Finite Group Databasebeta

← D9×#951 #13151911

Subgroups — D9×#951 #13151911

Order 1728.

834 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#41×D9C211yesmaximal, characteristic
2C2×D9×Dic6C211yesmaximal, characteristic, complemented
3#11721866C211yesmaximal, characteristic, complemented
4C9×#951C211yesmaximal, characteristic, complemented
5#11721219C212yesmaximal, complemented
6D9×#216C212yesmaximal, complemented
7#11721181C212yesmaximal, complemented
8D9×#214C212yesmaximal, complemented
9#11720893C211yesmaximal, characteristic, complemented
10#11720869C211yesmaximal, characteristic, complemented
11D9×#208C211yesmaximal, characteristic, complemented
12D3×#951—31nomaximal
13D9×#137—31nomaximal
14C3×C9×#41C2211yescharacteristic
15C2×#6226C2211yescharacteristic, complemented
16C3×#3701C2211yescharacteristic
17C3×Q8×D9C2211yescharacteristic
18C3×#3699—21no—
19C3×#3699C2211yescharacteristic
20C3×D4×D9—21no—
21C3×D4×D9C2211yescharacteristic
22C3×#3697—21no—
23C3×#3697C2211yescharacteristic
24C2×C3×C4×D9—21no—
25C2×C3×C4×D9C2211yescharacteristic
26C2×C9×Dic6C2211yescharacteristic, complemented
27C2×Dic3×D9—21no—
28C2×#6189—21no—
29D9×Dic6—21no—
30D9×Dic6—21no—
31D9×Dic6C2212yes—
32D9×Dic6C2212yescomplemented
33#88547C2212yescomplemented
34#88543C2212yescomplemented
35#88532C2211yescharacteristic, complemented
36C9×#216C2212yescomplemented
37C9×#214C2212yescomplemented
38C9×#208C2211yescharacteristic, complemented
39#88427C2212yescomplemented
40#88425C2212yescomplemented
41#88418C2212yescomplemented
42#88415C2212yescomplemented
43#88413C2212yescomplemented
44D9×C3⋊C8C2212yes—
45C3×D3×#41—31no—
46C2×D3×Dic6—31no—
47#64160—31no—
48C3×#951D311yescharacteristic
49#63943—32no—
50D3×#216—32no—
51#63929—32no—
52D3×#214—32no—
53#63831—31no—
54#63827—31no—
55D3×#208—31no—
56#41×D9D311yescharacteristic
57C2×Q8×D9—31no—
58C9×#137—31no—
59#63511—31no—
60#63511—61no—
61#63505—31no—
62#63480—32no—
63Q16×D9—32no—
64#63477—32no—
65SD16×D9—32no—
66#63471—31no—
67M16×D9—31no—
68C22×C3×D9—21no—
69C22×C3×D9D411yescharacteristic
70C3×Q8×C9C2311yescharacteristic
71C3×D4×C9—21no—
72C3×D4×C9C2311yescharacteristic
73C2×C3×C4×C9—21no—
74C2×C3×C4×C9C2311yescharacteristic
75C2×C3⋊Dic9—21no—
76#6226—21no—
77#6226C2312yes—
78C3×#611—21no—
79C3×#611—21no—
80C3×#611—21no—
81C2×C9×Dic3—21no—
82C2×C3×Dic9—21no—
83C2×C3×Dic9D411yescharacteristic
84C3×D36—21no—
85C3×D36C2311yescharacteristic
86C3×C4×D9—21no—
87C3×C4×D9—21no—
88C3×C4×D9—21no—
89C3×C4×D9D411yescharacteristic
90C3×C4×D9D411yescharacteristic
91C3×C4×D9C2311yescharacteristic
92C3×C4×D9C2311yescharacteristic
93C9×Dic6—21no—
94C9×Dic6C2312yes—
95C3×Dic18—21no—
96C3×Dic18C2311yescharacteristic
97Dic3×D9—22no—
98Dic3×D9—22no—
99#6189—22no—
100#6189—22no—
101C3⋊(C9⋊C8)C2312yes—
102C9×C3⋊C8C2312yes—
103C2×#951—91no—
104D3×#137—91no—
105C32×#41C2×D311yescharacteristic
106C2×#634—31no—
107C3×#239—31no—
108C3×D3×Q8—31no—
109C3×#237—31no—
110C3×#237—61no—
111C3×D3×D4—31no—
112C3×D3×D4—61no—
113C3×#235—31no—
114C3×#235—61no—
115C2×C3×C4×D3—31no—
116C2×C3×C4×D3—61no—
117C2×C3×Dic6C2×D311yescharacteristic
118C2×#627—61no—
119C2×D3×Dic3—61no—
120D3×Dic6—32no—
121D3×Dic6—32no—
122D3×Dic6—61no—
123D3×Dic6—61no—
124#3756—32no—
125#3754—32no—
126#3748—31no—
127C3×#216C2×D312yes—
128C3×#214C2×D312yes—
129C3×#208C2×D311yescharacteristic
130#3719—32no—
131#3718—32no—
132#3716—32no—
133#3715—32no—
134#3711—32no—
135D3×C3⋊C8—32no—
136C9×#41—21no—
137C9×#41C2×D311yescharacteristic
138C2×Q8×C9—31no—
139#3701C2×D311yescharacteristic
140Q8×D9—32no—
141Q8×D9—32no—
142Q8×D9—61no—
143Q8×D9—61no—
144Q8×D9C2×D311yescharacteristic
145#3699—21no—
146#3699C2×D311yescharacteristic
147D4×D9—21no—
148D4×D9C2×D311yescharacteristic
149#3697—21no—
150#3697C2×D311yescharacteristic
151C2×C4×D9—21no—
152C2×C4×D9—61no—
153C2×C4×D9C2×D311yescharacteristic
154C2×Dic18—31no—
155C2×Dic18—61no—
156C2×Dic18—61no—
157C9×Q16—32no—
158C9×SD16—32no—
159C9×M16—31no—
160#3675—32no—
161#3674—32no—
162#3674—32no—
163#3674—62no—
164#3672—32no—
165#3672—62no—
166#3667—31no—
167#3667—61no—
168#3664—32no—
169#3663—32no—
170C8×D9—32no—
171Dic36—32no—
172C22×C3×C9—21no—
173C22×C3×C9C2×D411yescharacteristic
174C2×C3×D9—22no—
175C2×C3×D9—21no—
176C2×C3×D9—42no—
177C2×C3×D9C2×D411yescharacteristic
178C2×C3×D9C2×D411yescharacteristic
179C3×C4×C9—21no—
180C3×C4×C9C2×D411yescharacteristic
181C3×C4×C9C2411yescharacteristic
182C3⋊Dic9—22no—
183C9×Dic3—22no—
184C3×Dic9—21no—
185C3×Dic9C2×D411yescharacteristic
186C3×Dic9C2×D411yescharacteristic
187C2×C3×#41—91no—
188D3×#41—31no—
189C2×D3×Q8—91no—
190C22×Dic6—91no—
191C3×#137—31no—
192#951—91no—
193#951—91no—
194#951—91no—
195#951—92no—
196#951—92no—
197#951—91no—
198#951—181no—
199#951D911yescharacteristic, direct product factor
200C2×#216—92no—
201#942—91no—
202C2×#214—92no—
203C2×#208—91no—
204#918—92no—
205D3×Q16—92no—
206#915—92no—
207D3×SD16—92no—
208#909—91no—
209D3×M16—91no—
210C22×C3×D3—31no—
211C22×C3×D3—61no—
212C32×Q8C22×D311yescharacteristic
213C32×D4—21no—
214C32×D4C22×D311yescharacteristic
215C2×C32×C4—21no—
216C2×C32×C4C22×D311yescharacteristic
217C2×C3⋊Dic3—61no—
218#634—32no—
219#634—61no—
220C3×#67—61no—
221C3×#67—61no—
222C3×#67—61no—
223C2×C3×Dic3—21no—
224C2×C3×Dic3—31no—
225C2×C3×Dic3—61no—
226C3×D12—31no—
227C3×D12—61no—
228C3×C4×D3—31no—
229C3×C4×D3—31no—
230C3×C4×D3—31no—
231C3×C4×D3—31no—
232C3×C4×D3—61no—
233C3×C4×D3—61no—
234C3×C4×D3—61no—
235C3×Dic6—21no—
236C3×Dic6—31no—
237C3×Dic6—61no—
238C3×Dic6C22×D312yes—
239#627—62no—
240#627—62no—
241D3×Dic3—62no—
242D3×Dic3—62no—
243C22×D9—21no—
244C22×D9#6711yescharacteristic
245C3⋊(C3⋊C8)—32no—
246C3×C3⋊C8C22×D312yes—
247Q8×C9—21no—
248Q8×C9—32no—
249Q8×C9—61no—
250Q8×C9C22×D311yescharacteristic
251D4×C9—21no—
252D4×C9—21no—
253D4×C9—41no—
254D4×C9C22×D311yescharacteristic
255C2×C4×C9—21no—
256C2×C4×C9—21no—
257C2×C4×C9—41no—
258C2×C4×C9—61no—
259C2×C4×C9C22×D311yescharacteristic
260#611—21no—
261#611—21no—
262#611—21no—
263C2×Dic9—21no—
264C2×Dic9—61no—
265C2×Dic9—121no—
266C2×Dic9#6711yescharacteristic
267D36—21no—
268D36C22×D311yescharacteristic
269C4×D9—21no—
270C4×D9—21no—
271C4×D9—21no—
272C4×D9—62no—
273C4×D9—62no—
274C4×D9#6711yescharacteristic
275C4×D9#6711yescharacteristic
276C4×D9C22×D311yescharacteristic
277C4×D9C22×D311yescharacteristic
278Dic18—21no—
279Dic18—32no—
280Dic18—62no—
281Dic18—61no—
282Dic18—62no—
283Dic18—62no—
284Dic18—121no—
285Dic18C22×D311yescharacteristic
286C8×C9—32no—
287C9⋊C8—32no—
288C9⋊C8—62no—
289C2×#137—271noSylow
290C2×C3×C9—21no—
291C2×C3×C9—41no—
292C2×C3×C9C22×D411yescharacteristic
293C3×D9—21no—
294C3×D9—41no—
295C3×D9#13712yescomplemented
296C3×#41—21no—
297C3×#41—91no—
298C3×#41—91no—
299C3×#41—91no—
300C3×#41—181no—
301C3×#41—181no—
302C3×#41C2×D911yescharacteristic
303C3×#41D3211yescharacteristic
304C2×C3×Q8—31no—
305C2×C3×Q8—91no—
306C2×C3×D4—91no—
307C2×C3×D4—181no—
308C22×C3×C4—91no—
309C22×C3×C4—181no—
310C22×Dic3—181no—
311#239—31no—
312D3×Q8—31no—
313D3×Q8—92no—
314D3×Q8—92no—
315D3×Q8—181no—
316D3×Q8—181no—
317#237—31no—
318#237—61no—
319D3×D4—31no—
320D3×D4—61no—
321#235—31no—
322#235—61no—
323C2×C4×D3—31no—
324C2×C4×D3—61no—
325C2×C4×D3—181no—
326C2×Dic6—91no—
327C2×Dic6—91no—
328C2×Dic6—92no—
329C2×Dic6—92no—
330C2×Dic6—181no—
331C2×Dic6—181no—
332C2×Dic6—181no—
333C2×Dic6—181no—
334C2×Dic6—181no—
335C2×Dic6C2×D911yescharacteristic, complemented
336C3×Q16—32no—
337C3×SD16—32no—
338C3×M16—31no—
339#216—92no—
340#216—92no—
341#216—92no—
342#216—92no—
343#216—92no—
344#216—182no—
345#216C2×D912yescomplemented
346#215—92no—
347#214—92no—
348#214—92no—
349#214—92no—
350#214—92no—
351#214—182no—
352#214C2×D912yescomplemented
353#208—91no—
354#208—91no—
355#208—92no—
356#208—181no—
357#208C2×D911yescharacteristic, complemented
358C2×C3⋊C8—92no—
359Dic12—92no—
360#204—92no—
361ΓL(1,𝔽25)—92no—
362D3×C8—92no—
363C22×C32—21no—
364C22×C32D3×D411yescharacteristic
365C2×C3×D3—31no—
366C2×C3×D3—31no—
367C2×C3×D3—62no—
368C2×C3×D3—61no—
369C2×C3×D3—122no—
370C32×C4—21no—
371C32×C4D3×D411yescharacteristic
372C32×C4C23×D311yescharacteristic
373C3⋊Dic3—62no—
374C3×Dic3—22no—
375C3×Dic3—31no—
376C3×Dic3—31no—
377C3×Dic3—61no—
378C22×C9—21no—
379C22×C9—21no—
380C22×C9—41no—
381C22×C9C2×#6711yescharacteristic
382C2×D9—22no—
383C2×D9—21no—
384C2×D9—42no—
385C2×D9C2×#6711yescharacteristic
386C2×D9C2×#6711yescharacteristic
387C4×C9—21no—
388C4×C9—21no—
389C4×C9—21no—
390C4×C9—41no—
391C4×C9—62no—
392C4×C9C2×#6711yescharacteristic
393C4×C9C23×D311yescharacteristic
394Dic9—21no—
395Dic9—62no—
396Dic9—122no—
397Dic9C2×#6711yescharacteristic
398Dic9C2×#6711yescharacteristic
399C2×#41—91no—
400C22×Q8—271no—
401#137—31no—
402#137—271no—
403#137—272no—
404#137—272no—
405#137—271no—
406#137—271no—
407C2×Q16—272no—
408C2×SD16—272no—
409C2×M16—271no—
410C3×C9C2×#13711yescharacteristic, complemented, Sylow
411C23×C3—91no—
412C23×C3—181no—
413C22×D3—31no—
414C22×D3—61no—
415C3×Q8—21no—
416C3×Q8—32no—
417C3×Q8—61no—
418C3×Q8—91no—
419C3×Q8—181no—
420C3×Q8C22×D911yescharacteristic
421C3×Q8C2×D3211yescharacteristic
422C3×D4—21no—
423C3×D4—21no—
424C3×D4—21no—
425C3×D4—41no—
426C3×D4—91no—
427C3×D4—181no—
428C3×D4—181no—
429C3×D4—181no—
430C3×D4—181no—
431C3×D4C22×D911yescharacteristic
432C3×D4C2×D3211yescharacteristic
433C2×C3×C4—21no—
434C2×C3×C4—21no—
435C2×C3×C4—21no—
436C2×C3×C4—41no—
437C2×C3×C4—61no—
438C2×C3×C4—91no—
439C2×C3×C4—91no—
440C2×C3×C4—91no—
441C2×C3×C4—91no—
442C2×C3×C4—91no—
443C2×C3×C4—181no—
444C2×C3×C4—181no—
445C2×C3×C4—181no—
446C2×C3×C4—181no—
447C2×C3×C4C22×D911yescharacteristic
448C2×C3×C4C2×D3211yescharacteristic
449#67—61no—
450#67—61no—
451#67—61no—
452C2×Dic3—21no—
453C2×Dic3—31no—
454C2×Dic3—61no—
455C2×Dic3—182no—
456C2×Dic3—182no—
457C2×Dic3—181no—
458C2×Dic3—181no—
459C2×Dic3—361no—
460D12—31no—
461D12—61no—
462C4×D3—31no—
463C4×D3—31no—
464C4×D3—31no—
465C4×D3—31no—
466C4×D3—61no—
467C4×D3—61no—
468C4×D3—61no—
469C4×D3—182no—
470C4×D3—182no—
471Dic6—21no—
472Dic6—31no—
473Dic6—61no—
474Dic6—92no—
475Dic6—92no—
476Dic6—182no—
477Dic6—182no—
478Dic6—182no—
479Dic6—182no—
480Dic6—182no—
481Dic6—181no—
482Dic6—181no—
483Dic6—361no—
484Dic6C22×D912yes—
485C3×C8—32no—
486C3⋊C8—92no—
487C3⋊C8—92no—
488C3⋊C8—182no—
489C3⋊C8C22×D912yes—
490C2×C32—21no—
491C2×C32—41no—
492C2×C32C2×D3×D411yescharacteristic
493C3×D3—32no—
494C3×D3—61no—
495C3×D3—121no—
496C2×C9—21no—
497C2×C9—21no—
498C2×C9—41no—
499C2×C9—41no—
500C2×C9—81no—
501C2×C9C22×#6711yescharacteristic
502D9—21no—
503D9—41no—
504D9#95112yesdirect product factor
505#41—91no—
506#41—91no—
507#41—91no—
508#41—181no—
509#41—181no—
510#41D3×D911yescharacteristic
511C2×Q8—31no—
512C2×Q8—91no—
513C2×Q8—272no—
514C2×Q8—272no—
515C2×Q8—271no—
516C2×Q8—541no—
517C2×Q8—541no—
518C2×Q8—541no—
519C2×D4—91no—
520C2×D4—181no—
521C22×C4—91no—
522C22×C4—181no—
523C22×C4—541no—
524Q16—32no—
525Q16—272no—
526Q16—272no—
527Q16—272no—
528SD16—32no—
529SD16—272no—
530SD16—272no—
531SD16—272no—
532M16—31no—
533M16—272no—
534M16—271no—
535C2×C8—272no—
536C22×C3—21no—
537C22×C3—21no—
538C22×C3—21no—
539C22×C3—41no—
540C22×C3—91no—
541C22×C3—91no—
542C22×C3—182no—
543C22×C3—181no—
544C22×C3—362no—
545C22×C3D4×D911yescharacteristic
546C22×C3D3×#6711yescharacteristic
547C2×D3—31no—
548C2×D3—31no—
549C2×D3—62no—
550C2×D3—61no—
551C2×D3—122no—
552C3×C4—21no—
553C3×C4—21no—
554C3×C4—21no—
555C3×C4—21no—
556C3×C4—41no—
557C3×C4—62no—
558C3×C4—91no—
559C3×C4—91no—
560C3×C4—181no—
561C3×C4D4×D911yescharacteristic
562C3×C4C23×D911yescharacteristic
563C3×C4D3×#6711yescharacteristic
564C3×C4C22×D3211yescharacteristic
565Dic3—22no—
566Dic3—31no—
567Dic3—31no—
568Dic3—61no—
569Dic3—182no—
570Dic3—182no—
571Dic3—362no—
572C32D3×#13711yescharacteristic
573C9—21no—
574C9C2×#95111yescharacteristic, complemented
575C23—91no—
576C23—181no—
577Q8—32no—
578Q8—61no—
579Q8—91no—
580Q8—181no—
581Q8—272no—
582Q8—542no—
583Q8—542no—
584Q8—541no—
585Q8C2×D3×D911yescharacteristic
586D4—21no—
587D4—91no—
588D4—181no—
589D4—181no—
590D4—181no—
591D4—181no—
592D4C2×D3×D911yescharacteristic
593C2×C4—21no—
594C2×C4—61no—
595C2×C4—91no—
596C2×C4—91no—
597C2×C4—91no—
598C2×C4—91no—
599C2×C4—91no—
600C2×C4—181no—
601C2×C4—181no—
602C2×C4—181no—
603C2×C4—181no—
604C2×C4—542no—
605C2×C4—542no—
606C2×C4—541no—
607C2×C4C2×D3×D911yescharacteristic
608C8—32no—
609C8—272no—
610C2×C3—21no—
611C2×C3—21no—
612C2×C3—21no—
613C2×C3—41no—
614C2×C3—41no—
615C2×C3—41no—
616C2×C3—81no—
617C2×C3—92no—
618C2×C3—181no—
619C2×C3—361no—
620C2×C3C2×D4×D911yescharacteristic
621C2×C3C2×D3×#6711yescharacteristic
622D3—32no—
623D3—61no—
624D3—121no—
625C22—21no—
626C22—91no—
627C22—91no—
628C22—182no—
629C22—181no—
630C22—362no—
631C22D9×#6711yescharacteristic
632C4—21no—
633C4—62no—
634C4—91no—
635C4—91no—
636C4—181no—
637C4—542no—
638C4D9×#6711yescharacteristic
639C4C22×D3×D911yescharacteristic
640C3—21no—
641C3D9×#13711yescharacteristic, complemented
642C3D3×#95111yescharacteristic
643C2—21no—
644C2—41no—
645C2—92no—
646C2—181no—
647C2—361no—
648C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC2C2×D9×#67
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC3×C9×#41C22
socleC2×C32C2×D3×D4
prime radicalC3×#41×D9C2
hypercenter#41D3×D9
hypocenterC3×C9C2×#137
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. C2×D9×Dic6C2semidirect
3. #11721866C2semidirect
4. C9×#951C2semidirect
5. #11721219C2semidirect
6. D9×#216C2semidirect
7. #11721181C2semidirect
8. D9×#214C2semidirect
9. #11720893C2semidirect
10. #11720869C2semidirect
11. D9×#208C2semidirect
15. C2×#6226C22semidirect
26. C2×C9×Dic6C22semidirect
32. D9×Dic6C22semidirect
33. #88547C22semidirect
34. #88543C22semidirect
35. #88532C22semidirect
36. C9×#216C22semidirect
37. C9×#214C22semidirect
38. C9×#208C22semidirect
39. #88427C22semidirect
40. #88425C22semidirect
41. #88418C22semidirect
42. #88415C22semidirect
43. #88413C22semidirect
199. #951D9direct (N × H)
295. C3×D9#137semidirect
335. C2×Dic6C2×D9semidirect
345. #216C2×D9semidirect
352. #214C2×D9semidirect
357. #208C2×D9semidirect
410. C3×C9C2×#137semidirect
504. D9#951direct (N × H)
574. C9C2×#951semidirect
641. C3D9×#137semidirect

Maximal subgroups

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

subgroupindex
C3×#41×D92
C2×D9×Dic62
#117218662
C9×#9512
#117212192
D9×#2162
#117211812
D9×#2142
#117208932
#117208692
D9×#2082
D3×#9513
D9×#1373