Finite Group Databasebeta

← C3×#10589721 #13152820

Subgroups — C3×#10589721 #13152820

Order 1728.

814 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#121C211yesmaximal, characteristic, complemented
2C3×#63502C211yesmaximal, characteristic, complemented
3C3×#63501C212yesmaximal, complemented
4C3×#63500C212yesmaximal, complemented
5C3×D4×Dic9C211yesmaximal, characteristic, complemented
6C3×#63492C211yesmaximal, characteristic, complemented
7C3×#63488C211yesmaximal, characteristic, complemented
8C3×#63455C211yesmaximal, characteristic, complemented
9C3×#63452C211yesmaximal, characteristic, complemented
10C3×#63448C212yesmaximal, complemented
11C3×#63445C212yesmaximal, complemented
12C3×#5619—31nomaximal
13#10589721C311yesmaximal, characteristic, direct product factor
14C2×C3×D4×C9C2211yescharacteristic
15C2×C3×D4×C9C2212yes—
16C22×C3×C4×C9C2211yescharacteristic
17C2×C3×#611C2212yes—
18C2×C3×#611C2212yes—
19C2×C3×#611C2212yes—
20C22×C3×Dic9C2212yes—
21C2×C3×D36C2211yescharacteristic
22C2×C3×C4×D9C2212yes—
23C3×C9×C4⋊C4C2211yescharacteristic
24C3×C9×C22⋊C4C2212yes—
25C3×#3676C2211yescharacteristic
26C3×#3676C2211yescharacteristic
27C3×#3676C2212yes—
28C3×(C2×D9)⋊C4C2212yes—
29C3×(C2×D9)⋊C4C2212yes—
30C3×C4⋊Dic9C2211yescharacteristic
31C3×C4⋊Dic9C2211yescharacteristic
32C3×C4⋊Dic9C2211yescharacteristic
33C3×C4⋊Dic9C2212yes—
34C3×Dic9⋊C4C2212yes—
35C3×C4×Dic9C2211yescharacteristic
36C3×C4×Dic9C2211yescharacteristic
37C32×#121D311yescharacteristic
38C3×#940—31no—
39C3×#939—32no—
40C3×#938—32no—
41C3×D4×Dic3—31no—
42C3×#930—31no—
43C3×#926—31no—
44C3×#893—31no—
45C3×#890—31no—
46C3×#886—32no—
47C3×#883—32no—
48C9×#121—21no—
49C9×#121C2×C311yescharacteristic, complemented
50#63502C2×C311yescharacteristic, complemented
51#63501C2×C312yescomplemented
52#63500C2×C312yescomplemented
53D4×Dic9C2×C311yescharacteristic, complemented
54#63492C2×C311yescharacteristic, complemented
55#63488C2×C311yescharacteristic, complemented
56#63455C2×C311yescharacteristic, complemented
57#63452C2×C311yescharacteristic, complemented
58#63448C2×C312yescomplemented
59#63445C2×C312yescomplemented
60C23×C3×C9C2312yes—
61C23×C3×C9C2311yescharacteristic
62C22×C3×D9C2312yes—
63C3×D4×C9—22no—
64C3×D4×C9—42no—
65C2×C3×C4×C9—22no—
66C2×C3×C4×C9C2312yes—
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3×C4×C9C2311yescharacteristic
69C3×#611—42no—
70C3×#611—42no—
71C3×#611—42no—
72C2×C3×Dic9—42no—
73C2×C3×Dic9C2311yescharacteristic
74C2×C3×Dic9C2311yescharacteristic
75C2×C3×Dic9C2312yes—
76C2×C3×Dic9C2311yescharacteristic
77C2×C3×Dic9C2311yescharacteristic
78C3×D36—22no—
79C3×C4×D9—42no—
80C3×#540—91no—
81#5619—31no—
82C2×C32×D4C2×D311yescharacteristic
83C2×C32×D4C2×D312yes—
84C22×C32×C4C2×D311yescharacteristic
85C2×C3×#67—32no—
86C2×C3×#67—32no—
87C2×C3×#67—32no—
88C22×C3×Dic3—32no—
89C2×C3×D12—31no—
90C2×C3×C4×D3—32no—
91C32×C4⋊C4C2×D311yescharacteristic
92C32×C22⋊C4C2×D312yes—
93C3×#217—32no—
94C3×#217—31no—
95C3×#217—31no—
96C3×(C2×D3)⋊C4—32no—
97C3×(C2×D3)⋊C4—32no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3—32no—
101C3×C4⋊Dic3—31no—
102C3×Dic3⋊C4—32no—
103C3×C4×Dic3—31no—
104C3×C4×Dic3—31no—
105C2×D4×C9—21no—
106C2×D4×C9—22no—
107C2×D4×C9C22×C311yescharacteristic
108C2×D4×C9C22×C312yes—
109C22×C4×C9—21no—
110C22×C4×C9C22×C311yescharacteristic
111C2×#611C22×C312yes—
112C2×#611C22×C312yes—
113C2×#611C22×C312yes—
114C22×Dic9C22×C312yes—
115C2×D36C22×C311yescharacteristic
116C2×C4×D9C22×C312yes—
117C9×C4⋊C4—21no—
118C9×C4⋊C4C22×C311yescharacteristic
119C9×C22⋊C4—22no—
120C9×C22⋊C4C22×C312yes—
121#3676C22×C311yescharacteristic
122#3676C22×C311yescharacteristic
123#3676C22×C312yes—
124(C2×D9)⋊C4C22×C312yes—
125(C2×D9)⋊C4C22×C312yes—
126C4⋊Dic9C22×C311yescharacteristic
127C4⋊Dic9C22×C311yescharacteristic
128C4⋊Dic9C22×C311yescharacteristic
129C4⋊Dic9C22×C312yes—
130Dic9⋊C4C22×C312yes—
131C4×Dic9C22×C311yescharacteristic
132C4×Dic9C22×C311yescharacteristic
133C22×C3×C9—21no—
134C22×C3×C9—22no—
135C22×C3×C9—21no—
136C22×C3×C9—22no—
137C22×C3×C9—21no—
138C22×C3×C9—22no—
139C22×C3×C9C2411yescharacteristic
140C2×C3×D9—22no—
141C2×C3×D9—22no—
142C2×C3×D9—22no—
143C3×C4×C9—21no—
144C3×C4×C9—22no—
145C3×C4×C9#4112yes—
146C3×Dic9—21no—
147C3×Dic9—21no—
148C3×Dic9—22no—
149C3×Dic9—21no—
150C3×Dic9—21no—
151C3×#127—91no—
152C3×#125—91no—
153C3×#123—92no—
154C3×#123—92no—
155C3×#121—21no—
156C3×#121—92no—
157C3×#121—92no—
158C3×#121—91no—
159C3×#121D911yescharacteristic, complemented
160C3×#121C3×D311yescharacteristic
161C3×C4×D4—91no—
162C3×C4×D4—91no—
163C3×#117—91no—
164#940—31no—
165#939—32no—
166#938—32no—
167D4×Dic3—31no—
168#930—31no—
169#926—31no—
170#893—31no—
171#890—31no—
172#886—32no—
173#883—32no—
174C23×C32C22×D311yescharacteristic
175C23×C32C22×D312yes—
176C22×C3×D3—32no—
177C32×D4—22no—
178C32×D4—42no—
179C2×C32×C4—22no—
180C2×C32×C4C22×D312yes—
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183C3×#67—122no—
184C3×#67—122no—
185C3×#67—122no—
186C2×C3×Dic3—31no—
187C2×C3×Dic3—31no—
188C2×C3×Dic3—31no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—32no—
191C2×C3×Dic3—122no—
192C3×D12—62no—
193C3×C4×D3—122no—
194C23×C9—21no—
195C23×C9—22no—
196C23×C9C23×C311yescharacteristic
197C23×C9C23×C312yes—
198C22×D9C23×C312yes—
199D4×C9—22no—
200D4×C9—42no—
201D4×C9—42no—
202D4×C9—44no—
203C2×C4×C9—22no—
204C2×C4×C9—21no—
205C2×C4×C9—21no—
206C2×C4×C9—22no—
207C2×C4×C9—42no—
208C2×C4×C9C23×C312yes—
209C2×C4×C9C23×C311yescharacteristic
210C2×C4×C9C23×C311yescharacteristic
211#611—42no—
212#611—42no—
213#611—42no—
214C2×Dic9—42no—
215C2×Dic9C23×C311yescharacteristic
216C2×Dic9C23×C311yescharacteristic
217C2×Dic9C23×C311yescharacteristic
218C2×Dic9C23×C311yescharacteristic
219C2×Dic9C23×C312yes—
220D36—22no—
221C4×D9—42no—
222#540—91noSylow
223C2×C3×C9—41no—
224C2×C3×C9—42no—
225C2×C3×C9C2×#4111yescharacteristic
226C2×C3×C9H(5,𝔽2)11yescharacteristic
227C2×C3×C9H(5,𝔽2)11yescharacteristic
228C3×D9—42no—
229C2×C3×D4—21no—
230C2×C3×D4—22no—
231C2×C3×D4—92no—
232C2×C3×D4—92no—
233C2×C3×D4—92no—
234C2×C3×D4—91no—
235C2×C3×D4C2×D911yescharacteristic
236C2×C3×D4C2×D912yes—
237C2×C3×D4C2×C3×D311yescharacteristic
238C2×C3×D4C2×C3×D312yes—
239C22×C3×C4—21no—
240C22×C3×C4—92no—
241C22×C3×C4—92no—
242C22×C3×C4C2×D911yescharacteristic
243C22×C3×C4C2×C3×D311yescharacteristic
244C2×#67—32no—
245C2×#67—32no—
246C2×#67—32no—
247C22×Dic3—32no—
248C2×D12—31no—
249C2×C4×D3—32no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—92no—
252C3×C4⋊C4—91no—
253C3×C4⋊C4—91no—
254C3×C4⋊C4—91no—
255C3×C4⋊C4—92no—
256C3×C4⋊C4C2×D911yescharacteristic
257C3×C4⋊C4C2×C3×D311yescharacteristic
258C3×C22⋊C4—22no—
259C3×C22⋊C4—91no—
260C3×C22⋊C4—91no—
261C3×C22⋊C4—92no—
262C3×C22⋊C4—92no—
263C3×C22⋊C4—92no—
264C3×C22⋊C4C2×D912yes—
265C3×C22⋊C4C2×C3×D312yes—
266C3×C42—91no—
267C3×C42—91no—
268#217—32no—
269#217—31no—
270#217—31no—
271(C2×D3)⋊C4—32no—
272(C2×D3)⋊C4—32no—
273C4⋊Dic3—31no—
274C4⋊Dic3—31no—
275C4⋊Dic3—32no—
276C4⋊Dic3—31no—
277Dic3⋊C4—32no—
278C4×Dic3—31no—
279C4×Dic3—31no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—22no—
283C22×C32—22no—
284C22×C32—22no—
285C22×C32—21no—
286C22×C32C23×D311yescharacteristic
287C2×C3×D3—62no—
288C2×C3×D3—62no—
289C2×C3×D3—62no—
290C32×C4—21no—
291C32×C4—22no—
292C32×C4#23712yes—
293C3×Dic3—61no—
294C3×Dic3—61no—
295C3×Dic3—61no—
296C3×Dic3—62no—
297C3×Dic3—61no—
298C22×C9—22no—
299C22×C9—21no—
300C22×C9—22no—
301C22×C9—22no—
302C22×C9—21no—
303C22×C9—22no—
304C22×C9—21no—
305C22×C9—21no—
306C22×C9—41no—
307C22×C9—41no—
308C22×C9—42no—
309C22×C9—42no—
310C22×C9—42no—
311C22×C9C24×C311yescharacteristic
312C2×D9—22no—
313C2×D9—22no—
314C2×D9—22no—
315C4×C9—21no—
316C4×C9—22no—
317C4×C9—22no—
318C4×C9—41no—
319C4×C9—42no—
320C4×C9C3×#4112yes—
321Dic9—21no—
322Dic9—21no—
323Dic9—22no—
324Dic9—21no—
325Dic9—21no—
326#127—91no—
327#125—91no—
328#123—92no—
329#123—92no—
330#121—92no—
331#121—92no—
332#121—91no—
333#121C3×D911yescharacteristic, complemented
334C4×D4—91no—
335C4×D4—91no—
336#117—91no—
337C3×C9#54011yescharacteristic, complemented, Sylow
338C23×C3—21no—
339C23×C3—22no—
340C23×C3—92no—
341C23×C3C22×D911yescharacteristic
342C23×C3C22×D912yes—
343C23×C3C22×C3×D311yescharacteristic
344C23×C3C22×C3×D312yes—
345C22×D3—32no—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—42no—
349C3×D4—42no—
350C3×D4—42no—
351C3×D4—44no—
352C3×D4—182no—
353C3×D4—362no—
354C3×D4—362no—
355C3×D4—362no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—42no—
362C2×C3×C4—91no—
363C2×C3×C4—91no—
364C2×C3×C4—91no—
365C2×C3×C4—91no—
366C2×C3×C4—92no—
367C2×C3×C4—362no—
368C2×C3×C4—362no—
369C2×C3×C4C22×D911yescharacteristic
370C2×C3×C4C22×D912yes—
371C2×C3×C4C22×D911yescharacteristic
372C2×C3×C4C22×C3×D312yes—
373C2×C3×C4C22×C3×D311yescharacteristic
374C2×C3×C4C22×C3×D311yescharacteristic
375#67—122no—
376#67—122no—
377#67—122no—
378C2×Dic3—31no—
379C2×Dic3—31no—
380C2×Dic3—31no—
381C2×Dic3—31no—
382C2×Dic3—32no—
383C2×Dic3—122no—
384D12—62no—
385C4×D3—122no—
386C2×C32—41no—
387C2×C32—42no—
388C2×C32C2×#23711yescharacteristic
389C2×C32#100411yescharacteristic
390C2×C32#100911yescharacteristic
391C3×D3—122no—
392C2×C9—21no—
393C2×C9—21no—
394C2×C9—21no—
395C2×C9—42no—
396C2×C9—41no—
397C2×C9—42no—
398C2×C9—82no—
399C2×C9C2×C3×#4111yescharacteristic
400C2×C9C3×H(5,𝔽2)11yescharacteristic
401C2×C9C3×H(5,𝔽2)11yescharacteristic
402D9—42no—
403C2×D4—92no—
404C2×D4—92no—
405C2×D4—92no—
406C2×D4—91no—
407C2×D4C2×C3×D911yescharacteristic
408C2×D4C2×C3×D912yes—
409C22×C4—92no—
410C22×C4—92no—
411C22×C4C2×C3×D911yescharacteristic
412C4⋊C4—91no—
413C4⋊C4—91no—
414C4⋊C4—91no—
415C4⋊C4—92no—
416C4⋊C4—92no—
417C4⋊C4C2×C3×D911yescharacteristic
418C22⋊C4—91no—
419C22⋊C4—91no—
420C22⋊C4—92no—
421C22⋊C4—92no—
422C22⋊C4—92no—
423C22⋊C4C2×C3×D912yes—
424C42—91no—
425C42—91no—
426C22×C3—22no—
427C22×C3—21no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—21no—
431C22×C3—22no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—22no—
435C22×C3—21no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—41no—
441C22×C3—41no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—182no—
446C22×C3—182no—
447C22×C3—182no—
448C22×C3C23×D911yescharacteristic
449C22×C3C23×C3×D311yescharacteristic
450C2×D3—62no—
451C2×D3—62no—
452C2×D3—62no—
453C3×C4—21no—
454C3×C4—22no—
455C3×C4—21no—
456C3×C4—22no—
457C3×C4—22no—
458C3×C4—41no—
459C3×C4—42no—
460C3×C4—181no—
461C3×C4—181no—
462C3×C4—181no—
463C3×C4—182no—
464C3×C4—181no—
465C3×C4#369912yes—
466C3×C4C3×#23712yes—
467Dic3—61no—
468Dic3—61no—
469Dic3—61no—
470Dic3—62no—
471Dic3—61no—
472C32#561911yescharacteristic
473C9—21no—
474C9C3×#54011yescharacteristic, complemented
475C23—92no—
476C23C22×C3×D911yescharacteristic
477C23C22×C3×D912yes—
478D4—22no—
479D4—42no—
480D4—182no—
481D4—362no—
482D4—362no—
483D4—362no—
484C2×C4—22no—
485C2×C4—91no—
486C2×C4—91no—
487C2×C4—91no—
488C2×C4—91no—
489C2×C4—92no—
490C2×C4—362no—
491C2×C4—362no—
492C2×C4C22×C3×D911yescharacteristic
493C2×C4C22×C3×D912yes—
494C2×C4C22×C3×D911yescharacteristic
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—42no—
499C2×C3—41no—
500C2×C3—42no—
501C2×C3—41no—
502C2×C3—42no—
503C2×C3—82no—
504C2×C3—362no—
505C2×C3C2×#369911yescharacteristic
506C2×C3#6371011yescharacteristic
507C2×C3#6371511yescharacteristic
508C2×C3C2×C3×#23711yescharacteristic
509C2×C3C3×#100411yescharacteristic
510C2×C3C3×#100911yescharacteristic
511D3—122no—
512C22—21no—
513C22—22no—
514C22—21no—
515C22—22no—
516C22—22no—
517C22—21no—
518C22—182no—
519C22—182no—
520C22—182no—
521C22C23×C3×D911yescharacteristic
522C4—21no—
523C4—22no—
524C4—181no—
525C4—181no—
526C4—181no—
527C4—182no—
528C4—181no—
529C4C3×#369912yes—
530C3—21no—
531C3#1058972111yescharacteristic, direct product factor
532C3C3×#561911yescharacteristic
533C2—41no—
534C2—42no—
535C2—362no—
536C2C2×C3×#369911yescharacteristic
537C2C3×#6371011yescharacteristic
538C2C3×#6371511yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#121C2
socleC22×C32C23×D3
hypercenterC3×#121D9
hypocenterC9C3×#540
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×#63502C2semidirect
3. C3×#63501C2semidirect
4. C3×#63500C2semidirect
5. C3×D4×Dic9C2semidirect
6. C3×#63492C2semidirect
7. C3×#63488C2semidirect
8. C3×#63455C2semidirect
9. C3×#63452C2semidirect
10. C3×#63448C2semidirect
11. C3×#63445C2semidirect
13. #10589721C3direct (N × H)
49. C9×#121C2×C3semidirect
50. #63502C2×C3semidirect
51. #63501C2×C3semidirect
52. #63500C2×C3semidirect
53. D4×Dic9C2×C3semidirect
54. #63492C2×C3semidirect
55. #63488C2×C3semidirect
56. #63455C2×C3semidirect
57. #63452C2×C3semidirect
58. #63448C2×C3semidirect
59. #63445C2×C3semidirect
159. C3×#121D9semidirect
333. #121C3×D9semidirect
337. C3×C9#540semidirect
474. C9C3×#540semidirect
531. C3#10589721direct (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×#635022
C3×#635012
C3×#635002
C3×D4×Dic92
C3×#634922
C3×#634882
C3×#634552
C3×#634522
C3×#634482
C3×#634452
C3×#56193
#105897213