Finite Group Databasebeta

← #13152015

Subgroups — #13152015

Order 1728.

1132 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×C3×#611C211yesmaximal, characteristic, complemented
2C2×#88662C211yesmaximal, characteristic, complemented
3#11721874C211yesmaximal, characteristic, complemented
4C9×#953C211yesmaximal, characteristic, complemented
5#11721289C212yesmaximal, complemented
6#11721285C211yesmaximal, characteristic, complemented
7#11721281C212yesmaximal, complemented
8#11721276C211yesmaximal, characteristic, complemented
9#11721257C212yesmaximal, complemented
10#11721256C212yesmaximal, complemented
11#11721245C211yesmaximal, characteristic, complemented
12#10595513—31nomaximal
13#10589694—31nomaximal
14C24×C3×C9C2211yescharacteristic
15C23×C3×D9C2211yescharacteristic
16C2×#6233C2211yescharacteristic
17C2×#6233C2212yescomplemented
18C2×C9×#67C2211yescharacteristic
19C2×C9×#67C2212yescomplemented
20C2×C3×#611—24no—
21C2×C3×#611C2212yes—
22C2×C3×#611C2212yes—
23C22×C3×Dic9C2211yescharacteristic
24C2×#6195C2211yescharacteristic, complemented
25C2×#6194C2212yescomplemented
26#88662—24no—
27C2×D3×Dic9C2211yescharacteristic, complemented
28C2×#6192C2212yescomplemented
29C2×(C3⋊D9)⋊C4C2211yescharacteristic, complemented
30C2×#6189C2211yescharacteristic, complemented
31#88551C2211yescharacteristic
32#88551C2212yescomplemented
33C9×#217C2211yescharacteristic
34C9×#217C2212yescomplemented
35(C2×D9)⋊Dic3C2212yescomplemented
36(C2×D9)⋊Dic3C2212yescomplemented
37Dic9⋊Dic3C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C22×C3×#67—31no—
40C2×#3804—31no—
41#64162—31no—
42C3×#953D311yescharacteristic
43#63979—32no—
44#63977—31no—
45#63976—31no—
46#63970—32no—
47#63959—32no—
48#63958—32no—
49#63955—31no—
50C22×#611D311yescharacteristic, complemented
51C2×#3699—31no—
52C9×#120—31no—
53#63515—31no—
54#63515—61no—
55#63502—32no—
56#63501—32no—
57#63499—31no—
58#63447—32no—
59#63446—32no—
60#63444—31no—
61#63439—31no—
62C23×C3×C9—22no—
63C23×C3×C9—42no—
64C23×C3×C9C2312yes—
65C23×C3×C9C2311yescharacteristic
66C22×C3⋊D9C2311yescharacteristic
67C22×D3×C9C2311yescharacteristic
68C22×C3×D9—22no—
69C22×C3×D9—42no—
70C22×C3×D9C2312yes—
71#6233—22no—
72#6233—42no—
73C2×C3⋊Dic9C2311yescharacteristic
74C2×C3⋊Dic9C2312yes—
75C9×#67—22no—
76C9×#67—42no—
77C3×#611—24no—
78C3×#611—24no—
79C2×C9×Dic3C2311yescharacteristic
80C2×C9×Dic3C2312yes—
81C2×C3×Dic9D414yescomplemented
82C2×C3×Dic9C2311yescharacteristic
83C2×C3×Dic9C2311yescharacteristic
84#6195—22no—
85#6194—42no—
86D3×Dic9—22no—
87#6192—42no—
88(C3⋊D9)⋊C4—22no—
89#6189—22no—
90#5805—91no—
91#5592—91no—
92C24×C32C2×D311yescharacteristic
93C23×C3×D3—31no—
94C2×#638—31no—
95C2×#638—32no—
96C2×C3×#67—32no—
97C2×C3×#67—32no—
98C2×C3×#67—64no—
99C2×C3×#67C2×D311yescharacteristic
100C2×C3×#67C2×D312yes—
101C22×C3×Dic3—31no—
102C2×#627—31no—
103C2×#626—31no—
104C2×#626—32no—
105C2×#625—32no—
106C2×#624—31no—
107#3804—64no—
108C2×D3×Dic3—31no—
109C24×C9—21no—
110C24×C9C2×D311yescharacteristic
111C23×D9C2×D311yescharacteristic
112#3757—31no—
113#3757—32no—
114C3×#217C2×D311yescharacteristic
115C3×#217C2×D312yes—
116Dic3⋊Dic3—32no—
117(C2×D3)⋊Dic3—32no—
118(C2×D3)⋊Dic3—32no—
119Dic32—32no—
120C2×D4×C9—31no—
121C2×D4×C9—32no—
122C2×#611—24no—
123C2×#611—31no—
124C2×#611—32no—
125C2×#611—32no—
126C2×#611—31no—
127C2×#611—61no—
128C2×#611—62no—
129C2×#611C2×D312yes—
130C2×#611C2×D312yes—
131C22×Dic9—31no—
132C22×Dic9C2×D311yescharacteristic
133#3699—64no—
134C2×D36—32no—
135C2×C4×D9—31no—
136C2×Dic18—31no—
137C9×C22⋊C4—31no—
138C9×C22⋊C4—32no—
139#3676—31no—
140#3676—32no—
141#3676—61no—
142#3676—62no—
143(C2×D9)⋊C4—32no—
144(C2×D9)⋊C4—32no—
145Dic9⋊C4—32no—
146C4×Dic9—32no—
147C22×C3×C9—22no—
148C22×C3×C9—22no—
149C22×C3×C9—22no—
150C22×C3×C9—22no—
151C22×C3×C9—44no—
152C22×C3×C9C2×D412yes—
153C22×C3×C9C2411yescharacteristic
154C2×C3⋊D9—21no—
155C2×C3⋊D9—22no—
156C2×D3×C9—21no—
157C2×D3×C9—22no—
158C2×C3×D9—22no—
159C2×C3×D9—22no—
160C2×C3×D9—22no—
161C2×C3×D9—44no—
162C3⋊Dic9—21no—
163C3⋊Dic9—22no—
164C9×Dic3—21no—
165C9×Dic3—22no—
166C3×Dic9C2×D412yes—
167C3×Dic9C2×D412yes—
168C22×C3×D4—91no—
169C22×#67—31no—
170C2×#237—91no—
171C2×#235—91no—
172C3×#120—31no—
173#953—91no—
174#953—91no—
175#953—91no—
176#953—91no—
177#953—181no—
178#953D911yescharacteristic, complemented
179#940—92no—
180#940—92no—
181#939—92no—
182#938—92no—
183#938—92no—
184#937—91no—
185#936—92no—
186#926—91no—
187#885—92no—
188#884—92no—
189#882—91no—
190#877—91no—
191C23×C32—22no—
192C23×C32—42no—
193C23×C32C22×D312yes—
194C23×C32C22×D311yescharacteristic
195C22×C3⋊D3—31no—
196C22×C3×D3—32no—
197C22×C3×D3—62no—
198C22×C3×D3—122no—
199C22×C3×D3C22×D311yescharacteristic
200#638—62no—
201#638—122no—
202C2×C3⋊Dic3—32no—
203C2×C3⋊Dic3—31no—
204C3×#67—22no—
205C3×#67—42no—
206C3×#67—64no—
207C3×#67—64no—
208C2×C3×Dic3—34no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3C22×D312yes—
212C2×C3×Dic3C22×D311yescharacteristic
213#627—62no—
214#626—62no—
215#626—122no—
216#625—122no—
217#624—62no—
218D3×Dic3—62no—
219C23×C9—22no—
220C23×C9—22no—
221C23×C9—21no—
222C23×C9—31no—
223C23×C9—42no—
224C23×C9—42no—
225C23×C9—44no—
226C23×C9C22×D312yes—
227C23×C9C22×D311yescharacteristic
228C22×D9—22no—
229C22×D9—31no—
230C22×D9—42no—
231C22×D9—61no—
232C22×D9C22×D312yes—
233D4×C9—62no—
234D4×C9—122no—
235C2×C4×C9—32no—
236C2×C4×C9—31no—
237#611—24no—
238#611—24no—
239#611—62no—
240#611—62no—
241#611—122no—
242#611—122no—
243#611—122no—
244#611—124no—
245C2×Dic9—32no—
246C2×Dic9—31no—
247C2×Dic9—62no—
248C2×Dic9—61no—
249C2×Dic9—62no—
250C2×Dic9#6714yescomplemented
251C2×Dic9C22×D311yescharacteristic
252C2×Dic9C22×D311yescharacteristic
253D36—122no—
254C4×D9—62no—
255Dic18—62no—
256#535—271noSylow
257C2×C3×C9—22no—
258C2×C3×C9—42no—
259C2×C3×C9C22×D411yescharacteristic
260C2×C3×C9H(5,𝔽2)12yes—
261C3⋊D9—41no—
262D3×C9—41no—
263C3×D9—42no—
264C24×C3—21no—
265C24×C3—91no—
266C24×C3C2×D911yescharacteristic
267C24×C3D3211yescharacteristic
268C23×D3—31no—
269C2×C3×D4—31no—
270C2×C3×D4—32no—
271C2×C3×D4—92no—
272C2×C3×D4—92no—
273C2×C3×D4—184no—
274C22×C3×C4—91no—
275C2×#67—32no—
276C2×#67—32no—
277C2×#67—64no—
278C2×#67—91no—
279C2×#67—91no—
280C2×#67—91no—
281C2×#67—92no—
282C2×#67—92no—
283C2×#67—92no—
284C2×#67—92no—
285C2×#67—92no—
286C2×#67—181no—
287C2×#67—182no—
288C2×#67C2×D911yescharacteristic
289C2×#67C2×D912yescomplemented
290C22×Dic3—31no—
291C22×Dic3—91no—
292#237—184no—
293#235—184no—
294C2×D12—91no—
295C2×D12—92no—
296C2×C4×D3—91no—
297C2×C4×D3—91no—
298C2×C4×D3—91no—
299C2×Dic6—91no—
300C2×Dic6—91no—
301C3×C22⋊C4—31no—
302C3×C22⋊C4—32no—
303#217—91no—
304#217—92no—
305#217—91no—
306#217—92no—
307#217—92no—
308#217—92no—
309#217—181no—
310#217—182no—
311#217C2×D911yescharacteristic
312#217C2×D912yescomplemented
313(C2×D3)⋊C4—92no—
314(C2×D3)⋊C4—92no—
315C4⋊Dic3—92no—
316Dic3⋊C4—92no—
317C4×Dic3—92no—
318C4×Dic3—92no—
319C22×C32—22no—
320C22×C32—22no—
321C22×C32—22no—
322C22×C32—22no—
323C22×C32—44no—
324C22×C32D3×D412yes—
325C22×C32C23×D311yescharacteristic
326C2×C3⋊D3—61no—
327C2×C3⋊D3—62no—
328C2×C3×D3—21no—
329C2×C3×D3—22no—
330C2×C3×D3—62no—
331C2×C3×D3—62no—
332C2×C3×D3—62no—
333C2×C3×D3—124no—
334C3⋊Dic3—61no—
335C3⋊Dic3—62no—
336C3×Dic3—21no—
337C3×Dic3—22no—
338C3×Dic3—32no—
339C3×Dic3—32no—
340C22×C9—22no—
341C22×C9—24no—
342C22×C9—22no—
343C22×C9—22no—
344C22×C9—22no—
345C22×C9—21no—
346C22×C9—22no—
347C22×C9—42no—
348C22×C9—42no—
349C22×C9—42no—
350C22×C9—44no—
351C22×C9—48no—
352C22×C9—62no—
353C22×C9—61no—
354C22×C9C2×#6712yes—
355C22×C9C23×D311yescharacteristic
356C2×D9—22no—
357C2×D9—22no—
358C2×D9—22no—
359C2×D9—44no—
360C2×D9—62no—
361C2×D9—61no—
362C2×D9—121no—
363C2×D9—122no—
364C4×C9—62no—
365C4×C9—61no—
366Dic9—62no—
367Dic9—61no—
368Dic9—121no—
369Dic9—122no—
370Dic9C2×#6712yes—
371Dic9C2×#6712yes—
372C2×#41—271no—
373C22×D4—91no—
374#127—272no—
375#124—272no—
376#121—272no—
377#121—272no—
378#120—31no—
379#120—271no—
380#120—271no—
381#120—271no—
382#117—271no—
383C3×C9#53511yescharacteristic, complemented, Sylow
384C23×C3—22no—
385C23×C3—22no—
386C23×C3—22no—
387C23×C3—21no—
388C23×C3—31no—
389C23×C3—42no—
390C23×C3—42no—
391C23×C3—42no—
392C23×C3—44no—
393C23×C3—92no—
394C23×C3—182no—
395C23×C3—362no—
396C23×C3C22×D912yes—
397C23×C3C22×D911yescharacteristic
398C23×C3C2×D3212yes—
399C23×C3C2×D3211yescharacteristic
400C22×D3—32no—
401C22×D3—62no—
402C22×D3—91no—
403C22×D3—91no—
404C22×D3—122no—
405C22×D3—181no—
406C22×D3C22×D911yescharacteristic
407C3×D4—62no—
408C3×D4—122no—
409C3×D4—184no—
410C3×D4—184no—
411C2×C3×C4—32no—
412C2×C3×C4—31no—
413C2×C3×C4—94no—
414C2×C3×C4—91no—
415C2×C3×C4—91no—
416#67—22no—
417#67—42no—
418#67—64no—
419#67—64no—
420#67—182no—
421#67—182no—
422#67—182no—
423#67—362no—
424#67—362no—
425#67—362no—
426#67—362no—
427#67—362no—
428#67—362no—
429#67—364no—
430C2×Dic3—31no—
431C2×Dic3—31no—
432C2×Dic3—34no—
433C2×Dic3—92no—
434C2×Dic3—91no—
435C2×Dic3—92no—
436C2×Dic3—91no—
437C2×Dic3—182no—
438C2×Dic3—181no—
439C2×Dic3—182no—
440C2×Dic3C22×D912yes—
441C2×Dic3C22×D911yescharacteristic
442D12—182no—
443D12—362no—
444C4×D3—182no—
445C4×D3—182no—
446C4×D3—182no—
447Dic6—182no—
448Dic6—182no—
449C2×C32—22no—
450C2×C32—42no—
451C2×C32C2×D3×D411yescharacteristic
452C2×C32#100412yes—
453C3⋊D3—121no—
454C3×D3—41no—
455C3×D3—122no—
456C2×C9—22no—
457C2×C9—21no—
458C2×C9—22no—
459C2×C9—42no—
460C2×C9—44no—
461C2×C9—42no—
462C2×C9—121no—
463C2×C9#100412yes—
464C2×C9C22×#6711yescharacteristic
465D9—42no—
466D9—121no—
467D9—241no—
468C24—91no—
469C24D3×D911yescharacteristic
470#41—544no—
471C2×Q8—271no—
472C2×D4—31no—
473C2×D4—32no—
474C2×D4—92no—
475C2×D4—92no—
476C2×D4—184no—
477C2×D4—272no—
478C2×D4—271no—
479C2×D4—271no—
480C2×D4—272no—
481C2×D4—272no—
482C22×C4—91no—
483C22×C4—271no—
484C22×C4—271no—
485C4⋊C4—272no—
486C22⋊C4—31no—
487C22⋊C4—32no—
488C22⋊C4—272no—
489C22⋊C4—272no—
490C22⋊C4—271no—
491C22⋊C4—272no—
492C42—272no—
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—24no—
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—22no—
501C22×C3—22no—
502C22×C3—21no—
503C22×C3—22no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—44no—
508C22×C3—44no—
509C22×C3—48no—
510C22×C3—61no—
511C22×C3—62no—
512C22×C3—182no—
513C22×C3—182no—
514C22×C3—182no—
515C22×C3—364no—
516C22×C3D4×D912yes—
517C22×C3C23×D911yescharacteristic
518C22×C3D3×#6712yes—
519C22×C3C22×D3211yescharacteristic
520C2×D3—21no—
521C2×D3—22no—
522C2×D3—62no—
523C2×D3—62no—
524C2×D3—62no—
525C2×D3—124no—
526C2×D3—181no—
527C2×D3—182no—
528C2×D3—181no—
529C2×D3—182no—
530C2×D3—361no—
531C2×D3—362no—
532C3×C4—61no—
533C3×C4—62no—
534C3×C4—92no—
535C3×C4—92no—
536Dic3—21no—
537Dic3—22no—
538Dic3—32no—
539Dic3—32no—
540Dic3—181no—
541Dic3—182no—
542Dic3—181no—
543Dic3—182no—
544Dic3—361no—
545Dic3—362no—
546C32#559211yescharacteristic
547C9—21no—
548C9#580511yescharacteristic, complemented
549C23—22no—
550C23—31no—
551C23—42no—
552C23—92no—
553C23—182no—
554C23—271no—
555C23—362no—
556C23C2×D3×D912yes—
557C23C2×D3×D911yescharacteristic
558Q8—542no—
559D4—62no—
560D4—122no—
561D4—184no—
562D4—184no—
563D4—542no—
564D4—542no—
565D4—1082no—
566D4—1082no—
567D4—1082no—
568C2×C4—32no—
569C2×C4—31no—
570C2×C4—91no—
571C2×C4—91no—
572C2×C4—94no—
573C2×C4—272no—
574C2×C4—271no—
575C2×C4—542no—
576C2×C4—542no—
577C2×C3—22no—
578C2×C3—22no—
579C2×C3—21no—
580C2×C3—22no—
581C2×C3—42no—
582C2×C3—44no—
583C2×C3—42no—
584C2×C3—42no—
585C2×C3—121no—
586C2×C3—362no—
587C2×C3C2×D4×D911yescharacteristic
588C2×C3#6371012yes—
589C2×C3C2×D3×#6711yescharacteristic
590C2×C3#6433012yes—
591D3—41no—
592D3—122no—
593D3—361no—
594D3—361no—
595D3—721no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—44no—
601C22—61no—
602C22—62no—
603C22—182no—
604C22—182no—
605C22—182no—
606C22—364no—
607C22—541no—
608C22—542no—
609C22D9×#6712yes—
610C22C22×D3×D911yescharacteristic
611C4—61no—
612C4—62no—
613C4—92no—
614C4—92no—
615C4—541no—
616C4—542no—
617C3—21no—
618C3#1058969411yescharacteristic, complemented
619C3#1059551311yescharacteristic
620C2—22no—
621C2—42no—
622C2—121no—
623C2—362no—
624C2—1081no—
625C2C2×D9×#6711yescharacteristic
626C2#1172253512yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC24×C3×C9C22
socleC22×C32C23×D3
hypercenterC24D3×D9
hypocenterC3×C9#535
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×C3×#611C2semidirect
2. C2×#88662C2semidirect
3. #11721874C2semidirect
4. C9×#953C2semidirect
5. #11721289C2semidirect
6. #11721285C2semidirect
7. #11721281C2semidirect
8. #11721276C2semidirect
9. #11721257C2semidirect
10. #11721256C2semidirect
11. #11721245C2semidirect
17. C2×#6233C22semidirect
19. C2×C9×#67C22semidirect
24. C2×#6195C22semidirect
25. C2×#6194C22semidirect
27. C2×D3×Dic9C22semidirect
28. C2×#6192C22semidirect
29. C2×(C3⋊D9)⋊C4C22semidirect
30. C2×#6189C22semidirect
32. #88551C22semidirect
34. C9×#217C22semidirect
35. (C2×D9)⋊Dic3C22semidirect
36. (C2×D9)⋊Dic3C22semidirect
37. Dic9⋊Dic3C22semidirect
38. Dic3×Dic9C22semidirect
50. C22×#611D3semidirect
81. C2×C3×Dic9D4semidirect
178. #953D9semidirect
250. C2×Dic9#67semidirect
289. C2×#67C2×D9semidirect
312. #217C2×D9semidirect
383. C3×C9#535semidirect
548. C9#5805semidirect
618. C3#10589694semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6112
C2×#886622
#117218742
C9×#9532
#117212892
#117212852
#117212812
#117212762
#117212572
#117212562
#117212452
#105955133
#105896943