Finite Group Databasebeta

← D9×#932 #13151104

Subgroups — D9×#932 #13151104

Order 1728.

962 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×D9C211yesmaximal, characteristic, complemented
2D9×#235C211yesmaximal, characteristic, complemented
3#11721833C211yesmaximal, characteristic, complemented
4C9×#932C211yesmaximal, characteristic, complemented
5#11721184C212yesmaximal, complemented
6D9×#214C212yesmaximal, complemented
7#11721176C212yesmaximal, complemented
8D9×#213C212yesmaximal, complemented
9#11720882C211yesmaximal, characteristic, complemented
10#11720868C211yesmaximal, characteristic, complemented
11D9×#208C211yesmaximal, characteristic, complemented
12D3×#932—31nomaximal
13D9×AGL(1,ℤ/8)—31nomaximal
14C23×C3×D9—21no—
15C2×C3×D4×C9C2211yescharacteristic, complemented
16#88736C2211yescharacteristic, complemented
17C2×C3×#611—21no—
18C3×D4×D9—21no—
19C3×D4×D9—21no—
20C3×D4×D9C2212yes—
21C3×D4×D9C2212yescomplemented
22C9×#235C2211yescharacteristic, complemented
23C2×C3×D36C2211yescharacteristic, complemented
24C2×C3×C4×D9C2211yescharacteristic
25D9×#67—21no—
26#88662—21no—
27D9×D12C2211yescharacteristic, complemented
28C4×D3×D9—21no—
29#88640—21no—
30#88638C2211yescharacteristic, complemented
31#88635C2211yescharacteristic, complemented
32D9×Dic6C2211yescharacteristic, complemented
33#88544C2212yescomplemented
34#88543C2212yescomplemented
35#88532C2211yescharacteristic, complemented
36C9×#214C2212yescomplemented
37C9×#213C2212yescomplemented
38C9×#208C2211yescharacteristic, complemented
39#88425C2212yescomplemented
40#88421C2212yescomplemented
41#88417C2212yescomplemented
42#88415C2212yescomplemented
43#88413C2212yescomplemented
44D9×C3⋊C8C2212yescomplemented
45C2×C3×D3×D4—31no—
46D3×#235—31no—
47#64141—31no—
48C3×#932D311yescharacteristic
49#63934—32no—
50#63932—32no—
51D3×#214—32no—
52D3×#213—32no—
53#63830—31no—
54#63823—31no—
55D3×#208—31no—
56#41×D9—31no—
57C2×D4×D9D311yescharacteristic, complemented
58C9×AGL(1,ℤ/8)—31no—
59#63512—31no—
60#63495—31no—
61#63495—61no—
62#63476—32no—
63SD16×D9—32no—
64#63473—32no—
65D8×D9—32no—
66#63470—31no—
67M16×D9—31no—
68C23×C3×C9—21no—
69C22×C3×D9—21no—
70C22×C3×D9—22no—
71C22×C3×D9—22no—
72C22×C3×D9—44no—
73C22×C3×D9D411yescharacteristic
74C2×D3×D9—21no—
75C3×D4×C9—21no—
76C3×D4×C9C2312yes—
77C2×C3×C4×C9C2311yescharacteristic
78#6233—21no—
79C4⋊(C3⋊D9)C2311yescharacteristic, complemented
80C4×C3⋊D9—21no—
81#6226C2311yescharacteristic, complemented
82C9×#67—21no—
83C3×#611—22no—
84C3×#611—22no—
85C2×C3×Dic9D411yescharacteristic
86C9×D12C2311yescharacteristic, complemented
87C4×D3×C9—21no—
88C3×D36—21no—
89C3×D36C2312yes—
90C3×C4×D9D411yescharacteristic
91C3×C4×D9D411yescharacteristic
92C3×C4×D9C2311yescharacteristic
93C3×C4×D9C2311yescharacteristic
94C9×Dic6C2311yescharacteristic, complemented
95#6195—21no—
96#6194—21no—
97D3×Dic9—21no—
98#6192—21no—
99(C3⋊D9)⋊C4—21no—
100Dic3×D9—21no—
101#6189—21no—
102C3⋊(C9⋊C8)C2312yescomplemented
103C9×C3⋊C8C2312yescomplemented
104C2×#932—91no—
105D3×AGL(1,ℤ/8)—91no—
106C23×C3×D3—61no—
107C2×C32×D4C2×D311yescharacteristic
108#3828—31no—
109C2×C3×#67—61no—
110C3×D3×D4—32no—
111C3×D3×D4—32no—
112C3×D3×D4—61no—
113C3×D3×D4—61no—
114C3×#235C2×D311yescharacteristic
115C2×C3×D12—31no—
116C2×C3×C4×D3—31no—
117D3×#67—61no—
118#3804—61no—
119D3×D12—31no—
120C4×D32—61no—
121#3799—31no—
122#3798—61no—
123#3795—31no—
124D3×Dic6—31no—
125C23×D9—21no—
126#3754—32no—
127#3753—32no—
128#3748—31no—
129C3×#214C2×D312yes—
130C3×#213C2×D312yes—
131C3×#208C2×D311yescharacteristic
132#3716—32no—
133#3715—32no—
134#3714—32no—
135#3713—32no—
136#3711—32no—
137D3×C3⋊C8—32no—
138C9×#41—31no—
139C2×D4×C9—21no—
140C2×D4×C9C2×D311yescharacteristic, complemented
141C2×#611—21no—
142#3701—31no—
143Q8×D9—31no—
144#3699—31no—
145#3699—61no—
146D4×D9—21no—
147D4×D9—21no—
148D4×D9—31no—
149D4×D9—61no—
150D4×D9C2×D312yes—
151D4×D9C2×D312yescomplemented
152#3697—31no—
153#3697—61no—
154#3697—61no—
155C2×D36C2×D311yescharacteristic, complemented
156C2×C4×D9—61no—
157C2×C4×D9C2×D311yescharacteristic
158C9×SD16—32no—
159C9×D8—32no—
160C9×M16—31no—
161#3675—32no—
162#3673—32no—
163#3673—32no—
164#3673—62no—
165#3672—32no—
166#3672—62no—
167#3667—31no—
168#3667—61no—
169D72—32no—
170#3664—32no—
171#3663—32no—
172C8×D9—32no—
173C22×C3×C9—22no—
174C22×C3×C9—42no—
175C22×C3×C9C2×D411yescharacteristic
176C2×C3⋊D9—21no—
177C2×D3×C9—21no—
178C2×C3×D9—22no—
179C2×C3×D9—22no—
180C2×C3×D9—44no—
181C2×C3×D9—42no—
182C2×C3×D9—44no—
183C2×C3×D9C2×D411yescharacteristic
184C2×C3×D9C2×D411yescharacteristic
185D3×D9—42no—
186C3×C4×C9C2×D411yescharacteristic
187C3×C4×C9C2411yescharacteristic
188C3⋊Dic9—21no—
189C9×Dic3—21no—
190C3×Dic9C2×D411yescharacteristic
191C3×Dic9C2×D411yescharacteristic
192C22×C3×D4—91no—
193D3×#41—91no—
194C2×D3×D4—31no—
195C2×#235—91no—
196C3×AGL(1,ℤ/8)—31no—
197#949—91no—
198C2×#214—92no—
199#932—91no—
200#932—91no—
201#932—91no—
202#932—92no—
203#932—92no—
204#932—91no—
205#932—181no—
206#932D911yescharacteristic, direct product factor
207C2×#213—92no—
208C2×#208—91no—
209#914—92no—
210D3×SD16—92no—
211#911—92no—
212D3×D8—92no—
213#908—91no—
214D3×M16—91no—
215C23×C32—21no—
216C22×C3×D3—31no—
217C22×C3×D3—62no—
218C22×C3×D3—62no—
219C22×C3×D3—61no—
220C22×C3×D3—124no—
221C2×D32—61no—
222C32×D4—21no—
223C32×D4C22×D312yes—
224C2×C32×C4C22×D311yescharacteristic
225#638—61no—
226C4⋊(C3⋊D3)—31no—
227C4×C3⋊D3—61no—
228#634—31no—
229C3×#67—21no—
230C3×#67—62no—
231C3×#67—62no—
232C2×C3×Dic3—31no—
233C3×D12—32no—
234C3×D12—61no—
235C3×D12C22×D311yescharacteristic
236C3×C4×D3—21no—
237C3×C4×D3—31no—
238C3×C4×D3—31no—
239C3×C4×D3—31no—
240C3×C4×D3—31no—
241C3×Dic6C22×D311yescharacteristic
242#627—61no—
243#626—61no—
244#626—61no—
245#625—61no—
246#624—61no—
247D3×Dic3—61no—
248D3×Dic3—61no—
249C23×C9—21no—
250C23×C9—41no—
251C22×D9—21no—
252C22×D9—22no—
253C22×D9—22no—
254C22×D9—44no—
255C22×D9—61no—
256C22×D9#6711yescharacteristic
257C3⋊(C3⋊C8)—32no—
258C3×C3⋊C8C22×D312yes—
259Q8×C9—31no—
260D4×C9—21no—
261D4×C9—22no—
262D4×C9—31no—
263D4×C9—41no—
264D4×C9—61no—
265D4×C9C22×D312yes—
266C2×C4×C9—21no—
267C2×C4×C9—61no—
268C2×C4×C9C22×D311yescharacteristic
269#611—22no—
270#611—22no—
271#611—61no—
272#611—61no—
273#611—61no—
274#611—121no—
275C2×Dic9—61no—
276C2×Dic9#6711yescharacteristic
277D36—21no—
278D36—31no—
279D36—61no—
280D36—61no—
281D36C22×D312yes—
282C4×D9—61no—
283C4×D9—61no—
284C4×D9—61no—
285C4×D9—121no—
286C4×D9#6711yescharacteristic
287C4×D9#6711yescharacteristic
288C4×D9C22×D311yescharacteristic
289C4×D9C22×D311yescharacteristic
290Dic18—31no—
291Dic18—61no—
292Dic18—61no—
293C8×C9—32no—
294C9⋊C8—32no—
295C9⋊C8—62no—
296C2×AGL(1,ℤ/8)—271noSylow
297C2×C3×C9—21no—
298C2×C3×C9—42no—
299C2×C3×C9C22×D411yescharacteristic
300C3⋊D9—41no—
301D3×C9—41no—
302C3×D9—21no—
303C3×D9—42no—
304C3×D9AGL(1,ℤ/8)12yescomplemented
305C24×C3—181no—
306C23×D3—61no—
307C3×#41—31no—
308C2×C3×D4—21no—
309C2×C3×D4—92no—
310C2×C3×D4—92no—
311C2×C3×D4—91no—
312C2×C3×D4—181no—
313C2×C3×D4—181no—
314C2×C3×D4—181no—
315C2×C3×D4C2×D911yescharacteristic, complemented
316C2×C3×D4D3211yescharacteristic
317C22×C3×C4—91no—
318C2×#67—61no—
319C2×#67—181no—
320#239—91no—
321D3×Q8—91no—
322#237—91no—
323#237—181no—
324D3×D4—32no—
325D3×D4—32no—
326D3×D4—61no—
327D3×D4—61no—
328D3×D4—91no—
329D3×D4—181no—
330#235—91no—
331#235—91no—
332#235—91no—
333#235—91no—
334#235—181no—
335#235—181no—
336#235—181no—
337#235—181no—
338#235C2×D911yescharacteristic, complemented
339C2×D12—31no—
340C2×D12—91no—
341C2×C4×D3—31no—
342C2×C4×D3—181no—
343C2×C4×D3—181no—
344C2×Dic6—91no—
345C3×SD16—32no—
346C3×D8—32no—
347C3×M16—31no—
348#215—92no—
349#214—92no—
350#214—92no—
351#214—92no—
352#214—92no—
353#214—182no—
354#214C2×D912yescomplemented
355#213—92no—
356#213—92no—
357#213—92no—
358#213—92no—
359#213—92no—
360#213—182no—
361#213C2×D912yescomplemented
362#208—91no—
363#208—91no—
364#208—92no—
365#208—181no—
366#208C2×D911yescharacteristic, complemented
367C2×C3⋊C8—92no—
368D24—92no—
369#204—92no—
370ΓL(1,𝔽25)—92no—
371D3×C8—92no—
372C22×C32—22no—
373C22×C32—42no—
374C22×C32D3×D411yescharacteristic
375C2×C3⋊D3—61no—
376C2×C3×D3—21no—
377C2×C3×D3—31no—
378C2×C3×D3—31no—
379C2×C3×D3—62no—
380C2×C3×D3—62no—
381C2×C3×D3—124no—
382C2×C3×D3—124no—
383C2×C3×D3—122no—
384D32—122no—
385C32×C4D3×D411yescharacteristic
386C32×C4C23×D311yescharacteristic
387C3⋊Dic3—61no—
388C3×Dic3—21no—
389C3×Dic3—31no—
390C3×Dic3—31no—
391C22×C9—22no—
392C22×C9—21no—
393C22×C9—42no—
394C22×C9—42no—
395C22×C9—61no—
396C22×C9—82no—
397C22×C9C2×#6711yescharacteristic
398C2×D9—22no—
399C2×D9—22no—
400C2×D9—44no—
401C2×D9—42no—
402C2×D9—44no—
403C2×D9—61no—
404C2×D9—121no—
405C2×D9—122no—
406C2×D9C2×#6711yescharacteristic
407C2×D9C2×#6711yescharacteristic
408C4×C9—21no—
409C4×C9—21no—
410C4×C9—61no—
411C4×C9C2×#6711yescharacteristic
412C4×C9C23×D311yescharacteristic
413Dic9—61no—
414Dic9—121no—
415Dic9C2×#6711yescharacteristic
416Dic9C2×#6711yescharacteristic
417C2×#41—271no—
418C22×D4—91no—
419AGL(1,ℤ/8)—31no—
420AGL(1,ℤ/8)—271no—
421AGL(1,ℤ/8)—272no—
422AGL(1,ℤ/8)—272no—
423AGL(1,ℤ/8)—271no—
424AGL(1,ℤ/8)—271no—
425C2×SD16—272no—
426C2×D8—272no—
427C2×M16—271no—
428C3×C9C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
429C23×C3—21no—
430C23×C3—21no—
431C23×C3—41no—
432C23×C3—91no—
433C23×C3—182no—
434C23×C3—182no—
435C23×C3—181no—
436C23×C3—364no—
437C22×D3—31no—
438C22×D3—62no—
439C22×D3—62no—
440C22×D3—61no—
441C22×D3—124no—
442C22×D3—181no—
443C22×D3—181no—
444C3×Q8—31no—
445C3×D4—21no—
446C3×D4—21no—
447C3×D4—22no—
448C3×D4—31no—
449C3×D4—41no—
450C3×D4—61no—
451C3×D4—92no—
452C3×D4—182no—
453C3×D4—182no—
454C3×D4—181no—
455C3×D4C22×D912yes—
456C3×D4C2×D3212yes—
457C2×C3×C4—21no—
458C2×C3×C4—61no—
459C2×C3×C4—91no—
460C2×C3×C4—91no—
461C2×C3×C4—91no—
462C2×C3×C4—91no—
463C2×C3×C4—91no—
464C2×C3×C4C22×D911yescharacteristic
465C2×C3×C4C2×D3211yescharacteristic
466#67—21no—
467#67—62no—
468#67—62no—
469#67—181no—
470#67—181no—
471#67—181no—
472#67—181no—
473#67—181no—
474#67—181no—
475#67—361no—
476C2×Dic3—31no—
477C2×Dic3—181no—
478C2×Dic3—181no—
479D12—32no—
480D12—61no—
481D12—91no—
482D12—91no—
483D12—181no—
484D12—181no—
485D12—181no—
486D12C22×D911yescharacteristic, complemented
487C4×D3—21no—
488C4×D3—31no—
489C4×D3—31no—
490C4×D3—31no—
491C4×D3—31no—
492C4×D3—181no—
493C4×D3—181no—
494C4×D3—181no—
495C4×D3—181no—
496C4×D3—181no—
497C4×D3—181no—
498C4×D3—361no—
499Dic6—91no—
500Dic6—91no—
501Dic6—181no—
502Dic6—181no—
503Dic6—181no—
504Dic6C22×D911yescharacteristic, complemented
505C3×C8—32no—
506C3⋊C8—92no—
507C3⋊C8—92no—
508C3⋊C8—182no—
509C3⋊C8C22×D912yescomplemented
510C2×C32—21no—
511C2×C32—42no—
512C2×C32C2×D3×D411yescharacteristic
513C3⋊D3—121no—
514C3×D3—32no—
515C3×D3—41no—
516C3×D3—61no—
517C3×D3—122no—
518C2×C9—21no—
519C2×C9—21no—
520C2×C9—41no—
521C2×C9—42no—
522C2×C9—82no—
523C2×C9—121no—
524C2×C9C22×#6711yescharacteristic
525D9—21no—
526D9—42no—
527D9—121no—
528D9—241no—
529D9#93212yesdirect product factor
530C24—181no—
531#41—31no—
532#41—271no—
533#41—271no—
534#41—271no—
535#41—541no—
536#41—541no—
537C2×Q8—271no—
538C2×D4—92no—
539C2×D4—92no—
540C2×D4—91no—
541C2×D4—181no—
542C2×D4—181no—
543C2×D4—181no—
544C2×D4—271no—
545C2×D4—541no—
546C2×D4D3×D911yescharacteristic, complemented
547C22×C4—91no—
548C22×C4—541no—
549SD16—32no—
550SD16—272no—
551SD16—272no—
552SD16—272no—
553D8—32no—
554D8—272no—
555D8—272no—
556D8—272no—
557M16—31no—
558M16—272no—
559M16—271no—
560C2×C8—272no—
561C22×C3—22no—
562C22×C3—22no—
563C22×C3—21no—
564C22×C3—42no—
565C22×C3—42no—
566C22×C3—42no—
567C22×C3—61no—
568C22×C3—82no—
569C22×C3—91no—
570C22×C3—91no—
571C22×C3—182no—
572C22×C3—182no—
573C22×C3—362no—
574C22×C3—364no—
575C22×C3—364no—
576C22×C3D4×D911yescharacteristic
577C22×C3D3×#6711yescharacteristic
578C2×D3—21no—
579C2×D3—31no—
580C2×D3—31no—
581C2×D3—62no—
582C2×D3—62no—
583C2×D3—124no—
584C2×D3—124no—
585C2×D3—122no—
586C2×D3—181no—
587C2×D3—181no—
588C2×D3—361no—
589C2×D3—362no—
590C2×D3—362no—
591C3×C4—21no—
592C3×C4—21no—
593C3×C4—61no—
594C3×C4—91no—
595C3×C4—91no—
596C3×C4D4×D911yescharacteristic
597C3×C4C23×D911yescharacteristic
598C3×C4D3×#6711yescharacteristic
599C3×C4C22×D3211yescharacteristic
600Dic3—21no—
601Dic3—31no—
602Dic3—31no—
603Dic3—181no—
604Dic3—181no—
605Dic3—361no—
606C32D3×AGL(1,ℤ/8)11yescharacteristic
607C9—21no—
608C9C2×#93211yescharacteristic, complemented
609C23—21no—
610C23—91no—
611C23—182no—
612C23—182no—
613C23—181no—
614C23—364no—
615C23—541no—
616Q8—31no—
617Q8—271no—
618Q8—541no—
619D4—21no—
620D4—31no—
621D4—61no—
622D4—92no—
623D4—182no—
624D4—182no—
625D4—181no—
626D4—271no—
627D4—541no—
628D4—541no—
629D4—541no—
630D4—541no—
631D4C2×D3×D912yes—
632C2×C4—61no—
633C2×C4—91no—
634C2×C4—91no—
635C2×C4—91no—
636C2×C4—91no—
637C2×C4—91no—
638C2×C4—541no—
639C2×C4—541no—
640C2×C4—541no—
641C2×C4—541no—
642C2×C4C2×D3×D911yescharacteristic
643C8—32no—
644C8—272no—
645C2×C3—21no—
646C2×C3—21no—
647C2×C3—21no—
648C2×C3—41no—
649C2×C3—42no—
650C2×C3—42no—
651C2×C3—82no—
652C2×C3—92no—
653C2×C3—121no—
654C2×C3—181no—
655C2×C3—362no—
656C2×C3C2×D4×D911yescharacteristic
657C2×C3C2×D3×#6711yescharacteristic
658D3—32no—
659D3—41no—
660D3—61no—
661D3—122no—
662D3—361no—
663D3—361no—
664D3—721no—
665C22—22no—
666C22—42no—
667C22—61no—
668C22—91no—
669C22—91no—
670C22—182no—
671C22—182no—
672C22—362no—
673C22—364no—
674C22—364no—
675C22—541no—
676C22—1082no—
677C22D9×#6711yescharacteristic
678C4—61no—
679C4—91no—
680C4—91no—
681C4—541no—
682C4D9×#6711yescharacteristic
683C4C22×D3×D911yescharacteristic
684C3—21no—
685C3D9×AGL(1,ℤ/8)11yescharacteristic, complemented
686C3D3×#93211yescharacteristic
687C2—21no—
688C2—42no—
689C2—92no—
690C2—121no—
691C2—181no—
692C2—362no—
693C2—1081no—
694C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC2C2×D9×#67
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC2×C3×D4×C9C22
socleC2×C32C2×D3×D4
hypercenterC2×D4D3×D9
hypocenterC3×C9C2×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×D9C2semidirect
2. D9×#235C2semidirect
3. #11721833C2semidirect
4. C9×#932C2semidirect
5. #11721184C2semidirect
6. D9×#214C2semidirect
7. #11721176C2semidirect
8. D9×#213C2semidirect
9. #11720882C2semidirect
10. #11720868C2semidirect
11. D9×#208C2semidirect
15. C2×C3×D4×C9C22semidirect
16. #88736C22semidirect
21. C3×D4×D9C22semidirect
22. C9×#235C22semidirect
23. C2×C3×D36C22semidirect
27. D9×D12C22semidirect
30. #88638C22semidirect
31. #88635C22semidirect
32. D9×Dic6C22semidirect
33. #88544C22semidirect
34. #88543C22semidirect
35. #88532C22semidirect
36. C9×#214C22semidirect
37. C9×#213C22semidirect
38. C9×#208C22semidirect
39. #88425C22semidirect
40. #88421C22semidirect
41. #88417C22semidirect
42. #88415C22semidirect
43. #88413C22semidirect
44. D9×C3⋊C8C22semidirect
57. C2×D4×D9D3semidirect
79. C4⋊(C3⋊D9)C23semidirect
81. #6226C23semidirect
86. C9×D12C23semidirect
94. C9×Dic6C23semidirect
102. C3⋊(C9⋊C8)C23semidirect
103. C9×C3⋊C8C23semidirect
140. C2×D4×C9C2×D3semidirect
151. D4×D9C2×D3semidirect
155. C2×D36C2×D3semidirect
206. #932D9direct (N × H)
304. C3×D9AGL(1,ℤ/8)semidirect
315. C2×C3×D4C2×D9semidirect
338. #235C2×D9semidirect
354. #214C2×D9semidirect
361. #213C2×D9semidirect
366. #208C2×D9semidirect
428. C3×C9C2×AGL(1,ℤ/8)semidirect
486. D12C22×D9semidirect
504. Dic6C22×D9semidirect
509. C3⋊C8C22×D9semidirect
529. D9#932direct (N × H)
546. C2×D4D3×D9semidirect
608. C9C2×#932semidirect
685. C3D9×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×D92
D9×#2352
#117218332
C9×#9322
#117211842
D9×#2142
#117211762
D9×#2132
#117208822
#117208682
D9×#2082
D3×#9323
D9×AGL(1,ℤ/8)3