×AGL(1,ℤ/9)12yes—
334C22×C3C22×AGL(1,ℤ/9)11yescharacteristic
335C22×C3C22×C3×C3⋊D311yescharacteristic
336C2×D3—94no—
337C2×D3—92no—
338C2×D3—94no—
339C2×D3—92no—
340C2×D3—94no—
341C2×D3—92no—
342C2×D3—92no—
343C2×D3—92no—
344C2×D3—92no—
345C3×C4—24no—
346C3×C4—22no—
347C3×C4—22no—
348C3×C4—62no—
349C3×C4—62no—
350C3×C4—122no—
351C3×C4—122no—
352C3×C4—272no—
353C3×C4—542no—
354C3×C4C4×AGL(1,ℤ/9)14yescomplemented
355C3×C4C4×AGL(1,ℤ/9)12yescomplemented
356C3×C4C3×C4×C3⋊D312yes—
357Dic3—94no—
358Dic3—92no—
359Dic3—92no—
360Dic3—184no—
361Dic3—182no—
362Dic3—182no—
363C32—21no—
364C32—32no—
365C32—61no—
366C32C3×D3×C4⋊C411yescharacteristic
367C32C4⋊C4×C3⋊D311yescharacteristic
368C9—23no—
369C9C3×D3×C4⋊C413yescomplemented
370C23—271no—
371C2×C4—271no—
372C2×C4—274no—
373C2×C4—272no—
374C2×C4—544no—
375C2×C4C2×C3⋊AGL(1,ℤ/9)12yes—
376C2×C4C2×C3⋊AGL(1,ℤ/9)11yescharacteristic
377C2×C3—61no—
378C2×C3—61no—
379C2×C3—61no—
380C2×C3—61no—
381C2×C3—61no—
382C2×C3—61no—
383C2×C3—274no—
384C2×C3C2×C4×AGL(1,ℤ/9)12yes—
385C2×C3C2×C4×AGL(1,ℤ/9)11yescharacteristic
386C2×C3D4×AGL(1,ℤ/9)12yes—
387C2×C3D4×AGL(1,ℤ/9)11yescharacteristic
388C2×C3Q8×AGL(1,ℤ/9)12yes—
389C2×C3Q8×AGL(1,ℤ/9)11yescharacteristic
390C2×C3C2×C3×C4×C3⋊D311yescharacteristic
391C2×C3C3×D4×C3⋊D311yescharacteristic
392C2×C3C3×Q8×C3⋊D311yescharacteristic
393D3—98no—
394D3—94no—
395D3—94no—
396C22—272no—
397C22—272no—
398C22—272no—
399C22C22×C3⋊AGL(1,ℤ/9)11yescharacteristic
400C4—22no—
401C4—272no—
402C4—542no—
403C4C4×C3⋊AGL(1,ℤ/9)12yescomplemented
404C3—61no—
405C3—61no—
406C3C4⋊C4×AGL(1,ℤ/9)12yescomplemented
407C3C4⋊C4×AGL(1,ℤ/9)11yescharacteristic, complemented
408C3C3×C4⋊C4×C3⋊D311yescharacteristic
409C2—274no—
410C2C2×C4×C3⋊AGL(1,ℤ/9)11yescharacteristic
411C2D4×C3⋊AGL(1,ℤ/9)11yescharacteristic
412C2Q8×C3⋊AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC22C22×C3⋊AGL(1,ℤ/9)
commutatorC2×C3×C9C22×C3×C4
FrattiniC22×C3C22×C3×C3⋊D3
FittingC3×C4⋊C4×3−1+2C2
socleC22×C32C22×C3×D3
prime radicalC22×C3⋊AGL(1,ℤ/9)C22
cosocleC22×C3×C9C23×C3
hypercenterC4⋊C4C3⋊AGL(1,ℤ/9)
hypocenterC3×C9C2×C3×C4⋊C4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C3×C4⋊C4×3−1+2C2semidirect
4. C4⋊(C3⋊(3−1+2⋊C4))C2semidirect
5. #13047947C2semidirect
7. C4⋊C4×C3⋊D9C3semidirect
15. C4×C3⋊AGL(1,ℤ/9)C4semidirect
26. C4⋊C4×3−1+2D3semidirect
28. C3×C9×C4⋊C4C2×C3semidirect
29. C4⋊(C3⋊Dic9)C2×C3semidirect
30. #88534C2×C3semidirect
38. C3×C4×3−1+2C2×C4semidirect
40. C3⋊(3−1+2⋊C4)C2×C4semidirect
66. C4×C3⋊D9C3×C4semidirect
74. C3⋊AGL(1,ℤ/9)C4⋊C4direct (N × H)
99. C9×C4⋊C4C3×D3semidirect
123. C4×3−1+2C4×D3semidirect
127. C3×C4×C9C2×C3×C4semidirect
129. C3⋊Dic9C2×C3×C4semidirect
136. C3×3−1+2C2×C4⋊C4semidirect
192. C3⋊D9C3×C4⋊C4semidirect
206. C3×C4⋊C4AGL(1,ℤ/9)semidirect
207. C3×C4⋊C4AGL(1,ℤ/9)semidirect
258. C4×C9C3×C4×D3semidirect
264. 3−1+2D3×C4⋊C4semidirect
266. C3×C9C2×C3×C4⋊C4semidirect
327. C4⋊C4C3⋊AGL(1,ℤ/9)direct (N × H)
354. C3×C4C4×AGL(1,ℤ/9)semidirect
355. C3×C4C4×AGL(1,ℤ/9)semidirect
369. C9C3×D3×C4⋊C4semidirect
403. C4C4×C3⋊AGL(1,ℤ/9)semidirect
406. C3C4⋊C4×AGL(1,ℤ/9)semidirect
407. C3C4⋊C4×AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊AGL(1,ℤ/9)2
C3×C4⋊C4×3−1+22
C4⋊(C3⋊(3−1+2⋊C4))2
#130479472
C3×C4⋊C4×C3⋊D33
C4⋊C4×C3⋊D93
C4⋊C4×AGL(1,ℤ/9)3