Finite Group Databasebeta

← C3×#10591386 #13158393

Subgroups — C3×#10591386 #13158393

Order 1728.

1442 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#625C211yesmaximal, characteristic
2C2×C3×#3750C211yesmaximal, characteristic, complemented
3C2×C32×(C2×D3)⋊C4C212yesmaximal, complemented
4C2×C3×(C2×D3)⋊Dic3C212yesmaximal, complemented
5C2×C3×Dic32C211yesmaximal, characteristic, complemented
6C3×#4957—32nomaximal
7#10591386C311yesmaximal, characteristic, direct product factor
8C23×C32×D3C2212yes—
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C2×C3×#625—22no—
14C2×C3×#625—22no—
15C2×C3×#625C414yescomplemented
16C3×#3750—22no—
17C32×(C2×D3)⋊C4—24no—
18C3×(C2×D3)⋊Dic3—24no—
19C3×Dic32—22no—
20C22×C3×#67—32no—
21C22×#625C2×C311yescharacteristic
22C2×C32×C22⋊C4—32no—
23C2×#3750C2×C311yescharacteristic, complemented
24C2×C3×#217—32no—
25C2×C3×(C2×D3)⋊C4—22no—
26C2×C3×(C2×D3)⋊C4—32no—
27C2×C3×(C2×D3)⋊C4D312yescomplemented
28C2×C3×(C2×D3)⋊C4C2×C312yescomplemented
29C2×C3×Dic3⋊C4—32no—
30C2×C3×Dic3⋊C4—61no—
31C2×C3×C4×Dic3—32no—
32C2×(C2×D3)⋊Dic3C2×C312yescomplemented
33C2×Dic32C2×C311yescharacteristic, complemented
34C23×C33C2311yescharacteristic
35C22×C32×D3—22no—
36C22×C32×D3—22no—
37C22×C32×D3—22no—
38C22×C32×D3—22no—
39C22×C32×D3—22no—
40C22×C32×D3—22no—
41C22×C32×D3C2×C414yes—
42C2×C33×C4—21no—
43C2×C33×C4—22no—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3—22no—
46C2×C3×C3⋊Dic3C2×C412yes—
47C2×C3×C3⋊Dic3D412yes—
48C2×C3×C3⋊Dic3D412yes—
49C2×C32×Dic3—22no—
50C2×C32×Dic3—22no—
51C2×C32×Dic3D414yescomplemented
52C3×#625—28no—
53C3×#390—91no—
54#4957—32no—
55C24×C32—32no—
56C23×C3×D3—22no—
57C23×C3×D3C2×D312yes—
58C23×C3×D3C22×C312yes—
59C22×C32×C4—21no—
60C22×C32×C4—22no—
61C22×C32×C4—32no—
62C22×C32×C4—41no—
63C22×C32×C4C2×D312yes—
64C22×C32×C4C22×C311yescharacteristic
65C22×C3⋊Dic3C22×C311yescharacteristic
66C22×C3⋊Dic3C22×C311yescharacteristic
67C2×C3×#67—38no—
68C2×C3×#67—64no—
69C2×C3×#67—64no—
70C22×C3×Dic3—22no—
71C22×C3×Dic3—32no—
72C22×C3×Dic3—32no—
73C22×C3×Dic3—61no—
74C22×C3×Dic3—61no—
75C22×C3×Dic3C2×D312yes—
76C22×C3×Dic3C22×C312yes—
77C2×#625—22no—
78C2×#625—22no—
79C2×#625C3×C414yescomplemented
80C32×C22⋊C4—64no—
81#3750—22no—
82C3×#217—64no—
83C3×(C2×D3)⋊C4—28no—
84C3×(C2×D3)⋊C4—24no—
85C3×(C2×D3)⋊C4—24no—
86C3×(C2×D3)⋊C4—64no—
87C3×Dic3⋊C4—64no—
88C3×Dic3⋊C4—64no—
89C3×C4×Dic3—64no—
90(C2×D3)⋊Dic3—24no—
91Dic32—22no—
92C22×C33C22×C411yescharacteristic
93C22×C33C2×D411yescharacteristic
94C22×C33C2×D411yescharacteristic
95C22×C33C2×D412yes—
96C22×C33#4112yes—
97C2×C32×D3—24no—
98C2×C32×D3—24no—
99C2×C32×D3—24no—
100C2×C32×D3—42no—
101C2×C32×D3—42no—
102C2×C32×D3—42no—
103C2×C32×D3—42no—
104C33×C4—41no—
105C3×C3⋊Dic3—41no—
106C3×C3⋊Dic3C22⋊C414yescomplemented
107C32×Dic3—24no—
108C22×C3×D4—91no—
109C22×#67—32no—
110C2×C3×C4⋊C4—91no—
111C2×C3×C22⋊C4—32no—
112C2×C3×C22⋊C4—32no—
113C2×C3×C22⋊C4—62no—
114C2×C3×C22⋊C4—92no—
115C2×C3×C42—91no—
116C2×#217—32no—
117C2×(C2×D3)⋊C4—32no—
118C2×(C2×D3)⋊C4C3×D312yescomplemented
119C2×Dic3⋊C4—32no—
120C2×Dic3⋊C4—61no—
121C2×C4×Dic3—32no—
122C23×C32—21no—
123C23×C32—22no—
124C23×C32—34no—
125C23×C32—41no—
126C23×C32—62no—
127C23×C32—62no—
128C23×C32—62no—
129C23×C32—62no—
130C23×C32—62no—
131C23×C32—62no—
132C23×C32C22×D312yes—
133C23×C32C23×C311yescharacteristic
134C22×C3×D3—24no—
135C22×C3×D3—24no—
136C22×C3×D3—22no—
137C22×C3×D3—22no—
138C22×C3×D3—22no—
139C22×C3×D3—22no—
140C22×C3×D3—22no—
141C22×C3×D3—22no—
142C22×C3×D3—22no—
143C22×C3×D3—22no—
144C22×C3×D3—22no—
145C22×C3×D3—22no—
146C22×C3×D3—22no—
147C22×C3×D3—22no—
148C22×C3×D3—24no—
149C22×C3×D3—42no—
150C22×C3×D3—42no—
151C22×C3×D3—42no—
152C22×C3×D3—42no—
153C22×C3×D3C4×D314yes—
154C22×C3×D3C2×C3×C414yes—
155C2×C32×C4—22no—
156C2×C32×C4—21no—
157C2×C32×C4—22no—
158C2×C32×C4—24no—
159C2×C32×C4—22no—
160C2×C32×C4—22no—
161C2×C32×C4—22no—
162C2×C32×C4—34no—
163C2×C32×C4—42no—
164C2×C32×C4—42no—
165C2×C32×C4—42no—
166C2×C32×C4—42no—
167C2×C32×C4—44no—
168C2×C32×C4—62no—
169C2×C32×C4—62no—
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3—22no—
172C2×C3⋊Dic3C2×C3×C412yes—
173C2×C3⋊Dic3C3×D412yes—
174C2×C3⋊Dic3C3×D412yes—
175C3×#67—616no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—24no—
181C2×C3×Dic3—34no—
182C2×C3×Dic3—34no—
183C2×C3×Dic3—34no—
184C2×C3×Dic3—42no—
185C2×C3×Dic3—42no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—62no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—122no—
194C2×C3×Dic3#6714yescomplemented
195C2×C3×Dic3C3×D414yescomplemented
196#625—28no—
197#390—91noSylow
198C2×C33C2×C22⋊C411yescharacteristic
199C2×C33C4×D412yes—
200C2×C33#12112yes—
201C2×C33#12411yescharacteristic
202C2×C33#12711yescharacteristic
203C32×D3—44no—
204C24×C3—32no—
205C24×C3—32no—
206C24×C3—62no—
207C23×D3C2×C3×D312yes—
208C2×C3×D4—94no—
209C2×C3×D4—182no—
210C2×C3×D4—182no—
211C22×C3×C4—21no—
212C22×C3×C4—22no—
213C22×C3×C4—32no—
214C22×C3×C4—32no—
215C22×C3×C4—41no—
216C22×C3×C4—62no—
217C22×C3×C4—91no—
218C22×C3×C4—91no—
219C22×C3×C4D3211yescharacteristic
220C22×C3×C4C2×C3×D312yes—
221C2×#67—38no—
222C2×#67—64no—
223C2×#67—64no—
224C22×Dic3—32no—
225C22×Dic3—32no—
226C22×Dic3—61no—
227C22×Dic3—61no—
228C22×Dic3C2×C3×D312yes—
229C3×C4⋊C4—182no—
230C3×C22⋊C4—64no—
231C3×C22⋊C4—64no—
232C3×C22⋊C4—68no—
233C3×C22⋊C4—184no—
234C3×C42—182no—
235#217—64no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—64no—
238Dic3⋊C4—64no—
239Dic3⋊C4—64no—
240C4×Dic3—64no—
241C22×C32—22no—
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—22no—
245C22×C32—21no—
246C22×C32—22no—
247C22×C32—22no—
248C22×C32—22no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32—22no—
252C22×C32—22no—
253C22×C32—41no—
254C22×C32—42no—
255C22×C32—41no—
256C22×C32—41no—
257C22×C32—42no—
258C22×C32—64no—
259C22×C32—64no—
260C22×C32—64no—
261C22×C32—122no—
262C22×C32—122no—
263C22×C32—122no—
264C22×C32—122no—
265C22×C32C2×C4×D312yes—
266C22×C32#23512yes—
267C22×C32D3×D412yes—
268C22×C32D3×D412yes—
269C22×C32D3×D412yes—
270C22×C32#23712yes—
271C22×C32C2×#6712yes—
272C22×C32C22×C3×C411yescharacteristic
273C22×C32C2×C3×D411yescharacteristic
274C22×C32C2×C3×D411yescharacteristic
275C22×C32C2×C3×D412yes—
276C22×C32C3×#4112yes—
277C2×C3×D3—28no—
278C2×C3×D3—24no—
279C2×C3×D3—24no—
280C2×C3×D3—24no—
281C2×C3×D3—24no—
282C2×C3×D3—24no—
283C2×C3×D3—24no—
284C2×C3×D3—44no—
285C2×C3×D3—44no—
286C2×C3×D3—44no—
287C2×C3×D3—44no—
288C2×C3×D3—44no—
289C2×C3×D3—44no—
290C2×C3×D3—42no—
291C2×C3×D3—42no—
292C2×C3×D3—42no—
293C2×C3×D3—42no—
294C2×C3×D3—42no—
295C2×C3×D3—42no—
296C2×C3×D3—42no—
297C2×C3×D3—42no—
298C32×C4—44no—
299C32×C4—42no—
300C32×C4—44no—
301C32×C4—42no—
302C32×C4—41no—
303C32×C4—64no—
304C3⋊Dic3—41no—
305C3⋊Dic3C3×C22⋊C414yescomplemented
306C3×Dic3—24no—
307C3×Dic3—24no—
308C3×Dic3—38no—
309C3×Dic3—44no—
310C3×Dic3—64no—
311C3×Dic3—122no—
312C3×Dic3—122no—
313C22×D4—91no—
314C2×C4⋊C4—91no—
315C2×C22⋊C4—32no—
316C2×C22⋊C4—92no—
317C2×C42—91no—
318C33#39011yescharacteristic, complemented, Sylow
319C23×C3—22no—
320C23×C3—21no—
321C23×C3—34no—
322C23×C3—34no—
323C23×C3—41no—
324C23×C3—64no—
325C23×C3—64no—
326C23×C3—64no—
327C23×C3—62no—
328C23×C3—62no—
329C23×C3—62no—
330C23×C3—62no—
331C23×C3—62no—
332C23×C3—62no—
333C23×C3—62no—
334C23×C3—62no—
335C23×C3—62no—
336C23×C3—62no—
337C23×C3—62no—
338C23×C3—62no—
339C23×C3—122no—
340C23×C3—122no—
341C23×C3—122no—
342C23×C3—122no—
343C23×C3C2×D3211yescharacteristic
344C23×C3C22×C3×D312yes—
345C22×D3—22no—
346C22×D3—22no—
347C22×D3—22no—
348C22×D3—22no—
349C22×D3—22no—
350C22×D3—22no—
351C22×D3C3×C4×D314yes—
352C3×D4—188no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—34no—
361C2×C3×C4—34no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—44no—
366C2×C3×C4—42no—
367C2×C3×C4—64no—
368C2×C3×C4—62no—
369C2×C3×C4—62no—
370C2×C3×C4—62no—
371C2×C3×C4—62no—
372C2×C3×C4—92no—
373C2×C3×C4—92no—
374C2×C3×C4—92no—
375C2×C3×C4—122no—
376C2×C3×C4—122no—
377C2×C3×C4—182no—
378C2×C3×C4—181no—
379#67—616no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—34no—
383C2×Dic3—34no—
384C2×Dic3—34no—
385C2×Dic3—62no—
386C2×Dic3—62no—
387C2×Dic3—62no—
388C2×Dic3—62no—
389C2×Dic3—62no—
390C2×Dic3—62no—
391C2×Dic3—62no—
392C2×Dic3—122no—
393C2×Dic3C3×#6714yescomplemented
394C2×C32—22no—
395C2×C32—22no—
396C2×C32—22no—
397C2×C32—22no—
398C2×C32—22no—
399C2×C32—22no—
400C2×C32—22no—
401C2×C32—22no—
402C2×C32—22no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—41no—
407C2×C32—42no—
408C2×C32—42no—
409C2×C32—41no—
410C2×C32—41no—
411C2×C32—124no—
412C2×C32D3×C22⋊C412yes—
413C2×C32#67⋊C412yes—
414C2×C32#88412yes—
415C2×C32#88512yes—
416C2×C32C4×#6712yes—
417C2×C32#93912yes—
418C2×C32#94012yes—
419C2×C32C2×C3×C22⋊C411yescharacteristic
420C2×C32C3×C4×D412yes—
421C2×C32C3×#12112yes—
422C2×C32C3×#12411yescharacteristic
423C2×C32C3×#12711yescharacteristic
424C3×D3—48no—
425C3×D3—44no—
426C3×D3—44no—
427C24—32no—
428C2×D4—94no—
429C2×D4—182no—
430C2×D4—182no—
431C22×C4—32no—
432C22×C4—91no—
433C22×C4—91no—
434C22×C4C3×D3211yescharacteristic
435C4⋊C4—182no—
436C22⋊C4—64no—
437C22⋊C4—184no—
438C42—182no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—22no—
442C22×C3—21no—
443C22×C3—22no—
444C22×C3—21no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—21no—
450C22×C3—22no—
451C22×C3—42no—
452C22×C3—41no—
453C22×C3—41no—
454C22×C3—42no—
455C22×C3—41no—
456C22×C3—68no—
457C22×C3—64no—
458C22×C3—64no—
459C22×C3—64no—
460C22×C3—64no—
461C22×C3—64no—
462C22×C3—64no—
463C22×C3—124no—
464C22×C3—124no—
465C22×C3—124no—
466C22×C3—124no—
467C22×C3—124no—
468C22×C3—124no—
469C22×C3—122no—
470C22×C3—122no—
471C22×C3—122no—
472C22×C3—122no—
473C22×C3—122no—
474C22×C3—122no—
475C22×C3—122no—
476C22×C3—122no—
477C22×C3#379512yes—
478C22×C3C4×D3211yescharacteristic
479C22×C3#380211yescharacteristic
480C22×C3D3×#6712yes—
481C22×C3#381111yescharacteristic
482C22×C3C2×C3×C4×D312yes—
483C22×C3C3×#23512yes—
484C22×C3C3×D3×D412yes—
485C22×C3C3×D3×D412yes—
486C22×C3C3×D3×D412yes—
487C22×C3C3×#23712yes—
488C22×C3C2×C3×#6712yes—
489C2×D3—24no—
490C2×D3—24no—
491C2×D3—24no—
492C2×D3—42no—
493C2×D3—42no—
494C2×D3—42no—
495C2×D3—42no—
496C3×C4—42no—
497C3×C4—44no—
498C3×C4—44no—
499C3×C4—42no—
500C3×C4—41no—
501C3×C4—64no—
502C3×C4—64no—
503C3×C4—94no—
504C3×C4—124no—
505C3×C4—361no—
506Dic3—24no—
507Dic3—38no—
508Dic3—64no—
509Dic3—122no—
510Dic3—122no—
511C32—22no—
512C32—21no—
513C32—41no—
514C32#495712yescomplemented
515C32C3×#39011yescharacteristic, complemented
516C23—34no—
517C23—62no—
518C23—62no—
519C23—62no—
520C23—62no—
521C23—62no—
522C23—62no—
523C23C2×C3×D3211yescharacteristic
524D4—188no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—34no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—92no—
531C2×C4—92no—
532C2×C4—92no—
533C2×C4—182no—
534C2×C4—181no—
535C2×C3—21no—
536C2×C3—22no—
537C2×C3—22no—
538C2×C3—22no—
539C2×C3—22no—
540C2×C3—22no—
541C2×C3—22no—
542C2×C3—22no—
543C2×C3—22no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—22no—
547C2×C3—41no—
548C2×C3—42no—
549C2×C3—42no—
550C2×C3—41no—
551C2×C3—41no—
552C2×C3—128no—
553C2×C3—124no—
554C2×C3—124no—
555C2×C3#6387912yes—
556C2×C3#6388512yes—
557C2×C3#6391311yescharacteristic
558C2×C3#6392111yescharacteristic
559C2×C3#6397911yescharacteristic
560C2×C3C3×D3×C22⋊C412yes—
561C2×C3C3×#67⋊C412yes—
562C2×C3C3×#88412yes—
563C2×C3C3×#88512yes—
564C2×C3C3×C4×#6712yes—
565C2×C3C3×#93912yes—
566C2×C3C3×#94012yes—
567D3—44no—
568C22—64no—
569C22—64no—
570C22—64no—
571C22—122no—
572C22—122no—
573C22—122no—
574C22—122no—
575C22C3×#379512yes—
576C22C3×C4×D3211yescharacteristic
577C22C3×#380211yescharacteristic
578C22C3×D3×#6712yes—
579C22C3×#381111yescharacteristic
580C4—41no—
581C4—64no—
582C4—94no—
583C4—361no—
584C3—22no—
585C3—21no—
586C3—41no—
587C3#1059138611yescharacteristic, direct product factor
588C3C3×#495712yescomplemented
589C2—124no—
590C2C3×#6387912yes—
591C2C3×#6388512yes—
592C2C3×#6391311yescharacteristic
593C2C3×#6392111yescharacteristic
594C2C3×#6397911yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC22×C3×#625C2
cosocleC23×C32C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#390
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3750C2semidirect
3. C2×C32×(C2×D3)⋊C4C2semidirect
4. C2×C3×(C2×D3)⋊Dic3C2semidirect
5. C2×C3×Dic32C2semidirect
7. #10591386C3direct (N × H)
15. C2×C3×#625C4semidirect
23. C2×#3750C2×C3semidirect
27. C2×C3×(C2×D3)⋊C4D3semidirect
28. C2×C3×(C2×D3)⋊C4C2×C3semidirect
32. C2×(C2×D3)⋊Dic3C2×C3semidirect
33. C2×Dic32C2×C3semidirect
51. C2×C32×Dic3D4semidirect
79. C2×#625C3×C4semidirect
106. C3×C3⋊Dic3C22⋊C4semidirect
118. C2×(C2×D3)⋊C4C3×D3semidirect
194. C2×C3×Dic3#67semidirect
195. C2×C3×Dic3C3×D4semidirect
305. C3⋊Dic3C3×C22⋊C4semidirect
318. C33#390semidirect
393. C2×Dic3C3×#67semidirect
514. C32#4957semidirect
515. C32C3×#390semidirect
587. C3#10591386direct (N × H)
588. C3C3×#4957semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6252
C2×C3×#37502
C2×C32×(C2×D3)⋊C42
C2×C3×(C2×D3)⋊Dic32
C2×C3×Dic322
C3×#49573
#105913863