Finite Group Databasebeta

← C3×#10595502 #13175094

Subgroups — C3×#10595502 #13175094

Order 1728.

1966 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#638C211yesmaximal, characteristic, complemented
2C2×C3×#3805C211yesmaximal, characteristic, complemented
3C32×#953C212yesmaximal, complemented
4C3×#63981C212yesmaximal, complemented
5C3×#63979C212yesmaximal, complemented
6C3×#63971C214yesmaximal, complemented
7C3×#63957C211yesmaximal, characteristic, complemented
8C3×#63953C212yesmaximal, complemented
9C3×#5592—32nomaximal
10#10595502C311yesmaximal, characteristic, direct product factor
11C24×C33C2211yescharacteristic
12C23×C3×C3⋊D3C2211yescharacteristic
13C2×C3×#638—24no—
14C2×C3×#638C2212yes—
15C2×C3×#638C2212yes—
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C2×C32×#67C2212yes—
18C2×C32×#67C2214yescomplemented
19C2×C3×#627C2211yescharacteristic, complemented
20C2×C3×#626C2214yescomplemented
21C2×C3×#625C2211yescharacteristic, complemented
22C3×#3805—24no—
23C2×C3×D3×Dic3C2212yescomplemented
24C32×#217C2212yes—
25C32×#217C2214yescomplemented
26C3×#3724C2212yescomplemented
27C3×#3722C2214yescomplemented
28C3×Dic32C2212yescomplemented
29C22×#638C2×C311yescharacteristic, complemented
30C22×C3×#67—32no—
31C22×C3×#67—61no—
32C2×C3×#237—32no—
33C2×#3805C2×C311yescharacteristic, complemented
34C32×#120—32no—
35C3×#953—22no—
36C3×#953D312yescomplemented
37C3×#953C2×C312yescomplemented
38C3×#940—34no—
39C3×#939—34no—
40C3×#937—32no—
41C3×#885—34no—
42C3×#884—34no—
43C3×#882—32no—
44C3×#877—32no—
45#63981C2×C312yescomplemented
46#63979C2×C312yescomplemented
47#63971C2×C314yescomplemented
48#63957C2×C311yescharacteristic, complemented
49#63953C2×C312yescomplemented
50C23×C33—22no—
51C23×C33—42no—
52C23×C33C2312yes—
53C23×C33C2311yescharacteristic
54C22×C3×C3⋊D3—22no—
55C22×C3×C3⋊D3—42no—
56C22×C3×C3⋊D3C2312yes—
57C22×C32×D3C2312yes—
58C3×#638—24no—
59C3×#638—24no—
60C2×C3×C3⋊Dic3D414yescomplemented
61C2×C3×C3⋊Dic3C2311yescharacteristic
62C2×C3×C3⋊Dic3C2311yescharacteristic
63C32×#67—24no—
64C32×#67—44no—
65C2×C32×Dic3C2314yes—
66C2×C32×Dic3C2312yes—
67C3×#627—22no—
68C3×#626—44no—
69C3×#625—22no—
70C3×D3×Dic3—24no—
71C3×#535—91no—
72#5592—32no—
73C24×C32—21no—
74C24×C32—22no—
75C24×C32—41no—
76C24×C32C2×D312yes—
77C24×C32C22×C311yescharacteristic
78C23×C3⋊D3C22×C311yescharacteristic
79C23×C3×D3—32no—
80C23×C3×D3—61no—
81C2×C32×D4—32no—
82C2×C32×D4—34no—
83C2×#638—24no—
84C2×#638C22×C312yes—
85C2×#638C22×C312yes—
86C22×C3⋊Dic3C22×C311yescharacteristic
87C2×C3×#67—22no—
88C2×C3×#67—24no—
89C2×C3×#67—34no—
90C2×C3×#67—34no—
91C2×C3×#67—32no—
92C2×C3×#67—34no—
93C2×C3×#67—68no—
94C2×C3×#67—62no—
95C2×C3×#67—62no—
96C2×C3×#67—68no—
97C2×C3×#67C2×D312yes—
98C2×C3×#67C2×D314yescomplemented
99C2×C3×#67C22×C312yes—
100C2×C3×#67C22×C314yescomplemented
101C22×C3×Dic3—32no—
102C22×C3×Dic3—32no—
103C22×C3×Dic3—61no—
104C3×#237—68no—
105C2×C3×D12—34no—
106C2×C3×C4×D3—32no—
107C2×C3×Dic6—32no—
108C2×#627C22×C311yescharacteristic, complemented
109C2×#626C22×C314yescomplemented
110C2×#625C22×C311yescharacteristic, complemented
111#3805—24no—
112C2×D3×Dic3C22×C312yescomplemented
113C32×C22⋊C4—32no—
114C32×C22⋊C4—34no—
115C3×#217—22no—
116C3×#217—24no—
117C3×#217C2×D312yes—
118C3×#217C2×D314yescomplemented
119C3×#217C22×C312yes—
120C3×#217C22×C314yescomplemented
121C3×(C2×D3)⋊C4—34no—
122C3×(C2×D3)⋊C4—34no—
123C3×Dic3⋊C4—34no—
124C3×C4×Dic3—34no—
125#3724C22×C312yescomplemented
126#3722C22×C314yescomplemented
127Dic32C22×C312yescomplemented
128C22×C33—22no—
129C22×C33—22no—
130C22×C33—24no—
131C22×C33—44no—
132C22×C33C2×D412yes—
133C22×C33C2411yescharacteristic
134C2×C3×C3⋊D3—22no—
135C2×C3×C3⋊D3—24no—
136C2×C3×C3⋊D3—44no—
137C2×C32×D3—22no—
138C2×C32×D3—24no—
139C3×C3⋊Dic3C2×D412yes—
140C3×C3⋊Dic3C2×D412yes—
141C32×Dic3—22no—
142C32×Dic3—24no—
143C2×C3×#41—91no—
144C22×C3×D4—91no—
145C22×#67—32no—
146C22×#67—61no—
147C2×#237—32no—
148C3×#127—92no—
149C3×#124—92no—
150C3×#121—94no—
151C3×#120—32no—
152C3×#120—32no—
153C3×#120—62no—
154C3×#120—92no—
155C3×#117—91no—
156#953C3×D312yescomplemented
157#940—34no—
158#939—34no—
159#937—32no—
160#885—34no—
161#884—34no—
162#882—32no—
163#877—32no—
164C23×C32—22no—
165C23×C32—24no—
166C23×C32—24no—
167C23×C32—21no—
168C23×C32—22no—
169C23×C32—24no—
170C23×C32—22no—
171C23×C32—32no—
172C23×C32—41no—
173C23×C32—42no—
174C23×C32—44no—
175C23×C32—44no—
176C23×C32—44no—
177C23×C32—42no—
178C23×C32—44no—
179C23×C32—48no—
180C23×C32—48no—
181C23×C32C22×D314yes—
182C23×C32C22×D312yes—
183C23×C32C23×C312yes—
184C23×C32C23×C311yescharacteristic
185C22×C3⋊D3—22no—
186C22×C3⋊D3—42no—
187C22×C3⋊D3C23×C312yes—
188C22×C3×D3—22no—
189C22×C3×D3—34no—
190C22×C3×D3—62no—
191C22×C3×D3—64no—
192C22×C3×D3—64no—
193C22×C3×D3—124no—
194C22×C3×D3—124no—
195C22×C3×D3C22×D312yes—
196C22×C3×D3C23×C312yes—
197C32×D4—64no—
198C32×D4—124no—
199C2×C32×C4—34no—
200C2×C32×C4—32no—
201#638—24no—
202#638—24no—
203C2×C3⋊Dic3C3×D414yescomplemented
204C2×C3⋊Dic3C23×C311yescharacteristic
205C2×C3⋊Dic3C23×C311yescharacteristic
206C3×#67—24no—
207C3×#67—24no—
208C3×#67—44no—
209C3×#67—44no—
210C3×#67—44no—
211C3×#67—48no—
212C3×#67—64no—
213C3×#67—68no—
214C3×#67—68no—
215C3×#67—68no—
216C3×#67—68no—
217C3×#67—124no—
218C2×C3×Dic3—24no—
219C2×C3×Dic3—22no—
220C2×C3×Dic3—38no—
221C2×C3×Dic3—32no—
222C2×C3×Dic3—32no—
223C2×C3×Dic3—64no—
224C2×C3×Dic3—61no—
225C2×C3×Dic3—61no—
226C2×C3×Dic3—64no—
227C2×C3×Dic3C22×D314yes—
228C2×C3×Dic3C22×D312yes—
229C2×C3×Dic3C23×C314yes—
230C2×C3×Dic3C23×C312yes—
231C3×D12—124no—
232C3×C4×D3—64no—
233C3×Dic6—64no—
234#627—22no—
235#626—44no—
236#625—22no—
237D3×Dic3—24no—
238#535—91noSylow
239C2×C33—22no—
240C2×C33—42no—
241C2×C33C22×D411yescharacteristic
242C2×C33H(5,𝔽2)12yes—
243C3×C3⋊D3—42no—
244C32×D3—42no—
245C24×C3—21no—
246C24×C3—22no—
247C24×C3—41no—
248C24×C3—91no—
249C24×C3D3211yescharacteristic
250C24×C3C2×C3×D312yes—
251C23×D3—32no—
252C23×D3—61no—
253C3×#41—184no—
254C2×C3×Q8—91no—
255C2×C3×D4—32no—
256C2×C3×D4—32no—
257C2×C3×D4—34no—
258C2×C3×D4—34no—
259C2×C3×D4—62no—
260C2×C3×D4—64no—
261C2×C3×D4—94no—
262C2×C3×D4—91no—
263C2×C3×D4—92no—
264C2×C3×D4—92no—
265C2×C3×D4—184no—
266C22×C3×C4—92no—
267C22×C3×C4—91no—
268C2×#67—34no—
269C2×#67—34no—
270C2×#67—32no—
271C2×#67—34no—
272C2×#67—62no—
273C2×#67—62no—
274C2×#67—68no—
275C2×#67—68no—
276C2×#67C2×C3×D312yes—
277C2×#67C2×C3×D314yescomplemented
278C22×Dic3—32no—
279C22×Dic3—32no—
280C22×Dic3—61no—
281#237—68no—
282C2×D12—34no—
283C2×C4×D3—32no—
284C2×Dic6—32no—
285C3×C4⋊C4—92no—
286C3×C22⋊C4—32no—
287C3×C22⋊C4—34no—
288C3×C22⋊C4—32no—
289C3×C22⋊C4—34no—
290C3×C22⋊C4—62no—
291C3×C22⋊C4—64no—
292C3×C22⋊C4—94no—
293C3×C42—92no—
294#217C2×C3×D312yes—
295#217C2×C3×D314yescomplemented
296(C2×D3)⋊C4—34no—
297(C2×D3)⋊C4—34no—
298Dic3⋊C4—34no—
299C4×Dic3—34no—
300C22×C32—22no—
301C22×C32—24no—
302C22×C32—22no—
303C22×C32—24no—
304C22×C32—24no—
305C22×C32—28no—
306C22×C32—24no—
307C22×C32—24no—
308C22×C32—24no—
309C22×C32—24no—
310C22×C32—21no—
311C22×C32—22no—
312C22×C32—22no—
313C22×C32—24no—
314C22×C32—42no—
315C22×C32—41no—
316C22×C32—44no—
317C22×C32—44no—
318C22×C32—44no—
319C22×C32—44no—
320C22×C32—44no—
321C22×C32—44no—
322C22×C32—48no—
323C22×C32—44no—
324C22×C32—48no—
325C22×C32—48no—
326C22×C32—416no—
327C22×C32—416no—
328C22×C32—62no—
329C22×C32—64no—
330C22×C32D3×D414yes—
331C22×C32C2×C3×D412yes—
332C22×C32C23×D312yes—
333C22×C32C24×C311yescharacteristic
334C2×C3⋊D3—22no—
335C2×C3⋊D3—24no—
336C2×C3⋊D3—44no—
337C2×C3×D3—22no—
338C2×C3×D3—24no—
339C2×C3×D3—22no—
340C2×C3×D3—24no—
341C2×C3×D3—42no—
342C2×C3×D3—44no—
343C2×C3×D3—64no—
344C2×C3×D3—64no—
345C2×C3×D3—64no—
346C2×C3×D3—64no—
347C2×C3×D3—124no—
348C2×C3×D3—128no—
349C2×C3×D3—128no—
350C32×C4—62no—
351C32×C4—64no—
352C3⋊Dic3C2×C3×D412yes—
353C3⋊Dic3C2×C3×D412yes—
354C3×Dic3—22no—
355C3×Dic3—24no—
356C3×Dic3—22no—
357C3×Dic3—24no—
358C3×Dic3—34no—
359C3×Dic3—34no—
360C3×Dic3—42no—
361C3×Dic3—44no—
362C3×Dic3—62no—
363C3×Dic3—62no—
364C2×#41—91no—
365C22×D4—91no—
366#127—92no—
367#124—92no—
368#121—94no—
369#120—32no—
370#120—92no—
371#117—91no—
372C33#53511yescharacteristic, complemented, Sylow
373C23×C3—24no—
374C23×C3—22no—
375C23×C3—24no—
376C23×C3—21no—
377C23×C3—24no—
378C23×C3—22no—
379C23×C3—22no—
380C23×C3—32no—
381C23×C3—32no—
382C23×C3—44no—
383C23×C3—44no—
384C23×C3—42no—
385C23×C3—44no—
386C23×C3—44no—
387C23×C3—48no—
388C23×C3—48no—
389C23×C3—42no—
390C23×C3—41no—
391C23×C3—62no—
392C23×C3—92no—
393C23×C3—182no—
394C23×C3—362no—
395C23×C3C2×D3212yes—
396C23×C3C2×D3211yescharacteristic
397C23×C3C22×C3×D314yes—
398C23×C3C22×C3×D312yes—
399C22×D3—34no—
400C22×D3—62no—
401C22×D3—64no—
402C22×D3—64no—
403C22×D3—124no—
404C22×D3—124no—
405C22×D3C22×C3×D312yes—
406C3×Q8—182no—
407C3×D4—64no—
408C3×D4—64no—
409C3×D4—124no—
410C3×D4—124no—
411C3×D4—124no—
412C3×D4—128no—
413C3×D4—184no—
414C3×D4—184no—
415C3×D4—182no—
416C3×D4—364no—
417C2×C3×C4—34no—
418C2×C3×C4—32no—
419C2×C3×C4—34no—
420C2×C3×C4—32no—
421C2×C3×C4—64no—
422C2×C3×C4—62no—
423C2×C3×C4—91no—
424C2×C3×C4—91no—
425C2×C3×C4—94no—
426C2×C3×C4—184no—
427#67—24no—
428#67—44no—
429#67—64no—
430#67—68no—
431#67—68no—
432#67—68no—
433#67—68no—
434#67—124no—
435C2×Dic3—32no—
436C2×Dic3—32no—
437C2×Dic3—38no—
438C2×Dic3—64no—
439C2×Dic3—61no—
440C2×Dic3—61no—
441C2×Dic3—64no—
442C2×Dic3C22×C3×D314yes—
443C2×Dic3C22×C3×D312yes—
444D12—124no—
445C4×D3—64no—
446Dic6—64no—
447C2×C32—22no—
448C2×C32—24no—
449C2×C32—24no—
450C2×C32—21no—
451C2×C32—22no—
452C2×C32—22no—
453C2×C32—24no—
454C2×C32—41no—
455C2×C32—42no—
456C2×C32—44no—
457C2×C32—44no—
458C2×C32—48no—
459C2×C32—44no—
460C2×C32—48no—
461C2×C32—44no—
462C2×C32—42no—
463C2×C32—122no—
464C2×C32C2×D3×D412yes—
465C2×C32#100414yes—
466C2×C32C22×C3×D411yescharacteristic
467C2×C32C3×H(5,𝔽2)12yes—
468C3⋊D3—42no—
469C3×D3—42no—
470C3×D3—42no—
471C3×D3—82no—
472C3×D3—124no—
473C3×D3—124no—
474C24—91no—
475C24C3×D3211yescharacteristic
476#41—184no—
477C2×Q8—91no—
478C2×D4—32no—
479C2×D4—34no—
480C2×D4—94no—
481C2×D4—91no—
482C2×D4—92no—
483C2×D4—92no—
484C2×D4—184no—
485C22×C4—92no—
486C22×C4—91no—
487C4⋊C4—92no—
488C22⋊C4—32no—
489C22⋊C4—34no—
490C22⋊C4—94no—
491C42—92no—
492C22×C3—24no—
493C22×C3—22no—
494C22×C3—24no—
495C22×C3—24no—
496C22×C3—28no—
497C22×C3—22no—
498C22×C3—24no—
499C22×C3—24no—
500C22×C3—24no—
501C22×C3—24no—
502C22×C3—21no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—24no—
506C22×C3—44no—
507C22×C3—44no—
508C22×C3—44no—
509C22×C3—44no—
510C22×C3—44no—
511C22×C3—44no—
512C22×C3—48no—
513C22×C3—48no—
514C22×C3—48no—
515C22×C3—44no—
516C22×C3—416no—
517C22×C3—416no—
518C22×C3—41no—
519C22×C3—42no—
520C22×C3—62no—
521C22×C3—64no—
522C22×C3—62no—
523C22×C3—64no—
524C22×C3—122no—
525C22×C3—124no—
526C22×C3—182no—
527C22×C3—184no—
528C22×C3—364no—
529C22×C3#381112yes—
530C22×C3C3×D3×D414yes—
531C22×C3C22×D3211yescharacteristic
532C22×C3C23×C3×D312yes—
533C2×D3—22no—
534C2×D3—24no—
535C2×D3—64no—
536C2×D3—64no—
537C2×D3—64no—
538C2×D3—64no—
539C2×D3—124no—
540C2×D3—128no—
541C2×D3—128no—
542C3×C4—62no—
543C3×C4—64no—
544C3×C4—62no—
545C3×C4—64no—
546C3×C4—92no—
547C3×C4—92no—
548C3×C4—122no—
549C3×C4—124no—
550Dic3—22no—
551Dic3—24no—
552Dic3—34no—
553Dic3—34no—
554Dic3—62no—
555Dic3—62no—
556C32—22no—
557C32—21no—
558C32—41no—
559C32#559212yescomplemented
560C32C3×#53511yescharacteristic, complemented
561C23—22no—
562C23—32no—
563C23—42no—
564C23—92no—
565C23—182no—
566C23—362no—
567C23C2×C3×D3212yes—
568C23C2×C3×D3211yescharacteristic
569Q8—182no—
570D4—64no—
571D4—124no—
572D4—184no—
573D4—184no—
574D4—182no—
575D4—364no—
576C2×C4—34no—
577C2×C4—32no—
578C2×C4—91no—
579C2×C4—91no—
580C2×C4—94no—
581C2×C4—184no—
582C2×C3—24no—
583C2×C3—24no—
584C2×C3—22no—
585C2×C3—21no—
586C2×C3—22no—
587C2×C3—24no—
588C2×C3—22no—
589C2×C3—44no—
590C2×C3—44no—
591C2×C3—44no—
592C2×C3—48no—
593C2×C3—48no—
594C2×C3—44no—
595C2×C3—42no—
596C2×C3—41no—
597C2×C3—42no—
598C2×C3—122no—
599C2×C3—122no—
600C2×C3—242no—
601C2×C3—362no—
602C2×C3C2×#381111yescharacteristic
603C2×C3#6433012yes—
604C2×C3C2×C3×D3×D412yes—
605C2×C3C3×#100414yes—
606D3—42no—
607D3—124no—
608D3—124no—
609C22—22no—
610C22—22no—
611C22—24no—
612C22—44no—
613C22—62no—
614C22—64no—
615C22—182no—
616C22—184no—
617C22—364no—
618C22C3×#381112yes—
619C22C22×C3×D3211yescharacteristic
620C4—62no—
621C4—64no—
622C4—92no—
623C4—92no—
624C3—21no—
625C3—22no—
626C3—41no—
627C3#1059550211yescharacteristic, direct product factor
628C3C3×#559212yescomplemented
629C2—22no—
630C2—42no—
631C2—122no—
632C2—362no—
633C2C2×C3×#381111yescharacteristic
634C2C3×#6433012yes—
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC24×C33C22
socleC22×C33C24
hypercenterC24×C3D32
hypocenterC32C3×#535
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×#638C2semidirect
2. C2×C3×#3805C2semidirect
3. C32×#953C2semidirect
4. C3×#63981C2semidirect
5. C3×#63979C2semidirect
6. C3×#63971C2semidirect
7. C3×#63957C2semidirect
8. C3×#63953C2semidirect
10. #10595502C3direct (N × H)
18. C2×C32×#67C22semidirect
19. C2×C3×#627C22semidirect
20. C2×C3×#626C22semidirect
21. C2×C3×#625C22semidirect
23. C2×C3×D3×Dic3C22semidirect
25. C32×#217C22semidirect
26. C3×#3724C22semidirect
27. C3×#3722C22semidirect
28. C3×Dic32C22semidirect
29. C22×#638C2×C3semidirect
33. C2×#3805C2×C3semidirect
36. C3×#953D3semidirect
37. C3×#953C2×C3semidirect
45. #63981C2×C3semidirect
46. #63979C2×C3semidirect
47. #63971C2×C3semidirect
48. #63957C2×C3semidirect
49. #63953C2×C3semidirect
60. C2×C3×C3⋊Dic3D4semidirect
98. C2×C3×#67C2×D3semidirect
100. C2×C3×#67C22×C3semidirect
108. C2×#627C22×C3semidirect
109. C2×#626C22×C3semidirect
110. C2×#625C22×C3semidirect
112. C2×D3×Dic3C22×C3semidirect
118. C3×#217C2×D3semidirect
120. C3×#217C22×C3semidirect
125. #3724C22×C3semidirect
126. #3722C22×C3semidirect
127. Dic32C22×C3semidirect
156. #953C3×D3semidirect
203. C2×C3⋊Dic3C3×D4semidirect
277. C2×#67C2×C3×D3semidirect
295. #217C2×C3×D3semidirect
372. C33#535semidirect
559. C32#5592semidirect
560. C32C3×#535semidirect
627. C3#10595502direct (N × H)
628. C3C3×#5592semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C2×C3×#38052
C32×#9532
C3×#639812
C3×#639792
C3×#639712
C3×#639572
C3×#639532
C3×#55923
#105955023