Finite Group Databasebeta

← #13177320

Subgroups — #13177320

Order 1728.

2388 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×D12C211yesmaximal, characteristic, complemented
2C2×D3×#638C211yesmaximal, characteristic, complemented
3#11723970C211yesmaximal, characteristic, complemented
4C3×#64138C211yesmaximal, characteristic, complemented
5#11723630C211yesmaximal, characteristic, complemented
6#11723629C211yesmaximal, characteristic, complemented
7D3×#3757C211yesmaximal, characteristic, complemented
8#11723540C212yesmaximal, complemented
9#11723475C212yesmaximal, complemented
10#11723474C212yesmaximal, complemented
11#11723473C212yesmaximal, complemented
12#10596275—31nomaximal
13#10595062—34nomaximal
14C23×C32×D3C2211yescharacteristic
15C23×C32×D3C2211yescharacteristic, complemented
16C22×D3×C3⋊D3C2211yescharacteristic, complemented
17C22×C33×C4C2211yescharacteristic, complemented
18C2×#6312C2211yescharacteristic
19C2×C3×#638C2211yescharacteristic
20C2×C32×D12—24no—
21C2×C32×D12C2212yes—
22C2×C32×D12C2212yescomplemented
23D3×#638—24no—
24C2×#6291C2211yescharacteristic, complemented
25C2×#6291C2212yescomplemented
26C2×#6290C2211yescharacteristic, complemented
27C2×#6290C2212yescomplemented
28C2×D3×C3⋊Dic3C2211yescharacteristic, complemented
29C2×D3×C3⋊Dic3C2212yescomplemented
30#88864C2211yescharacteristic, complemented
31#88859C2212yescomplemented
32#88857C2212yescomplemented
33C3×#3757C2211yescharacteristic, complemented
34C3×#3752C2212yescomplemented
35C3×#3750C2212yescomplemented
36(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
37(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C22×C32×D4—31no—
40C22×#638—31no—
41C22×C3×D12D314yescomplemented
42C2×D3×#67—34no—
43#64162—31no—
44#64162—31no—
45C2×#3757—31no—
46#64148—32no—
47#64148—32no—
48#64146—32no—
49#64144—32no—
50#64138—31no—
51#64138—34no—
52#64138—64no—
53#64138D311yescharacteristic, complemented
54C3×#929—34no—
55#63977—34no—
56#63976—34no—
57D3×#217—34no—
58#63922—38no—
59#63886—38no—
60#63885—38no—
61#63884—38no—
62C23×C33C2311yescharacteristic
63C22×C3⋊(C3⋊D3)C2311yescharacteristic
64C22×C3×C3⋊D3C2311yescharacteristic
65C22×C32×D3—22no—
66C22×C32×D3—42no—
67C22×C32×D3—42no—
68C22×C32×D3D414yescomplemented
69C22×C32×D3C2312yes—
70C22×C32×D3C2311yescharacteristic
71C22×C32×D3C2311yescharacteristic
72C2×D3×C3⋊D3—22no—
73C2×D3×C3⋊D3—42no—
74C2×C33×C4—22no—
75C2×C33×C4C2312yes—
76#6312—22no—
77C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
78C2×C3⋊(C3⋊Dic3)C2312yes—
79C3×#638—22no—
80C2×C3×C3⋊Dic3C2311yescharacteristic
81C2×C3×C3⋊Dic3C2312yes—
82C32×D12—24no—
83C32×D12—24no—
84#6291—22no—
85#6291—42no—
86#6290—22no—
87#6290—42no—
88D3×C3⋊Dic3—22no—
89D3×C3⋊Dic3—42no—
90#5804—94no—
91#5653—91no—
92C24×C32—31no—
93C24×C32—31no—
94C23×C3⋊D3—31no—
95C23×C3×D3C2×D314yes—
96C23×C3×D3C2×D314yescomplemented
97C22×D32—34no—
98C2×C32×D4—32no—
99C2×C32×D4—32no—
100C2×C32×D4—64no—
101C22×C32×C4—24no—
102C22×C32×C4C2×D314yescomplemented
103C22×C32×C4C2×D311yescharacteristic, complemented
104C2×#638—31no—
105C2×#638—34no—
106C2×#638—31no—
107C2×#638—32no—
108C2×#638—31no—
109C2×#638—32no—
110C2×#638—64no—
111C2×#638—64no—
112C2×#638C2×D311yescharacteristic
113C22×C3⋊Dic3—31no—
114C22×C3⋊Dic3—32no—
115C2×C3×#67—34no—
116C2×C3×D12—216no—
117C2×C3×D12C2×D318yes—
118C2×C3×D12C2×D318yescomplemented
119D3×#67—616no—
120C2×#626—34no—
121C2×#626—38no—
122C2×#625—34no—
123C2×#625—38no—
124C2×D3×Dic3—34no—
125C2×D3×Dic3—38no—
126#3757—31no—
127#3757—34no—
128#3757—32no—
129#3757—32no—
130#3757—32no—
131#3757—64no—
132#3757C2×D311yescharacteristic, complemented
133#3752—32no—
134#3752—38no—
135#3752—68no—
136#3752C2×D312yescomplemented
137#3750—32no—
138#3750—38no—
139#3750—68no—
140#3750C2×D312yescomplemented
141C3×#217—34no—
142C3×(C2×D3)⋊C4—38no—
143C3×Dic3⋊C4—38no—
144(C2×D3)⋊Dic3—38no—
145(C2×D3)⋊Dic3—38no—
146(C2×D3)⋊Dic3—38no—
147C22×C33—22no—
148C22×C33C2×D412yes—
149C22×C33C2411yescharacteristic
150C2×C3⋊(C3⋊D3)—21no—
151C2×C3⋊(C3⋊D3)—22no—
152C2×C3×C3⋊D3—21no—
153C2×C3×C3⋊D3—22no—
154C2×C32×D3—22no—
155C2×C32×D3—22no—
156C2×C32×D3—22no—
157C2×C32×D3—22no—
158C2×C32×D3—22no—
159C2×C32×D3—44no—
160C2×C32×D3—44no—
161C2×C32×D3C2×D412yes—
162C2×C32×D3C2×D412yes—
163D3×C3⋊D3—44no—
164C33×C4—22no—
165C3⋊(C3⋊Dic3)—21no—
166C3⋊(C3⋊Dic3)—22no—
167C3×C3⋊Dic3—21no—
168C3×C3⋊Dic3—22no—
169C22×C3×D4—34no—
170C22×#67—94no—
171C2×D3×D4—91no—
172C22×D12C3⋊D311yescharacteristic, complemented
173C3×#123—91no—
174#953—94no—
175#953—94no—
176C2×#217—94no—
177#939—98no—
178#939—98no—
179#937—98no—
180#937—91no—
181#935—98no—
182#929—34no—
183#929—94no—
184#929—91no—
185#929—184no—
186#895—92no—
187#894—92no—
188#884—92no—
189#882—91no—
190#882—92no—
191D3×C22⋊C4—91no—
192C23×C32—24no—
193C23×C32—31no—
194C23×C32—31no—
195C23×C32—34no—
196C23×C32—32no—
197C23×C32—62no—
198C23×C32—122no—
199C23×C32—122no—
200C23×C32C22×D314yes—
201C23×C32C22×D311yescharacteristic
202C22×C3⋊D3—31no—
203C22×C3⋊D3—34no—
204C22×C3⋊D3—64no—
205C22×C3⋊D3—62no—
206C22×C3⋊D3—122no—
207C22×C3⋊D3C22×D311yescharacteristic
208C22×C3×D3—28no—
209C22×C3×D3—34no—
210C22×C3×D3—48no—
211C22×C3×D3—48no—
212C22×C3×D3#67116yescomplemented
213C22×C3×D3C22×D318yes—
214C22×C3×D3C22×D314yes—
215C22×C3×D3C22×D314yes—
216C2×D32—68no—
217C2×D32—128no—
218C32×D4—64no—
219C32×D4—64no—
220C2×C32×C4—22no—
221C2×C32×C4—28no—
222C2×C32×C4—28no—
223C2×C32×C4—48no—
224C2×C32×C4C22×D318yes—
225C2×C32×C4C22×D312yes—
226#638—22no—
227#638—62no—
228#638—62no—
229#638—62no—
230#638—68no—
231#638—128no—
232#638—122no—
233#638—122no—
234C2×C3⋊Dic3—31no—
235C2×C3⋊Dic3—32no—
236C2×C3⋊Dic3—34no—
237C2×C3⋊Dic3—38no—
238C2×C3⋊Dic3—64no—
239C2×C3⋊Dic3—68no—
240C2×C3⋊Dic3—62no—
241C2×C3⋊Dic3—122no—
242C2×C3⋊Dic3C22×D311yescharacteristic
243C2×C3⋊Dic3C22×D312yes—
244C3×#67—68no—
245C2×C3×Dic3—34no—
246C2×C3×Dic3—38no—
247C3×D12—216no—
248C3×D12—216no—
249#626—68no—
250#626—128no—
251#625—68no—
252#625—128no—
253D3×Dic3—68no—
254D3×Dic3—128no—
255#534—271noSylow
256C2×C33—22no—
257C2×C33C22×D411yescharacteristic
258C2×C33H(5,𝔽2)12yes—
259C3⋊(C3⋊D3)—41no—
260C3×C3⋊D3—41no—
261C32×D3—22no—
262C32×D3—22no—
263C32×D3—42no—
264C24×C3—34no—
265C24×C3—34no—
266C23×D3—94no—
267C23×D3—91no—
268C23×D3C2×C3⋊D311yescharacteristic
269C23×D3C2×C3⋊D311yescharacteristic, complemented
270C2×C3×D4—38no—
271C2×C3×D4—38no—
272C2×C3×D4—616no—
273C2×C3×D4—91no—
274C22×C3×C4—24no—
275C22×C3×C4D3214yescomplemented
276C22×C3×C4C2×C3⋊D311yescharacteristic, complemented
277C2×#67—34no—
278C2×#67—94no—
279C2×#67—98no—
280C2×#67—94no—
281C2×#67—94no—
282C2×#67—91no—
283C2×#67—91no—
284C2×#67—98no—
285C2×#67—92no—
286C2×#67—184no—
287C2×#67—1816no—
288C22×Dic3—94no—
289C22×Dic3—98no—
290D3×D4—184no—
291C2×D12—24no—
292C2×D12—91no—
293C2×D12—92no—
294C2×D12C2×C3⋊D312yes—
295C2×D12C2×C3⋊D312yescomplemented
296C2×C4×D3—91no—
297C2×C4×D3—92no—
298C3×C4⋊C4—92no—
299C3×C22⋊C4—91no—
300C3×C22⋊C4—92no—
301#217—34no—
302#217—98no—
303#217—98no—
304#217—98no—
305#217—94no—
306#217—91no—
307#217—184no—
308(C2×D3)⋊C4—38no—
309(C2×D3)⋊C4—92no—
310(C2×D3)⋊C4—92no—
311(C2×D3)⋊C4—92no—
312(C2×D3)⋊C4—98no—
313(C2×D3)⋊C4—92no—
314(C2×D3)⋊C4—188no—
315Dic3⋊C4—38no—
316Dic3⋊C4—98no—
317Dic3⋊C4—92no—
318Dic3⋊C4—188no—
319C22×C32—22no—
320C22×C32—24no—
321C22×C32—28no—
322C22×C32—28no—
323C22×C32—32no—
324C22×C32—32no—
325C22×C32—48no—
326C22×C32—62no—
327C22×C32—62no—
328C22×C32—62no—
329C22×C32—62no—
330C22×C32—62no—
331C22×C32—124no—
332C22×C32—124no—
333C22×C32D3×D412yes—
334C22×C32C2×#6718yes—
335C22×C32C23×D314yes—
336C22×C32C23×D311yescharacteristic
337C2×C3⋊D3—21no—
338C2×C3⋊D3—22no—
339C2×C3⋊D3—61no—
340C2×C3⋊D3—62no—
341C2×C3⋊D3—68no—
342C2×C3⋊D3—64no—
343C2×C3⋊D3—124no—
344C2×C3⋊D3—128no—
345C2×C3⋊D3—124no—
346C2×C3×D3—28no—
347C2×C3×D3—28no—
348C2×C3×D3—28no—
349C2×C3×D3—28no—
350C2×C3×D3—28no—
351C2×C3×D3—416no—
352C2×C3×D3—416no—
353C2×C3×D3—68no—
354C2×C3×D3—64no—
355C2×C3×D3C2×#6718yes—
356C2×C3×D3C2×#6718yes—
357D32—1216no—
358C32×C4—22no—
359C32×C4—28no—
360C32×C4—48no—
361C3⋊Dic3—21no—
362C3⋊Dic3—22no—
363C3⋊Dic3—61no—
364C3⋊Dic3—62no—
365C3⋊Dic3—68no—
366C3⋊Dic3—64no—
367C3⋊Dic3—128no—
368C3⋊Dic3—124no—
369C3×Dic3—68no—
370C3×Dic3—64no—
371C22×D4—31no—
372C22×D4—271no—
373#123—91no—
374#123—272no—
375#123—271no—
376#121—272no—
377#121—272no—
378#120—272no—
379#120—271no—
380#120—271no—
381C2×C22⋊C4—271no—
382C33#53411yescharacteristic, complemented, Sylow
383C23×C3—24no—
384C23×C3—34no—
385C23×C3—34no—
386C23×C3—316no—
387C23×C3—38no—
388C23×C3—68no—
389C23×C3—91no—
390C23×C3—128no—
391C23×C3—128no—
392C23×C3C2×D3214yes—
393C23×C3C22×C3⋊D311yescharacteristic
394C22×D3—22no—
395C22×D3—34no—
396C22×D3—42no—
397C22×D3—42no—
398C22×D3—94no—
399C22×D3—91no—
400C22×D3—184no—
401C22×D3—188no—
402C22×D3—182no—
403C22×D3—362no—
404C22×D3—368no—
405C22×D3#63814yescomplemented
406C22×D3C22×C3⋊D312yes—
407C22×D3C22×C3⋊D311yescharacteristic
408C22×D3C22×C3⋊D311yescharacteristic
409C3×D4—616no—
410C3×D4—616no—
411C3×D4—182no—
412C2×C3×C4—28no—
413C2×C3×C4—22no—
414C2×C3×C4—28no—
415C2×C3×C4—48no—
416C2×C3×C4—92no—
417C2×C3×C4—91no—
418C2×C3×C4C2×D3218yes—
419C2×C3×C4C22×C3⋊D312yes—
420#67—68no—
421#67—188no—
422#67—188no—
423#67—188no—
424#67—182no—
425#67—182no—
426#67—368no—
427#67—368no—
428#67—362no—
429#67—368no—
430C2×Dic3—34no—
431C2×Dic3—38no—
432C2×Dic3—94no—
433C2×Dic3—98no—
434C2×Dic3—91no—
435C2×Dic3—92no—
436C2×Dic3—184no—
437C2×Dic3—188no—
438C2×Dic3—188no—
439C2×Dic3—368no—
440D12—24no—
441D12—24no—
442D12—182no—
443D12—362no—
444C4×D3—182no—
445C4×D3—362no—
446C2×C32—22no—
447C2×C32—28no—
448C2×C32—24no—
449C2×C32—28no—
450C2×C32—48no—
451C2×C32—62no—
452C2×C32—62no—
453C2×C32—122no—
454C2×C32C2×D3×D411yescharacteristic
455C2×C32#100418yes—
456C2×C32#100912yes—
457C2×C32C22×#6714yes—
458C3⋊D3—41no—
459C3⋊D3—121no—
460C3⋊D3—124no—
461C3⋊D3—244no—
462C3×D3—28no—
463C3×D3—28no—
464C3×D3—48no—
465C3×D3—124no—
466C24—31no—
467C24—31no—
468C24—271no—
469C2×D4—32no—
470C2×D4—32no—
471C2×D4—64no—
472C2×D4—91no—
473C2×D4—271no—
474C2×D4—272no—
475C2×D4—271no—
476C2×D4—271no—
477C2×D4—272no—
478C2×D4—544no—
479C22×C4—271no—
480C22×C4—272no—
481C22×C4D3×C3⋊D311yescharacteristic, complemented
482C4⋊C4—92no—
483C4⋊C4—272no—
484C22⋊C4—91no—
485C22⋊C4—92no—
486C22⋊C4—272no—
487C22⋊C4—272no—
488C22⋊C4—272no—
489C22⋊C4—271no—
490C22⋊C4—272no—
491C22×C3—28no—
492C22×C3—22no—
493C22×C3—24no—
494C22×C3—28no—
495C22×C3—38no—
496C22×C3—38no—
497C22×C3—48no—
498C22×C3—68no—
499C22×C3—68no—
500C22×C3—68no—
501C22×C3—68no—
502C22×C3—68no—
503C22×C3—1216no—
504C22×C3—1216no—
505C22×C3—182no—
506C22×C3—181no—
507C22×C3D3×#6718yes—
508C22×C3C2×#63812yes—
509C22×C3C22×D3214yes—
510C22×C3C23×C3⋊D311yescharacteristic
511C2×D3—22no—
512C2×D3—22no—
513C2×D3—22no—
514C2×D3—22no—
515C2×D3—22no—
516C2×D3—44no—
517C2×D3—44no—
518C2×D3—68no—
519C2×D3—64no—
520C2×D3—188no—
521C2×D3—184no—
522C2×D3—182no—
523C2×D3—181no—
524C2×D3—364no—
525C2×D3—368no—
526C2×D3—364no—
527C2×D3—3616no—
528C2×D3C2×#63812yes—
529C2×D3C2×#63812yes—
530C3×C4—28no—
531C3×C4—22no—
532C3×C4—48no—
533C3×C4—182no—
534C3×C4—181no—
535Dic3—68no—
536Dic3—64no—
537Dic3—188no—
538Dic3—184no—
539Dic3—182no—
540Dic3—181no—
541Dic3—364no—
542Dic3—368no—
543C32—24no—
544C32#565311yescharacteristic, complemented
545C32#580414yescomplemented
546C23—31no—
547C23—31no—
548C23—34no—
549C23—32no—
550C23—62no—
551C23—91no—
552C23—122no—
553C23—122no—
554C23—271no—
555C23—542no—
556C23—1082no—
557C23C2×D3×C3⋊D311yescharacteristic
558D4—64no—
559D4—64no—
560D4—182no—
561D4—542no—
562D4—542no—
563D4—542no—
564D4—1082no—
565D4—1082no—
566C2×C4—22no—
567C2×C4—92no—
568C2×C4—91no—
569C2×C4—272no—
570C2×C4—271no—
571C2×C4—542no—
572C2×C4—1082no—
573C2×C4C2×D3×C3⋊D312yes—
574C2×C3—22no—
575C2×C3—28no—
576C2×C3—28no—
577C2×C3—24no—
578C2×C3—48no—
579C2×C3—68no—
580C2×C3—68no—
581C2×C3—128no—
582C2×C3—361no—
583C2×C3#6430718yes—
584C2×C3C2×D3×#6714yes—
585C2×C3#6436112yes—
586C2×C3C22×#63811yescharacteristic
587D3—22no—
588D3—22no—
589D3—42no—
590D3—124no—
591D3—361no—
592D3—364no—
593D3—724no—
594C22—22no—
595C22—32no—
596C22—32no—
597C22—62no—
598C22—62no—
599C22—62no—
600C22—62no—
601C22—62no—
602C22—124no—
603C22—124no—
604C22—182no—
605C22—181no—
606C22—542no—
607C22—541no—
608C22—1084no—
609C22D3×#63812yes—
610C22C22×D3×C3⋊D311yescharacteristic
611C4—22no—
612C4—182no—
613C4—181no—
614C4—542no—
615C4—541no—
616C3—24no—
617C3#1059506214yescomplemented
618C3#1059627511yescharacteristic, complemented
619C2—22no—
620C2—62no—
621C2—62no—
622C2—122no—
623C2—361no—
624C2—1081no—
625C2#1172457912yes—
626C2C2×D3×#63811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#534
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C32×D12C2semidirect
2. C2×D3×#638C2semidirect
3. #11723970C2semidirect
4. C3×#64138C2semidirect
5. #11723630C2semidirect
6. #11723629C2semidirect
7. D3×#3757C2semidirect
8. #11723540C2semidirect
9. #11723475C2semidirect
10. #11723474C2semidirect
11. #11723473C2semidirect
15. C23×C32×D3C22semidirect
16. C22×D3×C3⋊D3C22semidirect
17. C22×C33×C4C22semidirect
22. C2×C32×D12C22semidirect
24. C2×#6291C22semidirect
25. C2×#6291C22semidirect
26. C2×#6290C22semidirect
27. C2×#6290C22semidirect
28. C2×D3×C3⋊Dic3C22semidirect
29. C2×D3×C3⋊Dic3C22semidirect
30. #88864C22semidirect
31. #88859C22semidirect
32. #88857C22semidirect
33. C3×#3757C22semidirect
34. C3×#3752C22semidirect
35. C3×#3750C22semidirect
36. (C2×D3)⋊(C3⋊Dic3)C22semidirect
37. (C2×D3)⋊(C3⋊Dic3)C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
41. C22×C3×D12D3semidirect
53. #64138D3semidirect
68. C22×C32×D3D4semidirect
96. C23×C3×D3C2×D3semidirect
102. C22×C32×C4C2×D3semidirect
103. C22×C32×C4C2×D3semidirect
118. C2×C3×D12C2×D3semidirect
132. #3757C2×D3semidirect
136. #3752C2×D3semidirect
140. #3750C2×D3semidirect
172. C22×D12C3⋊D3semidirect
212. C22×C3×D3#67semidirect
269. C23×D3C2×C3⋊D3semidirect
275. C22×C3×C4D32semidirect
276. C22×C3×C4C2×C3⋊D3semidirect
295. C2×D12C2×C3⋊D3semidirect
382. C33#534semidirect
405. C22×D3#638semidirect
481. C22×C4D3×C3⋊D3semidirect
544. C32#5653semidirect
545. C32#5804semidirect
617. C3#10595062semidirect
618. C3#10596275semidirect

Maximal subgroups

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

subgroupindex
C22×C32×D122
C2×D3×#6382
#117239702
C3×#641382
#117236302
#117236292
D3×#37572
#117235402
#117234752
#117234742
#117234732
#105962753
#105950623