Finite Group Databasebeta

← C3×#10589768 #13153021

Subgroups — C3×#10589768 #13153021

Order 1728.

834 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×#123C211yesmaximal, characteristic, complemented
2C3×#63502C211yesmaximal, characteristic, complemented
3C3×#63499C211yesmaximal, characteristic, complemented
4C3×#63492C212yesmaximal, complemented
5C3×#63460C212yesmaximal, complemented
6C3×#63457C212yesmaximal, complemented
7C3×#63446C212yesmaximal, complemented
8C3×#63445C212yesmaximal, complemented
9C3×#63444C212yesmaximal, complemented
10C3×#5666—31nomaximal
11#10589768C311yesmaximal, characteristic, direct product factor
12C23×C3×D9C2211yescharacteristic
13C2×C3×D4×C9C2211yescharacteristic
14C22×C3×C4×C9C2211yescharacteristic, complemented
15C2×C3×#611C2212yes—
16C2×C3×#611C2212yes—
17C2×C3×#611C2212yescomplemented
18C2×C3×D36C2212yes—
19C2×C3×D36C2212yes—
20C2×C3×D36C2211yescharacteristic, complemented
21C2×C3×C4×D9C2212yescomplemented
22C3×C9×C4⋊C4C2212yescomplemented
23C3×C9×C22⋊C4C2212yes—
24C3×C9×C22⋊C4C2211yescharacteristic, complemented
25C3×#3676C2211yescharacteristic
26C3×(C2×D9)⋊C4C2212yes—
27C3×(C2×D9)⋊C4C2212yes—
28C3×(C2×D9)⋊C4C2212yes—
29C3×(C2×D9)⋊C4C2212yescomplemented
30C3×C4⋊Dic9C2212yescomplemented
31C3×Dic9⋊C4C2212yescomplemented
32C3×C4×Dic9C2211yescharacteristic, complemented
33C32×#123D311yescharacteristic
34C3×#940—31no—
35C3×#937—31no—
36C3×#930—32no—
37C3×#898—32no—
38C3×#895—32no—
39C3×#884—32no—
40C3×#883—32no—
41C3×#882—32no—
42C9×#123—21no—
43C9×#123C2×C311yescharacteristic, complemented
44#63502C2×C311yescharacteristic, complemented
45#63499C2×C311yescharacteristic, complemented
46#63492C2×C312yescomplemented
47#63460C2×C312yescomplemented
48#63457C2×C312yescomplemented
49#63446C2×C312yescomplemented
50#63445C2×C312yescomplemented
51#63444C2×C312yescomplemented
52C23×C3×C9C2311yescharacteristic
53C23×C3×C9C2311yescharacteristic
54C22×C3×D9—41no—
55C22×C3×D9—42no—
56C22×C3×D9C2312yes—
57C22×C3×D9C2312yes—
58C3×D4×C9—41no—
59C2×C3×C4×C9—41no—
60C2×C3×C4×C9C2312yes—
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C3×#611—42no—
64C3×#611—42no—
65C3×#611—42no—
66C2×C3×Dic9C2312yes—
67C2×C3×Dic9C2312yes—
68C3×D36—42no—
69C3×D36—42no—
70C3×D36—41no—
71C3×C4×D9—42no—
72C3×#560—91no—
73#5666—31no—
74C23×C3×D3—31no—
75C2×C32×D4C2×D311yescharacteristic
76C22×C32×C4C2×D311yescharacteristic
77C2×C3×#67—32no—
78C2×C3×#67—32no—
79C2×C3×#67—32no—
80C2×C3×D12—32no—
81C2×C3×D12—32no—
82C2×C3×D12—31no—
83C2×C3×C4×D3—32no—
84C23×D9C22×C311yescharacteristic
85C32×C4⋊C4C2×D312yes—
86C32×C22⋊C4C2×D312yes—
87C32×C22⋊C4C2×D311yescharacteristic
88C3×#217—31no—
89C3×(C2×D3)⋊C4—32no—
90C3×(C2×D3)⋊C4—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×(C2×D3)⋊C4—32no—
93C3×C4⋊Dic3—32no—
94C3×Dic3⋊C4—32no—
95C3×C4×Dic3—31no—
96C2×D4×C9—21no—
97C2×D4×C9C22×C311yescharacteristic
98C22×C4×C9—21no—
99C22×C4×C9C22×C311yescharacteristic, complemented
100C2×#611C22×C312yes—
101C2×#611C22×C312yes—
102C2×#611C22×C312yescomplemented
103C2×D36C22×C312yes—
104C2×D36C22×C312yes—
105C2×D36C22×C311yescharacteristic, complemented
106C2×C4×D9C22×C312yescomplemented
107C9×C4⋊C4—22no—
108C9×C4⋊C4C22×C312yescomplemented
109C9×C22⋊C4—22no—
110C9×C22⋊C4—21no—
111C9×C22⋊C4C22×C312yes—
112C9×C22⋊C4C22×C311yescharacteristic, complemented
113#3676C22×C311yescharacteristic
114(C2×D9)⋊C4C22×C312yes—
115(C2×D9)⋊C4C22×C312yes—
116(C2×D9)⋊C4C22×C312yes—
117(C2×D9)⋊C4C22×C312yescomplemented
118C4⋊Dic9C22×C312yescomplemented
119Dic9⋊C4C22×C312yescomplemented
120C4×Dic9C22×C311yescharacteristic, complemented
121C22×C3×C9—21no—
122C22×C3×C9—22no—
123C22×C3×C9—22no—
124C22×C3×C9—21no—
125C22×C3×C9C2411yescharacteristic
126C2×C3×D9—22no—
127C2×C3×D9—22no—
128C2×C3×D9—22no—
129C2×C3×D9—22no—
130C2×C3×D9—22no—
131C2×C3×D9—22no—
132C2×C3×D9—44no—
133C3×C4×C9—22no—
134C3×C4×C9—22no—
135C3×C4×C9—21no—
136C3×Dic9—22no—
137C3×Dic9—22no—
138C3×#127—91no—
139C3×#126—92no—
140C3×#123—21no—
141C3×#123—92no—
142C3×#123D911yescharacteristic, complemented
143C3×#123C3×D311yescharacteristic
144C3×#121—92no—
145C3×#121—92no—
146C3×#121—92no—
147C3×#120—91no—
148C3×#120—92no—
149#940—31no—
150#937—31no—
151#930—32no—
152#898—32no—
153#895—32no—
154#884—32no—
155#883—32no—
156#882—32no—
157C23×C32C22×D311yescharacteristic
158C23×C32C22×D311yescharacteristic
159C22×C3×D3—32no—
160C22×C3×D3—32no—
161C22×C3×D3—121no—
162C22×C3×D3—122no—
163C32×D4—41no—
164C2×C32×C4—41no—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D312yes—
167C2×C32×C4C22×D312yes—
168C3×#67—122no—
169C3×#67—122no—
170C3×#67—122no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C3×D12—122no—
174C3×D12—121no—
175C3×D12—122no—
176C3×C4×D3—122no—
177C23×C9—21no—
178C23×C9—21no—
179C23×C9C23×C311yescharacteristic
180C23×C9C23×C311yescharacteristic
181C22×D9—41no—
182C22×D9—42no—
183C22×D9C23×C312yes—
184C22×D9C23×C312yes—
185D4×C9—41no—
186D4×C9—42no—
187C2×C4×C9—22no—
188C2×C4×C9—22no—
189C2×C4×C9—21no—
190C2×C4×C9—41no—
191C2×C4×C9—42no—
192C2×C4×C9C23×C311yescharacteristic
193C2×C4×C9C23×C312yes—
194C2×C4×C9C23×C312yes—
195#611—42no—
196#611—42no—
197#611—42no—
198C2×Dic9C23×C312yes—
199C2×Dic9C23×C312yes—
200D36—42no—
201D36—42no—
202D36—41no—
203C4×D9—42no—
204#560—91noSylow
205C2×C3×C9—41no—
206C2×C3×C9—41no—
207C2×C3×C9H(5,𝔽2)11yescharacteristic
208C2×C3×C9H(5,𝔽2)12yes—
209C3×D9—42no—
210C3×D9—42no—
211C24×C3—91no—
212C23×D3—31no—
213C2×C3×D4—21no—
214C2×C3×D4—92no—
215C2×C3×D4—92no—
216C2×C3×D4—92no—
217C2×C3×D4—92no—
218C2×C3×D4—92no—
219C2×C3×D4—91no—
220C2×C3×D4C2×D911yescharacteristic
221C2×C3×D4C2×C3×D311yescharacteristic
222C22×C3×C4—21no—
223C22×C3×C4—92no—
224C22×C3×C4C2×D911yescharacteristic, complemented
225C22×C3×C4C2×C3×D311yescharacteristic
226C2×#67—32no—
227C2×#67—32no—
228C2×#67—32no—
229C2×D12—32no—
230C2×D12—32no—
231C2×D12—31no—
232C2×C4×D3—32no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4C2×D912yescomplemented
237C3×C4⋊C4C2×C3×D312yes—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—91no—
241C3×C22⋊C4—92no—
242C3×C22⋊C4—92no—
243C3×C22⋊C4—92no—
244C3×C22⋊C4—92no—
245C3×C22⋊C4C2×D912yes—
246C3×C22⋊C4C2×D911yescharacteristic, complemented
247C3×C22⋊C4C2×C3×D312yes—
248C3×C22⋊C4C2×C3×D311yescharacteristic
249C3×C42—91no—
250#217—31no—
251(C2×D3)⋊C4—32no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—32no—
254(C2×D3)⋊C4—32no—
255C4⋊Dic3—32no—
256Dic3⋊C4—32no—
257C4×Dic3—31no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—22no—
261C22×C32—22no—
262C22×C32C23×D311yescharacteristic
263C2×C3×D3—62no—
264C2×C3×D3—62no—
265C2×C3×D3—62no—
266C2×C3×D3—62no—
267C2×C3×D3—62no—
268C2×C3×D3—62no—
269C2×C3×D3—124no—
270C32×C4—22no—
271C32×C4—21no—
272C32×C4—22no—
273C3×Dic3—62no—
274C3×Dic3—62no—
275C22×C9—22no—
276C22×C9—22no—
277C22×C9—22no—
278C22×C9—21no—
279C22×C9—21no—
280C22×C9—21no—
281C22×C9—42no—
282C22×C9—41no—
283C22×C9—42no—
284C22×C9C24×C311yescharacteristic
285C2×D9—22no—
286C2×D9—22no—
287C2×D9—22no—
288C2×D9—22no—
289C2×D9—22no—
290C2×D9—22no—
291C2×D9—44no—
292C4×C9—22no—
293C4×C9—22no—
294C4×C9—21no—
295C4×C9—42no—
296C4×C9—42no—
297C4×C9—41no—
298Dic9—22no—
299Dic9—22no—
300#127—91no—
301#126—92no—
302#123—92no—
303#123C3×D911yescharacteristic, complemented
304#121—92no—
305#121—92no—
306#121—92no—
307#120—91no—
308#120—92no—
309C3×C9#56011yescharacteristic, complemented, Sylow
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—92no—
313C23×C3—92no—
314C23×C3—361no—
315C23×C3—362no—
316C23×C3C22×D911yescharacteristic
317C23×C3C22×D911yescharacteristic
318C23×C3C22×C3×D311yescharacteristic
319C23×C3C22×C3×D311yescharacteristic
320C22×D3—32no—
321C22×D3—32no—
322C22×D3—121no—
323C22×D3—122no—
324C3×D4—41no—
325C3×D4—41no—
326C3×D4—42no—
327C3×D4—362no—
328C3×D4—361no—
329C3×D4—362no—
330C3×D4—362no—
331C3×D4—362no—
332C3×D4—362no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—92no—
340C2×C3×C4—92no—
341C2×C3×C4—362no—
342C2×C3×C4C22×D911yescharacteristic
343C2×C3×C4C22×D912yes—
344C2×C3×C4C22×D912yes—
345C2×C3×C4C22×C3×D311yescharacteristic
346C2×C3×C4C22×C3×D312yes—
347C2×C3×C4C22×C3×D312yes—
348#67—122no—
349#67—122no—
350#67—122no—
351C2×Dic3—32no—
352C2×Dic3—32no—
353D12—122no—
354D12—121no—
355D12—122no—
356C4×D3—122no—
357C2×C32—41no—
358C2×C32—41no—
359C2×C32#100411yescharacteristic
360C2×C32#100912yes—
361C3×D3—122no—
362C3×D3—122no—
363C2×C9—21no—
364C2×C9—22no—
365C2×C9—42no—
366C2×C9—41no—
367C2×C9—41no—
368C2×C9—81no—
369C2×C9C3×H(5,𝔽2)11yescharacteristic
370C2×C9C3×H(5,𝔽2)12yes—
371D9—42no—
372D9—42no—
373C24—91no—
374C2×D4—92no—
375C2×D4—92no—
376C2×D4—92no—
377C2×D4—92no—
378C2×D4—92no—
379C2×D4—91no—
380C2×D4C2×C3×D911yescharacteristic
381C22×C4—92no—
382C22×C4C2×C3×D911yescharacteristic, complemented
383C4⋊C4—92no—
384C4⋊C4—92no—
385C4⋊C4C2×C3×D912yescomplemented
386C22⋊C4—91no—
387C22⋊C4—92no—
388C22⋊C4—92no—
389C22⋊C4—92no—
390C22⋊C4—92no—
391C22⋊C4C2×C3×D912yes—
392C22⋊C4C2×C3×D911yescharacteristic, complemented
393C42—91no—
394C22×C3—22no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—21no—
404C22×C3—42no—
405C22×C3—41no—
406C22×C3—42no—
407C22×C3—182no—
408C22×C3—182no—
409C22×C3—182no—
410C22×C3—182no—
411C22×C3—182no—
412C22×C3—182no—
413C22×C3—364no—
414C22×C3C23×D911yescharacteristic
415C22×C3C23×C3×D311yescharacteristic
416C2×D3—62no—
417C2×D3—62no—
418C2×D3—62no—
419C2×D3—62no—
420C2×D3—62no—
421C2×D3—62no—
422C2×D3—124no—
423C3×C4—22no—
424C3×C4—21no—
425C3×C4—22no—
426C3×C4—22no—
427C3×C4—21no—
428C3×C4—22no—
429C3×C4—42no—
430C3×C4—41no—
431C3×C4—42no—
432C3×C4—182no—
433C3×C4—182no—
434Dic3—62no—
435Dic3—62no—
436C32#566611yescharacteristic
437C9—21no—
438C9C3×#56011yescharacteristic, complemented
439C23—92no—
440C23—92no—
441C23—361no—
442C23—362no—
443C23C22×C3×D911yescharacteristic
444C23C22×C3×D911yescharacteristic
445D4—41no—
446D4—362no—
447D4—361no—
448D4—362no—
449D4—362no—
450D4—362no—
451D4—362no—
452C2×C4—41no—
453C2×C4—92no—
454C2×C4—92no—
455C2×C4—362no—
456C2×C4C22×C3×D911yescharacteristic
457C2×C4C22×C3×D912yes—
458C2×C4C22×C3×D912yes—
459C2×C3—21no—
460C2×C3—22no—
461C2×C3—42no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3—41no—
466C2×C3—81no—
467C2×C3—362no—
468C2×C3—362no—
469C2×C3#6371011yescharacteristic
470C2×C3#6371512yes—
471C2×C3C3×#100411yescharacteristic
472C2×C3C3×#100912yes—
473D3—122no—
474D3—122no—
475C22—21no—
476C22—21no—
477C22—22no—
478C22—22no—
479C22—182no—
480C22—182no—
481C22—182no—
482C22—182no—
483C22—182no—
484C22—182no—
485C22—364no—
486C22C23×C3×D911yescharacteristic
487C4—22no—
488C4—21no—
489C4—22no—
490C4—182no—
491C4—182no—
492C3—21no—
493C3#1058976811yescharacteristic, direct product factor
494C3C3×#566611yescharacteristic
495C2—41no—
496C2—41no—
497C2—362no—
498C2—362no—
499C2C3×#6371011yescharacteristic
500C2C3×#6371512yes—
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#123C2
socleC22×C32C23×D3
hypercenterC3×#123D9
hypocenterC9C3×#560
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#123

Complements

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

NH ≅ G/Ntype
1. C3×C9×#123C2semidirect
2. C3×#63502C2semidirect
3. C3×#63499C2semidirect
4. C3×#63492C2semidirect
5. C3×#63460C2semidirect
6. C3×#63457C2semidirect
7. C3×#63446C2semidirect
8. C3×#63445C2semidirect
9. C3×#63444C2semidirect
11. #10589768C3direct (N × H)
14. C22×C3×C4×C9C22semidirect
17. C2×C3×#611C22semidirect
20. C2×C3×D36C22semidirect
21. C2×C3×C4×D9C22semidirect
22. C3×C9×C4⋊C4C22semidirect
24. C3×C9×C22⋊C4C22semidirect
29. C3×(C2×D9)⋊C4C22semidirect
30. C3×C4⋊Dic9C22semidirect
31. C3×Dic9⋊C4C22semidirect
32. C3×C4×Dic9C22semidirect
43. C9×#123C2×C3semidirect
44. #63502C2×C3semidirect
45. #63499C2×C3semidirect
46. #63492C2×C3semidirect
47. #63460C2×C3semidirect
48. #63457C2×C3semidirect
49. #63446C2×C3semidirect
50. #63445C2×C3semidirect
51. #63444C2×C3semidirect
99. C22×C4×C9C22×C3semidirect
102. C2×#611C22×C3semidirect
105. C2×D36C22×C3semidirect
106. C2×C4×D9C22×C3semidirect
108. C9×C4⋊C4C22×C3semidirect
112. C9×C22⋊C4C22×C3semidirect
117. (C2×D9)⋊C4C22×C3semidirect
118. C4⋊Dic9C22×C3semidirect
119. Dic9⋊C4C22×C3semidirect
120. C4×Dic9C22×C3semidirect
142. C3×#123D9semidirect
224. C22×C3×C4C2×D9semidirect
236. C3×C4⋊C4C2×D9semidirect
246. C3×C22⋊C4C2×D9semidirect
303. #123C3×D9semidirect
309. C3×C9#560semidirect
382. C22×C4C2×C3×D9semidirect
385. C4⋊C4C2×C3×D9semidirect
392. C22⋊C4C2×C3×D9semidirect
438. C9C3×#560semidirect
493. C3#10589768direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×#1232
C3×#635022
C3×#634992
C3×#634922
C3×#634602
C3×#634572
C3×#634462
C3×#634452
C3×#634442
C3×#56663
#105897683