Finite Group Databasebeta

← C3×#10589616 #13152411

Subgroups — C3×#10589616 #13152411

Order 1728.

806 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
2C3×#63492C211yesmaximal, characteristic, complemented
3C3×#63492C212yesmaximal, complemented
4C3×C4×#611C211yesmaximal, characteristic, complemented
5C3×C4×#611C212yesmaximal, complemented
6C3×#63460C212yesmaximal, complemented
7C3×#63457C211yesmaximal, characteristic, complemented
8C3×#63456C212yesmaximal, complemented
9C3×#63455C211yesmaximal, characteristic, complemented
10C3×#63454C211yesmaximal, characteristic, complemented
11C3×#63452C211yesmaximal, characteristic, complemented
12C3×#5514—31nomaximal
13#10589616C311yesmaximal, characteristic, direct product factor
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3×C4×C9C2212yes—
16C2×C3×#611C2211yescharacteristic
17C2×C3×#611C2212yes—
18C2×C3×D36C2211yescharacteristic
19C2×C3×D36C2212yes—
20C2×C3×C4×D9C2211yescharacteristic
21C2×C3×C4×D9C2212yes—
22C3×C9×C4⋊C4C2211yescharacteristic
23C3×C9×C4⋊C4C2211yescharacteristic
24C3×C9×C4⋊C4C2212yes—
25C3×#3676C2211yescharacteristic
26C3×(C2×D9)⋊C4C2211yescharacteristic
27C3×(C2×D9)⋊C4C2212yes—
28C3×(C2×D9)⋊C4C2212yes—
29C3×(C2×D9)⋊C4C2212yes—
30C3×(C2×D9)⋊C4C2212yes—
31C3×C4⋊Dic9C2211yescharacteristic
32C3×C4⋊Dic9C2212yes—
33C3×Dic9⋊C4C2212yes—
34C3×Dic9⋊C4C2211yescharacteristic
35C3×C4×Dic9C2211yescharacteristic
36C3×C4×Dic9C2212yes—
37C2×C32×C4⋊C4D311yescharacteristic
38C3×#930—31no—
39C3×#930—32no—
40C3×C4×#67—31no—
41C3×C4×#67—32no—
42C3×#898—32no—
43C3×#895—31no—
44C3×#894—32no—
45C3×#893—31no—
46C3×#892—31no—
47C3×#890—31no—
48C2×C9×C4⋊C4—21no—
49C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
50#63492C2×C311yescharacteristic, complemented
51#63492C2×C312yescomplemented
52C4×#611C2×C311yescharacteristic, complemented
53C4×#611C2×C312yescomplemented
54#63460C2×C312yescomplemented
55#63457C2×C311yescharacteristic, complemented
56#63456C2×C312yescomplemented
57#63455C2×C311yescharacteristic, complemented
58#63454C2×C311yescharacteristic, complemented
59#63452C2×C311yescharacteristic, complemented
60C23×C3×C9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
62C22×C3×D9C2312yes—
63C2×C3×C4×C9—22no—
64C2×C3×C4×C9—42no—
65C2×C3×C4×C9C2312yes—
66C2×C3×C4×C9C2312yes—
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3×C4×C9C2311yescharacteristic
69C3×#611—22no—
70C3×#611—24no—
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C2×C3×Dic9C2312yes—
75C3×D36—22no—
76C3×D36—42no—
77C3×C4×D9—22no—
78C3×C4×D9—42no—
79C3×#553—91no—
80#5514—31no—
81C22×C32×C4C2×D311yescharacteristic
82C22×C32×C4C2×D312yes—
83C2×C3×#67—31no—
84C2×C3×#67—32no—
85C2×C3×D12—31no—
86C2×C3×D12—32no—
87C2×C3×C4×D3—31no—
88C2×C3×C4×D3—32no—
89C32×C4⋊C4C2×D311yescharacteristic
90C32×C4⋊C4C2×D311yescharacteristic
91C32×C4⋊C4C2×D312yes—
92C3×#217—31no—
93C3×(C2×D3)⋊C4—32no—
94C3×(C2×D3)⋊C4—32no—
95C3×(C2×D3)⋊C4—31no—
96C3×(C2×D3)⋊C4—32no—
97C3×(C2×D3)⋊C4—32no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—32no—
100C3×Dic3⋊C4—32no—
101C3×Dic3⋊C4—31no—
102C3×C4×Dic3—31no—
103C3×C4×Dic3—32no—
104C22×C4×C9—21no—
105C22×C4×C9—22no—
106C22×C4×C9C22×C311yescharacteristic
107C22×C4×C9C22×C312yes—
108C2×#611C22×C311yescharacteristic
109C2×#611C22×C312yes—
110C2×D36C22×C311yescharacteristic
111C2×D36C22×C312yes—
112C2×C4×D9C22×C311yescharacteristic
113C2×C4×D9C22×C312yes—
114C9×C4⋊C4—21no—
115C9×C4⋊C4—21no—
116C9×C4⋊C4—22no—
117C9×C4⋊C4C22×C311yescharacteristic
118C9×C4⋊C4C22×C311yescharacteristic
119C9×C4⋊C4C22×C312yes—
120#3676C22×C311yescharacteristic
121(C2×D9)⋊C4C22×C311yescharacteristic
122(C2×D9)⋊C4C22×C312yes—
123(C2×D9)⋊C4C22×C312yes—
124(C2×D9)⋊C4C22×C312yes—
125(C2×D9)⋊C4C22×C312yes—
126C4⋊Dic9C22×C311yescharacteristic
127C4⋊Dic9C22×C312yes—
128Dic9⋊C4C22×C312yes—
129Dic9⋊C4C22×C311yescharacteristic
130C4×Dic9C22×C311yescharacteristic
131C4×Dic9C22×C312yes—
132C22×C3×C9—21no—
133C22×C3×C9—21no—
134C22×C3×C9#4112yes—
135C22×C3×C9C2411yescharacteristic
136C2×C3×D9—21no—
137C2×C3×D9—22no—
138C2×C3×D9—21no—
139C2×C3×D9—22no—
140C2×C3×D9—22no—
141C2×C3×D9—21no—
142C3×C4×C9—21no—
143C3×C4×C9—22no—
144C3×C4×C9—22no—
145C3×C4×C9#4112yes—
146C3×Dic9—21no—
147C3×Dic9—21no—
148C3×Dic9—22no—
149C3×Dic9—21no—
150C3×#126—92no—
151C3×#125—91no—
152C3×#123—92no—
153C3×#121—92no—
154C3×#121—91no—
155C3×#121—91no—
156C3×C4×D4—92no—
157C3×C4×D4—91no—
158C3×C4×D4—91no—
159C3×#117—91no—
160C2×C3×C4⋊C4—21no—
161C2×C3×C4⋊C4D911yescharacteristic, complemented
162C2×C3×C4⋊C4C3×D311yescharacteristic
163#930—31no—
164#930—32no—
165C4×#67—31no—
166C4×#67—32no—
167#898—32no—
168#895—31no—
169#894—32no—
170#893—31no—
171#892—31no—
172#890—31no—
173C23×C32C22×D311yescharacteristic
174C22×C3×D3—31no—
175C22×C3×D3—32no—
176C2×C32×C4—22no—
177C2×C32×C4—42no—
178C2×C32×C4C22×D312yes—
179C2×C32×C4C22×D312yes—
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182C3×#67—62no—
183C3×#67—64no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—31no—
187C2×C3×Dic3—32no—
188C3×D12—62no—
189C3×D12—122no—
190C3×C4×D3—62no—
191C3×C4×D3—122no—
192C23×C9—21no—
193C23×C9C23×C311yescharacteristic
194C22×D9C23×C311yescharacteristic
195C22×D9C23×C312yes—
196C2×C4×C9—24no—
197C2×C4×C9—22no—
198C2×C4×C9—21no—
199C2×C4×C9—21no—
200C2×C4×C9—22no—
201C2×C4×C9—22no—
202C2×C4×C9—42no—
203C2×C4×C9—44no—
204C2×C4×C9C23×C312yes—
205C2×C4×C9C23×C312yes—
206C2×C4×C9C23×C311yescharacteristic
207C2×C4×C9C23×C311yescharacteristic
208#611—22no—
209#611—24no—
210C2×Dic9C23×C311yescharacteristic
211C2×Dic9C23×C311yescharacteristic
212C2×Dic9C23×C311yescharacteristic
213C2×Dic9C23×C312yes—
214D36—22no—
215D36—42no—
216C4×D9—22no—
217C4×D9—42no—
218#553—91noSylow
219C2×C3×C9—22no—
220C2×C3×C9C2×#4111yescharacteristic
221C2×C3×C9C2×#4111yescharacteristic
222C2×C3×C9H(5,𝔽2)11yescharacteristic
223C3×D9—41no—
224C3×D9—42no—
225C2×C3×D4—92no—
226C2×C3×D4—91no—
227C2×C3×D4—91no—
228C2×C3×D4—92no—
229C22×C3×C4—21no—
230C22×C3×C4—22no—
231C22×C3×C4—91no—
232C22×C3×C4—92no—
233C22×C3×C4C2×D911yescharacteristic
234C22×C3×C4C2×D912yes—
235C22×C3×C4C2×C3×D311yescharacteristic
236C22×C3×C4C2×C3×D312yes—
237C2×#67—31no—
238C2×#67—32no—
239C2×D12—31no—
240C2×D12—32no—
241C2×C4×D3—31no—
242C2×C4×D3—32no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—92no—
247C3×C4⋊C4—91no—
248C3×C4⋊C4—91no—
249C3×C4⋊C4—92no—
250C3×C4⋊C4C2×D911yescharacteristic
251C3×C4⋊C4C2×D911yescharacteristic
252C3×C4⋊C4C2×D912yes—
253C3×C4⋊C4C2×C3×D311yescharacteristic
254C3×C4⋊C4C2×C3×D311yescharacteristic
255C3×C4⋊C4C2×C3×D312yes—
256C3×C22⋊C4—91no—
257C3×C22⋊C4—92no—
258C3×C22⋊C4—92no—
259C3×C22⋊C4—92no—
260C3×C22⋊C4—91no—
261C3×C22⋊C4—92no—
262C3×C42—92no—
263C3×C42—91no—
264#217—31no—
265(C2×D3)⋊C4—32no—
266(C2×D3)⋊C4—32no—
267(C2×D3)⋊C4—31no—
268(C2×D3)⋊C4—32no—
269(C2×D3)⋊C4—32no—
270C4⋊Dic3—31no—
271C4⋊Dic3—32no—
272Dic3⋊C4—32no—
273Dic3⋊C4—31no—
274C4×Dic3—31no—
275C4×Dic3—32no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32#23912yes—
279C22×C32C23×D311yescharacteristic
280C2×C3×D3—61no—
281C2×C3×D3—62no—
282C2×C3×D3—62no—
283C2×C3×D3—61no—
284C2×C3×D3—61no—
285C2×C3×D3—62no—
286C32×C4—21no—
287C32×C4—22no—
288C32×C4—22no—
289C32×C4#23512yes—
290C3×Dic3—61no—
291C3×Dic3—61no—
292C3×Dic3—61no—
293C3×Dic3—62no—
294C22×C9—22no—
295C22×C9—22no—
296C22×C9—21no—
297C22×C9—21no—
298C22×C9—21no—
299C22×C9—22no—
300C22×C9C3×#4112yes—
301C22×C9C24×C311yescharacteristic
302C2×D9—21no—
303C2×D9—22no—
304C2×D9—21no—
305C2×D9—22no—
306C2×D9—22no—
307C2×D9—21no—
308C4×C9—22no—
309C4×C9—21no—
310C4×C9—22no—
311C4×C9—22no—
312C4×C9—22no—
313C4×C9—42no—
314C4×C9—42no—
315C4×C9C3×#4112yes—
316Dic9—21no—
317Dic9—21no—
318Dic9—22no—
319Dic9—21no—
320#126—92no—
321#125—91no—
322#123—92no—
323#121—92no—
324#121—91no—
325#121—91no—
326C4×D4—92no—
327C4×D4—91no—
328C4×D4—91no—
329#117—91no—
330C2×C4⋊C4C3×D911yescharacteristic, complemented
331C3×C9#55311yescharacteristic, complemented, Sylow
332C23×C3—21no—
333C23×C3—91no—
334C23×C3—92no—
335C23×C3C22×D911yescharacteristic
336C23×C3C22×C3×D311yescharacteristic
337C22×D3—31no—
338C22×D3—32no—
339C3×D4—184no—
340C3×D4—182no—
341C3×D4—182no—
342C3×D4—362no—
343C2×C3×C4—24no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—44no—
353C2×C3×C4—91no—
354C2×C3×C4—92no—
355C2×C3×C4—91no—
356C2×C3×C4—91no—
357C2×C3×C4—182no—
358C2×C3×C4—362no—
359C2×C3×C4C22×D912yes—
360C2×C3×C4C22×D911yescharacteristic
361C2×C3×C4C22×D912yes—
362C2×C3×C4C22×D911yescharacteristic
363C2×C3×C4C22×C3×D312yes—
364C2×C3×C4C22×C3×D312yes—
365C2×C3×C4C22×C3×D311yescharacteristic
366C2×C3×C4C22×C3×D311yescharacteristic
367#67—62no—
368#67—64no—
369C2×Dic3—31no—
370C2×Dic3—31no—
371C2×Dic3—31no—
372C2×Dic3—32no—
373D12—62no—
374D12—122no—
375C4×D3—62no—
376C4×D3—122no—
377C2×C32—22no—
378C2×C32C2×#23511yescharacteristic
379C2×C32#100411yescharacteristic
380C2×C32C2×#23911yescharacteristic
381C3×D3—121no—
382C3×D3—122no—
383C2×C9—24no—
384C2×C9—22no—
385C2×C9—21no—
386C2×C9—21no—
387C2×C9—21no—
388C2×C9C2×C3×#4111yescharacteristic
389C2×C9C2×C3×#4111yescharacteristic
390C2×C9C3×H(5,𝔽2)11yescharacteristic
391D9—41no—
392D9—42no—
393C2×D4—92no—
394C2×D4—91no—
395C2×D4—91no—
396C2×D4—92no—
397C22×C4—91no—
398C22×C4—92no—
399C22×C4C2×C3×D911yescharacteristic
400C22×C4C2×C3×D912yes—
401C4⋊C4—91no—
402C4⋊C4—92no—
403C4⋊C4—91no—
404C4⋊C4—92no—
405C4⋊C4C2×C3×D911yescharacteristic
406C4⋊C4C2×C3×D911yescharacteristic
407C4⋊C4C2×C3×D912yes—
408C22⋊C4—91no—
409C22⋊C4—92no—
410C22⋊C4—92no—
411C22⋊C4—92no—
412C22⋊C4—91no—
413C22⋊C4—92no—
414C42—92no—
415C42—91no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—22no—
424C22×C3—181no—
425C22×C3—182no—
426C22×C3—182no—
427C22×C3—181no—
428C22×C3—181no—
429C22×C3—182no—
430C22×C3#370112yes—
431C22×C3C23×D911yescharacteristic
432C22×C3C3×#23912yes—
433C22×C3C23×C3×D311yescharacteristic
434C2×D3—61no—
435C2×D3—62no—
436C2×D3—62no—
437C2×D3—61no—
438C2×D3—61no—
439C2×D3—62no—
440C3×C4—22no—
441C3×C4—21no—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—21no—
445C3×C4—22no—
446C3×C4—22no—
447C3×C4—22no—
448C3×C4—42no—
449C3×C4—42no—
450C3×C4—181no—
451C3×C4—181no—
452C3×C4—182no—
453C3×C4—181no—
454C3×C4#369712yes—
455C3×C4C3×#23512yes—
456Dic3—61no—
457Dic3—61no—
458Dic3—61no—
459Dic3—62no—
460C32#551411yescharacteristic
461C9—21no—
462C9C3×#55311yescharacteristic, complemented
463C23—91no—
464C23—92no—
465C23C22×C3×D911yescharacteristic
466D4—184no—
467D4—182no—
468D4—182no—
469D4—362no—
470C2×C4—22no—
471C2×C4—42no—
472C2×C4—91no—
473C2×C4—92no—
474C2×C4—91no—
475C2×C4—91no—
476C2×C4—182no—
477C2×C4—362no—
478C2×C4C22×C3×D912yes—
479C2×C4C22×C3×D911yescharacteristic
480C2×C4C22×C3×D912yes—
481C2×C4C22×C3×D911yescharacteristic
482C2×C3—24no—
483C2×C3—22no—
484C2×C3—22no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—361no—
489C2×C3—362no—
490C2×C3C2×#369711yescharacteristic
491C2×C3#6371011yescharacteristic
492C2×C3C2×#370111yescharacteristic
493C2×C3C2×C3×#23511yescharacteristic
494C2×C3C3×#100411yescharacteristic
495C2×C3C2×C3×#23911yescharacteristic
496D3—121no—
497D3—122no—
498C22—21no—
499C22—21no—
500C22—181no—
501C22—182no—
502C22—182no—
503C22—181no—
504C22—181no—
505C22—182no—
506C22C3×#370112yes—
507C22C23×C3×D911yescharacteristic
508C4—21no—
509C4—22no—
510C4—22no—
511C4—181no—
512C4—181no—
513C4—182no—
514C4—181no—
515C4C3×#369712yes—
516C3—21no—
517C3#1058961611yescharacteristic, direct product factor
518C3C3×#551411yescharacteristic
519C2—22no—
520C2—361no—
521C2—362no—
522C2C2×C3×#369711yescharacteristic
523C2C3×#6371011yescharacteristic
524C2C2×C3×#370111yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C3×C9×C4⋊C4C2
socleC22×C32C23×D3
hypercenterC2×C3×C4⋊C4D9
hypocenterC9C3×#553
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C9×C4⋊C4C2semidirect
2. C3×#63492C2semidirect
3. C3×#63492C2semidirect
4. C3×C4×#611C2semidirect
5. C3×C4×#611C2semidirect
6. C3×#63460C2semidirect
7. C3×#63457C2semidirect
8. C3×#63456C2semidirect
9. C3×#63455C2semidirect
10. C3×#63454C2semidirect
11. C3×#63452C2semidirect
13. #10589616C3direct (N × H)
49. C2×C9×C4⋊C4C2×C3semidirect
50. #63492C2×C3semidirect
51. #63492C2×C3semidirect
52. C4×#611C2×C3semidirect
53. C4×#611C2×C3semidirect
54. #63460C2×C3semidirect
55. #63457C2×C3semidirect
56. #63456C2×C3semidirect
57. #63455C2×C3semidirect
58. #63454C2×C3semidirect
59. #63452C2×C3semidirect
161. C2×C3×C4⋊C4D9semidirect
330. C2×C4⋊C4C3×D9semidirect
331. C3×C9#553semidirect
462. C9C3×#553semidirect
517. C3#10589616direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C9×C4⋊C42
C3×#634922
C3×C4×#6112
C3×#634602
C3×#634572
C3×#634562
C3×#634552
C3×#634542
C3×#634522
C3×#55143
#105896163