Finite Group Databasebeta

← C3×#10595501 #13175093

Subgroups — C3×#10595501 #13175093

Order 1728.

1950 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#3805C211yesmaximal, characteristic, complemented
3C32×#953C212yesmaximal, complemented
4C3×#63982C212yesmaximal, complemented
5C3×#63980C212yesmaximal, complemented
6C3×#67⋊Dic3C214yesmaximal, complemented
7C3×#63962C212yesmaximal, complemented
8C3×#63957C211yesmaximal, characteristic, complemented
9C3×#5586—32nomaximal
10#10595501C311yesmaximal, characteristic, direct product factor
11C24×C33C2211yescharacteristic
12C22×C3×C3⋊Dic3—24no—
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C22×C3×C3⋊Dic3C2212yes—
15C22×C3×C3⋊Dic3C2212yes—
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C2×C32×#67C2212yes—
18C2×C32×#67C2214yes—
19C2×C3×#627C2211yescharacteristic, complemented
20C2×C3×#625C2211yescharacteristic, complemented
21C3×#3805—24no—
22C2×C3×D3×Dic3C2212yescomplemented
23C32×#217C2212yes—
24C32×#217C2214yes—
25C3×#3724C2212yescomplemented
26C3×Dic3⋊Dic3C2214yescomplemented
27C3×(C2×D3)⋊Dic3C2214yescomplemented
28C3×Dic32C2212yescomplemented
29C23×C3⋊Dic3C2×C311yescharacteristic, complemented
30C23×C3×Dic3—32no—
31C23×C3×Dic3—61no—
32C2×C3×#237—32no—
33C2×#3805C2×C311yescharacteristic, complemented
34C32×#120—32no—
35C3×#953—22no—
36C3×#953D312yescomplemented
37C3×#953C2×C312yescomplemented
38C3×#939—34no—
39C3×#935—32no—
40C3×D4×Dic3—34no—
41C3×#886—32no—
42C3×#67⋊C4—34no—
43C3×#878—34no—
44C3×#877—32no—
45#63982C2×C312yescomplemented
46#63980C2×C312yescomplemented
47#67⋊Dic3C2×C314yescomplemented
48#63962C2×C312yescomplemented
49#63957C2×C311yescharacteristic, complemented
50C23×C33—22no—
51C23×C33—24no—
52C23×C33C2311yescharacteristic
53C23×C33C2312yes—
54C22×C32×D3C2312yes—
55C2×C3×C3⋊Dic3—22no—
56C2×C3×C3⋊Dic3—24no—
57C2×C3×C3⋊Dic3—24no—
58C2×C3×C3⋊Dic3D414yescomplemented
59C2×C3×C3⋊Dic3C2312yes—
60C2×C3×C3⋊Dic3C2311yescharacteristic
61C2×C3×C3⋊Dic3C2311yescharacteristic
62C32×#67—24no—
63C32×#67—28no—
64C2×C32×Dic3C2312yes—
65C2×C32×Dic3C2314yes—
66C3×#627—22no—
67C3×#625—22no—
68C3×D3×Dic3—24no—
69C3×#525—91no—
70#5586—32no—
71C24×C32—21no—
72C24×C32—22no—
73C24×C32—41no—
74C24×C32C2×D312yes—
75C24×C32C22×C311yescharacteristic
76C2×C32×D4—34no—
77C2×C32×D4—32no—
78C22×C3⋊Dic3—24no—
79C22×C3⋊Dic3C22×C311yescharacteristic
80C22×C3⋊Dic3C22×C312yes—
81C22×C3⋊Dic3C22×C312yes—
82C22×C3⋊Dic3C22×C311yescharacteristic
83C2×C3×#67—24no—
84C2×C3×#67—22no—
85C2×C3×#67—32no—
86C2×C3×#67C2×D314yes—
87C2×C3×#67C2×D312yes—
88C2×C3×#67C22×C312yes—
89C2×C3×#67C22×C314yes—
90C22×C3×Dic3—32no—
91C22×C3×Dic3—34no—
92C22×C3×Dic3—34no—
93C22×C3×Dic3—32no—
94C22×C3×Dic3—32no—
95C22×C3×Dic3—68no—
96C22×C3×Dic3—61no—
97C22×C3×Dic3—62no—
98C22×C3×Dic3—62no—
99C22×C3×Dic3—61no—
100C22×C3×Dic3—68no—
101C3×#237—68no—
102C2×C3×C4×D3—32no—
103C2×C3×Dic6—32no—
104C2×#627C22×C311yescharacteristic, complemented
105C2×#625C22×C311yescharacteristic, complemented
106#3805—24no—
107C2×D3×Dic3C22×C312yescomplemented
108C32×C22⋊C4—32no—
109C32×C22⋊C4—34no—
110C3×#217—22no—
111C3×#217—24no—
112C3×#217—34no—
113C3×#217C2×D312yes—
114C3×#217C2×D314yes—
115C3×#217C22×C312yes—
116C3×#217C22×C314yes—
117C3×(C2×D3)⋊C4—34no—
118C3×C4⋊Dic3—34no—
119C3×Dic3⋊C4—34no—
120C3×Dic3⋊C4—34no—
121C3×C4×Dic3—34no—
122#3724C22×C312yescomplemented
123Dic3⋊Dic3C22×C314yescomplemented
124(C2×D3)⋊Dic3C22×C314yescomplemented
125Dic32C22×C312yescomplemented
126C22×C33—28no—
127C22×C33—22no—
128C22×C33—22no—
129C22×C33—22no—
130C22×C33C2×D412yes—
131C22×C33#4114yescomplemented
132C22×C33C2411yescharacteristic
133C2×C32×D3—22no—
134C2×C32×D3—24no—
135C3×C3⋊Dic3—22no—
136C3×C3⋊Dic3C2×D412yes—
137C3×C3⋊Dic3C2×D412yes—
138C32×Dic3—22no—
139C32×Dic3—24no—
140C2×C3×#41—91no—
141C23×C3×C4—91no—
142C23×Dic3—32no—
143C23×Dic3—61no—
144C2×#237—32no—
145C3×#123—92no—
146C3×C22⋊Q8—92no—
147C3×#121—92no—
148C3×#120—32no—
149C3×#120—32no—
150C3×#120—62no—
151C3×C4×D4—94no—
152C3×#117—91no—
153#953C3×D312yescomplemented
154#939—34no—
155#935—32no—
156D4×Dic3—34no—
157#886—32no—
158#67⋊C4—34no—
159#878—34no—
160#877—32no—
161C23×C32—22no—
162C23×C32—24no—
163C23×C32—28no—
164C23×C32—24no—
165C23×C32—24no—
166C23×C32—28no—
167C23×C32—22no—
168C23×C32—21no—
169C23×C32—22no—
170C23×C32—24no—
171C23×C32—32no—
172C23×C32—41no—
173C23×C32—42no—
174C23×C32—44no—
175C23×C32—48no—
176C23×C32—44no—
177C23×C32—48no—
178C23×C32C22×D312yes—
179C23×C32C22×D314yes—
180C23×C32C23×C311yescharacteristic
181C23×C32C23×C312yes—
182C22×C3×D3—22no—
183C22×C3×D3C22×D312yes—
184C22×C3×D3C23×C312yes—
185C32×D4—64no—
186C32×D4—68no—
187C2×C32×C4—34no—
188C2×C32×C4—32no—
189C2×C3⋊Dic3—22no—
190C2×C3⋊Dic3—24no—
191C2×C3⋊Dic3—24no—
192C2×C3⋊Dic3C3×D414yescomplemented
193C2×C3⋊Dic3C23×C312yes—
194C2×C3⋊Dic3C23×C311yescharacteristic
195C2×C3⋊Dic3C23×C311yescharacteristic
196C3×#67—24no—
197C3×#67—28no—
198C3×#67—24no—
199C3×#67—28no—
200C3×#67—44no—
201C3×#67—48no—
202C3×#67—64no—
203C2×C3×Dic3—24no—
204C2×C3×Dic3—22no—
205C2×C3×Dic3—38no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—34no—
209C2×C3×Dic3—64no—
210C2×C3×Dic3—61no—
211C2×C3×Dic3—61no—
212C2×C3×Dic3—62no—
213C2×C3×Dic3—64no—
214C2×C3×Dic3—68no—
215C2×C3×Dic3—68no—
216C2×C3×Dic3—68no—
217C2×C3×Dic3—68no—
218C2×C3×Dic3—64no—
219C2×C3×Dic3—64no—
220C2×C3×Dic3C22×D314yes—
221C2×C3×Dic3C22×D312yes—
222C2×C3×Dic3C23×C314yes—
223C2×C3×Dic3C23×C312yes—
224C3×C4×D3—64no—
225C3×Dic6—64no—
226#627—22no—
227#625—22no—
228D3×Dic3—24no—
229#525—91noSylow
230C2×C33—22no—
231C2×C33—24no—
232C2×C33C22×D411yescharacteristic
233C2×C33C2×#4112yes—
234C32×D3—42no—
235C24×C3—21no—
236C24×C3—22no—
237C24×C3—41no—
238C24×C3D3211yescharacteristic
239C24×C3C2×C3×D312yes—
240C3×#41—184no—
241C2×C3×Q8—91no—
242C2×C3×D4—34no—
243C2×C3×D4—32no—
244C2×C3×D4—34no—
245C2×C3×D4—32no—
246C2×C3×D4—64no—
247C2×C3×D4—62no—
248C2×C3×D4—91no—
249C22×C3×C4—92no—
250C22×C3×C4—91no—
251C22×C3×C4—92no—
252C22×C3×C4—92no—
253C22×C3×C4—91no—
254C22×C3×C4—184no—
255C2×#67—32no—
256C2×#67C2×C3×D314yes—
257C2×#67C2×C3×D312yes—
258C22×Dic3—32no—
259C22×Dic3—34no—
260C22×Dic3—34no—
261C22×Dic3—32no—
262C22×Dic3—32no—
263C22×Dic3—61no—
264C22×Dic3—62no—
265C22×Dic3—62no—
266C22×Dic3—61no—
267C22×Dic3—68no—
268C22×Dic3—68no—
269#237—68no—
270C2×C4×D3—32no—
271C2×Dic6—32no—
272C3×C4⋊C4—92no—
273C3×C4⋊C4—94no—
274C3×C22⋊C4—32no—
275C3×C22⋊C4—34no—
276C3×C22⋊C4—32no—
277C3×C22⋊C4—34no—
278C3×C22⋊C4—62no—
279C3×C22⋊C4—64no—
280C3×C22⋊C4—94no—
281C3×C42—92no—
282#217—34no—
283#217C2×C3×D312yes—
284#217C2×C3×D314yes—
285(C2×D3)⋊C4—34no—
286C4⋊Dic3—34no—
287Dic3⋊C4—34no—
288Dic3⋊C4—34no—
289C4×Dic3—34no—
290C22×C32—28no—
291C22×C32—22no—
292C22×C32—22no—
293C22×C32—22no—
294C22×C32—24no—
295C22×C32—216no—
296C22×C32—216no—
297C22×C32—24no—
298C22×C32—24no—
299C22×C32—24no—
300C22×C32—24no—
301C22×C32—24no—
302C22×C32—21no—
303C22×C32—22no—
304C22×C32—24no—
305C22×C32—22no—
306C22×C32—24no—
307C22×C32—28no—
308C22×C32—42no—
309C22×C32—44no—
310C22×C32—41no—
311C22×C32—44no—
312C22×C32—44no—
313C22×C32—44no—
314C22×C32—44no—
315C22×C32—44no—
316C22×C32—44no—
317C22×C32—416no—
318C22×C32—416no—
319C22×C32—62no—
320C22×C32—64no—
321C22×C32D3×D414yes—
322C22×C32#23718yescomplemented
323C22×C32C2×C3×D412yes—
324C22×C32C3×#4114yescomplemented
325C22×C32C23×D312yes—
326C22×C32C24×C311yescharacteristic
327C2×C3×D3—22no—
328C2×C3×D3—24no—
329C2×C3×D3—22no—
330C2×C3×D3—24no—
331C2×C3×D3—42no—
332C2×C3×D3—44no—
333C32×C4—62no—
334C32×C4—64no—
335C3⋊Dic3—22no—
336C3⋊Dic3C2×C3×D412yes—
337C3⋊Dic3C2×C3×D412yes—
338C3×Dic3—22no—
339C3×Dic3—24no—
340C3×Dic3—22no—
341C3×Dic3—24no—
342C3×Dic3—34no—
343C3×Dic3—34no—
344C3×Dic3—42no—
345C3×Dic3—44no—
346C3×Dic3—62no—
347C3×Dic3—62no—
348C3×Dic3—64no—
349C3×Dic3—64no—
350C2×#41—91no—
351C23×C4—91no—
352#123—92no—
353C22⋊Q8—92no—
354#121—92no—
355#120—32no—
356C4×D4—94no—
357#117—91no—
358C33#52511yescharacteristic, complemented, Sylow
359C23×C3—24no—
360C23×C3—28no—
361C23×C3—24no—
362C23×C3—22no—
363C23×C3—24no—
364C23×C3—28no—
365C23×C3—22no—
366C23×C3—21no—
367C23×C3—24no—
368C23×C3—22no—
369C23×C3—32no—
370C23×C3—32no—
371C23×C3—48no—
372C23×C3—44no—
373C23×C3—48no—
374C23×C3—44no—
375C23×C3—42no—
376C23×C3—41no—
377C23×C3—62no—
378C23×C3C2×D3212yes—
379C23×C3C2×D3211yescharacteristic
380C23×C3C22×C3×D312yes—
381C23×C3C22×C3×D314yes—
382C22×D3C22×C3×D312yes—
383C3×Q8—182no—
384C3×D4—68no—
385C3×D4—64no—
386C3×D4—64no—
387C3×D4—68no—
388C3×D4—128no—
389C3×D4—124no—
390C3×D4—182no—
391C2×C3×C4—34no—
392C2×C3×C4—32no—
393C2×C3×C4—34no—
394C2×C3×C4—32no—
395C2×C3×C4—64no—
396C2×C3×C4—62no—
397C2×C3×C4—91no—
398C2×C3×C4—91no—
399C2×C3×C4—94no—
400C2×C3×C4—92no—
401C2×C3×C4—184no—
402C2×C3×C4—184no—
403C2×C3×C4—184no—
404C2×C3×C4—182no—
405#67—24no—
406#67—28no—
407#67—64no—
408C2×Dic3—32no—
409C2×Dic3—32no—
410C2×Dic3—38no—
411C2×Dic3—34no—
412C2×Dic3—64no—
413C2×Dic3—61no—
414C2×Dic3—61no—
415C2×Dic3—62no—
416C2×Dic3—64no—
417C2×Dic3—68no—
418C2×Dic3—68no—
419C2×Dic3—68no—
420C2×Dic3—68no—
421C2×Dic3—64no—
422C2×Dic3—64no—
423C2×Dic3C22×C3×D314yes—
424C2×Dic3C22×C3×D312yes—
425C4×D3—64no—
426Dic6—64no—
427C2×C32—22no—
428C2×C32—24no—
429C2×C32—28no—
430C2×C32—24no—
431C2×C32—24no—
432C2×C32—28no—
433C2×C32—22no—
434C2×C32—21no—
435C2×C32—22no—
436C2×C32—24no—
437C2×C32—41no—
438C2×C32—42no—
439C2×C32—44no—
440C2×C32—48no—
441C2×C32—44no—
442C2×C32—48no—
443C2×C32—122no—
444C2×C32C2×D3×D412yes—
445C2×C32C2×#23714yes—
446C2×C32C22×C3×D411yescharacteristic
447C2×C32C2×C3×#4112yes—
448C3×D3—42no—
449C3×D3—42no—
450C3×D3—82no—
451C24C3×D3211yescharacteristic
452#41—184no—
453C2×Q8—91no—
454C2×D4—34no—
455C2×D4—32no—
456C2×D4—91no—
457C22×C4—92no—
458C22×C4—91no—
459C22×C4—92no—
460C22×C4—92no—
461C22×C4—91no—
462C22×C4—184no—
463C4⋊C4—92no—
464C4⋊C4—94no—
465C22⋊C4—32no—
466C22⋊C4—34no—
467C22⋊C4—94no—
468C42—92no—
469C22×C3—24no—
470C22×C3—28no—
471C22×C3—216no—
472C22×C3—216no—
473C22×C3—24no—
474C22×C3—24no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—24no—
479C22×C3—24no—
480C22×C3—24no—
481C22×C3—21no—
482C22×C3—22no—
483C22×C3—24no—
484C22×C3—28no—
485C22×C3—22no—
486C22×C3—24no—
487C22×C3—44no—
488C22×C3—44no—
489C22×C3—44no—
490C22×C3—416no—
491C22×C3—416no—
492C22×C3—44no—
493C22×C3—44no—
494C22×C3—44no—
495C22×C3—41no—
496C22×C3—44no—
497C22×C3—42no—
498C22×C3—64no—
499C22×C3—62no—
500C22×C3—62no—
501C22×C3—64no—
502C22×C3—124no—
503C22×C3—122no—
504C22×C3#380514yescomplemented
505C22×C3#381112yes—
506C22×C3C3×D3×D414yes—
507C22×C3C3×#23718yescomplemented
508C22×C3C22×D3211yescharacteristic
509C22×C3C23×C3×D312yes—
510C2×D3—22no—
511C2×D3—24no—
512C3×C4—64no—
513C3×C4—62no—
514C3×C4—62no—
515C3×C4—64no—
516C3×C4—92no—
517C3×C4—92no—
518C3×C4—124no—
519C3×C4—122no—
520C3×C4—182no—
521Dic3—22no—
522Dic3—24no—
523Dic3—34no—
524Dic3—34no—
525Dic3—62no—
526Dic3—62no—
527Dic3—64no—
528Dic3—64no—
529C32—22no—
530C32—21no—
531C32—41no—
532C32#558612yescomplemented
533C32C3×#52511yescharacteristic, complemented
534C23—24no—
535C23—22no—
536C23—32no—
537C23C2×C3×D3212yes—
538C23C2×C3×D3211yescharacteristic
539Q8—182no—
540D4—68no—
541D4—64no—
542D4—182no—
543C2×C4—34no—
544C2×C4—32no—
545C2×C4—91no—
546C2×C4—91no—
547C2×C4—94no—
548C2×C4—92no—
549C2×C4—184no—
550C2×C4—184no—
551C2×C4—184no—
552C2×C4—182no—
553C2×C3—28no—
554C2×C3—24no—
555C2×C3—24no—
556C2×C3—28no—
557C2×C3—24no—
558C2×C3—22no—
559C2×C3—24no—
560C2×C3—22no—
561C2×C3—21no—
562C2×C3—22no—
563C2×C3—48no—
564C2×C3—44no—
565C2×C3—48no—
566C2×C3—44no—
567C2×C3—42no—
568C2×C3—41no—
569C2×C3—122no—
570C2×C3—122no—
571C2×C3—242no—
572C2×C3C2×#380512yes—
573C2×C3C2×#381111yescharacteristic
574C2×C3C2×C3×D3×D412yes—
575C2×C3C2×C3×#23714yes—
576D3—42no—
577C22—28no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—64no—
582C22—62no—
583C22C3×#380514yescomplemented
584C22C3×#381112yes—
585C22C22×C3×D3211yescharacteristic
586C4—64no—
587C4—62no—
588C4—92no—
589C4—92no—
590C4—182no—
591C3—21no—
592C3—22no—
593C3—41no—
594C3#1059550111yescharacteristic, direct product factor
595C3C3×#558612yescomplemented
596C2—24no—
597C2—22no—
598C2—122no—
599C2C2×C3×#380512yes—
600C2C2×C3×#381111yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC24×C33C22
socleC22×C33C24
hypercenterC24×C3D32
hypocenterC32C3×#525
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
1. C23×C3×C3⋊Dic3C2semidirect
2. C2×C3×#3805C2semidirect
3. C32×#953C2semidirect
4. C3×#63982C2semidirect
5. C3×#63980C2semidirect
6. C3×#67⋊Dic3C2semidirect
7. C3×#63962C2semidirect
8. C3×#63957C2semidirect
10. #10595501C3direct (N × H)
19. C2×C3×#627C22semidirect
20. C2×C3×#625C22semidirect
22. C2×C3×D3×Dic3C22semidirect
25. C3×#3724C22semidirect
26. C3×Dic3⋊Dic3C22semidirect
27. C3×(C2×D3)⋊Dic3C22semidirect
28. C3×Dic32C22semidirect
29. C23×C3⋊Dic3C2×C3semidirect
33. C2×#3805C2×C3semidirect
36. C3×#953D3semidirect
37. C3×#953C2×C3semidirect
45. #63982C2×C3semidirect
46. #63980C2×C3semidirect
47. #67⋊Dic3C2×C3semidirect
48. #63962C2×C3semidirect
49. #63957C2×C3semidirect
58. C2×C3×C3⋊Dic3D4semidirect
104. C2×#627C22×C3semidirect
105. C2×#625C22×C3semidirect
107. C2×D3×Dic3C22×C3semidirect
122. #3724C22×C3semidirect
123. Dic3⋊Dic3C22×C3semidirect
124. (C2×D3)⋊Dic3C22×C3semidirect
125. Dic32C22×C3semidirect
131. C22×C33#41semidirect
153. #953C3×D3semidirect
192. C2×C3⋊Dic3C3×D4semidirect
322. C22×C32#237semidirect
324. C22×C32C3×#41semidirect
358. C33#525semidirect
504. C22×C3#3805semidirect
507. C22×C3C3×#237semidirect
532. C32#5586semidirect
533. C32C3×#525semidirect
583. C22C3×#3805semidirect
594. C3#10595501direct (N × H)
595. C3C3×#5586semidirect

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C2×C3×#38052
C32×#9532
C3×#639822
C3×#639802
C3×#67⋊Dic32
C3×#639622
C3×#639572
C3×#55863
#105955013