Finite Group Databasebeta

← #13178106

Subgroups — #13178106

Order 1728.

2596 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×#67C211yesmaximal, characteristic, complemented
2C2×D3×#638C211yesmaximal, characteristic, complemented
3#11723994C211yesmaximal, characteristic, complemented
4C3×#64162C211yesmaximal, characteristic, complemented
5#11723643C212yesmaximal, complemented
6#11723616C212yesmaximal, complemented
7#11723615C212yesmaximal, complemented
8#11723614C212yesmaximal, complemented
9D3×#3757C211yesmaximal, characteristic, complemented
10#11723612C211yesmaximal, characteristic, complemented
11#11723605C211yesmaximal, characteristic, complemented
12#10596275—31nomaximal
13#10595524—34nomaximal
14C24×C33C2211yescharacteristic, complemented
15C23×C32×D3C2211yescharacteristic
16C22×D3×C3⋊D3C2211yescharacteristic, complemented
17C2×#6312C2211yescharacteristic
18C2×#6312C2212yescomplemented
19C2×C3×#638C2211yescharacteristic
20C2×C3×#638C2212yescomplemented
21C2×C32×#67—24no—
22C2×C32×#67C2212yes—
23C2×C32×#67C2212yescomplemented
24C22×C32×Dic3C2211yescharacteristic, complemented
25D3×#638—24no—
26C2×#6291C2211yescharacteristic, complemented
27C2×#6290C2211yescharacteristic, complemented
28C2×D3×C3⋊Dic3C2211yescharacteristic, complemented
29C2×D3×C3⋊Dic3C2212yescomplemented
30#88864C2211yescharacteristic, complemented
31#88864C2212yescomplemented
32C3×#3757C2211yescharacteristic, complemented
33C3×#3757C2212yescomplemented
34#88807C2212yescomplemented
35#88806C2212yescomplemented
36#88804C2212yescomplemented
37#88803C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C22×C32×D4—31no—
40C22×#638—31no—
41C22×C3×#67D314yescomplemented
42C2×D3×#67—34no—
43#64162—31no—
44#64162—34no—
45#64162—64no—
46#64162D311yescharacteristic, complemented
47C2×#3757—31no—
48#64148—32no—
49#64148—32no—
50#64146—32no—
51#64144—32no—
52#64138—31no—
53#64138—31no—
54C3×#953—34no—
55#63983—38no—
56#63971—38no—
57#63970—38no—
58#63969—38no—
59D3×#217—34no—
60#63966—34no—
61#63961—34no—
62C23×C33—22no—
63C23×C33—42no—
64C23×C33C2312yes—
65C23×C33C2311yescharacteristic
66C22×C3⋊(C3⋊D3)C2311yescharacteristic
67C22×C3×C3⋊D3C2311yescharacteristic
68C22×C32×D3—42no—
69C22×C32×D3D414yescomplemented
70C22×C32×D3C2311yescharacteristic
71C22×C32×D3C2311yescharacteristic
72C2×D3×C3⋊D3—22no—
73C2×D3×C3⋊D3—42no—
74#6312—22no—
75#6312—42no—
76C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
77C2×C3⋊(C3⋊Dic3)C2312yes—
78C3×#638—22no—
79C3×#638—42no—
80C2×C3×C3⋊Dic3C2311yescharacteristic
81C2×C3×C3⋊Dic3C2312yes—
82C32×#67—24no—
83C32×#67—24no—
84C2×C32×Dic3—22no—
85C2×C32×Dic3C2312yes—
86#6291—22no—
87#6290—22no—
88D3×C3⋊Dic3—22no—
89D3×C3⋊Dic3—42no—
90#5804—94no—
91#5589—91no—
92C24×C32—24no—
93C24×C32—31no—
94C24×C32C2×D314yescomplemented
95C24×C32C2×D311yescharacteristic, complemented
96C23×C3⋊D3—31no—
97C23×C3×D3C2×D314yes—
98C22×D32—34no—
99C2×C32×D4—32no—
100C2×C32×D4—32no—
101C2×C32×D4—64no—
102C22×C32×C4—31no—
103C2×#638—31no—
104C2×#638—34no—
105C2×#638—31no—
106C2×#638—32no—
107C2×#638—38no—
108C2×#638—31no—
109C2×#638—64no—
110C2×#638—64no—
111C2×#638—68no—
112C2×#638C2×D311yescharacteristic
113C2×#638C2×D312yescomplemented
114C22×C3⋊Dic3—31no—
115C22×C3⋊Dic3—32no—
116C2×C3×#67—216no—
117C2×C3×#67—34no—
118C2×C3×#67—38no—
119C2×C3×#67C2×D318yes—
120C2×C3×#67C2×D318yescomplemented
121C22×C3×Dic3C2×D314yescomplemented
122D3×#67—616no—
123C2×#626—34no—
124C2×#625—34no—
125C2×D3×Dic3—34no—
126C2×D3×Dic3—38no—
127#3757—31no—
128#3757—32no—
129#3757—34no—
130#3757—38no—
131#3757—32no—
132#3757—64no—
133#3757—68no—
134#3757C2×D311yescharacteristic, complemented
135#3757C2×D312yescomplemented
136#3752—32no—
137#3752—32no—
138#3750—32no—
139#3750—32no—
140C3×#217—34no—
141C3×#217—38no—
142#3724—38no—
143Dic3⋊Dic3—38no—
144#3722—38no—
145(C2×D3)⋊Dic3—38no—
146(C2×D3)⋊Dic3—38no—
147C22×C33—22no—
148C22×C33—22no—
149C22×C33—22no—
150C22×C33—22no—
151C22×C33—44no—
152C22×C33C2×D412yes—
153C22×C33C2411yescharacteristic
154C2×C3⋊(C3⋊D3)—21no—
155C2×C3⋊(C3⋊D3)—22no—
156C2×C3×C3⋊D3—21no—
157C2×C3×C3⋊D3—22no—
158C2×C32×D3—22no—
159C2×C32×D3—22no—
160C2×C32×D3—44no—
161C2×C32×D3C2×D412yes—
162C2×C32×D3C2×D412yes—
163D3×C3⋊D3—44no—
164C3⋊(C3⋊Dic3)—21no—
165C3⋊(C3⋊Dic3)—22no—
166C3×C3⋊Dic3—21no—
167C3×C3⋊Dic3—22no—
168C32×Dic3—22no—
169C22×C3×D4—34no—
170C22×#67—94no—
171C22×#67C3⋊D311yescharacteristic, complemented
172C2×D3×D4—91no—
173C3×#120—91no—
174#953—34no—
175#953—94no—
176#953—91no—
177#953—184no—
178C2×#217—94no—
179#939—98no—
180#939—98no—
181#938—92no—
182#937—98no—
183#937—92no—
184#935—98no—
185#935—91no—
186#929—94no—
187#929—94no—
188#886—91no—
189#884—92no—
190#883—92no—
191D3×C22⋊C4—91no—
192C23×C32—22no—
193C23×C32—28no—
194C23×C32—24no—
195C23×C32—28no—
196C23×C32—31no—
197C23×C32—31no—
198C23×C32—34no—
199C23×C32—48no—
200C23×C32—42no—
201C23×C32—48no—
202C23×C32—416no—
203C23×C32—122no—
204C23×C32C22×D318yes—
205C23×C32C22×D314yes—
206C23×C32C22×D312yes—
207C23×C32C22×D311yescharacteristic
208C22×C3⋊D3—31no—
209C22×C3⋊D3—34no—
210C22×C3⋊D3—64no—
211C22×C3⋊D3—62no—
212C22×C3⋊D3—122no—
213C22×C3⋊D3C22×D311yescharacteristic
214C22×C3×D3—34no—
215C22×C3×D3—48no—
216C22×C3×D3#67116yescomplemented
217C22×C3×D3C22×D314yes—
218C22×C3×D3C22×D314yes—
219C2×D32—68no—
220C2×D32—128no—
221C32×D4—64no—
222C32×D4—64no—
223C2×C32×C4—32no—
224C2×C32×C4—62no—
225#638—22no—
226#638—42no—
227#638—62no—
228#638—62no—
229#638—62no—
230#638—68no—
231#638—128no—
232#638—122no—
233#638—128no—
234#638—1216no—
235C2×C3⋊Dic3—31no—
236C2×C3⋊Dic3—32no—
237C2×C3⋊Dic3—38no—
238C2×C3⋊Dic3—34no—
239C2×C3⋊Dic3—64no—
240C2×C3⋊Dic3—68no—
241C2×C3⋊Dic3—62no—
242C2×C3⋊Dic3—122no—
243C2×C3⋊Dic3C22×D311yescharacteristic
244C2×C3⋊Dic3C22×D312yes—
245C3×#67—216no—
246C3×#67—216no—
247C3×#67—68no—
248C3×#67—128no—
249C2×C3×Dic3—28no—
250C2×C3×Dic3—38no—
251C2×C3×Dic3—34no—
252C2×C3×Dic3C22×D318yes—
253#626—68no—
254#625—68no—
255D3×Dic3—68no—
256D3×Dic3—128no—
257#534—271noSylow
258C2×C33—22no—
259C2×C33—42no—
260C2×C33C22×D411yescharacteristic
261C2×C33H(5,𝔽2)12yes—
262C3⋊(C3⋊D3)—41no—
263C3×C3⋊D3—41no—
264C32×D3—22no—
265C32×D3—22no—
266C24×C3—24no—
267C24×C3—34no—
268C24×C3D3214yescomplemented
269C24×C3C2×C3⋊D311yescharacteristic, complemented
270C23×D3—94no—
271C23×D3—91no—
272C23×D3C2×C3⋊D311yescharacteristic
273C2×C3×D4—38no—
274C2×C3×D4—38no—
275C2×C3×D4—616no—
276C2×C3×D4—91no—
277C2×C3×D4—92no—
278C22×C3×C4—34no—
279C2×#67—24no—
280C2×#67—34no—
281C2×#67—38no—
282C2×#67—94no—
283C2×#67—94no—
284C2×#67—94no—
285C2×#67—91no—
286C2×#67—91no—
287C2×#67—98no—
288C2×#67—92no—
289C2×#67—184no—
290C2×#67—188no—
291C2×#67—1816no—
292C2×#67C2×C3⋊D312yes—
293C2×#67C2×C3⋊D312yescomplemented
294C22×Dic3—94no—
295C22×Dic3—98no—
296C22×Dic3C2×C3⋊D311yescharacteristic, complemented
297D3×D4—184no—
298C2×D12—91no—
299C2×C4×D3—91no—
300C2×C4×D3—92no—
301C3×C22⋊C4—91no—
302C3×C22⋊C4—92no—
303#217—34no—
304#217—38no—
305#217—98no—
306#217—94no—
307#217—98no—
308#217—91no—
309#217—92no—
310#217—92no—
311#217—184no—
312#217—188no—
313(C2×D3)⋊C4—98no—
314(C2×D3)⋊C4—98no—
315(C2×D3)⋊C4—92no—
316(C2×D3)⋊C4—92no—
317C4⋊Dic3—92no—
318Dic3⋊C4—98no—
319Dic3⋊C4—98no—
320Dic3⋊C4—92no—
321C22×C32—22no—
322C22×C32—216no—
323C22×C32—22no—
324C22×C32—22no—
325C22×C32—28no—
326C22×C32—22no—
327C22×C32—24no—
328C22×C32—28no—
329C22×C32—28no—
330C22×C32—28no—
331C22×C32—28no—
332C22×C32—32no—
333C22×C32—32no—
334C22×C32—48no—
335C22×C32—48no—
336C22×C32—48no—
337C22×C32—44no—
338C22×C32—416no—
339C22×C32—432no—
340C22×C32—62no—
341C22×C32—62no—
342C22×C32—124no—
343C22×C32D3×D412yes—
344C22×C32C2×#6718yes—
345C22×C32C23×D314yes—
346C22×C32C23×D311yescharacteristic
347C2×C3⋊D3—21no—
348C2×C3⋊D3—22no—
349C2×C3⋊D3—61no—
350C2×C3⋊D3—62no—
351C2×C3⋊D3—68no—
352C2×C3⋊D3—64no—
353C2×C3⋊D3—124no—
354C2×C3⋊D3—128no—
355C2×C3⋊D3—124no—
356C2×C3×D3—28no—
357C2×C3×D3—28no—
358C2×C3×D3—416no—
359C2×C3×D3—68no—
360C2×C3×D3—64no—
361C2×C3×D3C2×#6718yes—
362C2×C3×D3C2×#6718yes—
363D32—1216no—
364C32×C4—62no—
365C3⋊Dic3—21no—
366C3⋊Dic3—22no—
367C3⋊Dic3—61no—
368C3⋊Dic3—62no—
369C3⋊Dic3—68no—
370C3⋊Dic3—64no—
371C3⋊Dic3—124no—
372C3⋊Dic3—128no—
373C3×Dic3—28no—
374C3×Dic3—68no—
375C3×Dic3—64no—
376C22×D4—31no—
377C22×D4—271no—
378#123—271no—
379#123—272no—
380#123—271no—
381#121—272no—
382#121—272no—
383#120—91no—
384#120—272no—
385#120—271no—
386C2×C22⋊C4—271no—
387C33#53411yescharacteristic, complemented, Sylow
388C23×C3—28no—
389C23×C3—22no—
390C23×C3—28no—
391C23×C3—24no—
392C23×C3—34no—
393C23×C3—34no—
394C23×C3—316no—
395C23×C3—48no—
396C23×C3—48no—
397C23×C3—42no—
398C23×C3—416no—
399C23×C3—91no—
400C23×C3—128no—
401C23×C3C2×D3218yes—
402C23×C3C2×D3214yes—
403C23×C3C22×C3⋊D312yes—
404C23×C3C22×C3⋊D311yescharacteristic
405C22×D3—34no—
406C22×D3—42no—
407C22×D3—94no—
408C22×D3—91no—
409C22×D3—184no—
410C22×D3—188no—
411C22×D3—182no—
412C22×D3—362no—
413C22×D3—368no—
414C22×D3#63814yescomplemented
415C22×D3C22×C3⋊D311yescharacteristic
416C22×D3C22×C3⋊D311yescharacteristic
417C3×D4—616no—
418C3×D4—616no—
419C3×D4—182no—
420C3×D4—362no—
421C2×C3×C4—38no—
422C2×C3×C4—68no—
423C2×C3×C4—91no—
424C2×C3×C4—92no—
425#67—24no—
426#67—24no—
427#67—68no—
428#67—128no—
429#67—188no—
430#67—188no—
431#67—188no—
432#67—182no—
433#67—182no—
434#67—368no—
435#67—368no—
436#67—362no—
437#67—3616no—
438C2×Dic3—22no—
439C2×Dic3—34no—
440C2×Dic3—38no—
441C2×Dic3—94no—
442C2×Dic3—98no—
443C2×Dic3—92no—
444C2×Dic3—91no—
445C2×Dic3—188no—
446C2×Dic3—184no—
447C2×Dic3—188no—
448C2×Dic3—368no—
449C2×Dic3C22×C3⋊D312yes—
450D12—182no—
451C4×D3—182no—
452C4×D3—362no—
453C2×C32—22no—
454C2×C32—24no—
455C2×C32—28no—
456C2×C32—28no—
457C2×C32—48no—
458C2×C32—416no—
459C2×C32—48no—
460C2×C32—42no—
461C2×C32—62no—
462C2×C32—62no—
463C2×C32C2×D3×D411yescharacteristic
464C2×C32#100418yes—
465C2×C32#100412yes—
466C2×C32C22×#6714yes—
467C3⋊D3—41no—
468C3⋊D3—121no—
469C3⋊D3—124no—
470C3⋊D3—244no—
471C3×D3—28no—
472C3×D3—28no—
473C3×D3—124no—
474C24—31no—
475C24—271no—
476C24D3×C3⋊D311yescharacteristic, complemented
477C2×D4—32no—
478C2×D4—32no—
479C2×D4—64no—
480C2×D4—91no—
481C2×D4—92no—
482C2×D4—271no—
483C2×D4—271no—
484C2×D4—271no—
485C2×D4—272no—
486C2×D4—544no—
487C22×C4—31no—
488C22×C4—271no—
489C22×C4—272no—
490C4⋊C4—272no—
491C4⋊C4—272no—
492C22⋊C4—91no—
493C22⋊C4—92no—
494C22⋊C4—272no—
495C22⋊C4—272no—
496C22⋊C4—272no—
497C22⋊C4—271no—
498C22⋊C4—272no—
499C22×C3—28no—
500C22×C3—22no—
501C22×C3—216no—
502C22×C3—28no—
503C22×C3—28no—
504C22×C3—28no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—24no—
509C22×C3—28no—
510C22×C3—38no—
511C22×C3—38no—
512C22×C3—48no—
513C22×C3—48no—
514C22×C3—48no—
515C22×C3—44no—
516C22×C3—416no—
517C22×C3—432no—
518C22×C3—68no—
519C22×C3—68no—
520C22×C3—1216no—
521C22×C3—182no—
522C22×C3—181no—
523C22×C3D3×#6718yes—
524C22×C3C2×#63812yes—
525C22×C3C22×D3214yes—
526C22×C3C23×C3⋊D311yescharacteristic
527C2×D3—22no—
528C2×D3—22no—
529C2×D3—44no—
530C2×D3—64no—
531C2×D3—68no—
532C2×D3—184no—
533C2×D3—188no—
534C2×D3—182no—
535C2×D3—181no—
536C2×D3—364no—
537C2×D3—368no—
538C2×D3—364no—
539C2×D3—3616no—
540C2×D3C2×#63812yes—
541C2×D3C2×#63812yes—
542C3×C4—68no—
543C3×C4—182no—
544C3×C4—181no—
545Dic3—22no—
546Dic3—64no—
547Dic3—68no—
548Dic3—184no—
549Dic3—188no—
550Dic3—182no—
551Dic3—181no—
552Dic3—364no—
553Dic3—368no—
554C32—24no—
555C32#558911yescharacteristic, complemented
556C32#580414yescomplemented
557C23—22no—
558C23—31no—
559C23—31no—
560C23—34no—
561C23—42no—
562C23—91no—
563C23—122no—
564C23—271no—
565C23—542no—
566C23—1082no—
567C23C2×D3×C3⋊D312yes—
568C23C2×D3×C3⋊D311yescharacteristic
569D4—64no—
570D4—64no—
571D4—182no—
572D4—362no—
573D4—542no—
574D4—542no—
575D4—542no—
576D4—1082no—
577C2×C4—32no—
578C2×C4—62no—
579C2×C4—92no—
580C2×C4—91no—
581C2×C4—272no—
582C2×C4—271no—
583C2×C4—542no—
584C2×C4—1082no—
585C2×C3—22no—
586C2×C3—28no—
587C2×C3—24no—
588C2×C3—28no—
589C2×C3—48no—
590C2×C3—416no—
591C2×C3—42no—
592C2×C3—48no—
593C2×C3—68no—
594C2×C3—68no—
595C2×C3—361no—
596C2×C3C2×D3×#6714yes—
597C2×C3#6433018yes—
598C2×C3#6436112yes—
599C2×C3C22×#63811yescharacteristic
600D3—22no—
601D3—22no—
602D3—124no—
603D3—361no—
604D3—364no—
605D3—724no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—32no—
611C22—32no—
612C22—44no—
613C22—62no—
614C22—62no—
615C22—124no—
616C22—181no—
617C22—182no—
618C22—541no—
619C22—542no—
620C22—1084no—
621C22D3×#63812yes—
622C22C22×D3×C3⋊D311yescharacteristic
623C4—62no—
624C4—181no—
625C4—182no—
626C4—541no—
627C4—542no—
628C3—24no—
629C3#1059552414yescomplemented
630C3#1059627511yescharacteristic, complemented
631C2—22no—
632C2—42no—
633C2—62no—
634C2—62no—
635C2—361no—
636C2—1081no—
637C2C2×D3×#63811yescharacteristic
638C2#1172461112yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC24×C33C22
hypercenterC24D3×C3⋊D3
hypocenterC33#534
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×C32×#67C2semidirect
2. C2×D3×#638C2semidirect
3. #11723994C2semidirect
4. C3×#64162C2semidirect
5. #11723643C2semidirect
6. #11723616C2semidirect
7. #11723615C2semidirect
8. #11723614C2semidirect
9. D3×#3757C2semidirect
10. #11723612C2semidirect
11. #11723605C2semidirect
14. C24×C33C22semidirect
16. C22×D3×C3⋊D3C22semidirect
18. C2×#6312C22semidirect
20. C2×C3×#638C22semidirect
23. C2×C32×#67C22semidirect
24. C22×C32×Dic3C22semidirect
26. C2×#6291C22semidirect
27. C2×#6290C22semidirect
28. C2×D3×C3⋊Dic3C22semidirect
29. C2×D3×C3⋊Dic3C22semidirect
30. #88864C22semidirect
31. #88864C22semidirect
32. C3×#3757C22semidirect
33. C3×#3757C22semidirect
34. #88807C22semidirect
35. #88806C22semidirect
36. #88804C22semidirect
37. #88803C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
41. C22×C3×#67D3semidirect
46. #64162D3semidirect
69. C22×C32×D3D4semidirect
94. C24×C32C2×D3semidirect
95. C24×C32C2×D3semidirect
113. C2×#638C2×D3semidirect
120. C2×C3×#67C2×D3semidirect
121. C22×C3×Dic3C2×D3semidirect
134. #3757C2×D3semidirect
135. #3757C2×D3semidirect
171. C22×#67C3⋊D3semidirect
216. C22×C3×D3#67semidirect
268. C24×C3D32semidirect
269. C24×C3C2×C3⋊D3semidirect
293. C2×#67C2×C3⋊D3semidirect
296. C22×Dic3C2×C3⋊D3semidirect
387. C33#534semidirect
414. C22×D3#638semidirect
476. C24D3×C3⋊D3semidirect
555. C32#5589semidirect
556. C32#5804semidirect
629. C3#10595524semidirect
630. C3#10596275semidirect

Maximal subgroups

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

subgroupindex
C22×C32×#672
C2×D3×#6382
#117239942
C3×#641622
#117236432
#117236162
#117236152
#117236142
D3×#37572
#117236122
#117236052
#105962753
#105955243