Finite Group Databasebeta

← #13162061

Subgroups — #13162061

Order 1728.

1778 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#638C211yesmaximal, characteristic
2C22×#6294C211yesmaximal, characteristic, complemented
3C2×C3×#3757C212yesmaximal, complemented
4C2×#88814C211yesmaximal, characteristic, complemented
5#11722082C212yesmaximal, complemented
6#10591722—31nomaximal
7#10591721—32nomaximal
8C24×C33C2211yescharacteristic
9C23×C3×C3⋊D3C2211yescharacteristic
10C2×C3×#638—22no—
11C2×C3×#638—22no—
12C2×C3×#638C414yescomplemented
13C22×C3×C3⋊Dic3C2212yes—
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C3×C3⋊Dic3C2212yes—
16C2×#6294—22no—
17C2×#6294—24no—
18C3×#3757—24no—
19#88814—22no—
20C22×#638D311yescharacteristic
21C22×C3×#67—32no—
22C22×C3×#67—61no—
23C22×#624—31no—
24C22×D3×Dic3—32no—
25C2×#3757D312yescomplemented
26C2×C3×#217—32no—
27C2×C3×#217—32no—
28C2×C3×#217—62no—
29C2×#3722—31no—
30C2×(C2×D3)⋊Dic3—32no—
31#63579—32no—
32#63579—32no—
33#63579—32no—
34C23×C33—22no—
35C23×C33—21no—
36C23×C33—21no—
37C23×C33—22no—
38C23×C33C2×C412yescomplemented
39C23×C33C2311yescharacteristic
40C22×C3×C3⋊D3—22no—
41C22×C3×C3⋊D3—22no—
42C22×C3×C3⋊D3C2×C412yes—
43C22×C3×C3⋊D3D412yes—
44C22×C3×C3⋊D3D412yes—
45C3×#638—28no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3—22no—
49C2×C3×C3⋊Dic3—22no—
50C2×C3×C3⋊Dic3—22no—
51C2×C3×C3⋊Dic3—22no—
52C2×C3×C3⋊Dic3—22no—
53C2×C3×C3⋊Dic3C2×C412yescomplemented
54C2×C3×C3⋊Dic3D414yescomplemented
55#6294—28no—
56#5220—91no—
57#4956—92no—
58C24×C32—22no—
59C24×C32—21no—
60C24×C32—41no—
61C24×C32C2×D312yes—
62C24×C32C2×D311yescharacteristic
63C23×C3⋊D3C2×D311yescharacteristic
64C23×C3×D3—32no—
65C23×C3×D3—61no—
66C2×#638—22no—
67C2×#638—22no—
68C2×#638Dic314yescomplemented
69C22×C3⋊Dic3C2×D312yes—
70C22×C3⋊Dic3C2×D311yescharacteristic
71C22×C3⋊Dic3C2×D312yes—
72C2×C3×#67—38no—
73C2×C3×#67—64no—
74C2×C3×#67—64no—
75C2×C3×#67—64no—
76C2×C3×#67—64no—
77C2×C3×#67—64no—
78C22×C3×Dic3—32no—
79C22×C3×Dic3—32no—
80C22×C3×Dic3—32no—
81C22×C3×Dic3—32no—
82C22×C3×Dic3—32no—
83C22×C3×Dic3—61no—
84C22×C3×Dic3—62no—
85C22×C3×Dic3—62no—
86C2×#624—64no—
87C2×#624—62no—
88C2×D3×Dic3—64no—
89C2×D3×Dic3—64no—
90C2×D3×Dic3—64no—
91#3757—24no—
92C3×#217—64no—
93C3×#217—64no—
94C3×#217—68no—
95#3722—62no—
96(C2×D3)⋊Dic3—64no—
97C22×C33—22no—
98C22×C33—24no—
99C22×C33—41no—
100C22×C33—42no—
101C22×C33—41no—
102C22×C33C22×C411yescharacteristic
103C22×C33C2×D411yescharacteristic
104C22×C33C2×D411yescharacteristic
105C22×C33C2×D412yes—
106C22×C33#4112yes—
107C2×C3×C3⋊D3—24no—
108C2×C3×C3⋊D3—22no—
109C2×C3×C3⋊D3—24no—
110C2×C3×C3⋊D3—22no—
111C2×C3×C3⋊D3C22⋊C414yescomplemented
112C3×C3⋊Dic3—22no—
113C3×C3⋊Dic3—24no—
114C3×C3⋊Dic3—42no—
115C22×C3×D4—91no—
116C22×#67—32no—
117C22×#67—61no—
118C23×Dic3—91no—
119C22×C4×D3—92no—
120C2×C3×C22⋊C4—92no—
121C2×#217—32no—
122C2×#217—32no—
123C2×#217—62no—
124C2×#217—91no—
125C2×(C2×D3)⋊C4—92no—
126C3⋊#95—92no—
127C3⋊#95—92no—
128C3⋊#95—92no—
129C23×C32—22no—
130C23×C32—22no—
131C23×C32—21no—
132C23×C32—21no—
133C23×C32—24no—
134C23×C32—24no—
135C23×C32—22no—
136C23×C32—22no—
137C23×C32—22no—
138C23×C32—22no—
139C23×C32—22no—
140C23×C32—22no—
141C23×C32—22no—
142C23×C32—22no—
143C23×C32—24no—
144C23×C32—22no—
145C23×C32—21no—
146C23×C32—22no—
147C23×C32—42no—
148C23×C32—41no—
149C23×C32—42no—
150C23×C32—42no—
151C23×C32—42no—
152C23×C32—42no—
153C23×C32—42no—
154C23×C32—42no—
155C23×C32—44no—
156C23×C32—42no—
157C23×C32—42no—
158C23×C32—44no—
159C23×C32C4×D314yescomplemented
160C23×C32C2×Dic312yescomplemented
161C23×C32C22×D312yes—
162C23×C32C22×D311yescharacteristic
163C22×C3⋊D3—22no—
164C22×C3⋊D3—22no—
165C22×C3⋊D3C2×Dic312yes—
166C22×C3⋊D3#6712yes—
167C22×C3⋊D3#6712yes—
168C22×C3×D3—34no—
169C22×C3×D3—34no—
170C22×C3×D3—34no—
171C22×C3×D3—62no—
172C22×C3×D3—62no—
173C22×C3×D3—62no—
174C22×C3×D3—62no—
175C22×C3×D3—62no—
176C22×C3×D3—62no—
177C22×C3×D3—62no—
178C22×C3×D3—122no—
179C22×C3×D3—122no—
180#638—28no—
181C2×C3⋊Dic3—21no—
182C2×C3⋊Dic3—21no—
183C2×C3⋊Dic3—22no—
184C2×C3⋊Dic3—22no—
185C2×C3⋊Dic3—22no—
186C2×C3⋊Dic3—22no—
187C2×C3⋊Dic3—22no—
188C2×C3⋊Dic3C2×Dic312yescomplemented
189C2×C3⋊Dic3#6714yescomplemented
190C3×#67—616no—
191C3×#67—616no—
192C2×C3×Dic3—34no—
193C2×C3×Dic3—34no—
194C2×C3×Dic3—34no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—64no—
197C2×C3×Dic3—62no—
198C2×C3×Dic3—62no—
199C2×C3×Dic3—62no—
200C2×C3×Dic3—62no—
201C2×C3×Dic3—62no—
202C2×C3×Dic3—62no—
203C2×C3×Dic3—62no—
204C2×C3×Dic3—62no—
205C2×C3×Dic3—64no—
206C2×C3×Dic3—62no—
207C2×C3×Dic3—62no—
208C2×C3×Dic3—62no—
209C2×C3×Dic3—62no—
210C2×C3×Dic3—62no—
211C2×C3×Dic3—62no—
212C2×C3×Dic3—122no—
213C2×C3×Dic3—122no—
214C2×C3×Dic3—122no—
215C2×C3×Dic3—122no—
216#624—68no—
217D3×Dic3—616no—
218#386—271noSylow
219C2×C33—42no—
220C2×C33C2×C22⋊C411yescharacteristic
221C2×C33C4×D412yes—
222C2×C33#12011yescharacteristic
223C2×C33#12112yes—
224C2×C33#12311yescharacteristic
225C3×C3⋊D3—24no—
226C24×C3—22no—
227C24×C3—21no—
228C24×C3—41no—
229C24×C3—91no—
230C24×C3D3212yes—
231C24×C3D3211yescharacteristic
232C23×D3—32no—
233C23×D3—61no—
234C2×C3×D4—94no—
235C2×C3×D4—182no—
236C2×C3×D4—182no—
237C22×C3×C4—91no—
238C22×C3×C4—92no—
239C22×C3×C4—92no—
240C2×#67—38no—
241C2×#67—64no—
242C2×#67—64no—
243C2×#67—64no—
244C2×#67—64no—
245C2×#67—64no—
246C22×Dic3—32no—
247C22×Dic3—32no—
248C22×Dic3—32no—
249C22×Dic3—32no—
250C22×Dic3—32no—
251C22×Dic3—61no—
252C22×Dic3—62no—
253C22×Dic3—62no—
254C22×Dic3—184no—
255C22×Dic3—182no—
256C2×C4×D3—184no—
257C2×C4×D3—184no—
258C2×C4×D3—184no—
259C3×C22⋊C4—184no—
260#217—64no—
261#217—64no—
262#217—68no—
263#217—182no—
264(C2×D3)⋊C4—184no—
265C22×C32—24no—
266C22×C32—22no—
267C22×C32—24no—
268C22×C32—28no—
269C22×C32—24no—
270C22×C32—24no—
271C22×C32—24no—
272C22×C32—22no—
273C22×C32—22no—
274C22×C32—22no—
275C22×C32—22no—
276C22×C32—21no—
277C22×C32—22no—
278C22×C32—22no—
279C22×C32—22no—
280C22×C32—22no—
281C22×C32—22no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—41no—
285C22×C32—42no—
286C22×C32—41no—
287C22×C32—41no—
288C22×C32—42no—
289C22×C32—44no—
290C22×C32—44no—
291C22×C32—44no—
292C22×C32—44no—
293C22×C32—48no—
294C22×C32—44no—
295C22×C32—44no—
296C22×C32—44no—
297C22×C32—44no—
298C22×C32—42no—
299C22×C32—44no—
300C22×C32—42no—
301C22×C32—44no—
302C22×C32—48no—
303C22×C32—44no—
304C22×C32—42no—
305C22×C32—42no—
306C22×C32—41no—
307C22×C32—42no—
308C22×C32—41no—
309C22×C32—42no—
310C22×C32—42no—
311C22×C32C2×C4×D312yes—
312C22×C32#23512yes—
313C22×C32D3×D412yes—
314C22×C32D3×D412yes—
315C22×C32D3×D412yes—
316C22×C32D3×D412yes—
317C22×C32#23712yes—
318C22×C32#23712yes—
319C22×C32C22×Dic311yescharacteristic
320C22×C32C2×#6712yes—
321C22×C32C2×#6711yescharacteristic
322C22×C32C2×#6711yescharacteristic
323C2×C3⋊D3—24no—
324C2×C3⋊D3—22no—
325C2×C3⋊D3—22no—
326C2×C3⋊D3—24no—
327C2×C3⋊D3#21714yescomplemented
328C2×C3×D3—38no—
329C2×C3×D3—64no—
330C2×C3×D3—64no—
331C2×C3×D3—64no—
332C2×C3×D3—64no—
333C2×C3×D3—64no—
334C2×C3×D3—64no—
335C2×C3×D3—64no—
336C2×C3×D3—124no—
337C2×C3×D3—124no—
338C2×C3×D3—122no—
339C2×C3×D3—122no—
340C3⋊Dic3—22no—
341C3⋊Dic3—24no—
342C3⋊Dic3—42no—
343C3×Dic3—64no—
344C3×Dic3—64no—
345C3×Dic3—64no—
346C3×Dic3—64no—
347C3×Dic3—124no—
348C3×Dic3—124no—
349C3×Dic3—122no—
350C3×Dic3—122no—
351C22×D4—91no—
352C23×C4—271no—
353C2×C22⋊C4—92no—
354C2×C22⋊C4—271no—
355#95—272no—
356C33#38611yescharacteristic, complemented, Sylow
357C23×C3—24no—
358C23×C3—22no—
359C23×C3—22no—
360C23×C3—24no—
361C23×C3—22no—
362C23×C3—22no—
363C23×C3—22no—
364C23×C3—22no—
365C23×C3—22no—
366C23×C3—22no—
367C23×C3—22no—
368C23×C3—22no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—24no—
372C23×C3—22no—
373C23×C3—22no—
374C23×C3—21no—
375C23×C3—42no—
376C23×C3—42no—
377C23×C3—42no—
378C23×C3—42no—
379C23×C3—42no—
380C23×C3—42no—
381C23×C3—44no—
382C23×C3—44no—
383C23×C3—42no—
384C23×C3—42no—
385C23×C3—42no—
386C23×C3—41no—
387C23×C3—92no—
388C23×C3—92no—
389C23×C3—92no—
390C23×C3—182no—
391C23×C3—182no—
392C23×C3D3×Dic314yescomplemented
393C23×C3#62412yescomplemented
394C23×C3C2×D3212yes—
395C23×C3C2×D3211yescharacteristic
396C22×D3—34no—
397C22×D3—34no—
398C22×D3—34no—
399C22×D3—62no—
400C22×D3—62no—
401C22×D3—62no—
402C22×D3—62no—
403C22×D3—62no—
404C22×D3—62no—
405C22×D3—62no—
406C22×D3—122no—
407C22×D3—122no—
408C3×D4—188no—
409C2×C3×C4—92no—
410C2×C3×C4—94no—
411C2×C3×C4—182no—
412C2×C3×C4—182no—
413C2×C3×C4—181no—
414C2×C3×C4—181no—
415C2×C3×C4—182no—
416C2×C3×C4—182no—
417C2×C3×C4—182no—
418#67—616no—
419#67—616no—
420C2×Dic3—34no—
421C2×Dic3—34no—
422C2×Dic3—34no—
423C2×Dic3—62no—
424C2×Dic3—64no—
425C2×Dic3—62no—
426C2×Dic3—62no—
427C2×Dic3—62no—
428C2×Dic3—62no—
429C2×Dic3—62no—
430C2×Dic3—62no—
431C2×Dic3—62no—
432C2×Dic3—62no—
433C2×Dic3—64no—
434C2×Dic3—62no—
435C2×Dic3—62no—
436C2×Dic3—62no—
437C2×Dic3—62no—
438C2×Dic3—62no—
439C2×Dic3—62no—
440C2×Dic3—122no—
441C2×Dic3—122no—
442C2×Dic3—122no—
443C2×Dic3—122no—
444C2×Dic3—188no—
445C4×D3—1816no—
446C2×C32—22no—
447C2×C32—22no—
448C2×C32—22no—
449C2×C32—22no—
450C2×C32—22no—
451C2×C32—22no—
452C2×C32—22no—
453C2×C32—22no—
454C2×C32—21no—
455C2×C32—21no—
456C2×C32—22no—
457C2×C32—21no—
458C2×C32—41no—
459C2×C32—42no—
460C2×C32—41no—
461C2×C32—42no—
462C2×C32—41no—
463C2×C32—48no—
464C2×C32—44no—
465C2×C32—48no—
466C2×C32—44no—
467C2×C32—42no—
468C2×C32D3×C22⋊C412yes—
469C2×C32#67⋊C412yes—
470C2×C32#88312yes—
471C2×C32#88412yes—
472C2×C32C4×#6712yes—
473C2×C32D4×Dic312yes—
474C2×C32#93511yescharacteristic
475C2×C32#93712yes—
476C2×C32#93711yescharacteristic
477C2×C32#93812yes—
478C2×C32#93912yes—
479C2×C32C2×#21711yescharacteristic
480C3⋊D3—24no—
481C3×D3—68no—
482C3×D3—124no—
483C24—91no—
484C24#111311yescharacteristic
485C2×D4—94no—
486C2×D4—182no—
487C2×D4—182no—
488C22×C4—91no—
489C22×C4—92no—
490C22×C4—92no—
491C22×C4—544no—
492C22×C4—542no—
493C22⋊C4—184no—
494C22⋊C4—542no—
495C22×C3—24no—
496C22×C3—28no—
497C22×C3—24no—
498C22×C3—24no—
499C22×C3—24no—
500C22×C3—24no—
501C22×C3—22no—
502C22×C3—21no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—21no—
509C22×C3—21no—
510C22×C3—22no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—44no—
515C22×C3—44no—
516C22×C3—44no—
517C22×C3—44no—
518C22×C3—48no—
519C22×C3—42no—
520C22×C3—44no—
521C22×C3—44no—
522C22×C3—44no—
523C22×C3—44no—
524C22×C3—44no—
525C22×C3—42no—
526C22×C3—44no—
527C22×C3—48no—
528C22×C3—44no—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—41no—
534C22×C3—42no—
535C22×C3—41no—
536C22×C3—41no—
537C22×C3—42no—
538C22×C3—41no—
539C22×C3—41no—
540C22×C3—42no—
541C22×C3—94no—
542C22×C3—184no—
543C22×C3—184no—
544C22×C3—182no—
545C22×C3—182no—
546C22×C3C2×D3×Dic312yes—
547C22×C3#380412yes—
548C22×C3#380412yes—
549C22×C3#380512yes—
550C22×C3C2×#62411yescharacteristic
551C22×C3D3×#6712yes—
552C22×C3D3×#6712yes—
553C22×C3D3×#6712yes—
554C22×C3D3×#6712yes—
555C22×C3#381111yescharacteristic
556C22×C3#381112yes—
557C22×C3#381111yescharacteristic
558C2×D3—38no—
559C2×D3—64no—
560C2×D3—64no—
561C2×D3—64no—
562C2×D3—64no—
563C2×D3—64no—
564C2×D3—64no—
565C2×D3—64no—
566C2×D3—122no—
567C2×D3—124no—
568C2×D3—124no—
569C2×D3—122no—
570C3×C4—182no—
571C3×C4—184no—
572C3×C4—362no—
573Dic3—64no—
574Dic3—64no—
575Dic3—64no—
576Dic3—64no—
577Dic3—124no—
578Dic3—124no—
579Dic3—122no—
580Dic3—122no—
581C32—22no—
582C32—21no—
583C32—41no—
584C32#495612yescomplemented
585C32#522011yescharacteristic, complemented
586C23—22no—
587C23—22no—
588C23—21no—
589C23—21no—
590C23—92no—
591C23—92no—
592C23—92no—
593C23—182no—
594C23—182no—
595C23#629412yescomplemented
596C23C2×#111311yescharacteristic
597D4—188no—
598C2×C4—92no—
599C2×C4—94no—
600C2×C4—182no—
601C2×C4—182no—
602C2×C4—181no—
603C2×C4—181no—
604C2×C4—182no—
605C2×C4—182no—
606C2×C4—182no—
607C2×C4—548no—
608C2×C3—21no—
609C2×C3—22no—
610C2×C3—21no—
611C2×C3—22no—
612C2×C3—22no—
613C2×C3—22no—
614C2×C3—22no—
615C2×C3—22no—
616C2×C3—22no—
617C2×C3—22no—
618C2×C3—22no—
619C2×C3—21no—
620C2×C3—44no—
621C2×C3—48no—
622C2×C3—48no—
623C2×C3—44no—
624C2×C3—42no—
625C2×C3—41no—
626C2×C3—41no—
627C2×C3—42no—
628C2×C3—42no—
629C2×C3—41no—
630C2×C3—184no—
631C2×C3#6395212yes—
632C2×C3#6395812yes—
633C2×C3D3×#21712yes—
634C2×C3#6396912yes—
635C2×C3#6397012yes—
636C2×C3#6397112yes—
637C2×C3Dic3×#6712yes—
638C2×C3#67⋊Dic312yes—
639C2×C3#6397411yescharacteristic
640C2×C3#6397511yescharacteristic
641C2×C3#6398312yes—
642C2×C3#6398311yescharacteristic
643D3—68no—
644D3—124no—
645C22—22no—
646C22—24no—
647C22—41no—
648C22—42no—
649C22—41no—
650C22—94no—
651C22—182no—
652C22—184no—
653C22—184no—
654C22—182no—
655C22C2×#629411yescharacteristic
656C22#8904612yes—
657C22#8904911yescharacteristic
658C22#8904911yescharacteristic
659C22#8904912yes—
660C4—182no—
661C4—184no—
662C4—362no—
663C3—21no—
664C3—22no—
665C3—41no—
666C3#1059172112yescomplemented
667C3#1059172211yescharacteristic, complemented
668C2—42no—
669C2—184no—
670C2#1172372311yescharacteristic
671C2#1172372411yescharacteristic
672C2#1172372512yes—
673C2#1172372612yes—
674C2#1172372911yescharacteristic
subgroupG/H
center, FrattiniC23C2×#1113
commutatorC22×C33C22×C4
FittingC24×C33C22
socle, cosocleC23×C33C23
prime radicalC22×C3×#638C2
hypercenterC24#1113
hypocenterC33#386
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C22×#6294C2semidirect
3. C2×C3×#3757C2semidirect
4. C2×#88814C2semidirect
5. #11722082C2semidirect
12. C2×C3×#638C4semidirect
25. C2×#3757D3semidirect
38. C23×C33C2×C4semidirect
53. C2×C3×C3⋊Dic3C2×C4semidirect
54. C2×C3×C3⋊Dic3D4semidirect
68. C2×#638Dic3semidirect
111. C2×C3×C3⋊D3C22⋊C4semidirect
159. C23×C32C4×D3semidirect
160. C23×C32C2×Dic3semidirect
188. C2×C3⋊Dic3C2×Dic3semidirect
189. C2×C3⋊Dic3#67semidirect
327. C2×C3⋊D3#217semidirect
356. C33#386semidirect
392. C23×C3D3×Dic3semidirect
393. C23×C3#624semidirect
584. C32#4956semidirect
585. C32#5220semidirect
595. C23#6294semidirect
666. C3#10591721semidirect
667. C3#10591722semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C22×#62942
C2×C3×#37572
C2×#888142
#117220822
#105917223
#105917213