Finite Group Databasebeta

← C3×#10589688 #13152747

Subgroups — C3×#10589688 #13152747

Order 1728.

1120 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×Dic9C211yesmaximal, characteristic, complemented
2C2×C3×#3699C211yesmaximal, characteristic, complemented
3C3×C9×#120C211yesmaximal, characteristic, complemented
4C3×#63515C211yesmaximal, characteristic, complemented
5C3×#63501C212yesmaximal, complemented
6C3×#63497C211yesmaximal, characteristic, complemented
7C3×D4×Dic9C212yesmaximal, complemented
8C3×#63448C211yesmaximal, characteristic, complemented
9C3×#611⋊C4C212yesmaximal, complemented
10C3×#63440C212yesmaximal, complemented
11C3×#63439C211yesmaximal, characteristic, complemented
12C3×#5586—31nomaximal
13#10589688C311yesmaximal, characteristic, direct product factor
14C24×C3×C9C2211yescharacteristic
15C2×C3×D4×C9C2211yescharacteristic
16C2×C3×D4×C9C2212yes—
17C2×C3×#611C2211yescharacteristic
18C2×C3×#611C2212yes—
19C2×C3×#611C2211yescharacteristic, complemented
20C22×C3×Dic9—24no—
21C22×C3×Dic9C2212yes—
22C22×C3×Dic9C2211yescharacteristic
23C22×C3×Dic9C2212yes—
24C22×C3×Dic9C2211yescharacteristic
25C22×C3×Dic9C2211yescharacteristic, complemented
26C3×#3699—24no—
27C2×C3×C4×D9C2211yescharacteristic, complemented
28C2×C3×Dic18C2211yescharacteristic, complemented
29C3×C9×C22⋊C4C2211yescharacteristic
30C3×C9×C22⋊C4C2212yes—
31C3×#3676C2212yes—
32C3×#3676C2211yescharacteristic
33C3×#3676C2212yescomplemented
34C3×(C2×D9)⋊C4C2212yescomplemented
35C3×C4⋊Dic9C2212yescomplemented
36C3×Dic9⋊C4C2212yescomplemented
37C3×Dic9⋊C4C2212yescomplemented
38C3×C4×Dic9C2212yescomplemented
39C23×C3×Dic3—31no—
40C2×C3×#237—31no—
41C32×#120D311yescharacteristic
42C3×#953—31no—
43C3×#939—32no—
44C3×#935—31no—
45C3×D4×Dic3—32no—
46C3×#886—31no—
47C3×#67⋊C4—32no—
48C3×#878—32no—
49C3×#877—31no—
50C23×Dic9C2×C311yescharacteristic, complemented
51C2×#3699C2×C311yescharacteristic, complemented
52C9×#120—21no—
53C9×#120C2×C311yescharacteristic, complemented
54#63515C2×C311yescharacteristic, complemented
55#63501C2×C312yescomplemented
56#63497C2×C311yescharacteristic, complemented
57D4×Dic9C2×C312yescomplemented
58#63448C2×C311yescharacteristic, complemented
59#611⋊C4C2×C312yescomplemented
60#63440C2×C312yescomplemented
61#63439C2×C311yescharacteristic, complemented
62C23×C3×C9—22no—
63C23×C3×C9—24no—
64C23×C3×C9C2311yescharacteristic
65C23×C3×C9C2311yescharacteristic
66C23×C3×C9C2312yes—
67C22×C3×D9C2311yescharacteristic
68C3×D4×C9—22no—
69C3×D4×C9—24no—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2312yes—
72C3×#611—22no—
73C3×#611—24no—
74C3×#611—22no—
75C2×C3×Dic9—22no—
76C2×C3×Dic9—22no—
77C2×C3×Dic9—24no—
78C2×C3×Dic9—24no—
79C2×C3×Dic9D414yescomplemented
80C2×C3×Dic9C2311yescharacteristic
81C2×C3×Dic9C2312yes—
82C2×C3×Dic9C2312yes—
83C2×C3×Dic9C2311yescharacteristic
84C2×C3×Dic9C2311yescharacteristic
85C3×C4×D9—22no—
86C3×Dic18—22no—
87C3×#525—91no—
88#5586—31no—
89C24×C32C2×D311yescharacteristic
90C2×C32×D4C2×D312yes—
91C2×C32×D4C2×D311yescharacteristic
92C2×C3×#67—32no—
93C2×C3×#67—31no—
94C2×C3×#67—31no—
95C22×C3×Dic3—31no—
96C22×C3×Dic3—32no—
97C22×C3×Dic3—32no—
98C22×C3×Dic3—31no—
99C22×C3×Dic3—31no—
100C22×C3×Dic3—64no—
101C3×#237—64no—
102C2×C3×C4×D3—31no—
103C2×C3×Dic6—31no—
104C24×C9—21no—
105C24×C9C22×C311yescharacteristic
106C32×C22⋊C4C2×D311yescharacteristic
107C32×C22⋊C4C2×D312yes—
108C3×#217—31no—
109C3×#217—32no—
110C3×#217—32no—
111C3×(C2×D3)⋊C4—32no—
112C3×C4⋊Dic3—32no—
113C3×Dic3⋊C4—32no—
114C3×Dic3⋊C4—32no—
115C3×C4×Dic3—32no—
116C2×D4×C9—22no—
117C2×D4×C9—21no—
118C2×D4×C9C22×C312yes—
119C2×D4×C9C22×C311yescharacteristic
120C2×#611C22×C312yes—
121C2×#611C22×C311yescharacteristic
122C2×#611C22×C311yescharacteristic, complemented
123C22×Dic9—24no—
124C22×Dic9C22×C312yes—
125C22×Dic9C22×C311yescharacteristic
126C22×Dic9C22×C312yes—
127C22×Dic9C22×C311yescharacteristic
128C22×Dic9C22×C311yescharacteristic, complemented
129#3699—24no—
130C2×C4×D9C22×C311yescharacteristic, complemented
131C2×Dic18C22×C311yescharacteristic, complemented
132C9×C22⋊C4—21no—
133C9×C22⋊C4—22no—
134C9×C22⋊C4C22×C311yescharacteristic
135C9×C22⋊C4C22×C312yes—
136#3676C22×C312yes—
137#3676C22×C311yescharacteristic
138#3676C22×C312yescomplemented
139(C2×D9)⋊C4C22×C312yescomplemented
140C4⋊Dic9C22×C312yescomplemented
141Dic9⋊C4C22×C312yescomplemented
142Dic9⋊C4C22×C312yescomplemented
143C4×Dic9C22×C312yescomplemented
144C22×C3×C9—28no—
145C22×C3×C9—22no—
146C22×C3×C9—22no—
147C22×C3×C9—22no—
148C22×C3×C9—21no—
149C22×C3×C9—22no—
150C22×C3×C9C2×D412yes—
151C22×C3×C9#4114yescomplemented
152C22×C3×C9C2411yescharacteristic
153C2×C3×D9—21no—
154C2×C3×D9—22no—
155C3×C4×C9—21no—
156C3×C4×C9—22no—
157C3×Dic9—21no—
158C3×Dic9—22no—
159C3×Dic9—22no—
160C3×Dic9C2×D412yes—
161C3×Dic9C2×D412yes—
162C2×C3×#41—91no—
163C23×C3×C4—91no—
164C23×Dic3—31no—
165C2×#237—31no—
166C3×#123—91no—
167C3×#123—91no—
168C3×C22⋊Q8—92no—
169C3×#121—92no—
170C3×#120—21no—
171C3×#120—91no—
172C3×#120D911yescharacteristic, complemented
173C3×#120C3×D311yescharacteristic
174C3×C4×D4—92no—
175C3×C4×D4—92no—
176C3×#117—91no—
177#953—31no—
178#939—32no—
179#935—31no—
180D4×Dic3—32no—
181#886—31no—
182#67⋊C4—32no—
183#878—32no—
184#877—31no—
185C23×C32—22no—
186C23×C32—24no—
187C23×C32C22×D311yescharacteristic
188C23×C32C22×D311yescharacteristic
189C23×C32C22×D312yes—
190C22×C3×D3—31no—
191C32×D4—22no—
192C32×D4—24no—
193C2×C32×C4C22×D312yes—
194C2×C32×C4C22×D311yescharacteristic
195C3×#67—62no—
196C3×#67—62no—
197C3×#67—64no—
198C2×C3×Dic3—34no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—62no—
205C2×C3×Dic3—64no—
206C2×C3×Dic3—64no—
207C2×C3×Dic3—62no—
208C3×C4×D3—62no—
209C3×Dic6—62no—
210C23×C9—22no—
211C23×C9—24no—
212C23×C9—21no—
213C23×C9—22no—
214C23×C9—21no—
215C23×C9—42no—
216C23×C9—44no—
217C23×C9C23×C311yescharacteristic
218C23×C9C23×C311yescharacteristic
219C23×C9C23×C312yes—
220C22×D9C23×C311yescharacteristic
221D4×C9—22no—
222D4×C9—24no—
223D4×C9—42no—
224D4×C9—44no—
225C2×C4×C9—22no—
226C2×C4×C9—21no—
227C2×C4×C9C23×C312yes—
228C2×C4×C9C23×C311yescharacteristic
229#611—22no—
230#611—24no—
231#611—22no—
232C2×Dic9—22no—
233C2×Dic9—22no—
234C2×Dic9—24no—
235C2×Dic9—24no—
236C2×Dic9C3×D414yescomplemented
237C2×Dic9C23×C312yes—
238C2×Dic9C23×C312yes—
239C2×Dic9C23×C311yescharacteristic
240C2×Dic9C23×C311yescharacteristic
241C2×Dic9C23×C311yescharacteristic
242C4×D9—22no—
243Dic18—22no—
244#525—91noSylow
245C2×C3×C9—22no—
246C2×C3×C9—24no—
247C2×C3×C9—41no—
248C2×C3×C9C22×D411yescharacteristic
249C2×C3×C9C2×#4112yes—
250C3×D9—41no—
251C24×C3—21no—
252C24×C3C2×D911yescharacteristic
253C24×C3C2×C3×D311yescharacteristic
254C3×#41—184no—
255C2×C3×Q8—91no—
256C2×C3×D4—22no—
257C2×C3×D4—21no—
258C2×C3×D4—92no—
259C2×C3×D4—91no—
260C2×C3×D4—91no—
261C2×C3×D4C2×D912yes—
262C2×C3×D4C2×D911yescharacteristic
263C2×C3×D4C2×C3×D312yes—
264C2×C3×D4C2×C3×D311yescharacteristic
265C22×C3×C4—91no—
266C22×C3×C4—92no—
267C22×C3×C4—91no—
268C22×C3×C4—92no—
269C22×C3×C4—91no—
270C22×C3×C4—91no—
271C22×C3×C4—184no—
272C2×#67—32no—
273C2×#67—31no—
274C2×#67—31no—
275C22×Dic3—31no—
276C22×Dic3—32no—
277C22×Dic3—32no—
278C22×Dic3—31no—
279C22×Dic3—31no—
280C22×Dic3—64no—
281#237—64no—
282C2×C4×D3—31no—
283C2×Dic6—31no—
284C3×C4⋊C4—92no—
285C3×C4⋊C4—92no—
286C3×C4⋊C4—92no—
287C3×C22⋊C4—21no—
288C3×C22⋊C4—22no—
289C3×C22⋊C4—91no—
290C3×C22⋊C4—92no—
291C3×C22⋊C4—92no—
292C3×C22⋊C4—92no—
293C3×C22⋊C4C2×D911yescharacteristic
294C3×C22⋊C4C2×D912yes—
295C3×C22⋊C4C2×C3×D311yescharacteristic
296C3×C22⋊C4C2×C3×D312yes—
297C3×C42—92no—
298#217—31no—
299#217—32no—
300#217—32no—
301(C2×D3)⋊C4—32no—
302C4⋊Dic3—32no—
303Dic3⋊C4—32no—
304Dic3⋊C4—32no—
305C4×Dic3—32no—
306C22×C32—28no—
307C22×C32—22no—
308C22×C32—22no—
309C22×C32—22no—
310C22×C32—21no—
311C22×C32—22no—
312C22×C32D3×D412yes—
313C22×C32#23714yes—
314C22×C32C23×D311yescharacteristic
315C2×C3×D3—61no—
316C2×C3×D3—62no—
317C32×C4—21no—
318C32×C4—22no—
319C3×Dic3—32no—
320C3×Dic3—32no—
321C3×Dic3—61no—
322C3×Dic3—62no—
323C3×Dic3—62no—
324C22×C9—28no—
325C22×C9—22no—
326C22×C9—22no—
327C22×C9—22no—
328C22×C9—21no—
329C22×C9—22no—
330C22×C9—21no—
331C22×C9—22no—
332C22×C9—24no—
333C22×C9—42no—
334C22×C9—42no—
335C22×C9—42no—
336C22×C9—41no—
337C22×C9—42no—
338C22×C9—48no—
339C22×C9C2×C3×D412yes—
340C22×C9C3×#4114yescomplemented
341C22×C9C24×C311yescharacteristic
342C2×D9—21no—
343C2×D9—22no—
344C4×C9—21no—
345C4×C9—22no—
346C4×C9—41no—
347C4×C9—42no—
348Dic9—21no—
349Dic9—22no—
350Dic9—22no—
351Dic9C2×C3×D412yes—
352Dic9C2×C3×D412yes—
353C2×#41—91no—
354C23×C4—91no—
355#123—91no—
356#123—91no—
357C22⋊Q8—92no—
358#121—92no—
359#120—91no—
360#120C3×D911yescharacteristic, complemented
361C4×D4—92no—
362C4×D4—92no—
363#117—91no—
364C3×C9#52511yescharacteristic, complemented, Sylow
365C23×C3—22no—
366C23×C3—24no—
367C23×C3—22no—
368C23×C3—24no—
369C23×C3—21no—
370C23×C3—22no—
371C23×C3—21no—
372C23×C3—42no—
373C23×C3—44no—
374C23×C3—91no—
375C23×C3C22×D911yescharacteristic
376C23×C3C22×D911yescharacteristic
377C23×C3C22×D912yes—
378C23×C3C22×C3×D311yescharacteristic
379C23×C3C22×C3×D311yescharacteristic
380C23×C3C22×C3×D312yes—
381C22×D3—31no—
382C3×Q8—182no—
383C3×D4—22no—
384C3×D4—24no—
385C3×D4—22no—
386C3×D4—24no—
387C3×D4—42no—
388C3×D4—44no—
389C3×D4—182no—
390C3×D4—182no—
391C3×D4—184no—
392C2×C3×C4—22no—
393C2×C3×C4—21no—
394C2×C3×C4—94no—
395C2×C3×C4—91no—
396C2×C3×C4—91no—
397C2×C3×C4—92no—
398C2×C3×C4—92no—
399C2×C3×C4—91no—
400C2×C3×C4—182no—
401C2×C3×C4—184no—
402C2×C3×C4—184no—
403C2×C3×C4—182no—
404C2×C3×C4—182no—
405C2×C3×C4C22×D912yes—
406C2×C3×C4C22×D911yescharacteristic
407C2×C3×C4C22×C3×D312yes—
408C2×C3×C4C22×C3×D311yescharacteristic
409#67—62no—
410#67—62no—
411#67—64no—
412C2×Dic3—31no—
413C2×Dic3—31no—
414C2×Dic3—34no—
415C2×Dic3—32no—
416C2×Dic3—32no—
417C2×Dic3—31no—
418C2×Dic3—62no—
419C2×Dic3—64no—
420C2×Dic3—64no—
421C2×Dic3—62no—
422C4×D3—62no—
423Dic6—62no—
424C2×C32—22no—
425C2×C32—24no—
426C2×C32—41no—
427C2×C32C2×D3×D411yescharacteristic
428C2×C32C2×#23712yes—
429C3×D3—121no—
430C2×C9—22no—
431C2×C9—24no—
432C2×C9—21no—
433C2×C9—22no—
434C2×C9—42no—
435C2×C9—44no—
436C2×C9—41no—
437C2×C9—81no—
438C2×C9C22×C3×D411yescharacteristic
439C2×C9C2×C3×#4112yes—
440D9—41no—
441C24C2×C3×D911yescharacteristic
442#41—184no—
443C2×Q8—91no—
444C2×D4—92no—
445C2×D4—91no—
446C2×D4—91no—
447C2×D4C2×C3×D912yes—
448C2×D4C2×C3×D911yescharacteristic
449C22×C4—91no—
450C22×C4—92no—
451C22×C4—91no—
452C22×C4—92no—
453C22×C4—91no—
454C22×C4—91no—
455C22×C4—184no—
456C4⋊C4—92no—
457C4⋊C4—92no—
458C4⋊C4—92no—
459C22⋊C4—91no—
460C22⋊C4—92no—
461C22⋊C4—92no—
462C22⋊C4—92no—
463C22⋊C4C2×C3×D911yescharacteristic
464C22⋊C4C2×C3×D912yes—
465C42—92no—
466C22×C3—28no—
467C22×C3—28no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—21no—
475C22×C3—22no—
476C22×C3—21no—
477C22×C3—22no—
478C22×C3—21no—
479C22×C3—22no—
480C22×C3—24no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—41no—
485C22×C3—42no—
486C22×C3—48no—
487C22×C3—181no—
488C22×C3—182no—
489C22×C3D4×D912yes—
490C22×C3#369914yescomplemented
491C22×C3C23×D911yescharacteristic
492C22×C3C3×D3×D412yes—
493C22×C3C3×#23714yes—
494C22×C3C23×C3×D311yescharacteristic
495C2×D3—61no—
496C2×D3—62no—
497C3×C4—21no—
498C3×C4—22no—
499C3×C4—21no—
500C3×C4—22no—
501C3×C4—41no—
502C3×C4—42no—
503C3×C4—92no—
504C3×C4—92no—
505C3×C4—182no—
506C3×C4—181no—
507C3×C4—182no—
508Dic3—32no—
509Dic3—32no—
510Dic3—61no—
511Dic3—62no—
512Dic3—62no—
513C32#558611yescharacteristic
514C9—21no—
515C9C3×#52511yescharacteristic, complemented
516C23—22no—
517C23—24no—
518C23—91no—
519C23C22×C3×D911yescharacteristic
520C23C22×C3×D911yescharacteristic
521C23C22×C3×D912yes—
522Q8—182no—
523D4—22no—
524D4—24no—
525D4—182no—
526D4—182no—
527D4—184no—
528C2×C4—91no—
529C2×C4—91no—
530C2×C4—94no—
531C2×C4—92no—
532C2×C4—92no—
533C2×C4—91no—
534C2×C4—182no—
535C2×C4—184no—
536C2×C4—184no—
537C2×C4—182no—
538C2×C4—182no—
539C2×C4C22×C3×D912yes—
540C2×C4C22×C3×D911yescharacteristic
541C2×C3—22no—
542C2×C3—24no—
543C2×C3—22no—
544C2×C3—24no—
545C2×C3—21no—
546C2×C3—22no—
547C2×C3—42no—
548C2×C3—44no—
549C2×C3—41no—
550C2×C3—41no—
551C2×C3—81no—
552C2×C3—361no—
553C2×C3C2×D4×D911yescharacteristic
554C2×C3C2×#369912yes—
555C2×C3C2×C3×D3×D411yescharacteristic
556C2×C3C2×C3×#23712yes—
557D3—121no—
558C22—28no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—21no—
563C22—22no—
564C22—181no—
565C22—182no—
566C22C3×D4×D912yes—
567C22C3×#369914yescomplemented
568C22C23×C3×D911yescharacteristic
569C4—21no—
570C4—22no—
571C4—92no—
572C4—92no—
573C4—182no—
574C4—181no—
575C4—182no—
576C3—21no—
577C3#1058968811yescharacteristic, direct product factor
578C3C3×#558611yescharacteristic
579C2—22no—
580C2—24no—
581C2—41no—
582C2—361no—
583C2C2×C3×D4×D911yescharacteristic
584C2C2×C3×#369912yes—
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#120C2
socleC22×C32C23×D3
hypercenterC3×#120D9
hypocenterC9C3×#525
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#120

Complements

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

NH ≅ G/Ntype
1. C23×C3×Dic9C2semidirect
2. C2×C3×#3699C2semidirect
3. C3×C9×#120C2semidirect
4. C3×#63515C2semidirect
5. C3×#63501C2semidirect
6. C3×#63497C2semidirect
7. C3×D4×Dic9C2semidirect
8. C3×#63448C2semidirect
9. C3×#611⋊C4C2semidirect
10. C3×#63440C2semidirect
11. C3×#63439C2semidirect
13. #10589688C3direct (N × H)
19. C2×C3×#611C22semidirect
25. C22×C3×Dic9C22semidirect
27. C2×C3×C4×D9C22semidirect
28. C2×C3×Dic18C22semidirect
33. C3×#3676C22semidirect
34. C3×(C2×D9)⋊C4C22semidirect
35. C3×C4⋊Dic9C22semidirect
36. C3×Dic9⋊C4C22semidirect
37. C3×Dic9⋊C4C22semidirect
38. C3×C4×Dic9C22semidirect
50. C23×Dic9C2×C3semidirect
51. C2×#3699C2×C3semidirect
53. C9×#120C2×C3semidirect
54. #63515C2×C3semidirect
55. #63501C2×C3semidirect
56. #63497C2×C3semidirect
57. D4×Dic9C2×C3semidirect
58. #63448C2×C3semidirect
59. #611⋊C4C2×C3semidirect
60. #63440C2×C3semidirect
61. #63439C2×C3semidirect
79. C2×C3×Dic9D4semidirect
122. C2×#611C22×C3semidirect
128. C22×Dic9C22×C3semidirect
130. C2×C4×D9C22×C3semidirect
131. C2×Dic18C22×C3semidirect
138. #3676C22×C3semidirect
139. (C2×D9)⋊C4C22×C3semidirect
140. C4⋊Dic9C22×C3semidirect
141. Dic9⋊C4C22×C3semidirect
142. Dic9⋊C4C22×C3semidirect
143. C4×Dic9C22×C3semidirect
151. C22×C3×C9#41semidirect
172. C3×#120D9semidirect
236. C2×Dic9C3×D4semidirect
340. C22×C9C3×#41semidirect
360. #120C3×D9semidirect
364. C3×C9#525semidirect
490. C22×C3#3699semidirect
515. C9C3×#525semidirect
567. C22C3×#3699semidirect
577. C3#10589688direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×Dic92
C2×C3×#36992
C3×C9×#1202
C3×#635152
C3×#635012
C3×#634972
C3×D4×Dic92
C3×#634482
C3×#611⋊C42
C3×#634402
C3×#634392
C3×#55863
#105896883