Finite Group Databasebeta

← #99×AGL(1,ℤ/9) #13141659

Subgroups — #99×AGL(1,ℤ/9) #13141659

Order 1728.

656 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×AGL(1,ℤ/9)C211yesmaximal, characteristic
2C22⋊C4×AGL(1,ℤ/9)C212yesmaximal, complemented
33−1+2×#99C211yesmaximal, characteristic, complemented
4#63393⋊C3C211yesmaximal, characteristic, complemented
5#63365⋊C3C212yesmaximal, complemented
6C3×D3×#99—31nomaximal
7D9×#99C311yesmaximal, characteristic, complemented
8C23×AGL(1,ℤ/9)C411yescharacteristic, complemented
9C23×AGL(1,ℤ/9)C2211yescharacteristic
10C2×D4×3−1+2C2211yescharacteristic
11C2×#611⋊C3C411yescharacteristic, complemented
12C2×#611⋊C3C2211yescharacteristic
13D4×AGL(1,ℤ/9)—22no—
14D4×AGL(1,ℤ/9)—22no—
15C2×C4⋊AGL(1,ℤ/9)C411yescharacteristic, complemented
16C2×C4×AGL(1,ℤ/9)—22no—
17C2×C4×AGL(1,ℤ/9)C411yescharacteristic, complemented
18C22⋊C4×3−1+2C2212yescomplemented
19#3676⋊C3C2212yescomplemented
20(C2×AGL(1,ℤ/9))⋊C4—22no—
21C2×C3×D3×D4—31no—
22C3×D3×C22⋊C4—32no—
23C2×D4×D9C2×C311yescharacteristic
24C32×#99D311yescharacteristic
25C3×#834—31no—
26C3×#806—32no—
27C22⋊C4×D9C2×C312yescomplemented
28C9×#99—21no—
29C9×#99C2×C311yescharacteristic, complemented
30#63393C2×C311yescharacteristic, complemented
31#63365C2×C312yescomplemented
32C23×3−1+2C2×C411yescharacteristic, complemented
33C23×3−1+2C2311yescharacteristic
34C22×AGL(1,ℤ/9)—21no—
35C22×AGL(1,ℤ/9)—21no—
36C22×AGL(1,ℤ/9)—42no—
37C22×AGL(1,ℤ/9)—42no—
38C22×AGL(1,ℤ/9)C2×C411yescharacteristic
39C22×AGL(1,ℤ/9)C2×C411yescharacteristic
40C22×AGL(1,ℤ/9)C2×C411yescharacteristic, complemented
41C22×AGL(1,ℤ/9)D412yes—
42C22×AGL(1,ℤ/9)D412yes—
43D4×3−1+2—22no—
44C2×C4×3−1+2—22no—
45C2×C4×3−1+2C2×C411yescharacteristic, complemented
46#611⋊C3—22no—
47#611⋊C3—22no—
48C2×3−1+2⋊C4—22no—
49C2×3−1+2⋊C4C2×C411yescharacteristic, complemented
50C4⋊AGL(1,ℤ/9)—22no—
51C4×AGL(1,ℤ/9)—21no—
52C4×AGL(1,ℤ/9)—21no—
53C4×AGL(1,ℤ/9)—44no—
54C2×C3×#99—91no—
55D3×#99—31no—
56C23×C3×D3—31no—
57C23×C3×D3—31no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C3×#67—31no—
60C2×C3×#67—31no—
61C3×D3×D4—62no—
62C3×D3×D4—62no—
63C2×C3×D12—31no—
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—62no—
66C23×D9C3×C411yescharacteristic, complemented
67C23×D9C22×C311yescharacteristic
68C32×C22⋊C4C2×D312yes—
69C3×#217—32no—
70C3×(C2×D3)⋊C4—62no—
71C2×D4×C9—21no—
72C2×D4×C9C22×C311yescharacteristic
73C2×#611C3×C411yescharacteristic, complemented
74C2×#611C22×C311yescharacteristic
75D4×D9—22no—
76D4×D9—22no—
77C2×D36C3×C411yescharacteristic, complemented
78C2×C4×D9—22no—
79C2×C4×D9C3×C411yescharacteristic, complemented
80C9×C22⋊C4—22no—
81C9×C22⋊C4C22×C312yescomplemented
82#3676C22×C312yescomplemented
83(C2×D9)⋊C4—22no—
84C22×3−1+2—21no—
85C22×3−1+2—41no—
86C22×3−1+2—41no—
87C22×3−1+2C22×C411yescharacteristic
88C22×3−1+2C2×D412yes—
89C2×AGL(1,ℤ/9)—24no—
90C2×AGL(1,ℤ/9)—22no—
91C2×AGL(1,ℤ/9)—21no—
92C2×AGL(1,ℤ/9)—41no—
93C2×AGL(1,ℤ/9)—42no—
94C2×AGL(1,ℤ/9)—42no—
95C2×AGL(1,ℤ/9)—42no—
96C2×AGL(1,ℤ/9)—41no—
97C2×AGL(1,ℤ/9)C22⋊C411yescharacteristic
98C2×AGL(1,ℤ/9)C22⋊C412yes—
99C2×AGL(1,ℤ/9)C22⋊C411yescharacteristic
100C4×3−1+2—21no—
101C4×3−1+2—42no—
1023−1+2⋊C4—21no—
1033−1+2⋊C4—42no—
104C22×C3×D4—91no—
105C2×D3×D4—31no—
106C2×C3×C22⋊C4—92no—
107D3×C22⋊C4—32no—
108C3×#99—31no—
109C3×#99—92no—
110C3×#99—91no—
111C3×#99C3×D311yescharacteristic
112#834—31no—
113#806—32no—
114C23×C32C4×D311yescharacteristic
115C23×C32C22×D311yescharacteristic
116C22×C3×D3—31no—
117C22×C3×D3—32no—
118C22×C3×D3—31no—
119C22×C3×D3—32no—
120C22×C3×D3—31no—
121C22×C3×D3—61no—
122C22×C3×D3—61no—
123C22×C3×D3—122no—
124C22×C3×D3—122no—
125C32×D4—22no—
126C2×C32×C4—22no—
127C2×C32×C4C4×D311yescharacteristic
128C3×#67—62no—
129C3×#67—62no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—62no—
132C3×D12—62no—
133C3×C4×D3—61no—
134C3×C4×D3—61no—
135C3×C4×D3—124no—
136C23×C9—21no—
137C23×C9—21no—
138C23×C9C2×C3×C411yescharacteristic, complemented
139C23×C9C23×C311yescharacteristic
140C22×D9—21no—
141C22×D9—21no—
142C22×D9—42no—
143C22×D9—42no—
144C22×D9C2×C3×C411yescharacteristic
145C22×D9C2×C3×C411yescharacteristic
146C22×D9C2×C3×C411yescharacteristic, complemented
147C22×D9C3×D412yes—
148C22×D9C3×D412yes—
149D4×C9—22no—
150D4×C9—42no—
151C2×C4×C9—22no—
152C2×C4×C9—21no—
153C2×C4×C9—42no—
154C2×C4×C9C2×C3×C411yescharacteristic, complemented
155#611—22no—
156#611—22no—
157C2×Dic9—22no—
158C2×Dic9C2×C3×C411yescharacteristic, complemented
159D36—22no—
160C4×D9—21no—
161C4×D9—21no—
162C4×D9—44no—
163C2×#99—91noSylow
164C2×3−1+2—22no—
165C2×3−1+2—21no—
166C2×3−1+2—41no—
167C2×3−1+2C2×C22⋊C411yescharacteristic
168AGL(1,ℤ/9)—21no—
169AGL(1,ℤ/9)—22no—
170AGL(1,ℤ/9)—41no—
171AGL(1,ℤ/9)#9912yesdirect product factor
172C24×C3—91no—
173C24×C3—91no—
174C23×D3—31no—
175C23×D3—31no—
176C2×C3×D4—31no—
177C2×C3×D4—91no—
178C2×C3×D4—91no—
179C2×C3×D4—91no—
180C2×C3×D4—182no—
181C2×C3×D4—182no—
182C2×C3×D4C2×C3×D311yescharacteristic
183C22×C3×C4—91no—
184C22×C3×C4—182no—
185C2×#67—31no—
186C2×#67—31no—
187D3×D4—62no—
188D3×D4—62no—
189C2×D12—31no—
190C2×C4×D3—31no—
191C2×C4×D3—62no—
192C3×C22⋊C4—32no—
193C3×C22⋊C4—92no—
194C3×C22⋊C4—182no—
195C3×C22⋊C4C2×C3×D312yes—
196#217—32no—
197(C2×D3)⋊C4—62no—
198C22×C32—21no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32C2×C4×D311yescharacteristic
202C22×C32D3×D412yes—
203C2×C3×D3—31no—
204C2×C3×D3—31no—
205C2×C3×D3—32no—
206C2×C3×D3—64no—
207C2×C3×D3—62no—
208C2×C3×D3—61no—
209C2×C3×D3—122no—
210C2×C3×D3—121no—
211C2×C3×D3—122no—
212C2×C3×D3—122no—
213C2×C3×D3—121no—
214C32×C4—21no—
215C32×C4—42no—
216C3×Dic3—61no—
217C3×Dic3—122no—
218C22×C9—21no—
219C22×C9—21no—
220C22×C9—22no—
221C22×C9—41no—
222C22×C9—41no—
223C22×C9—41no—
224C22×C9—81no—
225C22×C9—81no—
226C22×C9C22×C3×C411yescharacteristic
227C22×C9C2×C3×D412yes—
228C2×D9—24no—
229C2×D9—22no—
230C2×D9—21no—
231C2×D9—41no—
232C2×D9—42no—
233C2×D9—42no—
234C2×D9—42no—
235C2×D9—41no—
236C2×D9C3×C22⋊C411yescharacteristic
237C2×D9C3×C22⋊C412yes—
238C2×D9C3×C22⋊C411yescharacteristic
239C4×C9—21no—
240C4×C9—41no—
241C4×C9—42no—
242C4×C9—82no—
243Dic9—21no—
244Dic9—42no—
245C22×D4—91no—
246C2×C22⋊C4—92no—
247#99—91no—
248#99—92no—
249#99AGL(1,ℤ/9)11yescharacteristic, direct product factor
2503−1+2C2×#9911yescharacteristic, complemented, Sylow
251C23×C3—31no—
252C23×C3—31no—
253C23×C3—91no—
254C23×C3—92no—
255C23×C3—92no—
256C23×C3—91no—
257C23×C3—91no—
258C23×C3—181no—
259C23×C3—181no—
260C23×C3—362no—
261C23×C3—362no—
262C23×C3C3×C4×D311yescharacteristic
263C23×C3C22×C3×D311yescharacteristic
264C22×D3—31no—
265C22×D3—32no—
266C22×D3—31no—
267C22×D3—32no—
268C22×D3—31no—
269C22×D3—61no—
270C22×D3—61no—
271C22×D3—122no—
272C22×D3—122no—
273C3×D4—22no—
274C3×D4—62no—
275C3×D4—182no—
276C3×D4—182no—
277C3×D4—182no—
278C2×C3×C4—22no—
279C2×C3×C4—31no—
280C2×C3×C4—62no—
281C2×C3×C4—91no—
282C2×C3×C4—181no—
283C2×C3×C4—181no—
284C2×C3×C4—182no—
285C2×C3×C4—364no—
286C2×C3×C4C3×C4×D311yescharacteristic
287#67—62no—
288#67—62no—
289C2×Dic3—31no—
290C2×Dic3—62no—
291D12—62no—
292C4×D3—61no—
293C4×D3—61no—
294C4×D3—124no—
295C2×C32—22no—
296C2×C32—21no—
297C2×C32—41no—
298C2×C32D3×C22⋊C411yescharacteristic
299C3×D3—32no—
300C3×D3—62no—
301C3×D3—61no—
302C3×D3—121no—
303C2×C9—22no—
304C2×C9—21no—
305C2×C9—21no—
306C2×C9—42no—
307C2×C9—41no—
308C2×C9—41no—
309C2×C9—81no—
310C2×C9C2×C3×C22⋊C411yescharacteristic
311D9—21no—
312D9—22no—
313D9—41no—
314D9C3×#9912yescomplemented
315C24—91no—
316C24—91no—
317C2×D4—91no—
318C2×D4—91no—
319C2×D4—91no—
320C2×D4—182no—
321C2×D4—182no—
322C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
323C22×C4—91no—
324C22×C4—182no—
325C22⋊C4—92no—
326C22⋊C4—182no—
327C22⋊C4C2×AGL(1,ℤ/9)12yescomplemented
328C22×C3—21no—
329C22×C3—31no—
330C22×C3—32no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—61no—
334C22×C3—91no—
335C22×C3—91no—
336C22×C3—92no—
337C22×C3—121no—
338C22×C3—121no—
339C22×C3—181no—
340C22×C3—184no—
341C22×C3—182no—
342C22×C3—362no—
343C22×C3—361no—
344C22×C3—361no—
345C22×C3—362no—
346C22×C3—362no—
347C22×C3C2×C3×C4×D311yescharacteristic
348C22×C3C3×D3×D412yes—
349C2×D3—31no—
350C2×D3—31no—
351C2×D3—32no—
352C2×D3—64no—
353C2×D3—62no—
354C2×D3—61no—
355C2×D3—122no—
356C2×D3—122no—
357C2×D3—121no—
358C2×D3—122no—
359C2×D3—121no—
360C3×C4—21no—
361C3×C4—42no—
362C3×C4—61no—
363C3×C4—122no—
364C3×C4—181no—
365C3×C4—362no—
366Dic3—61no—
367Dic3—122no—
368C32D3×#9911yescharacteristic
369C9—21no—
370C9C2×C3×#9911yescharacteristic, complemented
371C23—91no—
372C23—92no—
373C23—91no—
374C23—92no—
375C23—91no—
376C23—181no—
377C23—181no—
378C23—362no—
379C23—362no—
380C23C4×AGL(1,ℤ/9)11yescharacteristic, complemented
381C23C22×AGL(1,ℤ/9)11yescharacteristic
382D4—22no—
383D4—182no—
384D4—182no—
385D4—182no—
386C2×C4—22no—
387C2×C4—91no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4—182no—
391C2×C4—364no—
392C2×C4C4×AGL(1,ℤ/9)11yescharacteristic, complemented
393C2×C3—22no—
394C2×C3—21no—
395C2×C3—31no—
396C2×C3—41no—
397C2×C3—62no—
398C2×C3—61no—
399C2×C3—92no—
400C2×C3—121no—
401C2×C3—181no—
402C2×C3—182no—
403C2×C3—361no—
404C2×C3C3×D3×C22⋊C411yescharacteristic
405D3—32no—
406D3—61no—
407D3—62no—
408D3—121no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—91no—
413C22—91no—
414C22—92no—
415C22—184no—
416C22—182no—
417C22—181no—
418C22—362no—
419C22—361no—
420C22—361no—
421C22—362no—
422C22—362no—
423C22C2×C4×AGL(1,ℤ/9)11yescharacteristic
424C22D4×AGL(1,ℤ/9)12yes—
425C4—21no—
426C4—42no—
427C4—181no—
428C4—362no—
429C3—31no—
430C3C3×D3×#9911yescharacteristic
431C2—22no—
432C2—21no—
433C2—41no—
434C2—92no—
435C2—181no—
436C2—182no—
437C2—361no—
438C2C22⋊C4×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC2C22⋊C4×AGL(1,ℤ/9)
commutatorC22×C9C22×C3×C4
FrattiniC23×C3C22×C3×D3
Fitting3−1+2×#99C2
socleC2×C3C3×D3×C22⋊C4
prime radicalC2×D4×AGL(1,ℤ/9)C2
cosocleC23×C9C23×C3
hypercenter#99AGL(1,ℤ/9)
hypocenterC9C2×C3×#99
LCSG ▷ C22×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
2. C22⋊C4×AGL(1,ℤ/9)C2semidirect
3. 3−1+2×#99C2semidirect
4. #63393⋊C3C2semidirect
5. #63365⋊C3C2semidirect
7. D9×#99C3semidirect
8. C23×AGL(1,ℤ/9)C4semidirect
11. C2×#611⋊C3C4semidirect
15. C2×C4⋊AGL(1,ℤ/9)C4semidirect
17. C2×C4×AGL(1,ℤ/9)C4semidirect
18. C22⋊C4×3−1+2C22semidirect
19. #3676⋊C3C22semidirect
27. C22⋊C4×D9C2×C3semidirect
29. C9×#99C2×C3semidirect
30. #63393C2×C3semidirect
31. #63365C2×C3semidirect
32. C23×3−1+2C2×C4semidirect
40. C22×AGL(1,ℤ/9)C2×C4semidirect
45. C2×C4×3−1+2C2×C4semidirect
49. C2×3−1+2⋊C4C2×C4semidirect
66. C23×D9C3×C4semidirect
73. C2×#611C3×C4semidirect
77. C2×D36C3×C4semidirect
79. C2×C4×D9C3×C4semidirect
81. C9×C22⋊C4C22×C3semidirect
82. #3676C22×C3semidirect
138. C23×C9C2×C3×C4semidirect
146. C22×D9C2×C3×C4semidirect
154. C2×C4×C9C2×C3×C4semidirect
158. C2×Dic9C2×C3×C4semidirect
171. AGL(1,ℤ/9)#99direct (N × H)
249. #99AGL(1,ℤ/9)direct (N × H)
250. 3−1+2C2×#99semidirect
314. D9C3×#99semidirect
327. C22⋊C4C2×AGL(1,ℤ/9)semidirect
370. C9C2×C3×#99semidirect
380. C23C4×AGL(1,ℤ/9)semidirect
392. C2×C4C4×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×D4×AGL(1,ℤ/9)2
C22⋊C4×AGL(1,ℤ/9)2
3−1+2×#992
#63393⋊C32
#63365⋊C32
C3×D3×#993
D9×#993