Finite Group Databasebeta

← C3×#10589746 #13152921

Subgroups — C3×#10589746 #13152921

Order 1728.

824 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×C22⋊Q8C211yesmaximal, characteristic, complemented
2C3×#63509C212yesmaximal, complemented
3C3×#63492C211yesmaximal, characteristic, complemented
4C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
5C3×#63460C212yesmaximal, complemented
6C3×#63457C212yesmaximal, complemented
7C3×#63456C212yesmaximal, complemented
8C3×#63444C212yesmaximal, complemented
9C3×#611⋊C4C212yesmaximal, complemented
10C3×#5644—31nomaximal
11#10589746C311yesmaximal, characteristic, direct product factor
12C23×C3×D9C2211yescharacteristic
13C2×C3×Q8×C9C2211yescharacteristic, complemented
14C22×C3×C4×C9C2211yescharacteristic
15C2×C3×#611C2212yes—
16C22×C3×Dic9C2211yescharacteristic
17C2×C3×D36C2212yes—
18C2×C3×D36C2212yes—
19C2×C3×D36C2211yescharacteristic, complemented
20C2×C3×C4×D9C2212yescomplemented
21C3×C9×C4⋊C4C2211yescharacteristic, complemented
22C3×C9×C4⋊C4C2212yescomplemented
23C3×C9×C22⋊C4C2212yes—
24C3×(C2×D9)⋊C4C2212yes—
25C3×(C2×D9)⋊C4C2212yes—
26C3×(C2×D9)⋊C4C2212yes—
27C3×(C2×D9)⋊C4C2212yes—
28C3×(C2×D9)⋊C4C2212yescomplemented
29C3×(C2×D9)⋊C4C2212yescomplemented
30C3×C4⋊Dic9C2211yescharacteristic, complemented
31C3×Dic9⋊C4C2212yescomplemented
32C3×C4×Dic9C2212yescomplemented
33C32×C22⋊Q8D311yescharacteristic
34C3×#947—32no—
35C3×#930—31no—
36C2×C3×(C2×D3)⋊C4—31no—
37C3×#898—32no—
38C3×#895—32no—
39C3×#894—32no—
40C3×#882—32no—
41C3×#67⋊C4—32no—
42C9×C22⋊Q8—21no—
43C9×C22⋊Q8C2×C311yescharacteristic, complemented
44#63509C2×C312yescomplemented
45#63492C2×C311yescharacteristic, complemented
46C2×(C2×D9)⋊C4C2×C311yescharacteristic, complemented
47#63460C2×C312yescomplemented
48#63457C2×C312yescomplemented
49#63456C2×C312yescomplemented
50#63444C2×C312yescomplemented
51#611⋊C4C2×C312yescomplemented
52C23×C3×C9C2311yescharacteristic
53C22×C3×D9—22no—
54C22×C3×D9—41no—
55C22×C3×D9—41no—
56C22×C3×D9C2312yes—
57C22×C3×D9C2312yes—
58C3×Q8×C9—41no—
59C2×C3×C4×C9—41no—
60C2×C3×C4×C9C2312yes—
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C3×#611—24no—
65C2×C3×Dic9—41no—
66C2×C3×Dic9C2312yes—
67C2×C3×Dic9C2312yes—
68C3×D36—42no—
69C3×D36—42no—
70C3×D36—41no—
71C3×C4×D9—42no—
72C3×#538—91no—
73#5644—31no—
74C23×C3×D3—31no—
75C2×C32×Q8C2×D311yescharacteristic
76C22×C32×C4C2×D311yescharacteristic
77C2×C3×#67—32no—
78C22×C3×Dic3—31no—
79C2×C3×D12—32no—
80C2×C3×D12—32no—
81C2×C3×D12—31no—
82C2×C3×C4×D3—32no—
83C23×D9C22×C311yescharacteristic
84C32×C4⋊C4C2×D311yescharacteristic
85C32×C4⋊C4C2×D312yes—
86C32×C22⋊C4C2×D312yes—
87C3×(C2×D3)⋊C4—32no—
88C3×(C2×D3)⋊C4—32no—
89C3×(C2×D3)⋊C4—32no—
90C3×(C2×D3)⋊C4—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×(C2×D3)⋊C4—32no—
93C3×C4⋊Dic3—31no—
94C3×Dic3⋊C4—32no—
95C3×C4×Dic3—32no—
96C2×Q8×C9—21no—
97C2×Q8×C9C22×C311yescharacteristic, complemented
98C22×C4×C9—21no—
99C22×C4×C9C22×C311yescharacteristic
100C2×#611C22×C312yes—
101C22×Dic9C22×C311yescharacteristic
102C2×D36C22×C312yes—
103C2×D36C22×C312yes—
104C2×D36C22×C311yescharacteristic, complemented
105C2×C4×D9C22×C312yescomplemented
106C9×C4⋊C4—21no—
107C9×C4⋊C4—22no—
108C9×C4⋊C4C22×C311yescharacteristic, complemented
109C9×C4⋊C4C22×C312yescomplemented
110C9×C22⋊C4—22no—
111C9×C22⋊C4C22×C312yes—
112(C2×D9)⋊C4C22×C312yes—
113(C2×D9)⋊C4C22×C312yes—
114(C2×D9)⋊C4C22×C312yes—
115(C2×D9)⋊C4C22×C312yes—
116(C2×D9)⋊C4C22×C312yescomplemented
117(C2×D9)⋊C4C22×C312yescomplemented
118C4⋊Dic9C22×C311yescharacteristic, complemented
119Dic9⋊C4C22×C312yescomplemented
120C4×Dic9C22×C312yescomplemented
121C22×C3×C9—21no—
122C22×C3×C9—21no—
123C22×C3×C9#4112yes—
124C22×C3×C9C2411yescharacteristic
125C2×C3×D9—22no—
126C2×C3×D9—22no—
127C2×C3×D9—22no—
128C2×C3×D9—22no—
129C2×C3×D9—22no—
130C2×C3×D9—22no—
131C2×C3×D9—44no—
132C3×C4×C9—21no—
133C3×C4×C9—22no—
134C3×C4×C9—21no—
135C3×C4×C9—22no—
136C3×Dic9—22no—
137C3×Dic9—22no—
138C3×#126—92no—
139C3×#124—92no—
140C3×#123—92no—
141C3×C22⋊Q8—21no—
142C3×C22⋊Q8D911yescharacteristic, complemented
143C3×C22⋊Q8C3×D311yescharacteristic
144C3×#121—91no—
145C3×#121—92no—
146C3×#120—92no—
147C3×C4×D4—92no—
148C2×C3×C22⋊C4—91no—
149#947—32no—
150#930—31no—
151C2×(C2×D3)⋊C4—31no—
152#898—32no—
153#895—32no—
154#894—32no—
155#882—32no—
156#67⋊C4—32no—
157C23×C32C22×D311yescharacteristic
158C22×C3×D3—32no—
159C22×C3×D3—32no—
160C22×C3×D3—62no—
161C22×C3×D3—121no—
162C22×C3×D3—121no—
163C32×Q8—41no—
164C2×C32×C4—41no—
165C2×C32×C4C22×D312yes—
166C2×C32×C4C22×D312yes—
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C3×#67—64no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—121no—
173C3×D12—121no—
174C3×D12—122no—
175C3×D12—122no—
176C3×C4×D3—122no—
177C23×C9—21no—
178C23×C9C23×C311yescharacteristic
179C22×D9—22no—
180C22×D9—41no—
181C22×D9—41no—
182C22×D9C23×C312yes—
183C22×D9C23×C312yes—
184Q8×C9—41no—
185Q8×C9—42no—
186C2×C4×C9—21no—
187C2×C4×C9—22no—
188C2×C4×C9—22no—
189C2×C4×C9—21no—
190C2×C4×C9—41no—
191C2×C4×C9—42no—
192C2×C4×C9C23×C312yes—
193C2×C4×C9C23×C312yes—
194C2×C4×C9C23×C311yescharacteristic
195C2×C4×C9C23×C311yescharacteristic
196#611—24no—
197C2×Dic9—41no—
198C2×Dic9C23×C312yes—
199C2×Dic9C23×C312yes—
200D36—42no—
201D36—42no—
202D36—41no—
203C4×D9—42no—
204#538—91noSylow
205C2×C3×C9—22no—
206C2×C3×C9C2×#4111yescharacteristic
207C2×C3×C9H(5,𝔽2)11yescharacteristic
208C2×C3×C9H(5,𝔽2)11yescharacteristic
209C3×D9—42no—
210C3×D9—42no—
211C24×C3—91no—
212C23×D3—31no—
213C2×C3×Q8—21no—
214C2×C3×Q8C2×D911yescharacteristic, complemented
215C2×C3×Q8C2×C3×D311yescharacteristic
216C2×C3×D4—92no—
217C2×C3×D4—92no—
218C2×C3×D4—92no—
219C2×C3×D4—91no—
220C22×C3×C4—21no—
221C22×C3×C4—91no—
222C22×C3×C4—92no—
223C22×C3×C4C2×D911yescharacteristic
224C22×C3×C4C2×C3×D311yescharacteristic
225C2×#67—32no—
226C22×Dic3—31no—
227C2×D12—32no—
228C2×D12—32no—
229C2×D12—31no—
230C2×C4×D3—32no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4—91no—
235C3×C4⋊C4C2×D912yescomplemented
236C3×C4⋊C4C2×D911yescharacteristic, complemented
237C3×C4⋊C4C2×C3×D311yescharacteristic
238C3×C4⋊C4C2×C3×D312yes—
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×D912yes—
247C3×C22⋊C4C2×C3×D312yes—
248C3×C42—92no—
249(C2×D3)⋊C4—32no—
250(C2×D3)⋊C4—32no—
251(C2×D3)⋊C4—32no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—32no—
254(C2×D3)⋊C4—32no—
255C4⋊Dic3—31no—
256Dic3⋊C4—32no—
257C4×Dic3—32no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32#23912yes—
261C22×C32C23×D311yescharacteristic
262C2×C3×D3—62no—
263C2×C3×D3—62no—
264C2×C3×D3—62no—
265C2×C3×D3—62no—
266C2×C3×D3—62no—
267C2×C3×D3—62no—
268C2×C3×D3—124no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—22no—
272C32×C4—22no—
273C3×Dic3—62no—
274C3×Dic3—62no—
275C22×C9—21no—
276C22×C9—21no—
277C22×C9—21no—
278C22×C9—22no—
279C22×C9—41no—
280C22×C9—41no—
281C22×C9C3×#4112yes—
282C22×C9C24×C311yescharacteristic
283C2×D9—22no—
284C2×D9—22no—
285C2×D9—22no—
286C2×D9—22no—
287C2×D9—22no—
288C2×D9—22no—
289C2×D9—44no—
290C4×C9—22no—
291C4×C9—22no—
292C4×C9—21no—
293C4×C9—22no—
294C4×C9—21no—
295C4×C9—42no—
296C4×C9—41no—
297C4×C9—42no—
298Dic9—22no—
299Dic9—22no—
300#126—92no—
301#124—92no—
302#123—92no—
303C22⋊Q8C3×D911yescharacteristic, complemented
304#121—91no—
305#121—92no—
306#120—92no—
307C4×D4—92no—
308C2×C22⋊C4—91no—
309C3×C9#53811yescharacteristic, complemented, Sylow
310C23×C3—21no—
311C23×C3—92no—
312C23×C3—92no—
313C23×C3—182no—
314C23×C3—361no—
315C23×C3—361no—
316C23×C3C22×D911yescharacteristic
317C23×C3C22×C3×D311yescharacteristic
318C22×D3—32no—
319C22×D3—32no—
320C22×D3—62no—
321C22×D3—121no—
322C22×D3—121no—
323C3×Q8—41no—
324C3×Q8—41no—
325C3×Q8—42no—
326C3×D4—184no—
327C3×D4—361no—
328C3×D4—362no—
329C3×D4—362no—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—42no—
337C2×C3×C4—92no—
338C2×C3×C4—92no—
339C2×C3×C4—362no—
340C2×C3×C4—361no—
341C2×C3×C4C22×D912yes—
342C2×C3×C4C22×D911yescharacteristic
343C2×C3×C4C22×D911yescharacteristic
344C2×C3×C4C22×D912yes—
345C2×C3×C4C22×C3×D312yes—
346C2×C3×C4C22×C3×D312yes—
347C2×C3×C4C22×C3×D311yescharacteristic
348C2×C3×C4C22×C3×D311yescharacteristic
349#67—64no—
350C2×Dic3—32no—
351C2×Dic3—32no—
352C2×Dic3—121no—
353D12—121no—
354D12—122no—
355D12—122no—
356C4×D3—122no—
357C2×C32—22no—
358C2×C32#100411yescharacteristic
359C2×C32C2×#23911yescharacteristic
360C2×C32#100911yescharacteristic
361C3×D3—122no—
362C3×D3—122no—
363C2×C9—22no—
364C2×C9—21no—
365C2×C9—21no—
366C2×C9—21no—
367C2×C9—42no—
368C2×C9C2×C3×#4111yescharacteristic
369C2×C9C3×H(5,𝔽2)11yescharacteristic
370C2×C9C3×H(5,𝔽2)11yescharacteristic
371D9—42no—
372D9—42no—
373C24—91no—
374C2×Q8C2×C3×D911yescharacteristic, complemented
375C2×D4—92no—
376C2×D4—92no—
377C2×D4—92no—
378C2×D4—91no—
379C22×C4—91no—
380C22×C4—92no—
381C22×C4C2×C3×D911yescharacteristic
382C4⋊C4—92no—
383C4⋊C4—91no—
384C4⋊C4C2×C3×D912yescomplemented
385C4⋊C4C2×C3×D911yescharacteristic, complemented
386C22⋊C4—92no—
387C22⋊C4—92no—
388C22⋊C4—92no—
389C22⋊C4—92no—
390C22⋊C4—92no—
391C22⋊C4—92no—
392C22⋊C4C2×C3×D912yes—
393C42—92no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—22no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—182no—
403C22×C3—182no—
404C22×C3—182no—
405C22×C3—182no—
406C22×C3—182no—
407C22×C3—182no—
408C22×C3—364no—
409C22×C3#370112yes—
410C22×C3C23×D911yescharacteristic
411C22×C3C3×#23912yes—
412C22×C3C23×C3×D311yescharacteristic
413C2×D3—62no—
414C2×D3—62no—
415C2×D3—62no—
416C2×D3—62no—
417C2×D3—62no—
418C2×D3—62no—
419C2×D3—124no—
420C3×C4—22no—
421C3×C4—21no—
422C3×C4—22no—
423C3×C4—21no—
424C3×C4—22no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—22no—
428C3×C4—22no—
429C3×C4—41no—
430C3×C4—42no—
431C3×C4—42no—
432C3×C4—182no—
433C3×C4—182no—
434Dic3—62no—
435Dic3—62no—
436C32#564411yescharacteristic
437C9—21no—
438C9C3×#53811yescharacteristic, complemented
439C23—92no—
440C23—92no—
441C23—182no—
442C23—361no—
443C23—361no—
444C23C22×C3×D911yescharacteristic
445Q8—41no—
446D4—184no—
447D4—361no—
448D4—362no—
449D4—362no—
450C2×C4—41no—
451C2×C4—92no—
452C2×C4—92no—
453C2×C4—362no—
454C2×C4—361no—
455C2×C4C22×C3×D912yes—
456C2×C4C22×C3×D911yescharacteristic
457C2×C4C22×C3×D911yescharacteristic
458C2×C4C22×C3×D912yes—
459C2×C3—22no—
460C2×C3—22no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—42no—
465C2×C3—362no—
466C2×C3—362no—
467C2×C3#6371011yescharacteristic
468C2×C3C2×#370111yescharacteristic
469C2×C3#6371511yescharacteristic
470C2×C3C3×#100411yescharacteristic
471C2×C3C2×C3×#23911yescharacteristic
472C2×C3C3×#100911yescharacteristic
473D3—122no—
474D3—122no—
475C22—21no—
476C22—21no—
477C22—182no—
478C22—182no—
479C22—182no—
480C22—182no—
481C22—182no—
482C22—182no—
483C22—364no—
484C22C3×#370112yes—
485C22C23×C3×D911yescharacteristic
486C4—21no—
487C4—21no—
488C4—22no—
489C4—22no—
490C4—182no—
491C4—182no—
492C3—21no—
493C3#1058974611yescharacteristic, direct product factor
494C3C3×#564411yescharacteristic
495C2—22no—
496C2—362no—
497C2—362no—
498C2C3×#6371011yescharacteristic
499C2C2×C3×#370111yescharacteristic
500C2C3×#6371511yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×C22⋊Q8C2
socleC22×C32C23×D3
hypercenterC3×C22⋊Q8D9
hypocenterC9C3×#538
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×C9×C22⋊Q8C2semidirect
2. C3×#63509C2semidirect
3. C3×#63492C2semidirect
4. C2×C3×(C2×D9)⋊C4C2semidirect
5. C3×#63460C2semidirect
6. C3×#63457C2semidirect
7. C3×#63456C2semidirect
8. C3×#63444C2semidirect
9. C3×#611⋊C4C2semidirect
11. #10589746C3direct (N × H)
13. C2×C3×Q8×C9C22semidirect
19. C2×C3×D36C22semidirect
20. C2×C3×C4×D9C22semidirect
21. C3×C9×C4⋊C4C22semidirect
22. C3×C9×C4⋊C4C22semidirect
28. C3×(C2×D9)⋊C4C22semidirect
29. C3×(C2×D9)⋊C4C22semidirect
30. C3×C4⋊Dic9C22semidirect
31. C3×Dic9⋊C4C22semidirect
32. C3×C4×Dic9C22semidirect
43. C9×C22⋊Q8C2×C3semidirect
44. #63509C2×C3semidirect
45. #63492C2×C3semidirect
46. C2×(C2×D9)⋊C4C2×C3semidirect
47. #63460C2×C3semidirect
48. #63457C2×C3semidirect
49. #63456C2×C3semidirect
50. #63444C2×C3semidirect
51. #611⋊C4C2×C3semidirect
97. C2×Q8×C9C22×C3semidirect
104. C2×D36C22×C3semidirect
105. C2×C4×D9C22×C3semidirect
108. C9×C4⋊C4C22×C3semidirect
109. C9×C4⋊C4C22×C3semidirect
116. (C2×D9)⋊C4C22×C3semidirect
117. (C2×D9)⋊C4C22×C3semidirect
118. C4⋊Dic9C22×C3semidirect
119. Dic9⋊C4C22×C3semidirect
120. C4×Dic9C22×C3semidirect
142. C3×C22⋊Q8D9semidirect
214. C2×C3×Q8C2×D9semidirect
235. C3×C4⋊C4C2×D9semidirect
236. C3×C4⋊C4C2×D9semidirect
303. C22⋊Q8C3×D9semidirect
309. C3×C9#538semidirect
374. C2×Q8C2×C3×D9semidirect
384. C4⋊C4C2×C3×D9semidirect
385. C4⋊C4C2×C3×D9semidirect
438. C9C3×#538semidirect
493. C3#10589746direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×C22⋊Q82
C3×#635092
C3×#634922
C2×C3×(C2×D9)⋊C42
C3×#634602
C3×#634572
C3×#634562
C3×#634442
C3×#611⋊C42
C3×#56443
#105897463