Finite Group Databasebeta

← D9×#908 #13149858

Subgroups — D9×#908 #13149858

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
1D9×#235C211yesmaximal, characteristic, complemented
2C2×D9×D12C211yesmaximal, characteristic, complemented
3#11721783C211yesmaximal, characteristic, complemented
4C9×#908C211yesmaximal, characteristic, complemented
5C3×M16×D9C211yesmaximal, characteristic, complemented
6#11720890C211yesmaximal, characteristic, complemented
7#11720885C211yesmaximal, characteristic, complemented
8#11720816C212yesmaximal, complemented
9#11720815C212yesmaximal, complemented
10D9×D24C212yesmaximal, complemented
11D9×#204C212yesmaximal, complemented
12D3×#908—31nomaximal
13D9×AGL(1,ℤ/8)—31nomaximal
14C22×D3×D9—21no—
15#88736C2211yescharacteristic, complemented
16C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
17C9×#235C2211yescharacteristic, complemented
18C2×C9×D12C2211yescharacteristic, complemented
19C2×C3×C4×D9C2211yescharacteristic
20D9×#67—21no—
21C2×#6195—21no—
22#88662—21no—
23D9×D12—21no—
24D9×D12—21no—
25D9×D12C2212yes—
26D9×D12C2212yescomplemented
27D9×D12C2211yescharacteristic, complemented
28C4×D3×D9—21no—
29#88640—21no—
30#88638C2211yescharacteristic, complemented
31#88635C2211yescharacteristic, complemented
32D9×Dic6C2211yescharacteristic, complemented
33C3×C9×M16C2211yescharacteristic, complemented
34#88525C2212yescomplemented
35#88524C2212yescomplemented
36C3×#3667C2211yescharacteristic, complemented
37C9×D24C2212yescomplemented
38C9×#204C2212yescomplemented
39C3×#3663C2212yescomplemented
40C3×C8×D9C2212yescomplemented
41#88426C2212yescomplemented
42#88424C2212yescomplemented
43#88422C2212yescomplemented
44#88422C2212yescomplemented
45D3×#235—31no—
46C2×D3×D12—31no—
47#64117—31no—
48C3×#908D311yescharacteristic
49C3×D3×M16—31no—
50#63830—31no—
51#63825—31no—
52#63795—32no—
53#63794—32no—
54D3×D24—32no—
55D3×#204—32no—
56#41×D9—31no—
57C2×D4×D9—31no—
58C9×AGL(1,ℤ/8)—31no—
59#63512—31no—
60#63495—31no—
61#63476—32no—
62SD16×D9—32no—
63#63473—32no—
64D8×D9—32no—
65#63470—31no—
66#63470—61no—
67M16×D9D311yescharacteristic, complemented
68C22×C3⋊D9—21no—
69C22×D3×C9—21no—
70C22×C3×D9D411yescharacteristic
71C2×D3×D9—22no—
72C2×D3×D9—22no—
73C2×D3×D9—21no—
74C2×D3×D9—44no—
75C2×C3×C4×C9C2311yescharacteristic
76#6233—21no—
77C4⋊(C3⋊D9)—21no—
78C4⋊(C3⋊D9)C2312yes—
79C4⋊(C3⋊D9)C2311yescharacteristic, complemented
80C4×C3⋊D9—21no—
81#6226C2311yescharacteristic, complemented
82C9×#67—21no—
83C2×C3×Dic9D411yescharacteristic
84C9×D12—21no—
85C9×D12C2312yes—
86C9×D12C2311yescharacteristic, complemented
87C4×D3×C9—21no—
88C3×C4×D9D411yescharacteristic
89C3×C4×D9D411yescharacteristic
90C3×C4×D9C2311yescharacteristic
91C3×C4×D9C2311yescharacteristic
92C9×Dic6C2311yescharacteristic, complemented
93#6195—21no—
94#6195—22no—
95#6195—22no—
96#6194—21no—
97D3×Dic9—21no—
98#6192—21no—
99(C3⋊D9)⋊C4—21no—
100Dic3×D9—21no—
101#6189—21no—
102C3×C8×C9C2312yescomplemented
103C3×C9⋊C8C2312yescomplemented
104D3×AGL(1,ℤ/8)—91no—
105C2×#908—91no—
106C22×D32—61no—
107#3828—31no—
108C2×C4⋊(C3⋊D3)—31no—
109C3×#235C2×D311yescharacteristic
110C2×C3×D12C2×D311yescharacteristic
111C2×C3×C4×D3—31no—
112D3×#67—61no—
113C2×#626—61no—
114#3804—61no—
115D3×D12—32no—
116D3×D12—32no—
117D3×D12—31no—
118D3×D12—61no—
119D3×D12—61no—
120C4×D32—61no—
121#3799—31no—
122#3798—61no—
123#3795—31no—
124D3×Dic6—31no—
125C23×D9—61no—
126C32×M16C2×D311yescharacteristic
127#3745—32no—
128#3744—32no—
129C3×#208—31no—
130C3×D24C2×D312yes—
131C3×#204C2×D312yes—
132C3×ΓL(1,𝔽25)—32no—
133C3×D3×C8—32no—
134#3717—32no—
135#3716—32no—
136#3714—32no—
137#3714—32no—
138C9×#41—31no—
139C2×D4×C9—31no—
140C2×#611—61no—
141#3701—31no—
142Q8×D9—31no—
143#3699—31no—
144#3699—61no—
145D4×D9—32no—
146D4×D9—32no—
147D4×D9—31no—
148D4×D9—61no—
149D4×D9—61no—
150D4×D9—61no—
151#3697—31no—
152#3697—61no—
153#3697—61no—
154C2×D36—31no—
155C2×D36—61no—
156C2×C4×D9—61no—
157C2×C4×D9C2×D311yescharacteristic
158C9×SD16—32no—
159C9×D8—32no—
160C9×M16—21no—
161C9×M16C2×D311yescharacteristic, complemented
162#3675—32no—
163#3673—32no—
164#3673—32no—
165#3672—32no—
166#3667C2×D311yescharacteristic, complemented
167D72—32no—
168D72—62no—
169#3664—32no—
170#3664—62no—
171#3663C2×D312yescomplemented
172C8×D9C2×D312yescomplemented
173C22×C3×C9C2×D411yescharacteristic
174C2×C3⋊D9—21no—
175C2×C3⋊D9—22no—
176C2×C3⋊D9—42no—
177C2×D3×C9—21no—
178C2×D3×C9—22no—
179C2×D3×C9—42no—
180C2×C3×D9—22no—
181C2×C3×D9C2×D411yescharacteristic
182C2×C3×D9C2×D411yescharacteristic
183D3×D9—44no—
184D3×D9—42no—
185D3×D9—44no—
186C3×C4×C9C2×D411yescharacteristic
187C3×C4×C9C2411yescharacteristic
188C3⋊Dic9—21no—
189C9×Dic3—21no—
190C3×Dic9C2×D411yescharacteristic
191C3×Dic9C2×D411yescharacteristic
192D3×#41—91no—
193C2×D3×D4—91no—
194C2×#235—91no—
195C22×D12—91no—
196C3×AGL(1,ℤ/8)—31no—
197C2×C3×M16—91no—
198#949—91no—
199#932—91no—
200#914—92no—
201D3×SD16—92no—
202#911—92no—
203D3×D8—92no—
204#908—91no—
205#908—91no—
206#908—91no—
207#908—91no—
208#908—92no—
209#908—92no—
210#908—181no—
211#908D911yescharacteristic, direct product factor
212D3×M16—31no—
213C2×D24—92no—
214C2×#204—92no—
215C22×C3⋊D3—61no—
216C22×C3×D3—21no—
217C22×C3×D3—31no—
218C2×D32—62no—
219C2×D32—62no—
220C2×D32—61no—
221C2×D32—124no—
222C2×C32×C4C22×D311yescharacteristic
223#638—61no—
224C4⋊(C3⋊D3)—32no—
225C4⋊(C3⋊D3)—31no—
226C4⋊(C3⋊D3)—61no—
227C4×C3⋊D3—61no—
228#634—31no—
229C3×#67—21no—
230C2×C3×Dic3—31no—
231C3×D12—21no—
232C3×D12C22×D312yes—
233C3×D12C22×D311yescharacteristic
234C3×C4×D3—21no—
235C3×C4×D3—31no—
236C3×C4×D3—31no—
237C3×C4×D3—31no—
238C3×C4×D3—31no—
239C3×Dic6C22×D311yescharacteristic
240#627—61no—
241#626—61no—
242#626—61no—
243#626—62no—
244#626—62no—
245#625—61no—
246#624—61no—
247D3×Dic3—61no—
248D3×Dic3—61no—
249C23×C9—61no—
250C22×D9—61no—
251C22×D9—61no—
252C22×D9—62no—
253C22×D9—62no—
254C22×D9—121no—
255C22×D9—124no—
256C22×D9D1211yescharacteristic
257C32×C8C22×D312yes—
258C3×C3⋊C8—32no—
259Q8×C9—31no—
260D4×C9—32no—
261D4×C9—31no—
262D4×C9—61no—
263D4×C9—61no—
264C2×C4×C9—21no—
265C2×C4×C9—61no—
266C2×C4×C9C22×D311yescharacteristic
267#611—61no—
268#611—61no—
269#611—61no—
270#611—62no—
271#611—62no—
272#611—121no—
273C2×Dic9—61no—
274C2×Dic9D1211yescharacteristic
275D36—32no—
276D36—31no—
277D36—62no—
278D36—61no—
279D36—61no—
280D36—61no—
281D36—121no—
282C4×D9—61no—
283C4×D9—61no—
284C4×D9—61no—
285C4×D9—121no—
286C4×D9D1211yescharacteristic
287C4×D9D1211yescharacteristic
288C4×D9C22×D311yescharacteristic
289C4×D9C22×D311yescharacteristic
290Dic18—31no—
291Dic18—61no—
292Dic18—61no—
293C8×C9—22no—
294C8×C9C22×D312yescomplemented
295C9⋊C8C22×D312yescomplemented
296C2×AGL(1,ℤ/8)—271noSylow
297C2×C3×C9—21no—
298C2×C3×C9C22×D411yescharacteristic
299C3⋊D9—42no—
300C3⋊D9—41no—
301D3×C9—42no—
302D3×C9—41no—
303C3×D9—21no—
304C3×D9AGL(1,ℤ/8)12yescomplemented
305C23×D3—181no—
306C23×D3—181no—
307C3×#41—31no—
308C2×C3×D4—31no—
309C22×C3×C4—91no—
310C2×#67—181no—
311C2×#67—181no—
312#239—91no—
313D3×Q8—91no—
314#237—91no—
315#237—181no—
316D3×D4—91no—
317D3×D4—92no—
318D3×D4—92no—
319D3×D4—181no—
320D3×D4—181no—
321D3×D4—181no—
322#235—91no—
323#235—91no—
324#235—91no—
325#235—91no—
326#235—181no—
327#235—181no—
328#235—181no—
329#235—181no—
330#235C2×D911yescharacteristic, complemented
331C2×D12—91no—
332C2×D12—91no—
333C2×D12—92no—
334C2×D12—92no—
335C2×D12—91no—
336C2×D12—181no—
337C2×D12—181no—
338C2×D12—181no—
339C2×D12—181no—
340C2×D12C2×D911yescharacteristic, complemented
341C2×C4×D3—31no—
342C2×C4×D3—181no—
343C2×C4×D3—181no—
344C2×Dic6—91no—
345C3×SD16—32no—
346C3×D8—32no—
347C3×M16—21no—
348C3×M16—91no—
349C3×M16—92no—
350C3×M16C2×D911yescharacteristic, complemented
351C3×M16D3211yescharacteristic
352C2×C3×C8—92no—
353#215—92no—
354#214—92no—
355#213—92no—
356#213—92no—
357#208—31no—
358D24—92no—
359D24—92no—
360D24—92no—
361D24—92no—
362D24—182no—
363D24C2×D912yescomplemented
364#204—92no—
365#204—92no—
366#204—92no—
367#204—92no—
368#204—182no—
369#204C2×D912yescomplemented
370ΓL(1,𝔽25)—32no—
371D3×C8—32no—
372C22×C32D3×D411yescharacteristic
373C2×C3⋊D3—61no—
374C2×C3⋊D3—62no—
375C2×C3⋊D3—122no—
376C2×C3×D3—21no—
377C2×C3×D3—22no—
378C2×C3×D3—31no—
379C2×C3×D3—31no—
380C2×C3×D3—42no—
381C2×C3×D3—62no—
382D32—122no—
383D32—124no—
384D32—124no—
385C32×C4D3×D411yescharacteristic
386C32×C4C23×D311yescharacteristic
387C3⋊Dic3—61no—
388C3×Dic3—21no—
389C3×Dic3—31no—
390C3×Dic3—31no—
391C22×C9—21no—
392C22×C9—61no—
393C22×C9—62no—
394C22×C9—122no—
395C22×C9C2×D1211yescharacteristic
396C2×D9—22no—
397C2×D9—61no—
398C2×D9—62no—
399C2×D9—121no—
400C2×D9—122no—
401C2×D9—124no—
402C2×D9—122no—
403C2×D9—122no—
404C2×D9—124no—
405C2×D9—242no—
406C2×D9C2×D1211yescharacteristic
407C2×D9C2×D1211yescharacteristic
408C4×C9—21no—
409C4×C9—21no—
410C4×C9—61no—
411C4×C9C2×D1211yescharacteristic
412C4×C9C23×D311yescharacteristic
413Dic9—61no—
414Dic9—121no—
415Dic9C2×D1211yescharacteristic
416Dic9C2×D1211yescharacteristic
417C2×#41—271no—
418C22×D4—271no—
419AGL(1,ℤ/8)—31no—
420AGL(1,ℤ/8)—272no—
421AGL(1,ℤ/8)—272no—
422AGL(1,ℤ/8)—271no—
423AGL(1,ℤ/8)—271no—
424AGL(1,ℤ/8)—271no—
425C2×SD16—272no—
426C2×D8—272no—
427C2×M16—91no—
428C3×C9C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
429C23×C3—61no—
430C23×C3—91no—
431C22×D3—21no—
432C22×D3—31no—
433C22×D3—181no—
434C22×D3—182no—
435C22×D3—182no—
436C22×D3—181no—
437C22×D3—182no—
438C22×D3—182no—
439C22×D3—181no—
440C22×D3—181no—
441C22×D3—361no—
442C22×D3—364no—
443C22×D3—364no—
444C3×Q8—31no—
445C3×D4—31no—
446C3×D4—32no—
447C3×D4—61no—
448C3×D4—61no—
449C2×C3×C4—21no—
450C2×C3×C4—61no—
451C2×C3×C4—91no—
452C2×C3×C4—91no—
453C2×C3×C4—91no—
454C2×C3×C4—91no—
455C2×C3×C4—91no—
456C2×C3×C4C22×D911yescharacteristic
457C2×C3×C4C2×D3211yescharacteristic
458#67—21no—
459#67—181no—
460#67—181no—
461#67—182no—
462#67—182no—
463#67—181no—
464#67—181no—
465#67—181no—
466#67—181no—
467#67—361no—
468C2×Dic3—31no—
469C2×Dic3—181no—
470C2×Dic3—181no—
471D12—21no—
472D12—92no—
473D12—91no—
474D12—92no—
475D12—91no—
476D12—182no—
477D12—181no—
478D12—181no—
479D12—182no—
480D12—182no—
481D12—181no—
482D12—181no—
483D12—181no—
484D12—361no—
485D12C22×D912yes—
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—22no—
506C3×C8—92no—
507C3×C8C22×D912yescomplemented
508C3×C8C2×D3212yes—
509C3⋊C8—32no—
510C2×C32—21no—
511C2×C32C2×D3×D411yescharacteristic
512C3⋊D3—122no—
513C3⋊D3—121no—
514C3×D3—32no—
515C3×D3—42no—
516C3×D3—41no—
517C3×D3—61no—
518C2×C9—21no—
519C2×C9—21no—
520C2×C9—41no—
521C2×C9—122no—
522C2×C9—121no—
523C2×C9C22×D1211yescharacteristic
524D9—21no—
525D9—121no—
526D9—122no—
527D9—241no—
528D9—242no—
529D9#90812yesdirect product factor
530C24—541no—
531#41—31no—
532#41—271no—
533#41—271no—
534#41—271no—
535#41—541no—
536#41—541no—
537C2×Q8—271no—
538C2×D4—31no—
539C2×D4—271no—
540C2×D4—272no—
541C2×D4—272no—
542C2×D4—271no—
543C2×D4—541no—
544C2×D4—541no—
545C2×D4—541no—
546C2×D4—541no—
547C22×C4—91no—
548C22×C4—541no—
549SD16—32no—
550SD16—272no—
551SD16—272no—
552SD16—272no—
553D8—32no—
554D8—272no—
555D8—272no—
556D8—272no—
557M16—91no—
558M16—92no—
559M16D3×D911yescharacteristic, complemented
560C2×C8—92no—
561C22×C3—21no—
562C22×C3—61no—
563C22×C3—62no—
564C22×C3—91no—
565C22×C3—91no—
566C22×C3—122no—
567C22×C3—182no—
568C22×C3D4×D911yescharacteristic
569C22×C3D3×D1211yescharacteristic
570C2×D3—21no—
571C2×D3—22no—
572C2×D3—31no—
573C2×D3—31no—
574C2×D3—42no—
575C2×D3—62no—
576C2×D3—181no—
577C2×D3—182no—
578C2×D3—181no—
579C2×D3—182no—
580C2×D3—361no—
581C2×D3—362no—
582C2×D3—364no—
583C2×D3—362no—
584C2×D3—364no—
585C2×D3—362no—
586C2×D3—364no—
587C2×D3—364no—
588C2×D3—362no—
589C2×D3—362no—
590C2×D3—722no—
591C3×C4—21no—
592C3×C4—21no—
593C3×C4—61no—
594C3×C4—91no—
595C3×C4—91no—
596C3×C4D4×D911yescharacteristic
597C3×C4C23×D911yescharacteristic
598C3×C4D3×D1211yescharacteristic
599C3×C4C22×D3211yescharacteristic
600Dic3—21no—
601Dic3—31no—
602Dic3—31no—
603Dic3—181no—
604Dic3—181no—
605Dic3—361no—
606C32D3×AGL(1,ℤ/8)11yescharacteristic
607C9—21no—
608C9C2×#90811yescharacteristic, complemented
609C23—61no—
610C23—91no—
611C23—541no—
612C23—542no—
613C23—542no—
614C23—541no—
615C23—1084no—
616Q8—31no—
617Q8—271no—
618Q8—541no—
619D4—31no—
620D4—32no—
621D4—61no—
622D4—61no—
623D4—271no—
624D4—272no—
625D4—541no—
626D4—542no—
627D4—542no—
628D4—541no—
629D4—541no—
630D4—541no—
631D4—541no—
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—92no—
644C8C2×D3×D912yescomplemented
645C2×C3—21no—
646C2×C3—21no—
647C2×C3—21no—
648C2×C3—41no—
649C2×C3—92no—
650C2×C3—122no—
651C2×C3—121no—
652C2×C3—181no—
653C2×C3C2×D4×D911yescharacteristic
654C2×C3C2×D3×D1211yescharacteristic
655D3—32no—
656D3—42no—
657D3—41no—
658D3—61no—
659D3—362no—
660D3—361no—
661D3—362no—
662D3—361no—
663D3—722no—
664D3—721no—
665C22—61no—
666C22—62no—
667C22—91no—
668C22—91no—
669C22—122no—
670C22—182no—
671C22—541no—
672C22—542no—
673C22—1082no—
674C22—1084no—
675C22—1082no—
676C22—1084no—
677C22D9×D1211yescharacteristic
678C4—61no—
679C4—91no—
680C4—91no—
681C4—541no—
682C4D9×D1211yescharacteristic
683C4C22×D3×D911yescharacteristic
684C3—21no—
685C3D9×AGL(1,ℤ/8)11yescharacteristic, complemented
686C3D3×#90811yescharacteristic
687C2—21no—
688C2—92no—
689C2—122no—
690C2—121no—
691C2—181no—
692C2—1082no—
693C2—1081no—
694C2C2×D9×D1211yescharacteristic
subgroupG/H
centerC2C2×D9×D12
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC3×C9×M16C22
socleC2×C32C2×D3×D4
hypercenterM16D3×D9
hypocenterC3×C9C2×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. D9×#235C2semidirect
2. C2×D9×D12C2semidirect
3. #11721783C2semidirect
4. C9×#908C2semidirect
5. C3×M16×D9C2semidirect
6. #11720890C2semidirect
7. #11720885C2semidirect
8. #11720816C2semidirect
9. #11720815C2semidirect
10. D9×D24C2semidirect
11. D9×#204C2semidirect
15. #88736C22semidirect
16. C2×C4⋊(C3⋊D9)C22semidirect
17. C9×#235C22semidirect
18. C2×C9×D12C22semidirect
26. D9×D12C22semidirect
27. D9×D12C22semidirect
30. #88638C22semidirect
31. #88635C22semidirect
32. D9×Dic6C22semidirect
33. C3×C9×M16C22semidirect
34. #88525C22semidirect
35. #88524C22semidirect
36. C3×#3667C22semidirect
37. C9×D24C22semidirect
38. C9×#204C22semidirect
39. C3×#3663C22semidirect
40. C3×C8×D9C22semidirect
41. #88426C22semidirect
42. #88424C22semidirect
43. #88422C22semidirect
44. #88422C22semidirect
67. M16×D9D3semidirect
79. C4⋊(C3⋊D9)C23semidirect
81. #6226C23semidirect
86. C9×D12C23semidirect
92. C9×Dic6C23semidirect
102. C3×C8×C9C23semidirect
103. C3×C9⋊C8C23semidirect
161. C9×M16C2×D3semidirect
166. #3667C2×D3semidirect
171. #3663C2×D3semidirect
172. C8×D9C2×D3semidirect
211. #908D9direct (N × H)
294. C8×C9C22×D3semidirect
295. C9⋊C8C22×D3semidirect
304. C3×D9AGL(1,ℤ/8)semidirect
330. #235C2×D9semidirect
340. C2×D12C2×D9semidirect
350. C3×M16C2×D9semidirect
363. D24C2×D9semidirect
369. #204C2×D9semidirect
428. C3×C9C2×AGL(1,ℤ/8)semidirect
486. D12C22×D9semidirect
504. Dic6C22×D9semidirect
507. C3×C8C22×D9semidirect
529. D9#908direct (N × H)
559. M16D3×D9semidirect
608. C9C2×#908semidirect
644. C8C2×D3×D9semidirect
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
D9×#2352
C2×D9×D122
#117217832
C9×#9082
C3×M16×D92
#117208902
#117208852
#117208162
#117208152
D9×D242
D9×#2042
D3×#9083
D9×AGL(1,ℤ/8)3