Finite Group Databasebeta

← C3×#10589720 #13152819

Subgroups — C3×#10589720 #13152819

Order 1728.

808 conjugacy classes of subgroups.

The subgroup lattice has 534 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×#63500C211yesmaximal, characteristic, complemented
4C3×D4×Dic9C212yesmaximal, complemented
5C3×D4×Dic9C211yesmaximal, characteristic, complemented
6C3×C4×#611C211yesmaximal, characteristic, complemented
7C2×C3×C4⋊Dic9C211yesmaximal, characteristic, complemented
8C3×#63454C211yesmaximal, characteristic, complemented
9C3×#63452C211yesmaximal, characteristic, complemented
10C3×#63448C212yesmaximal, complemented
11C3×#63441C212yesmaximal, complemented
12C3×#5618—31nomaximal
13#10589720C311yesmaximal, characteristic, direct product factor
14C2×C3×D4×C9C2211yescharacteristic
15C2×C3×D4×C9C2212yes—
16C22×C3×C4×C9C2211yescharacteristic
17C2×C3×#611C2211yescharacteristic
18C2×C3×#611C2212yes—
19C22×C3×Dic9C2212yes—
20C22×C3×Dic9C2212yes—
21C2×C3×C4×D9C2211yescharacteristic
22C3×C9×C4⋊C4C2211yescharacteristic
23C3×C9×C22⋊C4C2212yes—
24C3×#3676C2211yescharacteristic
25C3×#3676C2212yes—
26C3×#3676C2212yes—
27C3×(C2×D9)⋊C4C2211yescharacteristic
28C3×(C2×D9)⋊C4C2212yes—
29C3×C4⋊Dic9C2211yescharacteristic
30C3×C4⋊Dic9C2211yescharacteristic
31C3×C4⋊Dic9C2212yes—
32C3×C4⋊Dic9C2212yes—
33C3×Dic9⋊C4C2211yescharacteristic
34C3×Dic9⋊C4C2212yes—
35C3×C4×Dic9C2211yescharacteristic
36C3×C4×Dic9C2212yes—
37C32×#121D311yescharacteristic
38C3×#939—32no—
39C3×#938—31no—
40C3×D4×Dic3—32no—
41C3×D4×Dic3—31no—
42C3×C4×#67—31no—
43C2×C3×C4⋊Dic3—31no—
44C3×#892—31no—
45C3×#890—31no—
46C3×#886—32no—
47C3×#879—32no—
48C9×#121—21no—
49C9×#121C2×C311yescharacteristic, complemented
50#63501C2×C312yescomplemented
51#63500C2×C311yescharacteristic, complemented
52D4×Dic9C2×C311yescharacteristic, complemented
53D4×Dic9C2×C312yescomplemented
54C4×#611C2×C311yescharacteristic, complemented
55C2×C4⋊Dic9C2×C311yescharacteristic, complemented
56#63454C2×C311yescharacteristic, complemented
57#63452C2×C311yescharacteristic, complemented
58#63448C2×C312yescomplemented
59#63441C2×C312yescomplemented
60C23×C3×C9C2312yes—
61C23×C3×C9C2311yescharacteristic
62C22×C3×D9C2311yescharacteristic
63C3×D4×C9—24no—
64C3×D4×C9—22no—
65C2×C3×C4×C9—22no—
66C2×C3×C4×C9C2312yes—
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3×C4×C9C2311yescharacteristic
69C3×#611—22no—
70C3×#611—42no—
71C2×C3×Dic9—42no—
72C2×C3×Dic9—42no—
73C2×C3×Dic9C2311yescharacteristic
74C2×C3×Dic9C2312yes—
75C2×C3×Dic9C2311yescharacteristic
76C2×C3×Dic9C2312yes—
77C2×C3×Dic9C2311yescharacteristic
78C3×C4×D9—22no—
79C3×#553—91no—
80#5618—31no—
81C2×C32×D4C2×D311yescharacteristic
82C2×C32×D4C2×D312yes—
83C22×C32×C4C2×D311yescharacteristic
84C2×C3×#67—31no—
85C2×C3×#67—32no—
86C22×C3×Dic3—32no—
87C22×C3×Dic3—32no—
88C2×C3×C4×D3—31no—
89C32×C4⋊C4C2×D311yescharacteristic
90C32×C22⋊C4C2×D312yes—
91C3×#217—32no—
92C3×#217—32no—
93C3×#217—31no—
94C3×(C2×D3)⋊C4—31no—
95C3×(C2×D3)⋊C4—32no—
96C3×C4⋊Dic3—32no—
97C3×C4⋊Dic3—32no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—31no—
100C3×Dic3⋊C4—31no—
101C3×Dic3⋊C4—32no—
102C3×C4×Dic3—32no—
103C3×C4×Dic3—31no—
104C2×D4×C9—21no—
105C2×D4×C9—22no—
106C2×D4×C9C22×C311yescharacteristic
107C2×D4×C9C22×C312yes—
108C22×C4×C9—21no—
109C22×C4×C9C22×C311yescharacteristic
110C2×#611C22×C311yescharacteristic
111C2×#611C22×C312yes—
112C22×Dic9C22×C312yes—
113C22×Dic9C22×C312yes—
114C2×C4×D9C22×C311yescharacteristic
115C9×C4⋊C4—21no—
116C9×C4⋊C4C22×C311yescharacteristic
117C9×C22⋊C4—22no—
118C9×C22⋊C4C22×C312yes—
119#3676C22×C311yescharacteristic
120#3676C22×C312yes—
121#3676C22×C312yes—
122(C2×D9)⋊C4C22×C311yescharacteristic
123(C2×D9)⋊C4C22×C312yes—
124C4⋊Dic9C22×C312yes—
125C4⋊Dic9C22×C311yescharacteristic
126C4⋊Dic9C22×C311yescharacteristic
127C4⋊Dic9C22×C312yes—
128Dic9⋊C4C22×C311yescharacteristic
129Dic9⋊C4C22×C312yes—
130C4×Dic9C22×C311yescharacteristic
131C4×Dic9C22×C312yes—
132C22×C3×C9—21no—
133C22×C3×C9—21no—
134C22×C3×C9—22no—
135C22×C3×C9—22no—
136C22×C3×C9—22no—
137C22×C3×C9#4112yes—
138C22×C3×C9C2411yescharacteristic
139C2×C3×D9—21no—
140C2×C3×D9—21no—
141C2×C3×D9—21no—
142C3×C4×C9—21no—
143C3×C4×C9—22no—
144C3×C4×C9#4112yes—
145C3×Dic9—21no—
146C3×Dic9—21no—
147C3×Dic9—21no—
148C3×Dic9—22no—
149C3×Dic9—22no—
150C3×#126—92no—
151C3×#125—91no—
152C3×#123—92no—
153C3×#121—21no—
154C3×#121—91no—
155C3×#121—92no—
156C3×#121D911yescharacteristic, complemented
157C3×#121C3×D311yescharacteristic
158C3×C4×D4—92no—
159C3×C4×D4—91no—
160C3×C4×D4—91no—
161C3×#117—91no—
162C2×C3×C4⋊C4—91no—
163#939—32no—
164#938—31no—
165D4×Dic3—32no—
166D4×Dic3—31no—
167C4×#67—31no—
168C2×C4⋊Dic3—31no—
169#892—31no—
170#890—31no—
171#886—32no—
172#879—32no—
173C23×C32C22×D312yes—
174C23×C32C22×D311yescharacteristic
175C22×C3×D3—31no—
176C32×D4—24no—
177C32×D4—22no—
178C2×C32×C4—22no—
179C2×C32×C4C22×D312yes—
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182C3×#67—62no—
183C3×#67—122no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—31no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—122no—
190C2×C3×Dic3—122no—
191C3×C4×D3—62no—
192C23×C9—21no—
193C23×C9—22no—
194C23×C9C23×C312yes—
195C23×C9C23×C311yescharacteristic
196C22×D9C23×C311yescharacteristic
197D4×C9—24no—
198D4×C9—22no—
199D4×C9—42no—
200D4×C9—44no—
201C2×C4×C9—22no—
202C2×C4×C9—21no—
203C2×C4×C9—21no—
204C2×C4×C9—22no—
205C2×C4×C9—42no—
206C2×C4×C9C23×C312yes—
207C2×C4×C9C23×C311yescharacteristic
208C2×C4×C9C23×C311yescharacteristic
209#611—22no—
210#611—42no—
211C2×Dic9—42no—
212C2×Dic9—42no—
213C2×Dic9C23×C311yescharacteristic
214C2×Dic9C23×C311yescharacteristic
215C2×Dic9C23×C311yescharacteristic
216C2×Dic9C23×C312yes—
217C2×Dic9C23×C312yes—
218C4×D9—22no—
219#553—91noSylow
220C2×C3×C9—22no—
221C2×C3×C9—42no—
222C2×C3×C9C2×#4111yescharacteristic
223C2×C3×C9C2×#4111yescharacteristic
224C2×C3×C9H(5,𝔽2)11yescharacteristic
225C3×D9—41no—
226C2×C3×D4—21no—
227C2×C3×D4—22no—
228C2×C3×D4—91no—
229C2×C3×D4—92no—
230C2×C3×D4C2×D912yes—
231C2×C3×D4C2×D911yescharacteristic
232C2×C3×D4C2×C3×D311yescharacteristic
233C2×C3×D4C2×C3×D312yes—
234C22×C3×C4—21no—
235C22×C3×C4—92no—
236C22×C3×C4—92no—
237C22×C3×C4—91no—
238C22×C3×C4C2×D911yescharacteristic
239C22×C3×C4C2×C3×D311yescharacteristic
240C2×#67—31no—
241C2×#67—32no—
242C22×Dic3—32no—
243C22×Dic3—32no—
244C2×C4×D3—31no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—92no—
248C3×C4⋊C4—91no—
249C3×C4⋊C4—91no—
250C3×C4⋊C4—92no—
251C3×C4⋊C4—92no—
252C3×C4⋊C4C2×D911yescharacteristic
253C3×C4⋊C4C2×C3×D311yescharacteristic
254C3×C22⋊C4—22no—
255C3×C22⋊C4—91no—
256C3×C22⋊C4—92no—
257C3×C22⋊C4—92no—
258C3×C22⋊C4—92no—
259C3×C22⋊C4—91no—
260C3×C22⋊C4C2×D912yes—
261C3×C22⋊C4C2×C3×D312yes—
262C3×C42—92no—
263C3×C42—91no—
264#217—32no—
265#217—32no—
266#217—31no—
267(C2×D3)⋊C4—31no—
268(C2×D3)⋊C4—32no—
269C4⋊Dic3—32no—
270C4⋊Dic3—32no—
271C4⋊Dic3—31no—
272C4⋊Dic3—31no—
273Dic3⋊C4—31no—
274Dic3⋊C4—32no—
275C4×Dic3—32no—
276C4×Dic3—31no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—22no—
280C22×C32—22no—
281C22×C32—22no—
282C22×C32#23712yes—
283C22×C32C23×D311yescharacteristic
284C2×C3×D3—61no—
285C2×C3×D3—61no—
286C2×C3×D3—61no—
287C32×C4—21no—
288C32×C4—22no—
289C32×C4#23712yes—
290C3×Dic3—61no—
291C3×Dic3—61no—
292C3×Dic3—62no—
293C3×Dic3—61no—
294C3×Dic3—62no—
295C22×C9—21no—
296C22×C9—21no—
297C22×C9—22no—
298C22×C9—22no—
299C22×C9—22no—
300C22×C9—21no—
301C22×C9—22no—
302C22×C9—41no—
303C22×C9—41no—
304C22×C9—42no—
305C22×C9—42no—
306C22×C9—42no—
307C22×C9C3×#4112yes—
308C22×C9C24×C311yescharacteristic
309C2×D9—21no—
310C2×D9—21no—
311C2×D9—21no—
312C4×C9—21no—
313C4×C9—22no—
314C4×C9—22no—
315C4×C9—41no—
316C4×C9—42no—
317C4×C9C3×#4112yes—
318Dic9—21no—
319Dic9—21no—
320Dic9—21no—
321Dic9—22no—
322Dic9—22no—
323#126—92no—
324#125—91no—
325#123—92no—
326#121—91no—
327#121—92no—
328#121C3×D911yescharacteristic, complemented
329C4×D4—92no—
330C4×D4—91no—
331C4×D4—91no—
332#117—91no—
333C2×C4⋊C4—91no—
334C3×C9#55311yescharacteristic, complemented, Sylow
335C23×C3—21no—
336C23×C3—22no—
337C23×C3—91no—
338C23×C3C22×D912yes—
339C23×C3C22×D911yescharacteristic
340C23×C3C22×C3×D312yes—
341C23×C3C22×C3×D311yescharacteristic
342C22×D3—31no—
343C3×D4—24no—
344C3×D4—24no—
345C3×D4—22no—
346C3×D4—22no—
347C3×D4—42no—
348C3×D4—44no—
349C3×D4—182no—
350C3×D4—362no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—42no—
357C2×C3×C4—91no—
358C2×C3×C4—91no—
359C2×C3×C4—92no—
360C2×C3×C4—91no—
361C2×C3×C4—92no—
362C2×C3×C4—182no—
363C2×C3×C4—362no—
364C2×C3×C4—362no—
365C2×C3×C4C22×D912yes—
366C2×C3×C4C22×D911yescharacteristic
367C2×C3×C4C22×D911yescharacteristic
368C2×C3×C4C22×C3×D312yes—
369C2×C3×C4C22×C3×D311yescharacteristic
370C2×C3×C4C22×C3×D311yescharacteristic
371#67—62no—
372#67—122no—
373C2×Dic3—31no—
374C2×Dic3—31no—
375C2×Dic3—32no—
376C2×Dic3—31no—
377C2×Dic3—32no—
378C2×Dic3—122no—
379C2×Dic3—122no—
380C4×D3—62no—
381C2×C32—22no—
382C2×C32—42no—
383C2×C32C2×#23711yescharacteristic
384C2×C32C2×#23711yescharacteristic
385C2×C32#100911yescharacteristic
386C3×D3—121no—
387C2×C9—22no—
388C2×C9—21no—
389C2×C9—21no—
390C2×C9—21no—
391C2×C9—42no—
392C2×C9—42no—
393C2×C9—82no—
394C2×C9C2×C3×#4111yescharacteristic
395C2×C9C2×C3×#4111yescharacteristic
396C2×C9C3×H(5,𝔽2)11yescharacteristic
397D9—41no—
398C2×D4—91no—
399C2×D4—92no—
400C2×D4C2×C3×D912yes—
401C2×D4C2×C3×D911yescharacteristic
402C22×C4—92no—
403C22×C4—92no—
404C22×C4—91no—
405C22×C4C2×C3×D911yescharacteristic
406C4⋊C4—91no—
407C4⋊C4—91no—
408C4⋊C4—92no—
409C4⋊C4—92no—
410C4⋊C4—91no—
411C4⋊C4—92no—
412C4⋊C4C2×C3×D911yescharacteristic
413C22⋊C4—91no—
414C22⋊C4—92no—
415C22⋊C4—92no—
416C22⋊C4—92no—
417C22⋊C4—91no—
418C22⋊C4C2×C3×D912yes—
419C42—92no—
420C42—91no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—21no—
432C22×C3—22no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—181no—
439C22×C3—181no—
440C22×C3—181no—
441C22×C3#369912yes—
442C22×C3C23×D911yescharacteristic
443C22×C3C3×#23712yes—
444C22×C3C23×C3×D311yescharacteristic
445C2×D3—61no—
446C2×D3—61no—
447C2×D3—61no—
448C3×C4—21no—
449C3×C4—22no—
450C3×C4—21no—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—41no—
454C3×C4—42no—
455C3×C4—181no—
456C3×C4—181no—
457C3×C4—182no—
458C3×C4—181no—
459C3×C4—182no—
460C3×C4#369912yes—
461C3×C4C3×#23712yes—
462Dic3—61no—
463Dic3—61no—
464Dic3—62no—
465Dic3—61no—
466Dic3—62no—
467C32#561811yescharacteristic
468C9—21no—
469C9C3×#55311yescharacteristic, complemented
470C23—91no—
471C23C22×C3×D912yes—
472C23C22×C3×D911yescharacteristic
473D4—24no—
474D4—22no—
475D4—182no—
476D4—362no—
477C2×C4—22no—
478C2×C4—91no—
479C2×C4—91no—
480C2×C4—92no—
481C2×C4—91no—
482C2×C4—92no—
483C2×C4—182no—
484C2×C4—362no—
485C2×C4—362no—
486C2×C4C22×C3×D912yes—
487C2×C4C22×C3×D911yescharacteristic
488C2×C4C22×C3×D911yescharacteristic
489C2×C3—22no—
490C2×C3—22no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—42no—
495C2×C3—42no—
496C2×C3—42no—
497C2×C3—82no—
498C2×C3—361no—
499C2×C3C2×#369911yescharacteristic
500C2×C3C2×#369911yescharacteristic
501C2×C3#6371511yescharacteristic
502C2×C3C2×C3×#23711yescharacteristic
503C2×C3C2×C3×#23711yescharacteristic
504C2×C3C3×#100911yescharacteristic
505D3—121no—
506C22—21no—
507C22—21no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—181no—
512C22—181no—
513C22—181no—
514C22C3×#369912yes—
515C22C23×C3×D911yescharacteristic
516C4—21no—
517C4—22no—
518C4—181no—
519C4—181no—
520C4—182no—
521C4—181no—
522C4—182no—
523C4C3×#369912yes—
524C3—21no—
525C3#1058972011yescharacteristic, direct product factor
526C3C3×#561811yescharacteristic
527C2—22no—
528C2—42no—
529C2—361no—
530C2C2×C3×#369911yescharacteristic
531C2C2×C3×#369911yescharacteristic
532C2C3×#6371511yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#121C2
socleC22×C32C23×D3
hypercenterC3×#121D9
hypocenterC9C3×#553
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×#63500C2semidirect
4. C3×D4×Dic9C2semidirect
5. C3×D4×Dic9C2semidirect
6. C3×C4×#611C2semidirect
7. C2×C3×C4⋊Dic9C2semidirect
8. C3×#63454C2semidirect
9. C3×#63452C2semidirect
10. C3×#63448C2semidirect
11. C3×#63441C2semidirect
13. #10589720C3direct (N × H)
49. C9×#121C2×C3semidirect
50. #63501C2×C3semidirect
51. #63500C2×C3semidirect
52. D4×Dic9C2×C3semidirect
53. D4×Dic9C2×C3semidirect
54. C4×#611C2×C3semidirect
55. C2×C4⋊Dic9C2×C3semidirect
56. #63454C2×C3semidirect
57. #63452C2×C3semidirect
58. #63448C2×C3semidirect
59. #63441C2×C3semidirect
156. C3×#121D9semidirect
328. #121C3×D9semidirect
334. C3×C9#553semidirect
469. C9C3×#553semidirect
525. C3#10589720direct (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×#635002
C3×D4×Dic92
C3×C4×#6112
C2×C3×C4⋊Dic92
C3×#634542
C3×#634522
C3×#634482
C3×#634412
C3×#56183
#105897203