Finite Group Databasebeta

← C3×D9×AGL(1,ℤ/9) #72243328

Subgroups — C3×D9×AGL(1,ℤ/9) #72243328

Order 2916.

556 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C3×#92299C211yesmaximal, characteristic, complemented
2C3×D9×3−1+2C211yesmaximal, characteristic, complemented
3C3×C9×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
4C3×D3×AGL(1,ℤ/9)—31nomaximal
5C32×D3×D9—31nomaximal
6D9×AGL(1,ℤ/9)C311yesmaximal, characteristic, direct product factor
7D9×AGL(1,ℤ/9)C311yesmaximal, characteristic, direct product factor
8D9×AGL(1,ℤ/9)C311yesmaximal, characteristic, direct product factor
9C3×D92C311yesmaximal, characteristic, complemented
10C3×C9×3−1+2C2211yescharacteristic, complemented, Sylow
11C3×C3⋊AGL(1,ℤ/9)—31no—
12C32×C3⋊D9—31no—
13C3×D3×3−1+2—31no—
14C32×AGL(1,ℤ/9)D311yescharacteristic
15C32×D3×C9—31no—
16C33×D9D311yescharacteristic
17#92299C2×C311yescharacteristic, complemented
18#92299C2×C311yescharacteristic, complemented
19#92299C2×C311yescharacteristic, complemented
20C3×C9⋊D9C2×C311yescharacteristic, complemented
21D9×3−1+2—21no—
22D9×3−1+2—21no—
23D9×3−1+2—21no—
24D9×3−1+2C2×C311yescharacteristic, complemented
25D9×3−1+2C2×C311yescharacteristic, complemented
26D9×3−1+2C2×C311yescharacteristic, complemented
27C9×AGL(1,ℤ/9)—21no—
28C9×AGL(1,ℤ/9)—21no—
29C9×AGL(1,ℤ/9)—21no—
30C9×AGL(1,ℤ/9)C2×C311yescharacteristic, complemented
31C9×AGL(1,ℤ/9)C2×C311yescharacteristic, complemented
32C9×AGL(1,ℤ/9)C2×C311yescharacteristic, complemented
33C3×D9⋊C9—21no—
34C3×#4169—61no—
35C3×C9×D9—21no—
36C3×C9×D9C2×C311yescharacteristic, complemented
37C3×C9×D9C2×C311yescharacteristic, complemented
38C32×D32—91no—
39C2×C3×AGL(1,ℤ/9)—91no—
40C2×C32×D9—91no—
41D3×AGL(1,ℤ/9)—31no—
42D3×AGL(1,ℤ/9)—31no—
43D3×AGL(1,ℤ/9)—31no—
44C3×D3×D9—31no—
45C3×D3×D9—31no—
46C3×D3×D9—31no—
47C3×D3×D9—31no—
48C3×D3×D9—31no—
49D92C3211yescharacteristic, complemented
50C32×3−1+2C2×D311yescharacteristic
51C33×C9C2×D311yescharacteristic
52C9×3−1+2—21no—
53C9×3−1+2—21no—
54C9×3−1+2—21no—
55C9×3−1+2—21no—
56C9×3−1+2—21no—
57C9×3−1+2—21no—
58C9×3−1+2—41no—
59C9×3−1+2—41no—
60C9×3−1+2—41no—
61C9×3−1+2C22×C311yescharacteristic, complemented
62C9×3−1+2C22×C311yescharacteristic, complemented
63C9×3−1+2C22×C311yescharacteristic, complemented
64C3×C9⋊C9—21no—
65C3×C9⋊C9—41no—
66C3×C32⋊C9—21no—
67C3×C92—21no—
68C3×C92C22×C311yescharacteristic, complemented
69C32×C3⋊D3—91no—
70C33×D3—31no—
71C33×D3—31no—
72C2×C3×3−1+2—91no—
73C2×C32×C9—91no—
74C3⋊AGL(1,ℤ/9)—31no—
75C3⋊AGL(1,ℤ/9)—31no—
76C3⋊AGL(1,ℤ/9)—31no—
77C3×C3⋊D9—31no—
78C3×C3⋊D9—31no—
79C3×C3⋊D9—31no—
80C3×C3⋊D9—31no—
81C3×C3⋊D9—31no—
82C3×C3⋊D9—61no—
83D3×3−1+2—31no—
84D3×3−1+2—31no—
85D3×3−1+2—31no—
86D3×3−1+2—61no—
87D3×3−1+2—61no—
88D3×3−1+2—61no—
89C3×AGL(1,ℤ/9)—21no—
90C3×AGL(1,ℤ/9)—21no—
91C3×AGL(1,ℤ/9)—21no—
92C3×AGL(1,ℤ/9)—21no—
93C3×AGL(1,ℤ/9)—91no—
94C3×AGL(1,ℤ/9)—181no—
95C3×AGL(1,ℤ/9)D911yescharacteristic, direct product factor
96C3×AGL(1,ℤ/9)C3×D311yescharacteristic
97C3×AGL(1,ℤ/9)C3×D311yescharacteristic
98C3×AGL(1,ℤ/9)C3×D311yescharacteristic
99C3×D3×C9—31no—
100C3×D3×C9—31no—
101C3×D3×C9—31no—
102C3×D3×C9—31no—
103C3×D3×C9—31no—
104C3×D3×C9—61no—
105C3×D3×C9—61no—
106C3×D3×C9—61no—
107C3×D3×C9—61no—
108C3×D3×C9—61no—
109C32×D9—61no—
110C32×D9—61no—
111C32×D9—91no—
112C32×D9—181no—
113C32×D9C3×D311yescharacteristic
114C32×D9C3×D311yescharacteristic
115C32×D9C3×D311yescharacteristic
116C32×D9C3×D311yescharacteristic
117C32×D9C3×D311yescharacteristic
118C9⋊D9C2×C3211yescharacteristic, complemented
119D9⋊C9—21no—
120D9⋊C9—21no—
121D9⋊C9—21no—
122#4169—61no—
123#4169—61no—
124#4169—61no—
125C9×D9—21no—
126C9×D9—21no—
127C9×D9—21no—
128C9×D9—21no—
129C9×D9—21no—
130C9×D9C2×C3211yescharacteristic, complemented
131C9×D9C2×C3211yescharacteristic, complemented
132C2×C32×D3—271no—
133C2×C32×D3—271no—
134C3×D32—91no—
135C3×D32—91no—
136C3×D32—91no—
137C3×D32—91no—
138C2×AGL(1,ℤ/9)—91no—
139C2×AGL(1,ℤ/9)—91no—
140C2×AGL(1,ℤ/9)—91no—
141C2×C3×D9—91no—
142C2×C3×D9—91no—
143C2×C3×D9—91no—
144C2×C3×D9—91no—
145C2×C3×D9—91no—
146D3×D9—31no—
147D3×D9—31no—
148C34D3211yescharacteristic
149C3×3−1+2—21no—
150C3×3−1+2—21no—
151C3×3−1+2—21no—
152C3×3−1+2—21no—
153C3×3−1+2—21no—
154C3×3−1+2—21no—
155C3×3−1+2—21no—
156C3×3−1+2—21no—
157C3×3−1+2—41no—
158C3×3−1+2—41no—
159C3×3−1+2—41no—
160C3×3−1+2—41no—
161C3×3−1+2C2×D911yescharacteristic, complemented
162C3×3−1+2C2×C3×D311yescharacteristic
163C3×3−1+2C2×C3×D311yescharacteristic
164C3×3−1+2C2×C3×D311yescharacteristic
165C32×C9—21no—
166C32×C9—21no—
167C32×C9—21no—
168C32×C9—21no—
169C32×C9—21no—
170C32×C9—21no—
171C32×C9—41no—
172C32×C9—61no—
173C32×C9—61no—
174C32×C9—121no—
175C32×C9—121no—
176C32×C9C2×C3×D311yescharacteristic
177C32×C9C2×C3×D311yescharacteristic
178C32×C9C2×C3×D311yescharacteristic
179C32×C9C2×C3×D311yescharacteristic
180C32×C9C2×C3×D311yescharacteristic
181C9⋊C9—21no—
182C9⋊C9—21no—
183C9⋊C9—21no—
184C9⋊C9—41no—
185C9⋊C9—41no—
186C9⋊C9—41no—
187C9⋊C9—41no—
188C9⋊C9—41no—
189C9⋊C9—41no—
190C9⋊C9—41no—
191C9⋊C9—41no—
192C9⋊C9—41no—
193C9⋊C9—41no—
194C9⋊C9—41no—
195C9⋊C9—41no—
196C32⋊C9—21no—
197C32⋊C9—21no—
198C32⋊C9—21no—
199C32⋊C9—41no—
200C32⋊C9—41no—
201C32⋊C9—41no—
202C92—21no—
203C92—21no—
204C92—21no—
205C92—21no—
206C92—21no—
207C92—41no—
208C92—41no—
209C92—41no—
210C92—41no—
211C92C22×C3211yescharacteristic, complemented
212C2×C33—91no—
213C2×C33—91no—
214C3×C3⋊D3—91no—
215C3×C3⋊D3—91no—
216C3×C3⋊D3—91no—
217C3×C3⋊D3—91no—
218C32×D3—31no—
219C32×D3—31no—
220C32×D3—31no—
221C32×D3—31no—
222C32×D3—31no—
223C32×D3—31no—
224C32×D3—31no—
225C32×D3—31no—
226C32×D3—31no—
227C32×D3—61no—
228C32×D3—61no—
229C32×D3—61no—
230C32×D3—61no—
231C32×D3—91no—
232C32×D3—181no—
233C32×D3—271no—
234C32×D3—271no—
235C32×D3—541no—
236C2×3−1+2—91no—
237C2×3−1+2—91no—
238C2×3−1+2—91no—
239C2×3−1+2—181no—
240C2×3−1+2—181no—
241C2×3−1+2—181no—
242C2×C3×C9—91no—
243C2×C3×C9—91no—
244C2×C3×C9—91no—
245C2×C3×C9—91no—
246C2×C3×C9—91no—
247C2×C3×C9—181no—
248C2×C3×C9—181no—
249C2×C3×C9—181no—
250C2×C3×C9—181no—
251C2×C3×C9—181no—
252C3⋊D9—31no—
253C3⋊D9—31no—
254C3⋊D9—61no—
255AGL(1,ℤ/9)—21no—
256AGL(1,ℤ/9)—21no—
257AGL(1,ℤ/9)—21no—
258AGL(1,ℤ/9)—91no—
259AGL(1,ℤ/9)—91no—
260AGL(1,ℤ/9)—91no—
261AGL(1,ℤ/9)—181no—
262AGL(1,ℤ/9)—181no—
263AGL(1,ℤ/9)—181no—
264AGL(1,ℤ/9)C3×D911yescharacteristic, direct product factor
265AGL(1,ℤ/9)C3×D911yescharacteristic, direct product factor
266AGL(1,ℤ/9)C3×D911yescharacteristic, direct product factor
267D3×C9—31no—
268D3×C9—31no—
269D3×C9—61no—
270D3×C9—61no—
271D3×C9—61no—
272D3×C9—61no—
273D3×C9—61no—
274D3×C9—61no—
275D3×C9—61no—
276D3×C9—61no—
277C3×D9—21no—
278C3×D9—21no—
279C3×D9—61no—
280C3×D9—61no—
281C3×D9—61no—
282C3×D9—91no—
283C3×D9—91no—
284C3×D9—91no—
285C3×D9—91no—
286C3×D9—91no—
287C3×D9—181no—
288C3×D9—181no—
289C3×D9—181no—
290C3×D9—181no—
291C3×D9—181no—
292C3×D9—541no—
293C3×D9C3×D911yescharacteristic, complemented
294C3×D9AGL(1,ℤ/9)11yescharacteristic, direct product factor
295C3×D9C32×D311yescharacteristic
296C3×D9C32×D311yescharacteristic
297C22×C32—811no—
298C2×C3×D3—271no—
299C2×C3×D3—271no—
300C2×C3×D3—271no—
301C2×C3×D3—271no—
302C2×C3×D3—271no—
303C2×C3×D3—271no—
304C2×C3×D3—271no—
305C2×C3×D3—271no—
306D32—91no—
307C2×D9—91no—
308C2×D9—91no—
309C33—21no—
310C33—21no—
311C33—21no—
312C33—21no—
313C33—31no—
314C33—61no—
315C33—61no—
316C33—121no—
317C33D3×D911yescharacteristic
318C33C3×D3211yescharacteristic
319C33C3×D3211yescharacteristic
320C33C3×D3211yescharacteristic
321C33C3×D3211yescharacteristic
3223−1+2—21no—
3233−1+2—21no—
3243−1+2—21no—
3253−1+2—21no—
3263−1+2—21no—
3273−1+2—21no—
3283−1+2—21no—
3293−1+2—21no—
3303−1+2—21no—
3313−1+2—41no—
3323−1+2—41no—
3333−1+2—41no—
3343−1+2—41no—
3353−1+2—41no—
3363−1+2—41no—
3373−1+2—41no—
3383−1+2—41no—
3393−1+2—41no—
3403−1+2—41no—
3413−1+2—41no—
3423−1+2—41no—
3433−1+2—41no—
3443−1+2—41no—
3453−1+2—41no—
3463−1+2C2×C3×D911yescharacteristic, complemented
3473−1+2C2×C3×D911yescharacteristic, complemented
3483−1+2C2×C3×D911yescharacteristic, complemented
349C3×C9—21no—
350C3×C9—21no—
351C3×C9—21no—
352C3×C9—21no—
353C3×C9—21no—
354C3×C9—21no—
355C3×C9—21no—
356C3×C9—21no—
357C3×C9—21no—
358C3×C9—21no—
359C3×C9—21no—
360C3×C9—21no—
361C3×C9—21no—
362C3×C9—21no—
363C3×C9—41no—
364C3×C9—41no—
365C3×C9—41no—
366C3×C9—41no—
367C3×C9—41no—
368C3×C9—41no—
369C3×C9—41no—
370C3×C9—41no—
371C3×C9—41no—
372C3×C9—41no—
373C3×C9—61no—
374C3×C9—61no—
375C3×C9—61no—
376C3×C9—61no—
377C3×C9—61no—
378C3×C9—61no—
379C3×C9—61no—
380C3×C9—121no—
381C3×C9—121no—
382C3×C9—121no—
383C3×C9—121no—
384C3×C9—121no—
385C3×C9—121no—
386C3×C9—121no—
387C3×C9—121no—
388C3×C9—121no—
389C3×C9—121no—
390C3×C9—122no—
391C3×C9C2×C3×D911yescharacteristic, complemented
392C3×C9C2×AGL(1,ℤ/9)11yescharacteristic, complemented
393C3×C9C2×C32×D311yescharacteristic
394C3×C9C2×C32×D311yescharacteristic
395C2×C32—91no—
396C2×C32—91no—
397C2×C32—91no—
398C2×C32—91no—
399C2×C32—91no—
400C2×C32—91no—
401C2×C32—91no—
402C2×C32—91no—
403C2×C32—91no—
404C2×C32—181no—
405C2×C32—181no—
406C2×C32—181no—
407C2×C32—181no—
408C2×C32—271no—
409C2×C32—541no—
410C2×C32—811no—
411C3⋊D3—91no—
412C3×D3—31no—
413C3×D3—31no—
414C3×D3—31no—
415C3×D3—31no—
416C3×D3—31no—
417C3×D3—31no—
418C3×D3—31no—
419C3×D3—61no—
420C3×D3—61no—
421C3×D3—61no—
422C3×D3—61no—
423C3×D3—61no—
424C3×D3—91no—
425C3×D3—91no—
426C3×D3—91no—
427C3×D3—271no—
428C3×D3—271no—
429C3×D3—271no—
430C3×D3—271no—
431C3×D3—271no—
432C3×D3—271no—
433C3×D3—271no—
434C3×D3—271no—
435C3×D3—541no—
436C3×D3—541no—
437C3×D3—541no—
438C3×D3—541no—
439C2×C9—91no—
440C2×C9—91no—
441C2×C9—181no—
442C2×C9—181no—
443C2×C9—181no—
444C2×C9—181no—
445C2×C9—181no—
446C2×C9—181no—
447C2×C9—181no—
448C2×C9—181no—
449D9—91no—
450D9—91no—
451D9—181no—
452D9—181no—
453D9—541no—
454D9C32×D911yescharacteristic, complemented
455D9C3×AGL(1,ℤ/9)11yescharacteristic, direct product factor
456C22×C3—811no—
457C22×C3—811no—
458C22×C3—811no—
459C22×C3—811no—
460C2×D3—271no—
461C2×D3—271no—
462C32—21no—
463C32—21no—
464C32—21no—
465C32—21no—
466C32—21no—
467C32—21no—
468C32—31no—
469C32—31no—
470C32—31no—
471C32—31no—
472C32—41no—
473C32—61no—
474C32—61no—
475C32—61no—
476C32—121no—
477C32—121no—
478C32—121no—
479C32—121no—
480C32—121no—
481C32—121no—
482C32—121no—
483C32—121no—
484C32—121no—
485C32C3×D3×D911yescharacteristic
486C32C3×D3×D911yescharacteristic
487C32C3×D3×D911yescharacteristic
488C32C3×D3×D911yescharacteristic
489C32D3×AGL(1,ℤ/9)11yescharacteristic
490C32C32×D3211yescharacteristic
491C9—21no—
492C9—21no—
493C9—21no—
494C9—21no—
495C9—21no—
496C9—21no—
497C9—21no—
498C9—41no—
499C9—41no—
500C9—41no—
501C9—41no—
502C9—41no—
503C9—61no—
504C9—61no—
505C9—61no—
506C9—121no—
507C9—121no—
508C9—121no—
509C9—121no—
510C9—121no—
511C9C2×C32×D911yescharacteristic, complemented
512C9C2×C3×AGL(1,ℤ/9)11yescharacteristic, complemented
513C2×C3—91no—
514C2×C3—91no—
515C2×C3—91no—
516C2×C3—91no—
517C2×C3—91no—
518C2×C3—91no—
519C2×C3—91no—
520C2×C3—181no—
521C2×C3—181no—
522C2×C3—181no—
523C2×C3—181no—
524C2×C3—181no—
525C2×C3—271no—
526C2×C3—271no—
527C2×C3—541no—
528C2×C3—811no—
529C2×C3—811no—
530C2×C3—811no—
531C2×C3—811no—
532D3—31no—
533D3—31no—
534D3—271no—
535D3—271no—
536D3—541no—
537C22—811noSylow
538C3—21no—
539C3—21no—
540C3—21no—
541C3—31no—
542C3—31no—
543C3—41no—
544C3—61no—
545C3—61no—
546C3—61no—
547C3—121no—
548C3D9×AGL(1,ℤ/9)11yescharacteristic, direct product factor
549C3C32×D3×D911yescharacteristic
550C3C3×D3×AGL(1,ℤ/9)11yescharacteristic
551C2—91no—
552C2—91no—
553C2—811no—
subgroupG/H
center, hypercenterC3D9×AGL(1,ℤ/9)
commutator, cosocle, hypocenterC92C22×C32
FrattiniC32C32×D32
FittingC3×C9×3−1+2C22
socleC33C3×D32
LCSG ▷ C92
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. C3×#92299C2semidirect
2. C3×D9×3−1+2C2semidirect
3. C3×C9×AGL(1,ℤ/9)C2semidirect
6. D9×AGL(1,ℤ/9)C3direct (N × H)
7. D9×AGL(1,ℤ/9)C3direct (N × H)
8. D9×AGL(1,ℤ/9)C3direct (N × H)
9. C3×D92C3semidirect
10. C3×C9×3−1+2C22semidirect
17. #92299C2×C3semidirect
18. #92299C2×C3semidirect
19. #92299C2×C3semidirect
20. C3×C9⋊D9C2×C3semidirect
24. D9×3−1+2C2×C3semidirect
25. D9×3−1+2C2×C3semidirect
26. D9×3−1+2C2×C3semidirect
30. C9×AGL(1,ℤ/9)C2×C3semidirect
31. C9×AGL(1,ℤ/9)C2×C3semidirect
32. C9×AGL(1,ℤ/9)C2×C3semidirect
36. C3×C9×D9C2×C3semidirect
37. C3×C9×D9C2×C3semidirect
49. D92C32semidirect
61. C9×3−1+2C22×C3semidirect
62. C9×3−1+2C22×C3semidirect
63. C9×3−1+2C22×C3semidirect
68. C3×C92C22×C3semidirect
95. C3×AGL(1,ℤ/9)D9direct (N × H)
118. C9⋊D9C2×C32semidirect
130. C9×D9C2×C32semidirect
131. C9×D9C2×C32semidirect
161. C3×3−1+2C2×D9semidirect
211. C92C22×C32semidirect
264. AGL(1,ℤ/9)C3×D9direct (N × H)
265. AGL(1,ℤ/9)C3×D9direct (N × H)
266. AGL(1,ℤ/9)C3×D9direct (N × H)
293. C3×D9C3×D9semidirect
294. C3×D9AGL(1,ℤ/9)direct (N × H)
346. 3−1+2C2×C3×D9semidirect
347. 3−1+2C2×C3×D9semidirect
348. 3−1+2C2×C3×D9semidirect
391. C3×C9C2×C3×D9semidirect
392. C3×C9C2×AGL(1,ℤ/9)semidirect
454. D9C32×D9semidirect
455. D9C3×AGL(1,ℤ/9)direct (N × H)
511. C9C2×C32×D9semidirect
512. C9C2×C3×AGL(1,ℤ/9)semidirect
548. C3D9×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
C3×#922992
C3×D9×3−1+22
C3×C9×AGL(1,ℤ/9)2
C3×D3×AGL(1,ℤ/9)3
C32×D3×D93
D9×AGL(1,ℤ/9)3
C3×D923