Finite Group Databasebeta

← C3×#10595037 #13174617

Subgroups — C3×#10595037 #13174617

Order 1728.

1206 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3750C211yesmaximal, characteristic, complemented
2C32×C4×#67C212yesmaximal, complemented
3C3×#63982C211yesmaximal, characteristic, complemented
4C3×#63980C211yesmaximal, characteristic
5C3×#63962C212yesmaximal, complemented
6C3×#63905C212yesmaximal, complemented
7C3×#63867C212yesmaximal, complemented
8C3×#63859C212yesmaximal, complemented
9C3×#63851C212yesmaximal, complemented
10C3×#5564—32nomaximal
11#10595037C311yesmaximal, characteristic, direct product factor
12C22×C33×C4C2211yescharacteristic
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C2×C32×#67C2212yes—
16C2×C32×C4×D3C2212yes—
17C2×C3×#627C2211yescharacteristic
18C2×C3×#625C2211yescharacteristic
19C3×#3750C2212yes—
20C3×#3750C2212yes—
21C32×#217C2212yes—
22C32×(C2×D3)⋊C4C2212yes—
23C32×Dic3⋊C4C2212yes—
24C32×C4×Dic3C2212yes—
25C3×#3724C2211yescharacteristic
26C3×#3724C2211yescharacteristic
27C3×Dic3⋊Dic3C2212yes—
28C3×Dic3⋊Dic3C2212yes—
29C3×Dic3⋊Dic3C2212yes—
30C3×(C2×D3)⋊Dic3C2212yes—
31C3×(C2×D3)⋊Dic3C2212yes—
32C3×(C2×D3)⋊Dic3C2212yes—
33C32×C4×D4—32no—
34C2×#3750C2×C311yescharacteristic, complemented
35C3×#939—32no—
36C3×#935—32no—
37C3×#929—32no—
38C3×C4×#67—22no—
39C3×C4×#67D312yescomplemented
40C3×C4×#67C2×C312yescomplemented
41C3×#924—32no—
42C2×C3×Dic3⋊C4—32no—
43C2×C3×Dic3⋊C4—61no—
44C3×#896—32no—
45C3×#889—32no—
46C3×#886—32no—
47C3×#883—32no—
48C3×#879—32no—
49C3×#878—32no—
50C3×#876—32no—
51C3×#870—32no—
52#63982C2×C311yescharacteristic, complemented
53#63980C2×C311yescharacteristic
54#63962C2×C312yescomplemented
55#63905C2×C312yescomplemented
56#63867C2×C312yescomplemented
57#63859C2×C312yescomplemented
58#63851C2×C312yescomplemented
59C23×C33C2311yescharacteristic
60C22×C32×D3C2312yes—
61C2×C33×C4—41no—
62C2×C33×C4C2312yes—
63C2×C3×C3⋊Dic3—22no—
64C2×C3×C3⋊Dic3—41no—
65C2×C3×C3⋊Dic3C2311yescharacteristic
66C2×C3×C3⋊Dic3C2311yescharacteristic
67C2×C3×C3⋊Dic3C2312yes—
68C32×#67—24no—
69C2×C32×Dic3C2312yes—
70C2×C32×Dic3C2312yes—
71C2×C32×Dic3C2312yes—
72C32×C4×D3—42no—
73C3×#627—41no—
74C3×#625—41no—
75C3×#539—91no—
76#5564—32no—
77C2×C32×D4—32no—
78C22×C32×C4—21no—
79C22×C32×C4—22no—
80C22×C32×C4—32no—
81C22×C32×C4—41no—
82C22×C32×C4C2×D312yes—
83C22×C32×C4C22×C311yescharacteristic
84C22×C3⋊Dic3C22×C311yescharacteristic
85C22×C3⋊Dic3C22×C311yescharacteristic
86C2×C3×#67—22no—
87C2×C3×#67—32no—
88C2×C3×#67C2×D312yes—
89C2×C3×#67C22×C312yes—
90C22×C3×Dic3—32no—
91C22×C3×Dic3—32no—
92C22×C3×Dic3—61no—
93C22×C3×Dic3—61no—
94C2×C3×C4×D3—22no—
95C2×C3×C4×D3C2×D312yes—
96C2×C3×C4×D3C22×C312yes—
97C2×C3×Dic6—32no—
98C2×#627C22×C311yescharacteristic
99C2×#625C22×C311yescharacteristic
100C32×C4⋊C4—32no—
101C32×C22⋊C4—32no—
102C32×C22⋊C4—32no—
103C32×C42—32no—
104#3750C22×C312yes—
105#3750C22×C312yes—
106C3×#217—22no—
107C3×#217—32no—
108C3×#217—32no—
109C3×#217—32no—
110C3×#217C2×D312yes—
111C3×#217C22×C312yes—
112C3×(C2×D3)⋊C4—22no—
113C3×(C2×D3)⋊C4—32no—
114C3×(C2×D3)⋊C4—32no—
115C3×(C2×D3)⋊C4—32no—
116C3×(C2×D3)⋊C4C2×D312yes—
117C3×(C2×D3)⋊C4C22×C312yes—
118C3×C4⋊Dic3—32no—
119C3×C4⋊Dic3—32no—
120C3×C4⋊Dic3—32no—
121C3×Dic3⋊C4—22no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—32no—
128C3×Dic3⋊C4—32no—
129C3×Dic3⋊C4—32no—
130C3×Dic3⋊C4—32no—
131C3×Dic3⋊C4—62no—
132C3×Dic3⋊C4—62no—
133C3×Dic3⋊C4C2×D312yes—
134C3×Dic3⋊C4C22×C312yes—
135C3×C4×Dic3—22no—
136C3×C4×Dic3C2×D312yes—
137C3×C4×Dic3C22×C312yes—
138#3724C22×C311yescharacteristic
139#3724C22×C311yescharacteristic
140Dic3⋊Dic3C22×C312yes—
141Dic3⋊Dic3C22×C312yes—
142Dic3⋊Dic3C22×C312yes—
143(C2×D3)⋊Dic3C22×C312yes—
144(C2×D3)⋊Dic3C22×C312yes—
145(C2×D3)⋊Dic3C22×C312yes—
146C22×C33—21no—
147C22×C33—21no—
148C22×C33#4112yes—
149C22×C33C2411yescharacteristic
150C2×C32×D3—22no—
151C2×C32×D3—22no—
152C2×C32×D3—22no—
153C33×C4—22no—
154C3×C3⋊Dic3—22no—
155C3×C3⋊Dic3—21no—
156C3×C3⋊Dic3—21no—
157C32×Dic3—22no—
158C32×Dic3—22no—
159C32×Dic3—22no—
160C3×#126—92no—
161C3×#125—92no—
162C3×#123—92no—
163C3×#123—92no—
164C3×C22⋊Q8—91no—
165C3×C22⋊Q8—92no—
166C3×#121—91no—
167C3×C4×D4—32no—
168C3×C4×D4—32no—
169C3×C4×D4—62no—
170C2×C3×C4⋊C4—91no—
171#939—32no—
172#935—32no—
173#929—32no—
174C4×#67C3×D312yescomplemented
175#924—32no—
176C2×Dic3⋊C4—32no—
177C2×Dic3⋊C4—61no—
178#896—32no—
179#889—32no—
180#886—32no—
181#883—32no—
182#879—32no—
183#878—32no—
184#876—32no—
185#870—32no—
186C23×C32—21no—
187C23×C32—22no—
188C23×C32—32no—
189C23×C32—41no—
190C23×C32C22×D312yes—
191C23×C32C23×C311yescharacteristic
192C22×C3×D3—22no—
193C22×C3×D3C22×D312yes—
194C22×C3×D3C23×C312yes—
195C32×D4—64no—
196C2×C32×C4—22no—
197C2×C32×C4—22no—
198C2×C32×C4—22no—
199C2×C32×C4—32no—
200C2×C32×C4—32no—
201C2×C32×C4—32no—
202C2×C32×C4—42no—
203C2×C32×C4—42no—
204C2×C32×C4—41no—
205C2×C32×C4—44no—
206C2×C32×C4—42no—
207C2×C32×C4—44no—
208C2×C32×C4—122no—
209C2×C32×C4C22×D312yes—
210C2×C32×C4C22×D312yes—
211C2×C32×C4C23×C312yes—
212C2×C3⋊Dic3—22no—
213C2×C3⋊Dic3—41no—
214C2×C3⋊Dic3C23×C312yes—
215C2×C3⋊Dic3C23×C311yescharacteristic
216C2×C3⋊Dic3C23×C311yescharacteristic
217C3×#67—28no—
218C3×#67—24no—
219C3×#67—24no—
220C3×#67—122no—
221C2×C3×Dic3—22no—
222C2×C3×Dic3—22no—
223C2×C3×Dic3—22no—
224C2×C3×Dic3—32no—
225C2×C3×Dic3—32no—
226C2×C3×Dic3—32no—
227C2×C3×Dic3—32no—
228C2×C3×Dic3—62no—
229C2×C3×Dic3—61no—
230C2×C3×Dic3—61no—
231C2×C3×Dic3—64no—
232C2×C3×Dic3—64no—
233C2×C3×Dic3—122no—
234C2×C3×Dic3—122no—
235C2×C3×Dic3C22×D312yes—
236C2×C3×Dic3C22×D312yes—
237C2×C3×Dic3C22×D312yes—
238C2×C3×Dic3C23×C312yes—
239C2×C3×Dic3C23×C312yes—
240C2×C3×Dic3C23×C312yes—
241C3×C4×D3—42no—
242C3×C4×D3—42no—
243C3×C4×D3—44no—
244C3×Dic6—122no—
245#627—41no—
246#625—41no—
247#539—91noSylow
248C2×C33—22no—
249C2×C33C2×#4111yescharacteristic
250C2×C33H(5,𝔽2)11yescharacteristic
251C2×C332−1+411yescharacteristic
252C32×D3—42no—
253C2×C3×Q8—91no—
254C2×C3×D4—32no—
255C2×C3×D4—32no—
256C2×C3×D4—62no—
257C2×C3×D4—91no—
258C22×C3×C4—21no—
259C22×C3×C4—22no—
260C22×C3×C4—32no—
261C22×C3×C4—32no—
262C22×C3×C4—41no—
263C22×C3×C4—62no—
264C22×C3×C4—91no—
265C22×C3×C4—91no—
266C22×C3×C4D3211yescharacteristic
267C22×C3×C4C2×C3×D312yes—
268C2×#67—32no—
269C2×#67C2×C3×D312yes—
270C22×Dic3—32no—
271C22×Dic3—32no—
272C22×Dic3—61no—
273C22×Dic3—61no—
274C2×C4×D3C2×C3×D312yes—
275C2×Dic6—32no—
276C3×C4⋊C4—32no—
277C3×C4⋊C4—32no—
278C3×C4⋊C4—62no—
279C3×C4⋊C4—91no—
280C3×C4⋊C4—91no—
281C3×C4⋊C4—92no—
282C3×C4⋊C4—92no—
283C3×C4⋊C4—92no—
284C3×C4⋊C4—92no—
285C3×C4⋊C4—92no—
286C3×C22⋊C4—32no—
287C3×C22⋊C4—32no—
288C3×C22⋊C4—32no—
289C3×C22⋊C4—32no—
290C3×C22⋊C4—62no—
291C3×C22⋊C4—62no—
292C3×C22⋊C4—92no—
293C3×C22⋊C4—92no—
294C3×C22⋊C4—92no—
295C3×C42—32no—
296C3×C42—32no—
297C3×C42—62no—
298#217—32no—
299#217—32no—
300#217—32no—
301#217C2×C3×D312yes—
302(C2×D3)⋊C4—32no—
303(C2×D3)⋊C4—32no—
304(C2×D3)⋊C4—32no—
305(C2×D3)⋊C4C2×C3×D312yes—
306C4⋊Dic3—32no—
307C4⋊Dic3—32no—
308C4⋊Dic3—32no—
309Dic3⋊C4—32no—
310Dic3⋊C4—32no—
311Dic3⋊C4—32no—
312Dic3⋊C4—32no—
313Dic3⋊C4—32no—
314Dic3⋊C4—32no—
315Dic3⋊C4—32no—
316Dic3⋊C4—32no—
317Dic3⋊C4—32no—
318Dic3⋊C4—62no—
319Dic3⋊C4—62no—
320Dic3⋊C4C2×C3×D312yes—
321C4×Dic3C2×C3×D312yes—
322C22×C32—21no—
323C22×C32—21no—
324C22×C32—22no—
325C22×C32—22no—
326C22×C32—22no—
327C22×C32—22no—
328C22×C32—21no—
329C22×C32—22no—
330C22×C32—22no—
331C22×C32—24no—
332C22×C32—42no—
333C22×C32—41no—
334C22×C32—42no—
335C22×C32—42no—
336C22×C32—42no—
337C22×C32—42no—
338C22×C32—62no—
339C22×C32—62no—
340C22×C32—62no—
341C22×C32#23514yes—
342C22×C32C3×#4112yes—
343C22×C32C23×D312yes—
344C22×C32C24×C311yescharacteristic
345C2×C3×D3—24no—
346C2×C3×D3—22no—
347C2×C3×D3—22no—
348C2×C3×D3—22no—
349C2×C3×D3—22no—
350C2×C3×D3—22no—
351C2×C3×D3—22no—
352C2×C3×D3—42no—
353C2×C3×D3—42no—
354C32×C4—22no—
355C32×C4—24no—
356C32×C4—22no—
357C32×C4—22no—
358C32×C4—42no—
359C32×C4—42no—
360C32×C4—44no—
361C32×C4—62no—
362C32×C4—62no—
363C32×C4—62no—
364C3⋊Dic3—22no—
365C3⋊Dic3—21no—
366C3⋊Dic3—21no—
367C3×Dic3—24no—
368C3×Dic3—22no—
369C3×Dic3—22no—
370C3×Dic3—22no—
371C3×Dic3—22no—
372C3×Dic3—22no—
373C3×Dic3—22no—
374C3×Dic3—42no—
375C3×Dic3—42no—
376C3×Dic3—64no—
377C3×Dic3—62no—
378C3×Dic3—62no—
379C3×Dic3—62no—
380C3×Dic3—62no—
381C3×Dic3—121no—
382C3×Dic3—121no—
383#126—92no—
384#125—92no—
385#123—92no—
386#123—92no—
387C22⋊Q8—91no—
388C22⋊Q8—92no—
389#121—91no—
390C4×D4—32no—
391C2×C4⋊C4—91no—
392C33#53911yescharacteristic, complemented, Sylow
393C23×C3—21no—
394C23×C3—22no—
395C23×C3—32no—
396C23×C3—32no—
397C23×C3—41no—
398C23×C3—62no—
399C23×C3C2×D3211yescharacteristic
400C23×C3C22×C3×D312yes—
401C22×D3C22×C3×D312yes—
402C3×Q8—361no—
403C3×D4—68no—
404C3×D4—64no—
405C3×D4—64no—
406C3×D4—361no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—32no—
411C2×C3×C4—32no—
412C2×C3×C4—32no—
413C2×C3×C4—32no—
414C2×C3×C4—32no—
415C2×C3×C4—32no—
416C2×C3×C4—41no—
417C2×C3×C4—42no—
418C2×C3×C4—44no—
419C2×C3×C4—42no—
420C2×C3×C4—44no—
421C2×C3×C4—42no—
422C2×C3×C4—62no—
423C2×C3×C4—62no—
424C2×C3×C4—62no—
425C2×C3×C4—92no—
426C2×C3×C4—91no—
427C2×C3×C4—91no—
428C2×C3×C4—122no—
429C2×C3×C4—122no—
430C2×C3×C4—124no—
431C2×C3×C4—182no—
432C2×C3×C4—361no—
433C2×C3×C4C2×D3212yes—
434C2×C3×C4C22×C3×D312yes—
435C2×C3×C4C22×C3×D312yes—
436#67—24no—
437#67—122no—
438C2×Dic3—32no—
439C2×Dic3—32no—
440C2×Dic3—32no—
441C2×Dic3—32no—
442C2×Dic3—62no—
443C2×Dic3—61no—
444C2×Dic3—61no—
445C2×Dic3—64no—
446C2×Dic3—64no—
447C2×Dic3—122no—
448C2×Dic3—122no—
449C2×Dic3C22×C3×D312yes—
450C2×Dic3C22×C3×D312yes—
451C2×Dic3C22×C3×D312yes—
452C4×D3—42no—
453Dic6—122no—
454C2×C32—22no—
455C2×C32—24no—
456C2×C32—24no—
457C2×C32—21no—
458C2×C32—21no—
459C2×C32—21no—
460C2×C32—22no—
461C2×C32—22no—
462C2×C32—22no—
463C2×C32—41no—
464C2×C32—41no—
465C2×C32—41no—
466C2×C32—44no—
467C2×C32—44no—
468C2×C32—122no—
469C2×C32C2×#23512yes—
470C2×C32#100412yes—
471C2×C32#101012yes—
472C2×C32C2×C3×#4111yescharacteristic
473C2×C32C3×H(5,𝔽2)11yescharacteristic
474C2×C32C3×2−1+411yescharacteristic
475C3×D3—44no—
476C3×D3—42no—
477C3×D3—42no—
478C2×Q8—91no—
479C2×D4—32no—
480C2×D4—91no—
481C22×C4—32no—
482C22×C4—91no—
483C22×C4—91no—
484C22×C4C3×D3211yescharacteristic
485C4⋊C4—32no—
486C4⋊C4—91no—
487C4⋊C4—92no—
488C4⋊C4—92no—
489C4⋊C4—91no—
490C4⋊C4—92no—
491C4⋊C4—92no—
492C4⋊C4—92no—
493C22⋊C4—32no—
494C22⋊C4—32no—
495C22⋊C4—92no—
496C22⋊C4—92no—
497C22⋊C4—92no—
498C42—32no—
499C22×C3—22no—
500C22×C3—22no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—21no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—24no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—41no—
514C22×C3—42no—
515C22×C3—64no—
516C22×C3—62no—
517C22×C3—62no—
518C22×C3—62no—
519C22×C3—62no—
520C22×C3—62no—
521C22×C3—62no—
522C22×C3—122no—
523C22×C3—122no—
524C22×C3#379812yes—
525C22×C3C3×#23514yes—
526C22×C3C22×D3211yescharacteristic
527C22×C3C23×C3×D312yes—
528C2×D3—22no—
529C2×D3—22no—
530C2×D3—22no—
531C3×C4—24no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—22no—
535C3×C4—42no—
536C3×C4—42no—
537C3×C4—44no—
538C3×C4—64no—
539C3×C4—62no—
540C3×C4—62no—
541C3×C4—62no—
542C3×C4—62no—
543C3×C4—62no—
544C3×C4—62no—
545C3×C4—122no—
546C3×C4—122no—
547C3×C4—182no—
548C3×C4—181no—
549C3×C4—181no—
550Dic3—22no—
551Dic3—22no—
552Dic3—22no—
553Dic3—64no—
554Dic3—62no—
555Dic3—62no—
556Dic3—62no—
557Dic3—62no—
558Dic3—121no—
559Dic3—121no—
560C32—22no—
561C32—21no—
562C32—41no—
563C32#556412yescomplemented
564C32C3×#53911yescharacteristic, complemented
565C23—32no—
566C23C2×C3×D3211yescharacteristic
567Q8—361no—
568D4—64no—
569D4—361no—
570C2×C4—32no—
571C2×C4—32no—
572C2×C4—32no—
573C2×C4—41no—
574C2×C4—92no—
575C2×C4—91no—
576C2×C4—91no—
577C2×C4—122no—
578C2×C4—182no—
579C2×C4—361no—
580C2×C4C2×C3×D3212yes—
581C2×C3—24no—
582C2×C3—24no—
583C2×C3—22no—
584C2×C3—21no—
585C2×C3—22no—
586C2×C3—22no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—22no—
590C2×C3—44no—
591C2×C3—44no—
592C2×C3—41no—
593C2×C3—41no—
594C2×C3—41no—
595C2×C3—124no—
596C2×C3—122no—
597C2×C3—122no—
598C2×C3#6429811yescharacteristic
599C2×C3C2×#379811yescharacteristic
600C2×C3#6433011yescharacteristic
601C2×C3C2×C3×#23512yes—
602C2×C3C3×#100412yes—
603C2×C3C3×#101012yes—
604D3—42no—
605C22—21no—
606C22—21no—
607C22—62no—
608C22—62no—
609C22—62no—
610C22C3×#379812yes—
611C22C22×C3×D3211yescharacteristic
612C4—22no—
613C4—62no—
614C4—62no—
615C4—62no—
616C4—181no—
617C4—181no—
618C4—182no—
619C3—21no—
620C3—22no—
621C3—41no—
622C3#1059503711yescharacteristic, direct product factor
623C3C3×#556412yescomplemented
624C2—22no—
625C2—122no—
626C2C3×#6429811yescharacteristic
627C2C2×C3×#379811yescharacteristic
628C2C3×#6433011yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC22×C33C24
prime radicalC3×#63980C2
hypercenterC22×C3×C4D32
hypocenterC32C3×#539
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3750C2semidirect
2. C32×C4×#67C2semidirect
3. C3×#63982C2semidirect
5. C3×#63962C2semidirect
6. C3×#63905C2semidirect
7. C3×#63867C2semidirect
8. C3×#63859C2semidirect
9. C3×#63851C2semidirect
11. #10595037C3direct (N × H)
34. C2×#3750C2×C3semidirect
39. C3×C4×#67D3semidirect
40. C3×C4×#67C2×C3semidirect
52. #63982C2×C3semidirect
54. #63962C2×C3semidirect
55. #63905C2×C3semidirect
56. #63867C2×C3semidirect
57. #63859C2×C3semidirect
58. #63851C2×C3semidirect
174. C4×#67C3×D3semidirect
392. C33#539semidirect
563. C32#5564semidirect
564. C32C3×#539semidirect
622. C3#10595037direct (N × H)
623. C3C3×#5564semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37502
C32×C4×#672
C3×#639822
C3×#639802
C3×#639622
C3×#639052
C3×#638672
C3×#638592
C3×#638512
C3×#55643
#105950373