Finite Group Databasebeta

← #13150535

Subgroups — #13150535

Order 1728.

1116 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×C9×D12C211yesmaximal, characteristic, complemented
2C2×D3×#611C211yesmaximal, characteristic, complemented
3#11721818C211yesmaximal, characteristic, complemented
4C3×#63491C211yesmaximal, characteristic, complemented
5#11721287C211yesmaximal, characteristic, complemented
6#11721286C211yesmaximal, characteristic, complemented
7D3×#3676C211yesmaximal, characteristic, complemented
8#11721050C212yesmaximal, complemented
9#11721017C212yesmaximal, complemented
10#11721016C212yesmaximal, complemented
11#11721015C212yesmaximal, complemented
12#10595062—31nomaximal
13#10589906—31nomaximal
14C23×D3×C9C2211yescharacteristic
15C23×D3×C9C2211yescharacteristic, complemented
16C22×D3×D9C2211yescharacteristic, complemented
17C22×C3×C4×C9C2211yescharacteristic, complemented
18C2×#6233C2211yescharacteristic
19C2×C3×#611C2211yescharacteristic
20C2×C9×D12—24no—
21C2×C9×D12C2212yes—
22C2×C9×D12C2212yescomplemented
23D3×#611—24no—
24C2×#6195C2211yescharacteristic, complemented
25C2×#6195C2212yescomplemented
26C2×#6194C2211yescharacteristic, complemented
27C2×#6194C2212yescomplemented
28C2×D3×Dic9C2211yescharacteristic, complemented
29C2×D3×Dic9C2212yescomplemented
30#88551C2211yescharacteristic, complemented
31#88536C2212yescomplemented
32#88534C2212yescomplemented
33C3×#3676C2211yescharacteristic, complemented
34C3×(C2×D9)⋊C4C2212yescomplemented
35C3×Dic9⋊C4C2212yescomplemented
36(C2×D3)⋊Dic9C2212yescomplemented
37(C2×D3)⋊Dic9C2212yescomplemented
38(C2×D3)⋊Dic9C2212yescomplemented
39C22×C3×D12D311yescharacteristic
40C2×D3×#67—31no—
41#64138—31no—
42C3×#929—31no—
43#63977—31no—
44#63976—31no—
45D3×#217—31no—
46#63922—32no—
47#63886—32no—
48#63885—32no—
49#63884—32no—
50C22×D4×C9—31no—
51C22×#611—31no—
52#63515—31no—
53#63515—31no—
54C2×#3676—31no—
55#63501—32no—
56#63501—32no—
57#63499—32no—
58#63497—32no—
59#63491—31no—
60#63491—61no—
61#63491D311yescharacteristic, complemented
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2311yescharacteristic
64C22×D3×C9—22no—
65C22×D3×C9—42no—
66C22×D3×C9—42no—
67C22×D3×C9D414yescomplemented
68C22×D3×C9C2312yes—
69C22×D3×C9C2311yescharacteristic
70C22×D3×C9C2311yescharacteristic
71C22×C3×D9C2311yescharacteristic
72C2×D3×D9—22no—
73C2×D3×D9—42no—
74C2×C3×C4×C9—22no—
75C2×C3×C4×C9C2312yes—
76#6233—22no—
77C2×C3⋊Dic9C2311yescharacteristic
78C2×C3⋊Dic9C2312yes—
79C3×#611—22no—
80C2×C3×Dic9C2311yescharacteristic
81C2×C3×Dic9C2312yes—
82C9×D12—24no—
83C9×D12—24no—
84#6195—22no—
85#6195—42no—
86#6194—22no—
87#6194—42no—
88D3×Dic9—22no—
89D3×Dic9—42no—
90#5804—91no—
91#5653—91no—
92C23×C3×D3C2×D311yescharacteristic
93C23×C3×D3C2×D311yescharacteristic
94C22×D32—31no—
95C22×C32×C4C2×D311yescharacteristic
96C2×#638—31no—
97C2×C3×#67—31no—
98C2×C3×D12—24no—
99C2×C3×D12C2×D312yes—
100C2×C3×D12C2×D312yes—
101D3×#67—64no—
102C2×#626—31no—
103C2×#626—32no—
104C2×#625—31no—
105C2×#625—32no—
106C2×D3×Dic3—31no—
107C2×D3×Dic3—32no—
108C24×C9—31no—
109C24×C9—31no—
110C23×D9—31no—
111#3757—31no—
112#3752—32no—
113#3750—32no—
114C3×#217—31no—
115C3×(C2×D3)⋊C4—32no—
116C3×Dic3⋊C4—32no—
117(C2×D3)⋊Dic3—32no—
118(C2×D3)⋊Dic3—32no—
119(C2×D3)⋊Dic3—32no—
120C2×D4×C9—32no—
121C2×D4×C9—32no—
122C2×D4×C9—64no—
123C22×C4×C9—21no—
124C22×C4×C9C2×D311yescharacteristic, complemented
125C2×#611—31no—
126C2×#611—31no—
127C2×#611—32no—
128C2×#611—31no—
129C2×#611—32no—
130C2×#611—64no—
131C2×#611—61no—
132C2×#611C2×D311yescharacteristic
133C22×Dic9—31no—
134C22×Dic9—32no—
135#3676—31no—
136#3676—32no—
137#3676—32no—
138#3676—32no—
139#3676—61no—
140#3676C2×D311yescharacteristic, complemented
141(C2×D9)⋊C4—32no—
142(C2×D9)⋊C4—62no—
143(C2×D9)⋊C4C2×D312yescomplemented
144Dic9⋊C4—32no—
145Dic9⋊C4—62no—
146Dic9⋊C4C2×D312yescomplemented
147C22×C3×C9—22no—
148C22×C3×C9C2×D412yes—
149C22×C3×C9C2411yescharacteristic
150C2×C3⋊D9—21no—
151C2×C3⋊D9—22no—
152C2×D3×C9—22no—
153C2×D3×C9—22no—
154C2×D3×C9—22no—
155C2×D3×C9—22no—
156C2×D3×C9—22no—
157C2×D3×C9—44no—
158C2×D3×C9—44no—
159C2×D3×C9C2×D412yes—
160C2×D3×C9C2×D412yes—
161C2×C3×D9—21no—
162C2×C3×D9—22no—
163D3×D9—44no—
164C3×C4×C9—22no—
165C3⋊Dic9—21no—
166C3⋊Dic9—22no—
167C3×Dic9—21no—
168C3×Dic9—22no—
169C22×C3×D4—31no—
170C22×#67—91no—
171C2×D3×D4—91no—
172C22×D12D911yescharacteristic, complemented
173C3×#123—91no—
174#953—91no—
175#953—91no—
176C2×#217—91no—
177#939—92no—
178#939—92no—
179#937—92no—
180#937—91no—
181#935—92no—
182#929—31no—
183#929—91no—
184#929—91no—
185#929—181no—
186#895—92no—
187#894—92no—
188#884—92no—
189#882—91no—
190#882—92no—
191D3×C22⋊C4—91no—
192C23×C32C22×D311yescharacteristic
193C22×C3⋊D3—31no—
194C22×C3×D3—22no—
195C22×C3×D3—31no—
196C22×C3×D3—42no—
197C22×C3×D3—42no—
198C22×C3×D3#6714yes—
199C22×C3×D3C22×D312yes—
200C22×C3×D3C22×D311yescharacteristic
201C22×C3×D3C22×D311yescharacteristic
202C2×D32—62no—
203C2×D32—122no—
204C2×C32×C4—22no—
205C2×C32×C4C22×D312yes—
206#638—62no—
207C2×C3⋊Dic3—31no—
208C2×C3⋊Dic3—32no—
209C3×#67—62no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—32no—
212C3×D12—24no—
213C3×D12—24no—
214#626—62no—
215#626—122no—
216#625—62no—
217#625—122no—
218D3×Dic3—62no—
219D3×Dic3—122no—
220C23×C9—21no—
221C23×C9—31no—
222C23×C9—31no—
223C23×C9—34no—
224C23×C9—32no—
225C23×C9—62no—
226C23×C9—122no—
227C23×C9—122no—
228C23×C9C22×D311yescharacteristic
229C22×D9—31no—
230C22×D9—61no—
231C22×D9—62no—
232C22×D9—122no—
233C22×D9C22×D311yescharacteristic
234D4×C9—64no—
235D4×C9—64no—
236C2×C4×C9—22no—
237C2×C4×C9—22no—
238C2×C4×C9—42no—
239C2×C4×C9C22×D312yes—
240#611—22no—
241#611—62no—
242#611—62no—
243#611—62no—
244#611—122no—
245#611—122no—
246#611—122no—
247C2×Dic9—31no—
248C2×Dic9—32no—
249C2×Dic9—61no—
250C2×Dic9—62no—
251C2×Dic9—62no—
252C2×Dic9—122no—
253C2×Dic9C22×D311yescharacteristic
254C2×Dic9C22×D312yes—
255#534—271noSylow
256C2×C3×C9—22no—
257C2×C3×C9C22×D411yescharacteristic
258C2×C3×C9H(5,𝔽2)12yes—
259C3⋊D9—41no—
260D3×C9—22no—
261D3×C9—22no—
262D3×C9—42no—
263C3×D9—41no—
264C24×C3—31no—
265C24×C3—31no—
266C23×D3—91no—
267C23×D3—91no—
268C23×D3C2×D911yescharacteristic
269C23×D3C2×D911yescharacteristic, complemented
270C2×C3×D4—32no—
271C2×C3×D4—32no—
272C2×C3×D4—64no—
273C2×C3×D4—91no—
274C22×C3×C4—21no—
275C22×C3×C4C2×D911yescharacteristic, complemented
276C22×C3×C4D3211yescharacteristic
277C2×#67—31no—
278C2×#67—91no—
279C2×#67—92no—
280C2×#67—91no—
281C2×#67—91no—
282C2×#67—91no—
283C2×#67—91no—
284C2×#67—92no—
285C2×#67—92no—
286C2×#67—181no—
287C2×#67—184no—
288C22×Dic3—91no—
289C22×Dic3—92no—
290D3×D4—184no—
291C2×D12—24no—
292C2×D12—91no—
293C2×D12—92no—
294C2×D12C2×D912yes—
295C2×D12C2×D912yescomplemented
296C2×C4×D3—91no—
297C2×C4×D3—92no—
298C3×C4⋊C4—92no—
299C3×C22⋊C4—91no—
300C3×C22⋊C4—92no—
301#217—31no—
302#217—92no—
303#217—92no—
304#217—92no—
305#217—91no—
306#217—91no—
307#217—181no—
308(C2×D3)⋊C4—32no—
309(C2×D3)⋊C4—92no—
310(C2×D3)⋊C4—92no—
311(C2×D3)⋊C4—92no—
312(C2×D3)⋊C4—92no—
313(C2×D3)⋊C4—92no—
314(C2×D3)⋊C4—182no—
315Dic3⋊C4—32no—
316Dic3⋊C4—92no—
317Dic3⋊C4—92no—
318Dic3⋊C4—182no—
319C22×C32—22no—
320C22×C32C2×#6712yes—
321C22×C32C23×D311yescharacteristic
322C2×C3⋊D3—62no—
323C2×C3⋊D3—61no—
324C2×C3×D3—22no—
325C2×C3×D3—22no—
326C2×C3×D3—22no—
327C2×C3×D3—22no—
328C2×C3×D3—22no—
329C2×C3×D3—44no—
330C2×C3×D3—44no—
331C2×C3×D3—62no—
332C2×C3×D3—61no—
333C2×C3×D3C2×#6712yes—
334C2×C3×D3C2×#6712yes—
335D32—124no—
336C32×C4—22no—
337C3⋊Dic3—62no—
338C3⋊Dic3—61no—
339C3×Dic3—62no—
340C3×Dic3—61no—
341C22×C9—22no—
342C22×C9—21no—
343C22×C9—22no—
344C22×C9—32no—
345C22×C9—32no—
346C22×C9—42no—
347C22×C9—62no—
348C22×C9—62no—
349C22×C9—62no—
350C22×C9—62no—
351C22×C9—62no—
352C22×C9—124no—
353C22×C9—124no—
354C22×C9D3×D412yes—
355C22×C9C23×D311yescharacteristic
356C2×D9—21no—
357C2×D9—22no—
358C2×D9—61no—
359C2×D9—62no—
360C2×D9—121no—
361C2×D9—122no—
362C2×D9—124no—
363C4×C9—22no—
364C4×C9—42no—
365Dic9—21no—
366Dic9—22no—
367Dic9—61no—
368Dic9—62no—
369Dic9—122no—
370Dic9—121no—
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—
382C3×C9#53411yescharacteristic, complemented, Sylow
383C23×C3—21no—
384C23×C3—31no—
385C23×C3—31no—
386C23×C3—34no—
387C23×C3—32no—
388C23×C3—62no—
389C23×C3—91no—
390C23×C3—122no—
391C23×C3—122no—
392C23×C3C22×D911yescharacteristic
393C23×C3C2×D3211yescharacteristic
394C22×D3—22no—
395C22×D3—31no—
396C22×D3—42no—
397C22×D3—42no—
398C22×D3—91no—
399C22×D3—91no—
400C22×D3—181no—
401C22×D3—182no—
402C22×D3—182no—
403C22×D3—362no—
404C22×D3—362no—
405C22×D3#61114yescomplemented
406C22×D3C22×D912yes—
407C22×D3C22×D911yescharacteristic
408C22×D3C22×D911yescharacteristic
409C3×D4—64no—
410C3×D4—64no—
411C3×D4—182no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—42no—
416C2×C3×C4—92no—
417C2×C3×C4—91no—
418C2×C3×C4C22×D912yes—
419C2×C3×C4C2×D3212yes—
420#67—62no—
421#67—182no—
422#67—182no—
423#67—182no—
424#67—182no—
425#67—182no—
426#67—362no—
427#67—362no—
428#67—362no—
429#67—362no—
430C2×Dic3—31no—
431C2×Dic3—32no—
432C2×Dic3—91no—
433C2×Dic3—92no—
434C2×Dic3—91no—
435C2×Dic3—92no—
436C2×Dic3—181no—
437C2×Dic3—182no—
438C2×Dic3—182no—
439C2×Dic3—362no—
440D12—24no—
441D12—24no—
442D12—182no—
443D12—362no—
444C4×D3—182no—
445C4×D3—362no—
446C2×C32—22no—
447C2×C32#100412yes—
448C2×C32C22×#6711yescharacteristic
449C3⋊D3—121no—
450C3×D3—22no—
451C3×D3—22no—
452C3×D3—42no—
453C3×D3—121no—
454C2×C9—22no—
455C2×C9—22no—
456C2×C9—21no—
457C2×C9—42no—
458C2×C9—62no—
459C2×C9—62no—
460C2×C9—122no—
461C2×C9C2×D3×D411yescharacteristic
462C2×C9#100912yes—
463D9—41no—
464D9—121no—
465D9—241no—
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×D911yescharacteristic, 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—22no—
492C22×C3—22no—
493C22×C3—21no—
494C22×C3—22no—
495C22×C3—32no—
496C22×C3—32no—
497C22×C3—42no—
498C22×C3—62no—
499C22×C3—62no—
500C22×C3—62no—
501C22×C3—62no—
502C22×C3—62no—
503C22×C3—124no—
504C22×C3—124no—
505C22×C3—182no—
506C22×C3—181no—
507C22×C3C2×#61112yes—
508C22×C3C23×D911yescharacteristic
509C22×C3D3×#6712yes—
510C22×C3C22×D3211yescharacteristic
511C2×D3—22no—
512C2×D3—22no—
513C2×D3—22no—
514C2×D3—22no—
515C2×D3—22no—
516C2×D3—44no—
517C2×D3—44no—
518C2×D3—62no—
519C2×D3—61no—
520C2×D3—182no—
521C2×D3—181no—
522C2×D3—182no—
523C2×D3—181no—
524C2×D3—361no—
525C2×D3—362no—
526C2×D3—364no—
527C2×D3—364no—
528C2×D3C2×#61112yes—
529C2×D3C2×#61112yes—
530C3×C4—22no—
531C3×C4—22no—
532C3×C4—42no—
533C3×C4—182no—
534C3×C4—181no—
535Dic3—62no—
536Dic3—61no—
537Dic3—182no—
538Dic3—181no—
539Dic3—182no—
540Dic3—181no—
541Dic3—361no—
542Dic3—362no—
543C32#580411yescharacteristic
544C9—21no—
545C9#565311yescharacteristic, complemented
546C23—31no—
547C23—31no—
548C23—34no—
549C23—32no—
550C23—62no—
551C23—91no—
552C23—122no—
553C23—122no—
554C23—271no—
555C23—542no—
556C23—1082no—
557C23C2×D3×D911yescharacteristic
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×D912yes—
574C2×C3—22no—
575C2×C3—22no—
576C2×C3—21no—
577C2×C3—22no—
578C2×C3—42no—
579C2×C3—62no—
580C2×C3—62no—
581C2×C3—122no—
582C2×C3—361no—
583C2×C3#6371012yes—
584C2×C3C22×#61111yescharacteristic
585C2×C3#6430712yes—
586C2×C3C2×D3×#6711yescharacteristic
587D3—22no—
588D3—22no—
589D3—42no—
590D3—121no—
591D3—361no—
592D3—361no—
593D3—721no—
594C22—22no—
595C22—32no—
596C22—32no—
597C22—62no—
598C22—62no—
599C22—62no—
600C22—62no—
601C22—62no—
602C22—124no—
603C22—124no—
604C22—181no—
605C22—182no—
606C22—541no—
607C22—542no—
608C22—1084no—
609C22D3×#61112yes—
610C22C22×D3×D911yescharacteristic
611C4—22no—
612C4—181no—
613C4—182no—
614C4—541no—
615C4—542no—
616C3—21no—
617C3#1058990611yescharacteristic, complemented
618C3#1059506211yescharacteristic
619C2—22no—
620C2—62no—
621C2—62no—
622C2—122no—
623C2—361no—
624C2—1081no—
625C2#1172247612yes—
626C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#534
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C9×D12C2semidirect
2. C2×D3×#611C2semidirect
3. #11721818C2semidirect
4. C3×#63491C2semidirect
5. #11721287C2semidirect
6. #11721286C2semidirect
7. D3×#3676C2semidirect
8. #11721050C2semidirect
9. #11721017C2semidirect
10. #11721016C2semidirect
11. #11721015C2semidirect
15. C23×D3×C9C22semidirect
16. C22×D3×D9C22semidirect
17. C22×C3×C4×C9C22semidirect
22. C2×C9×D12C22semidirect
24. C2×#6195C22semidirect
25. C2×#6195C22semidirect
26. C2×#6194C22semidirect
27. C2×#6194C22semidirect
28. C2×D3×Dic9C22semidirect
29. C2×D3×Dic9C22semidirect
30. #88551C22semidirect
31. #88536C22semidirect
32. #88534C22semidirect
33. C3×#3676C22semidirect
34. C3×(C2×D9)⋊C4C22semidirect
35. C3×Dic9⋊C4C22semidirect
36. (C2×D3)⋊Dic9C22semidirect
37. (C2×D3)⋊Dic9C22semidirect
38. (C2×D3)⋊Dic9C22semidirect
61. #63491D3semidirect
67. C22×D3×C9D4semidirect
124. C22×C4×C9C2×D3semidirect
140. #3676C2×D3semidirect
143. (C2×D9)⋊C4C2×D3semidirect
146. Dic9⋊C4C2×D3semidirect
172. C22×D12D9semidirect
269. C23×D3C2×D9semidirect
275. C22×C3×C4C2×D9semidirect
295. C2×D12C2×D9semidirect
382. C3×C9#534semidirect
405. C22×D3#611semidirect
481. C22×C4D3×D9semidirect
545. C9#5653semidirect
617. C3#10589906semidirect

Maximal subgroups

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

subgroupindex
C22×C9×D122
C2×D3×#6112
#117218182
C3×#634912
#117212872
#117212862
D3×#36762
#117210502
#117210172
#117210162
#117210152
#105950623
#105899063