Finite Group Databasebeta

← C3×#10589719 #13152818

Subgroups — C3×#10589719 #13152818

Order 1728.

844 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#121C211yesmaximal, characteristic, complemented
2C3×#63501C212yesmaximal, complemented
3C3×#63499C212yesmaximal, complemented
4C3×D4×Dic9C212yesmaximal, complemented
5C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
6C3×#63486C211yesmaximal, characteristic, complemented
7C3×#63460C212yesmaximal, complemented
8C3×#63448C212yesmaximal, complemented
9C3×#63447C212yesmaximal, complemented
10C3×#5617—31nomaximal
11#10589719C311yesmaximal, characteristic, direct product factor
12C23×C3×D9C2211yescharacteristic
13C2×C3×D4×C9C2212yes—
14C2×C3×D4×C9C2211yescharacteristic, complemented
15C22×C3×C4×C9C2211yescharacteristic
16C2×C3×#611C2212yes—
17C2×C3×#611C2212yes—
18C22×C3×Dic9C2211yescharacteristic
19C22×C3×Dic9C2212yescomplemented
20C2×C3×Dic18C2211yescharacteristic, complemented
21C3×C9×C4⋊C4C2211yescharacteristic, complemented
22C3×C9×C22⋊C4C2212yescomplemented
23C3×#3676C2212yes—
24C3×#3676C2212yescomplemented
25C3×(C2×D9)⋊C4C2212yes—
26C3×(C2×D9)⋊C4C2212yes—
27C3×(C2×D9)⋊C4C2212yes—
28C3×(C2×D9)⋊C4C2212yes—
29C3×C4⋊Dic9C2211yescharacteristic, complemented
30C3×C4⋊Dic9C2212yescomplemented
31C3×Dic9⋊C4C2212yescomplemented
32C3×C4×Dic9C2212yescomplemented
33C32×#121D311yescharacteristic
34C3×#939—32no—
35C3×#937—32no—
36C3×D4×Dic3—32no—
37C2×C3×(C2×D3)⋊C4—31no—
38C3×#924—31no—
39C3×#898—32no—
40C3×#886—32no—
41C3×#885—32no—
42C9×#121—21no—
43C9×#121C2×C311yescharacteristic, complemented
44#63501C2×C312yescomplemented
45#63499C2×C312yescomplemented
46D4×Dic9C2×C312yescomplemented
47C2×(C2×D9)⋊C4C2×C311yescharacteristic, complemented
48#63486C2×C311yescharacteristic, complemented
49#63460C2×C312yescomplemented
50#63448C2×C312yescomplemented
51#63447C2×C312yescomplemented
52C23×C3×C9C2312yes—
53C23×C3×C9C2311yescharacteristic
54C22×C3×D9—22no—
55C22×C3×D9—41no—
56C22×C3×D9—41no—
57C22×C3×D9C2312yes—
58C3×D4×C9—24no—
59C3×D4×C9—41no—
60C2×C3×C4×C9—41no—
61C2×C3×C4×C9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C3×#611—42no—
65C3×#611—42no—
66C2×C3×Dic9—41no—
67C2×C3×Dic9—42no—
68C2×C3×Dic9C2312yes—
69C2×C3×Dic9C2312yes—
70C2×C3×Dic9C2312yes—
71C3×Dic18—41no—
72C3×#538—91no—
73#5617—31no—
74C23×C3×D3—31no—
75C2×C32×D4C2×D312yes—
76C2×C32×D4C2×D311yescharacteristic
77C22×C32×C4C2×D311yescharacteristic
78C2×C3×#67—32no—
79C2×C3×#67—32no—
80C22×C3×Dic3—32no—
81C22×C3×Dic3—31no—
82C2×C3×Dic6—31no—
83C23×D9C22×C311yescharacteristic
84C32×C4⋊C4C2×D311yescharacteristic
85C32×C22⋊C4C2×D312yes—
86C3×#217—32no—
87C3×#217—32no—
88C3×(C2×D3)⋊C4—32no—
89C3×(C2×D3)⋊C4—32no—
90C3×(C2×D3)⋊C4—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—32no—
94C3×Dic3⋊C4—32no—
95C3×C4×Dic3—32no—
96C2×D4×C9—22no—
97C2×D4×C9—21no—
98C2×D4×C9C22×C312yes—
99C2×D4×C9C22×C311yescharacteristic, complemented
100C22×C4×C9—21no—
101C22×C4×C9C22×C311yescharacteristic
102C2×#611C22×C312yes—
103C2×#611C22×C312yes—
104C22×Dic9C22×C311yescharacteristic
105C22×Dic9C22×C312yescomplemented
106C2×Dic18C22×C311yescharacteristic, complemented
107C9×C4⋊C4—21no—
108C9×C4⋊C4C22×C311yescharacteristic, complemented
109C9×C22⋊C4—22no—
110C9×C22⋊C4C22×C312yescomplemented
111#3676C22×C312yes—
112#3676C22×C312yescomplemented
113(C2×D9)⋊C4C22×C312yes—
114(C2×D9)⋊C4C22×C312yes—
115(C2×D9)⋊C4C22×C312yes—
116(C2×D9)⋊C4C22×C312yes—
117C4⋊Dic9C22×C312yescomplemented
118C4⋊Dic9C22×C311yescharacteristic, complemented
119Dic9⋊C4C22×C312yescomplemented
120C4×Dic9C22×C312yescomplemented
121C22×C3×C9—21no—
122C22×C3×C9—21no—
123C22×C3×C9—22no—
124C22×C3×C9—22no—
125C22×C3×C9—22no—
126C22×C3×C9#4112yes—
127C22×C3×C9C2411yescharacteristic
128C2×C3×D9—22no—
129C2×C3×D9—22no—
130C2×C3×D9—22no—
131C2×C3×D9—44no—
132C3×C4×C9—22no—
133C3×C4×C9—21no—
134C3×C4×C9—21no—
135C3×Dic9—22no—
136C3×Dic9—22no—
137C3×Dic9—22no—
138C3×#126—92no—
139C3×#124—92no—
140C3×#123—92no—
141C3×C22⋊Q8—91no—
142C3×#121—21no—
143C3×#121—92no—
144C3×#121D911yescharacteristic, complemented
145C3×#121C3×D311yescharacteristic
146C3×#120—92no—
147C3×C4×D4—92no—
148C2×C3×C22⋊C4—91no—
149#939—32no—
150#937—32no—
151D4×Dic3—32no—
152C2×(C2×D3)⋊C4—31no—
153#924—31no—
154#898—32no—
155#886—32no—
156#885—32no—
157C23×C32C22×D312yes—
158C23×C32C22×D311yescharacteristic
159C22×C3×D3—32no—
160C22×C3×D3—62no—
161C22×C3×D3—121no—
162C22×C3×D3—121no—
163C32×D4—24no—
164C32×D4—41no—
165C2×C32×C4—41no—
166C2×C32×C4C22×D312yes—
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C3×#67—122no—
170C3×#67—122no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—121no—
175C2×C3×Dic3—122no—
176C3×Dic6—121no—
177C23×C9—21no—
178C23×C9—22no—
179C23×C9C23×C312yes—
180C23×C9C23×C311yescharacteristic
181C22×D9—22no—
182C22×D9—41no—
183C22×D9—41no—
184C22×D9C23×C312yes—
185D4×C9—24no—
186D4×C9—44no—
187D4×C9—41no—
188D4×C9—42no—
189C2×C4×C9—21no—
190C2×C4×C9—21no—
191C2×C4×C9—22no—
192C2×C4×C9—41no—
193C2×C4×C9—42no—
194C2×C4×C9C23×C312yes—
195C2×C4×C9C23×C311yescharacteristic
196C2×C4×C9C23×C311yescharacteristic
197#611—42no—
198#611—42no—
199C2×Dic9—41no—
200C2×Dic9—42no—
201C2×Dic9C23×C312yes—
202C2×Dic9C23×C312yes—
203C2×Dic9C23×C312yes—
204Dic18—41no—
205#538—91noSylow
206C2×C3×C9—22no—
207C2×C3×C9—42no—
208C2×C3×C9C2×#4111yescharacteristic
209C2×C3×C9H(5,𝔽2)11yescharacteristic
210C2×C3×C9H(5,𝔽2)11yescharacteristic
211C3×D9—42no—
212C24×C3—91no—
213C23×D3—31no—
214C2×C3×Q8—91no—
215C2×C3×D4—22no—
216C2×C3×D4—21no—
217C2×C3×D4—92no—
218C2×C3×D4—92no—
219C2×C3×D4C2×D912yes—
220C2×C3×D4C2×D911yescharacteristic, complemented
221C2×C3×D4C2×C3×D312yes—
222C2×C3×D4C2×C3×D311yescharacteristic
223C22×C3×C4—21no—
224C22×C3×C4—91no—
225C22×C3×C4—92no—
226C22×C3×C4C2×D911yescharacteristic
227C22×C3×C4C2×C3×D311yescharacteristic
228C2×#67—32no—
229C2×#67—32no—
230C22×Dic3—32no—
231C22×Dic3—31no—
232C2×Dic6—31no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—91no—
237C3×C4⋊C4C2×D911yescharacteristic, complemented
238C3×C4⋊C4C2×C3×D311yescharacteristic
239C3×C22⋊C4—22no—
240C3×C22⋊C4—92no—
241C3×C22⋊C4—92no—
242C3×C22⋊C4—92no—
243C3×C22⋊C4—92no—
244C3×C22⋊C4—92no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4C2×D912yescomplemented
247C3×C22⋊C4C2×C3×D312yes—
248C3×C42—92no—
249#217—32no—
250#217—32no—
251(C2×D3)⋊C4—32no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—32no—
254(C2×D3)⋊C4—32no—
255C4⋊Dic3—31no—
256C4⋊Dic3—32no—
257Dic3⋊C4—32no—
258C4×Dic3—32no—
259C22×C32—21no—
260C22×C32—21no—
261C22×C32—22no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32#23712yes—
265C22×C32C23×D311yescharacteristic
266C2×C3×D3—62no—
267C2×C3×D3—62no—
268C2×C3×D3—62no—
269C2×C3×D3—124no—
270C32×C4—21no—
271C32×C4—22no—
272C32×C4—21no—
273C3×Dic3—62no—
274C3×Dic3—62no—
275C3×Dic3—62no—
276C22×C9—21no—
277C22×C9—21no—
278C22×C9—22no—
279C22×C9—22no—
280C22×C9—22no—
281C22×C9—21no—
282C22×C9—22no—
283C22×C9—41no—
284C22×C9—41no—
285C22×C9—42no—
286C22×C9—42no—
287C22×C9—42no—
288C22×C9C3×#4112yes—
289C22×C9C24×C311yescharacteristic
290C2×D9—22no—
291C2×D9—22no—
292C2×D9—22no—
293C2×D9—44no—
294C4×C9—22no—
295C4×C9—22no—
296C4×C9—21no—
297C4×C9—21no—
298C4×C9—41no—
299C4×C9—42no—
300Dic9—22no—
301Dic9—22no—
302Dic9—22no—
303#126—92no—
304#124—92no—
305#123—92no—
306C22⋊Q8—91no—
307#121—92no—
308#121C3×D911yescharacteristic, complemented
309#120—92no—
310C4×D4—92no—
311C2×C22⋊C4—91no—
312C3×C9#53811yescharacteristic, complemented, Sylow
313C23×C3—21no—
314C23×C3—22no—
315C23×C3—92no—
316C23×C3—182no—
317C23×C3—361no—
318C23×C3—361no—
319C23×C3C22×D912yes—
320C23×C3C22×D911yescharacteristic
321C23×C3C22×C3×D312yes—
322C23×C3C22×C3×D311yescharacteristic
323C22×D3—32no—
324C22×D3—62no—
325C22×D3—121no—
326C22×D3—121no—
327C3×Q8—361no—
328C3×D4—24no—
329C3×D4—24no—
330C3×D4—44no—
331C3×D4—41no—
332C3×D4—41no—
333C3×D4—42no—
334C3×D4—362no—
335C3×D4—362no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—42no—
342C2×C3×C4—92no—
343C2×C3×C4—92no—
344C2×C3×C4—92no—
345C2×C3×C4—361no—
346C2×C3×C4—362no—
347C2×C3×C4C22×D912yes—
348C2×C3×C4C22×D911yescharacteristic
349C2×C3×C4C22×D911yescharacteristic
350C2×C3×C4C22×C3×D312yes—
351C2×C3×C4C22×C3×D311yescharacteristic
352C2×C3×C4C22×C3×D311yescharacteristic
353#67—122no—
354#67—122no—
355C2×Dic3—32no—
356C2×Dic3—32no—
357C2×Dic3—32no—
358C2×Dic3—121no—
359C2×Dic3—122no—
360Dic6—121no—
361C2×C32—22no—
362C2×C32—42no—
363C2×C32C2×#23711yescharacteristic
364C2×C32#100411yescharacteristic
365C2×C32#100911yescharacteristic
366C3×D3—122no—
367C2×C9—22no—
368C2×C9—21no—
369C2×C9—21no—
370C2×C9—21no—
371C2×C9—42no—
372C2×C9—42no—
373C2×C9—82no—
374C2×C9C2×C3×#4111yescharacteristic
375C2×C9C3×H(5,𝔽2)11yescharacteristic
376C2×C9C3×H(5,𝔽2)11yescharacteristic
377D9—42no—
378C24—91no—
379C2×Q8—91no—
380C2×D4—92no—
381C2×D4—92no—
382C2×D4C2×C3×D912yes—
383C2×D4C2×C3×D911yescharacteristic, complemented
384C22×C4—91no—
385C22×C4—92no—
386C22×C4C2×C3×D911yescharacteristic
387C4⋊C4—92no—
388C4⋊C4—92no—
389C4⋊C4—91no—
390C4⋊C4C2×C3×D911yescharacteristic, complemented
391C22⋊C4—92no—
392C22⋊C4—92no—
393C22⋊C4—92no—
394C22⋊C4—92no—
395C22⋊C4—92no—
396C22⋊C4—92no—
397C22⋊C4C2×C3×D912yescomplemented
398C42—92no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—21no—
410C22×C3—22no—
411C22×C3—41no—
412C22×C3—41no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—182no—
417C22×C3—182no—
418C22×C3—182no—
419C22×C3—364no—
420C22×C3#369912yes—
421C22×C3C23×D911yescharacteristic
422C22×C3C3×#23712yes—
423C22×C3C23×C3×D311yescharacteristic
424C2×D3—62no—
425C2×D3—62no—
426C2×D3—62no—
427C2×D3—124no—
428C3×C4—22no—
429C3×C4—22no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—22no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—41no—
436C3×C4—42no—
437C3×C4—182no—
438C3×C4—182no—
439C3×C4—182no—
440Dic3—62no—
441Dic3—62no—
442Dic3—62no—
443C32#561711yescharacteristic
444C9—21no—
445C9C3×#53811yescharacteristic, complemented
446C23—92no—
447C23—182no—
448C23—361no—
449C23—361no—
450C23C22×C3×D912yes—
451C23C22×C3×D911yescharacteristic
452Q8—361no—
453D4—24no—
454D4—41no—
455D4—362no—
456D4—362no—
457C2×C4—41no—
458C2×C4—92no—
459C2×C4—92no—
460C2×C4—92no—
461C2×C4—361no—
462C2×C4—362no—
463C2×C4C22×C3×D912yes—
464C2×C4C22×C3×D911yescharacteristic
465C2×C4C22×C3×D911yescharacteristic
466C2×C3—22no—
467C2×C3—22no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—42no—
472C2×C3—42no—
473C2×C3—42no—
474C2×C3—82no—
475C2×C3—362no—
476C2×C3C2×#369911yescharacteristic
477C2×C3#6371011yescharacteristic
478C2×C3#6371511yescharacteristic
479C2×C3C2×C3×#23711yescharacteristic
480C2×C3C3×#100411yescharacteristic
481C2×C3C3×#100911yescharacteristic
482D3—122no—
483C22—21no—
484C22—21no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—182no—
489C22—182no—
490C22—182no—
491C22—364no—
492C22C3×#369912yes—
493C22C23×C3×D911yescharacteristic
494C4—22no—
495C4—21no—
496C4—21no—
497C4—182no—
498C4—182no—
499C4—182no—
500C3—21no—
501C3#1058971911yescharacteristic, direct product factor
502C3C3×#561711yescharacteristic
503C2—22no—
504C2—42no—
505C2—362no—
506C2C2×C3×#369911yescharacteristic
507C2C3×#6371011yescharacteristic
508C2C3×#6371511yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#121C2
socleC22×C32C23×D3
hypercenterC3×#121D9
hypocenterC9C3×#538
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
1. C3×C9×#121C2semidirect
2. C3×#63501C2semidirect
3. C3×#63499C2semidirect
4. C3×D4×Dic9C2semidirect
5. C2×C3×(C2×D9)⋊C4C2semidirect
6. C3×#63486C2semidirect
7. C3×#63460C2semidirect
8. C3×#63448C2semidirect
9. C3×#63447C2semidirect
11. #10589719C3direct (N × H)
14. C2×C3×D4×C9C22semidirect
19. C22×C3×Dic9C22semidirect
20. C2×C3×Dic18C22semidirect
21. C3×C9×C4⋊C4C22semidirect
22. C3×C9×C22⋊C4C22semidirect
24. C3×#3676C22semidirect
29. C3×C4⋊Dic9C22semidirect
30. C3×C4⋊Dic9C22semidirect
31. C3×Dic9⋊C4C22semidirect
32. C3×C4×Dic9C22semidirect
43. C9×#121C2×C3semidirect
44. #63501C2×C3semidirect
45. #63499C2×C3semidirect
46. D4×Dic9C2×C3semidirect
47. C2×(C2×D9)⋊C4C2×C3semidirect
48. #63486C2×C3semidirect
49. #63460C2×C3semidirect
50. #63448C2×C3semidirect
51. #63447C2×C3semidirect
99. C2×D4×C9C22×C3semidirect
105. C22×Dic9C22×C3semidirect
106. C2×Dic18C22×C3semidirect
108. C9×C4⋊C4C22×C3semidirect
110. C9×C22⋊C4C22×C3semidirect
112. #3676C22×C3semidirect
117. C4⋊Dic9C22×C3semidirect
118. C4⋊Dic9C22×C3semidirect
119. Dic9⋊C4C22×C3semidirect
120. C4×Dic9C22×C3semidirect
144. C3×#121D9semidirect
220. C2×C3×D4C2×D9semidirect
237. C3×C4⋊C4C2×D9semidirect
246. C3×C22⋊C4C2×D9semidirect
308. #121C3×D9semidirect
312. C3×C9#538semidirect
383. C2×D4C2×C3×D9semidirect
390. C4⋊C4C2×C3×D9semidirect
397. C22⋊C4C2×C3×D9semidirect
445. C9C3×#538semidirect
501. C3#10589719direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×#1212
C3×#635012
C3×#634992
C3×D4×Dic92
C2×C3×(C2×D9)⋊C42
C3×#634862
C3×#634602
C3×#634482
C3×#634472
C3×#56173
#105897193