Finite Group Databasebeta

← C3×#10589713 #13152814

Subgroups — C3×#10589713 #13152814

Order 1728.

970 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3699C212yesmaximal, complemented
2C3×C9×#121C211yesmaximal, characteristic, complemented
3C3×#63500C211yesmaximal, characteristic, complemented
4C3×#63497C212yesmaximal, complemented
5C3×D4×Dic9C211yesmaximal, characteristic, complemented
6C3×C4×#611C211yesmaximal, characteristic, complemented
7C2×C3×C4⋊Dic9C211yesmaximal, characteristic, complemented
8C3×C4⋊C4×D9C211yesmaximal, characteristic, complemented
9C3×#63450C211yesmaximal, characteristic, complemented
10C3×#63445C212yesmaximal, complemented
11C3×#63440C212yesmaximal, complemented
12C3×#5611—31nomaximal
13#10589713C311yesmaximal, characteristic, direct product factor
14C2×C3×D4×C9C2212yes—
15C2×C3×D4×C9C2211yescharacteristic
16C22×C3×C4×C9C2211yescharacteristic
17C2×C3×#611C2212yes—
18C2×C3×#611C2211yescharacteristic
19C22×C3×Dic9C2212yes—
20C22×C3×Dic9C2212yes—
21C3×#3699—28no—
22C2×C3×C4×D9C2211yescharacteristic
23C2×C3×C4×D9C2212yes—
24C2×C3×Dic18C2212yes—
25C3×C9×C4⋊C4C2211yescharacteristic
26C3×C9×C22⋊C4C2212yes—
27C3×#3676C2211yescharacteristic
28C3×#3676C2212yes—
29C3×#3676C2212yes—
30C3×(C2×D9)⋊C4C2211yescharacteristic
31C3×C4⋊Dic9C2211yescharacteristic
32C3×C4⋊Dic9C2211yescharacteristic
33C3×C4⋊Dic9C2212yes—
34C3×Dic9⋊C4C2211yescharacteristic
35C3×Dic9⋊C4C2212yes—
36C3×Dic9⋊C4C2212yes—
37C3×C4×Dic9C2211yescharacteristic
38C2×C3×#237—32no—
39C32×#121D311yescharacteristic
40C3×#938—31no—
41C3×#935—32no—
42C3×D4×Dic3—31no—
43C3×C4×#67—31no—
44C2×C3×C4⋊Dic3—31no—
45C3×D3×C4⋊C4—31no—
46C3×#888—31no—
47C3×#883—32no—
48C3×#878—32no—
49C2×#3699C2×C312yescomplemented
50C9×#121—21no—
51C9×#121C2×C311yescharacteristic, complemented
52#63500C2×C311yescharacteristic, complemented
53#63497C2×C312yescomplemented
54D4×Dic9C2×C311yescharacteristic, complemented
55C4×#611C2×C311yescharacteristic, complemented
56C2×C4⋊Dic9C2×C311yescharacteristic, complemented
57C4⋊C4×D9C2×C311yescharacteristic, complemented
58#63450C2×C311yescharacteristic, complemented
59#63445C2×C312yescomplemented
60#63440C2×C312yescomplemented
61C23×C3×C9C2312yes—
62C23×C3×C9C2311yescharacteristic
63C22×C3×D9C2311yescharacteristic
64C3×D4×C9—24no—
65C3×D4×C9—22no—
66C2×C3×C4×C9—22no—
67C2×C3×C4×C9C2312yes—
68C2×C3×C4×C9C2311yescharacteristic
69C2×C3×C4×C9C2311yescharacteristic
70C3×#611—24no—
71C3×#611D414yescomplemented
72C2×C3×Dic9—24no—
73C2×C3×Dic9—24no—
74C2×C3×Dic9C2312yes—
75C2×C3×Dic9C2312yes—
76C2×C3×Dic9C2311yescharacteristic
77C2×C3×Dic9C2311yescharacteristic
78C2×C3×Dic9C2311yescharacteristic
79C3×C4×D9—24no—
80C3×C4×D9—22no—
81C3×Dic18—24no—
82C3×#547—91no—
83#5611—31no—
84C2×C32×D4C2×D311yescharacteristic
85C2×C32×D4C2×D312yes—
86C22×C32×C4C2×D311yescharacteristic
87C2×C3×#67—31no—
88C2×C3×#67—32no—
89C22×C3×Dic3—32no—
90C22×C3×Dic3—32no—
91C3×#237—68no—
92C2×C3×C4×D3—32no—
93C2×C3×C4×D3—31no—
94C2×C3×Dic6—32no—
95C32×C4⋊C4C2×D311yescharacteristic
96C32×C22⋊C4C2×D312yes—
97C3×#217—32no—
98C3×#217—32no—
99C3×#217—31no—
100C3×(C2×D3)⋊C4—31no—
101C3×C4⋊Dic3—32no—
102C3×C4⋊Dic3—31no—
103C3×C4⋊Dic3—31no—
104C3×Dic3⋊C4—31no—
105C3×Dic3⋊C4—32no—
106C3×Dic3⋊C4—32no—
107C3×C4×Dic3—31no—
108C2×D4×C9—21no—
109C2×D4×C9—22no—
110C2×D4×C9C22×C311yescharacteristic
111C2×D4×C9C22×C312yes—
112C22×C4×C9—21no—
113C22×C4×C9C22×C311yescharacteristic
114C2×#611C22×C312yes—
115C2×#611C22×C311yescharacteristic
116C22×Dic9C22×C312yes—
117C22×Dic9C22×C312yes—
118#3699—28no—
119C2×C4×D9C22×C311yescharacteristic
120C2×C4×D9C22×C312yes—
121C2×Dic18C22×C312yes—
122C9×C4⋊C4—21no—
123C9×C4⋊C4C22×C311yescharacteristic
124C9×C22⋊C4—22no—
125C9×C22⋊C4C22×C312yes—
126#3676C22×C311yescharacteristic
127#3676C22×C312yes—
128#3676C22×C312yes—
129(C2×D9)⋊C4C22×C311yescharacteristic
130C4⋊Dic9C22×C311yescharacteristic
131C4⋊Dic9C22×C311yescharacteristic
132C4⋊Dic9C22×C312yes—
133Dic9⋊C4C22×C311yescharacteristic
134Dic9⋊C4C22×C312yes—
135Dic9⋊C4C22×C312yes—
136C4×Dic9C22×C311yescharacteristic
137C22×C3×C9—21no—
138C22×C3×C9—21no—
139C22×C3×C9—22no—
140C22×C3×C9—22no—
141C22×C3×C9—22no—
142C22×C3×C9C2×D412yes—
143C22×C3×C9C2411yescharacteristic
144C2×C3×D9—21no—
145C2×C3×D9—21no—
146C2×C3×D9C2×D412yes—
147C3×C4×C9—22no—
148C3×C4×C9—21no—
149C3×C4×C9#4112yescomplemented
150C3×Dic9—22no—
151C3×Dic9—22no—
152C3×Dic9—21no—
153C3×Dic9—21no—
154C3×Dic9C2×D412yes—
155C2×C3×#41—92no—
156C2×#237—32no—
157C3×C4⋊Q8—91no—
158C3×#123—92no—
159C3×#123—92no—
160C3×C22⋊Q8—92no—
161C3×#121—21no—
162C3×#121—91no—
163C3×#121D911yescharacteristic, complemented
164C3×#121C3×D311yescharacteristic
165C3×C4×D4—91no—
166C3×C4×D4—91no—
167C2×C3×C4⋊C4—91no—
168C2×C3×C4⋊C4—91no—
169#938—31no—
170#935—32no—
171D4×Dic3—31no—
172C4×#67—31no—
173C2×C4⋊Dic3—31no—
174D3×C4⋊C4—31no—
175#888—31no—
176#883—32no—
177#878—32no—
178C23×C32C22×D312yes—
179C23×C32C22×D311yescharacteristic
180C22×C3×D3—31no—
181C32×D4—24no—
182C32×D4—22no—
183C2×C32×C4—22no—
184C2×C32×C4C22×D312yes—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C3×#67—34no—
188C3×#67—64no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—64no—
195C2×C3×Dic3—64no—
196C3×C4×D3—64no—
197C3×C4×D3—62no—
198C3×Dic6—64no—
199C23×C9—21no—
200C23×C9—22no—
201C23×C9C23×C312yes—
202C23×C9C23×C311yescharacteristic
203C22×D9C23×C311yescharacteristic
204D4×C9—24no—
205D4×C9—22no—
206D4×C9—42no—
207D4×C9—44no—
208C2×C4×C9—22no—
209C2×C4×C9—21no—
210C2×C4×C9—21no—
211C2×C4×C9—22no—
212C2×C4×C9—42no—
213C2×C4×C9C23×C312yes—
214C2×C4×C9C23×C311yescharacteristic
215C2×C4×C9C23×C311yescharacteristic
216#611—24no—
217#611C3×D414yescomplemented
218C2×Dic9—24no—
219C2×Dic9—24no—
220C2×Dic9C23×C311yescharacteristic
221C2×Dic9C23×C311yescharacteristic
222C2×Dic9C23×C312yes—
223C2×Dic9C23×C312yes—
224C2×Dic9C23×C311yescharacteristic
225C4×D9—24no—
226C4×D9—22no—
227Dic18—24no—
228#547—91noSylow
229C2×C3×C9—22no—
230C2×C3×C9—42no—
231C2×C3×C9C22×D411yescharacteristic
232C2×C3×C9C2×#4111yescharacteristic
233C2×C3×C92−1+411yescharacteristic
234C3×D9—22no—
235C3×#41—188no—
236C2×C3×Q8—92no—
237C2×C3×D4—21no—
238C2×C3×D4—22no—
239C2×C3×D4—91no—
240C2×C3×D4—92no—
241C2×C3×D4C2×D911yescharacteristic
242C2×C3×D4C2×D912yes—
243C2×C3×D4C2×C3×D311yescharacteristic
244C2×C3×D4C2×C3×D312yes—
245C22×C3×C4—21no—
246C22×C3×C4—92no—
247C22×C3×C4—92no—
248C22×C3×C4—91no—
249C22×C3×C4—92no—
250C22×C3×C4C2×D911yescharacteristic
251C22×C3×C4C2×C3×D311yescharacteristic
252C2×#67—31no—
253C2×#67—32no—
254C22×Dic3—32no—
255C22×Dic3—32no—
256#237—68no—
257C2×C4×D3—32no—
258C2×C4×D3—31no—
259C2×Dic6—32no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—92no—
262C3×C4⋊C4—91no—
263C3×C4⋊C4—92no—
264C3×C4⋊C4—91no—
265C3×C4⋊C4—91no—
266C3×C4⋊C4—92no—
267C3×C4⋊C4C2×D911yescharacteristic
268C3×C4⋊C4C2×C3×D311yescharacteristic
269C3×C22⋊C4—22no—
270C3×C22⋊C4—91no—
271C3×C22⋊C4—92no—
272C3×C22⋊C4—92no—
273C3×C22⋊C4—91no—
274C3×C22⋊C4C2×D912yes—
275C3×C22⋊C4C2×C3×D312yes—
276C3×C42—91no—
277#217—32no—
278#217—32no—
279#217—31no—
280(C2×D3)⋊C4—31no—
281C4⋊Dic3—32no—
282C4⋊Dic3—31no—
283C4⋊Dic3—31no—
284Dic3⋊C4—31no—
285Dic3⋊C4—32no—
286Dic3⋊C4—32no—
287C4×Dic3—31no—
288C22×C32—21no—
289C22×C32—21no—
290C22×C32—22no—
291C22×C32—22no—
292C22×C32—22no—
293C22×C32D3×D412yes—
294C22×C32C23×D311yescharacteristic
295C2×C3×D3—32no—
296C2×C3×D3—61no—
297C2×C3×D3—61no—
298C32×C4—22no—
299C32×C4—21no—
300C32×C4#23712yes—
301C3×Dic3—32no—
302C3×Dic3—62no—
303C3×Dic3—62no—
304C3×Dic3—61no—
305C3×Dic3—61no—
306C22×C9—21no—
307C22×C9—21no—
308C22×C9—22no—
309C22×C9—22no—
310C22×C9—22no—
311C22×C9—21no—
312C22×C9—22no—
313C22×C9—41no—
314C22×C9—41no—
315C22×C9—42no—
316C22×C9—42no—
317C22×C9—42no—
318C22×C9C2×C3×D412yes—
319C22×C9C24×C311yescharacteristic
320C2×D9—21no—
321C2×D9—21no—
322C2×D9C2×C3×D412yes—
323C4×C9—22no—
324C4×C9—21no—
325C4×C9—22no—
326C4×C9—41no—
327C4×C9—42no—
328C4×C9C3×#4112yescomplemented
329Dic9—22no—
330Dic9—22no—
331Dic9—21no—
332Dic9—21no—
333Dic9C2×C3×D412yes—
334C2×#41—92no—
335C4⋊Q8—91no—
336#123—92no—
337#123—92no—
338C22⋊Q8—92no—
339#121—91no—
340#121C3×D911yescharacteristic, complemented
341C4×D4—91no—
342C4×D4—91no—
343C2×C4⋊C4—91no—
344C2×C4⋊C4—91no—
345C3×C9#54711yescharacteristic, complemented, Sylow
346C23×C3—21no—
347C23×C3—22no—
348C23×C3—91no—
349C23×C3C22×D912yes—
350C23×C3C22×D911yescharacteristic
351C23×C3C22×C3×D312yes—
352C23×C3C22×C3×D311yescharacteristic
353C22×D3—31no—
354C3×Q8—184no—
355C3×D4—24no—
356C3×D4—24no—
357C3×D4—22no—
358C3×D4—22no—
359C3×D4—42no—
360C3×D4—44no—
361C3×D4—94no—
362C3×D4—184no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—42no—
369C2×C3×C4—91no—
370C2×C3×C4—91no—
371C2×C3×C4—91no—
372C2×C3×C4—92no—
373C2×C3×C4—92no—
374C2×C3×C4—184no—
375C2×C3×C4—184no—
376C2×C3×C4—184no—
377C2×C3×C4—182no—
378C2×C3×C4C22×D911yescharacteristic
379C2×C3×C4C22×D912yes—
380C2×C3×C4C22×D911yescharacteristic
381C2×C3×C4C22×C3×D312yes—
382C2×C3×C4C22×C3×D311yescharacteristic
383C2×C3×C4C22×C3×D311yescharacteristic
384#67—34no—
385#67—64no—
386C2×Dic3—31no—
387C2×Dic3—31no—
388C2×Dic3—31no—
389C2×Dic3—32no—
390C2×Dic3—32no—
391C2×Dic3—64no—
392C2×Dic3—64no—
393C4×D3—64no—
394C4×D3—62no—
395Dic6—64no—
396C2×C32—22no—
397C2×C32—42no—
398C2×C32C2×D3×D411yescharacteristic
399C2×C32C2×#23711yescharacteristic
400C2×C32#101011yescharacteristic
401C3×D3—62no—
402C2×C9—22no—
403C2×C9—21no—
404C2×C9—21no—
405C2×C9—21no—
406C2×C9—42no—
407C2×C9—42no—
408C2×C9—82no—
409C2×C9C22×C3×D411yescharacteristic
410C2×C9C2×C3×#4111yescharacteristic
411C2×C9C3×2−1+411yescharacteristic
412D9—22no—
413#41—188no—
414C2×Q8—92no—
415C2×D4—91no—
416C2×D4—92no—
417C2×D4C2×C3×D911yescharacteristic
418C2×D4C2×C3×D912yes—
419C22×C4—92no—
420C22×C4—92no—
421C22×C4—91no—
422C22×C4—92no—
423C22×C4C2×C3×D911yescharacteristic
424C4⋊C4—91no—
425C4⋊C4—91no—
426C4⋊C4—92no—
427C4⋊C4—92no—
428C4⋊C4—91no—
429C4⋊C4—92no—
430C4⋊C4C2×C3×D911yescharacteristic
431C22⋊C4—91no—
432C22⋊C4—92no—
433C22⋊C4—92no—
434C22⋊C4—91no—
435C22⋊C4C2×C3×D912yes—
436C42—91no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—21no—
448C22×C3—22no—
449C22×C3—41no—
450C22×C3—41no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—92no—
455C22×C3—181no—
456C22×C3—181no—
457C22×C3D4×D912yes—
458C22×C3C23×D911yescharacteristic
459C22×C3C3×D3×D412yes—
460C22×C3C23×C3×D311yescharacteristic
461C2×D3—32no—
462C2×D3—61no—
463C2×D3—61no—
464C3×C4—22no—
465C3×C4—21no—
466C3×C4—22no—
467C3×C4—21no—
468C3×C4—22no—
469C3×C4—41no—
470C3×C4—42no—
471C3×C4—92no—
472C3×C4—182no—
473C3×C4—182no—
474C3×C4—181no—
475C3×C4—181no—
476C3×C4#369912yescomplemented
477C3×C4C3×#23712yes—
478Dic3—32no—
479Dic3—62no—
480Dic3—62no—
481Dic3—61no—
482Dic3—61no—
483C32#561111yescharacteristic
484C9—21no—
485C9C3×#54711yescharacteristic, complemented
486C23—91no—
487C23C22×C3×D912yes—
488C23C22×C3×D911yescharacteristic
489Q8—184no—
490D4—24no—
491D4—22no—
492D4—94no—
493D4—184no—
494C2×C4—22no—
495C2×C4—91no—
496C2×C4—91no—
497C2×C4—91no—
498C2×C4—92no—
499C2×C4—92no—
500C2×C4—184no—
501C2×C4—184no—
502C2×C4—184no—
503C2×C4—182no—
504C2×C4C22×C3×D911yescharacteristic
505C2×C4C22×C3×D912yes—
506C2×C4C22×C3×D911yescharacteristic
507C2×C3—22no—
508C2×C3—22no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—42no—
513C2×C3—42no—
514C2×C3—42no—
515C2×C3—82no—
516C2×C3—182no—
517C2×C3C2×D4×D911yescharacteristic
518C2×C3C2×#369911yescharacteristic
519C2×C3#6371611yescharacteristic
520C2×C3C2×C3×D3×D411yescharacteristic
521C2×C3C2×C3×#23711yescharacteristic
522C2×C3C3×#101011yescharacteristic
523D3—62no—
524C22—21no—
525C22—21no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—92no—
530C22—181no—
531C22—181no—
532C22C3×D4×D912yes—
533C22C23×C3×D911yescharacteristic
534C4—22no—
535C4—21no—
536C4—92no—
537C4—182no—
538C4—182no—
539C4—181no—
540C4—181no—
541C4C3×#369912yescomplemented
542C3—21no—
543C3#1058971311yescharacteristic, direct product factor
544C3C3×#561111yescharacteristic
545C2—22no—
546C2—42no—
547C2—182no—
548C2C2×C3×D4×D911yescharacteristic
549C2C2×C3×#369911yescharacteristic
550C2C3×#6371611yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#121C2
socleC22×C32C23×D3
hypercenterC3×#121D9
hypocenterC9C3×#547
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3699C2semidirect
2. C3×C9×#121C2semidirect
3. C3×#63500C2semidirect
4. C3×#63497C2semidirect
5. C3×D4×Dic9C2semidirect
6. C3×C4×#611C2semidirect
7. C2×C3×C4⋊Dic9C2semidirect
8. C3×C4⋊C4×D9C2semidirect
9. C3×#63450C2semidirect
10. C3×#63445C2semidirect
11. C3×#63440C2semidirect
13. #10589713C3direct (N × H)
49. C2×#3699C2×C3semidirect
51. C9×#121C2×C3semidirect
52. #63500C2×C3semidirect
53. #63497C2×C3semidirect
54. D4×Dic9C2×C3semidirect
55. C4×#611C2×C3semidirect
56. C2×C4⋊Dic9C2×C3semidirect
57. C4⋊C4×D9C2×C3semidirect
58. #63450C2×C3semidirect
59. #63445C2×C3semidirect
60. #63440C2×C3semidirect
71. C3×#611D4semidirect
149. C3×C4×C9#41semidirect
163. C3×#121D9semidirect
217. #611C3×D4semidirect
328. C4×C9C3×#41semidirect
340. #121C3×D9semidirect
345. C3×C9#547semidirect
476. C3×C4#3699semidirect
485. C9C3×#547semidirect
541. C4C3×#3699semidirect
543. C3#10589713direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#36992
C3×C9×#1212
C3×#635002
C3×#634972
C3×D4×Dic92
C3×C4×#6112
C2×C3×C4⋊Dic92
C3×C4⋊C4×D92
C3×#634502
C3×#634452
C3×#634402
C3×#56113
#105897133