Finite Group Databasebeta

← D9×#907 #13149844

Subgroups — D9×#907 #13149844

Order 1728.

860 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D9×#235C211yesmaximal, characteristic
2#11721782C211yesmaximal, characteristic, complemented
3C9×#907C211yesmaximal, characteristic, complemented
4C3×M16×D9C211yesmaximal, characteristic, complemented
5#11720878C211yesmaximal, characteristic, complemented
6#11720862C211yesmaximal, characteristic, complemented
7C2×D9×C3⋊C8C211yesmaximal, characteristic, complemented
8#11720809C212yesmaximal, complemented
9#11720808C212yesmaximal, complemented
10D9×ΓL(1,𝔽25)C212yesmaximal, complemented
11D3×C8×D9C212yesmaximal, complemented
12D3×#907—31nomaximal
13D9×#131—31nomaximal
14#88736C2211yescharacteristic
15C9×#235C2211yescharacteristic
16C2×C3×C4×D9C2211yescharacteristic
17D9×#67C412yes—
18#88662C412yes—
19D9×D12C411yescharacteristic
20C4×D3×D9C2212yes—
21#88640C2212yes—
22#88638C411yescharacteristic
23#88635C411yescharacteristic
24D9×Dic6C411yescharacteristic
25C3×C9×M16C2211yescharacteristic, complemented
26C2×C3⋊(C9⋊C8)C2211yescharacteristic, complemented
27#88523C2212yescomplemented
28C8×C3⋊D9C2212yescomplemented
29C2×C9×C3⋊C8C2211yescharacteristic, complemented
30C3×#3667C2211yescharacteristic, complemented
31C9×ΓL(1,𝔽25)C2212yescomplemented
32D3×C8×C9C2212yescomplemented
33C3×#3663C2212yescomplemented
34C3×C8×D9C2212yes—
35#88420C2212yescomplemented
36D3×C9⋊C8C2212yescomplemented
37#88414C2212yescomplemented
38(C3⋊D9)⋊C8C2212yescomplemented
39D9×C3⋊C8C2211yescharacteristic
40D9×C3⋊C8C2211yescharacteristic
41D9×C3⋊C8C2211yescharacteristic, complemented
42D9×C3⋊C8C2211yescharacteristic, complemented
43D3×#235—31no—
44#64116—31no—
45C3×#907D311yescharacteristic
46C3×D3×M16—31no—
47#63817—31no—
48#63815—31no—
49C2×D3×C3⋊C8—31no—
50#63805—32no—
51#63804—32no—
52D3×ΓL(1,𝔽25)—32no—
53D32×C8—32no—
54#41×D9—31no—
55C9×#131—31no—
56#63510—31no—
57#63469—31no—
58#63469—32no—
59#63469—61no—
60M16×D9—32no—
61M16×D9D311yescharacteristic, complemented
62#63464—31no—
63#63464—32no—
64C2×C8×D9—31no—
65C2×C8×D9—32no—
66C22×C3×D9C2×C411yescharacteristic
67C2×D3×D9C2×C412yes—
68C2×C3×C4×C9C2311yescharacteristic
69#6233C2×C412yes—
70C4⋊(C3⋊D9)C2×C411yescharacteristic
71C4×C3⋊D9C2312yes—
72#6226C2×C411yescharacteristic
73C9×#67C2×C412yes—
74C2×C3×Dic9C2×C411yescharacteristic
75C9×D12C2×C411yescharacteristic
76C4×D3×C9C2312yes—
77C3×C4×D9C2×C411yescharacteristic
78C3×C4×D9C2×C411yescharacteristic
79C3×C4×D9C2311yescharacteristic
80C3×C4×D9C2311yescharacteristic
81C9×Dic6C2×C411yescharacteristic
82#6195C2×C412yes—
83#6194C2×C412yes—
84D3×Dic9C2×C412yes—
85#6192C2×C412yes—
86(C3⋊D9)⋊C4C2×C412yes—
87Dic3×D9C2×C412yes—
88#6189C2×C412yes—
89C3×C8×C9C2312yes—
90C3⋊(C9⋊C8)C2311yescharacteristic
91C3⋊(C9⋊C8)C2311yescharacteristic
92C9×C3⋊C8C2311yescharacteristic
93C9×C3⋊C8C2311yescharacteristic
94C3×C9⋊C8C2312yes—
95D3×#131—91no—
96C2×#907—91no—
97#3828—31no—
98C3×#235C2×D311yescharacteristic
99C2×C3×C4×D3—31no—
100D3×#67—32no—
101#3804—32no—
102D3×D12—31no—
103C4×D32—32no—
104#3799—31no—
105#3798—32no—
106#3795—31no—
107D3×Dic6—31no—
108C32×M16C2×D311yescharacteristic
109C2×C3⋊(C3⋊C8)—31no—
110#3743—32no—
111C8×C3⋊D3—32no—
112C3×#208—31no—
113C2×C3×C3⋊C8C2×D311yescharacteristic
114C3×ΓL(1,𝔽25)—32no—
115C3×ΓL(1,𝔽25)C2×D312yes—
116C3×D3×C8—32no—
117C3×D3×C8C2×D312yes—
118#3712—32no—
119#3711—32no—
120#3710—32no—
121D3×C3⋊C8—32no—
122D3×C3⋊C8—31no—
123D3×C3⋊C8—31no—
124D3×C3⋊C8—31no—
125D3×C3⋊C8—31no—
126C9×#41—31no—
127#3701—31no—
128Q8×D9—31no—
129#3699—32no—
130#3699—31no—
131D4×D9—32no—
132D4×D9—31no—
133#3697—31no—
134#3697—32no—
135#3697—61no—
136C2×C4×D9—32no—
137C2×C4×D9C2×D311yescharacteristic
138C9×M16—21no—
139C9×M16—32no—
140C9×M16C2×D311yescharacteristic, complemented
141C2×C8×C9—31no—
142C2×C8×C9—32no—
143#3667—32no—
144#3667C2×D311yescharacteristic, complemented
145C2×C9⋊C8—31no—
146C2×C9⋊C8—32no—
147C2×C9⋊C8—61no—
148#3663—32no—
149#3663—32no—
150#3663—62no—
151#3663C2×D312yescomplemented
152C8×D9—31no—
153C8×D9—31no—
154C8×D9—31no—
155C8×D9—31no—
156C8×D9—32no—
157C8×D9—32no—
158C8×D9—62no—
159C8×D9C2×D312yes—
160C22×C3×C9C22×C411yescharacteristic
161C2×C3⋊D9C22×C412yes—
162C2×D3×C9C22×C412yes—
163C2×C3×D9—22no—
164C2×C3×D9C22×C411yescharacteristic
165C2×C3×D9C22×C411yescharacteristic
166D3×D9—24no—
167C3×C4×C9C22×C411yescharacteristic
168C3×C4×C9C2411yescharacteristic
169C3⋊Dic9C22×C412yes—
170C9×Dic3C22×C412yes—
171C3×Dic9C22×C411yescharacteristic
172C3×Dic9C22×C411yescharacteristic
173D3×#41—91no—
174C2×#235—91no—
175C3×#131—31no—
176C2×C3×M16—91no—
177#948—91no—
178C22×C3⋊C8—91no—
179#907—92no—
180#907—91no—
181#907—91no—
182#907—91no—
183#907—91no—
184#907—92no—
185#907—92no—
186#907—181no—
187#907D911yescharacteristic, direct product factor
188D3×M16—31no—
189D3×M16—92no—
190#901—92no—
191#901—91no—
192C2×ΓL(1,𝔽25)—92no—
193C2×D3×C8—92no—
194C2×D3×C8—91no—
195C2×D3×C8—92no—
196C22×C3×D3—31no—
197C2×D32—32no—
198C2×C32×C4C22×D311yescharacteristic
199#638—32no—
200C4⋊(C3⋊D3)—31no—
201C4×C3⋊D3—32no—
202#634—31no—
203C3×#67C4×D312yes—
204C2×C3×Dic3—31no—
205C3×D12C4×D311yescharacteristic
206C3×C4×D3—31no—
207C3×C4×D3—31no—
208C3×C4×D3—31no—
209C3×C4×D3—31no—
210C3×C4×D3C22×D312yes—
211C3×Dic6C4×D311yescharacteristic
212#627—32no—
213#626—32no—
214#626—32no—
215#625—32no—
216#624—32no—
217D3×Dic3—32no—
218D3×Dic3—32no—
219C22×D9—32no—
220C22×D9C4×D311yescharacteristic
221C32×C8C22×D312yes—
222C3⋊(C3⋊C8)—31no—
223C3⋊(C3⋊C8)—31no—
224C3×C3⋊C8—32no—
225C3×C3⋊C8C22×D311yescharacteristic
226C3×C3⋊C8C22×D311yescharacteristic
227Q8×C9—31no—
228D4×C9—32no—
229D4×C9—31no—
230C2×C4×C9—21no—
231C2×C4×C9—32no—
232C2×C4×C9C22×D311yescharacteristic
233#611—32no—
234#611—32no—
235#611—32no—
236#611—62no—
237C2×Dic9—32no—
238C2×Dic9C4×D311yescharacteristic
239D36—32no—
240D36—31no—
241D36—61no—
242C4×D9—32no—
243C4×D9—32no—
244C4×D9—32no—
245C4×D9—62no—
246C4×D9C4×D311yescharacteristic
247C4×D9C4×D311yescharacteristic
248C4×D9C22×D311yescharacteristic
249C4×D9C22×D311yescharacteristic
250Dic18—31no—
251Dic18—32no—
252Dic18—61no—
253C8×C9—22no—
254C8×C9—31no—
255C8×C9—31no—
256C8×C9C22×D312yes—
257C9⋊C8—31no—
258C9⋊C8—31no—
259C9⋊C8—61no—
260C9⋊C8—61no—
261C9⋊C8C22×D312yes—
262C2×#131—271noSylow
263C2×C3×C9—21no—
264C2×C3×C9C23×C411yescharacteristic
265C3⋊D9—22no—
266D3×C9—22no—
267C3×D9—21no—
268C3×D9#13112yescomplemented
269C3×#41—31no—
270C22×C3×C4—91no—
271C2×#67—92no—
272#239—91no—
273D3×Q8—91no—
274#237—91no—
275#237—92no—
276D3×D4—91no—
277D3×D4—92no—
278#235—92no—
279#235—91no—
280#235—91no—
281#235—92no—
282#235—91no—
283#235—92no—
284#235—91no—
285#235—181no—
286#235C2×D911yescharacteristic
287C2×D12—91no—
288C2×C4×D3—31no—
289C2×C4×D3—92no—
290C2×C4×D3—92no—
291C2×Dic6—91no—
292C3×M16—21no—
293C3×M16—32no—
294C3×M16—91no—
295C3×M16—92no—
296C3×M16C2×D911yescharacteristic, complemented
297C3×M16D3211yescharacteristic
298C2×C3×C8—31no—
299C2×C3×C8—32no—
300C2×C3×C8—92no—
301#208—31no—
302#208—92no—
303C2×C3⋊C8—92no—
304C2×C3⋊C8—91no—
305C2×C3⋊C8—91no—
306C2×C3⋊C8—91no—
307C2×C3⋊C8—91no—
308C2×C3⋊C8—91no—
309C2×C3⋊C8—91no—
310C2×C3⋊C8—181no—
311C2×C3⋊C8C2×D911yescharacteristic, complemented
312ΓL(1,𝔽25)—32no—
313ΓL(1,𝔽25)—92no—
314ΓL(1,𝔽25)—92no—
315ΓL(1,𝔽25)—92no—
316ΓL(1,𝔽25)—92no—
317ΓL(1,𝔽25)—92no—
318ΓL(1,𝔽25)—182no—
319ΓL(1,𝔽25)C2×D912yescomplemented
320D3×C8—32no—
321D3×C8—92no—
322D3×C8—91no—
323D3×C8—91no—
324D3×C8—91no—
325D3×C8—91no—
326D3×C8—92no—
327D3×C8—92no—
328D3×C8—92no—
329D3×C8—92no—
330D3×C8—182no—
331D3×C8C2×D912yescomplemented
332C22×C32C2×C4×D311yescharacteristic
333C2×C3⋊D3—32no—
334C2×C3×D3—31no—
335C2×C3×D3—31no—
336C2×C3×D3—62no—
337C2×C3×D3C2×C4×D312yes—
338D32—64no—
339C32×C4C2×C4×D311yescharacteristic
340C32×C4C23×D311yescharacteristic
341C3⋊Dic3—32no—
342C3×Dic3—31no—
343C3×Dic3—31no—
344C3×Dic3C2×C4×D312yes—
345C22×C9—21no—
346C22×C9—32no—
347C22×C9C2×C4×D311yescharacteristic
348C2×D9—22no—
349C2×D9—32no—
350C2×D9—62no—
351C2×D9—64no—
352C2×D9C2×C4×D311yescharacteristic
353C2×D9C2×C4×D311yescharacteristic
354C4×C9—21no—
355C4×C9—21no—
356C4×C9—32no—
357C4×C9C2×C4×D311yescharacteristic
358C4×C9C23×D311yescharacteristic
359Dic9—32no—
360Dic9—62no—
361Dic9C2×C4×D311yescharacteristic
362Dic9C2×C4×D311yescharacteristic
363C2×#41—271no—
364#131—31no—
365#131—271no—
366#131—272no—
367#131—272no—
368#131—271no—
369#131—271no—
370C2×M16—91no—
371C2×M16—272no—
372C22×C8—272no—
373C22×C8—271no—
374C3×C9C2×#13111yescharacteristic, complemented, Sylow
375C23×C3—91no—
376C22×D3—31no—
377C22×D3—92no—
378C22×D3—92no—
379C3×Q8—31no—
380C3×D4—32no—
381C3×D4—31no—
382C2×C3×C4—21no—
383C2×C3×C4—32no—
384C2×C3×C4—91no—
385C2×C3×C4—91no—
386C2×C3×C4—91no—
387C2×C3×C4—91no—
388C2×C3×C4—91no—
389C2×C3×C4C22×D911yescharacteristic
390C2×C3×C4C2×D3211yescharacteristic
391#67—92no—
392#67—92no—
393#67—92no—
394#67—92no—
395#67—92no—
396#67—92no—
397#67—182no—
398#67C4×D912yes—
399C2×Dic3—31no—
400C2×Dic3—92no—
401C2×Dic3—92no—
402D12—92no—
403D12—92no—
404D12—91no—
405D12—91no—
406D12—181no—
407D12C4×D911yescharacteristic
408C4×D3—31no—
409C4×D3—31no—
410C4×D3—31no—
411C4×D3—31no—
412C4×D3—92no—
413C4×D3—92no—
414C4×D3—92no—
415C4×D3—92no—
416C4×D3—92no—
417C4×D3—92no—
418C4×D3—182no—
419C4×D3C22×D912yes—
420Dic6—92no—
421Dic6—91no—
422Dic6—91no—
423Dic6—92no—
424Dic6—181no—
425Dic6C4×D911yescharacteristic
426C3×C8—22no—
427C3×C8—31no—
428C3×C8—31no—
429C3×C8—92no—
430C3×C8C22×D912yes—
431C3×C8C2×D3212yes—
432C3⋊C8—32no—
433C3⋊C8—91no—
434C3⋊C8—91no—
435C3⋊C8—91no—
436C3⋊C8—91no—
437C3⋊C8—181no—
438C3⋊C8—181no—
439C3⋊C8C22×D911yescharacteristic
440C3⋊C8C22×D911yescharacteristic
441C2×C32—21no—
442C2×C32C22×C4×D311yescharacteristic
443C3⋊D3—62no—
444C3×D3—22no—
445C3×D3—32no—
446C3×D3—61no—
447C2×C9—21no—
448C2×C9—21no—
449C2×C9—41no—
450C2×C9—62no—
451C2×C9C22×C4×D311yescharacteristic
452D9—21no—
453D9—62no—
454D9—122no—
455D9#90712yesdirect product factor
456#41—31no—
457#41—272no—
458#41—272no—
459#41—271no—
460#41—271no—
461#41—271no—
462C2×Q8—271no—
463C2×D4—271no—
464C2×D4—272no—
465C22×C4—91no—
466C22×C4—272no—
467M16—32no—
468M16—91no—
469M16—92no—
470M16—272no—
471M16—272no—
472M16—272no—
473M16D3×D911yescharacteristic, complemented
474C2×C8—31no—
475C2×C8—32no—
476C2×C8—92no—
477C2×C8—272no—
478C2×C8—271no—
479C2×C8—271no—
480C2×C8—271no—
481C2×C8—271no—
482C2×C8—272no—
483C2×C8—271no—
484C2×C8—272no—
485C22×C3—21no—
486C22×C3—32no—
487C22×C3—91no—
488C22×C3—91no—
489C22×C3—182no—
490C22×C3C2×C4×D911yescharacteristic
491C22×C3C4×D3211yescharacteristic
492C2×D3—31no—
493C2×D3—31no—
494C2×D3—62no—
495C2×D3—92no—
496C2×D3—92no—
497C2×D3—182no—
498C2×D3—184no—
499C2×D3—184no—
500C2×D3C2×C4×D912yes—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—32no—
504C3×C4—91no—
505C3×C4—91no—
506C3×C4C2×C4×D911yescharacteristic
507C3×C4C23×D911yescharacteristic
508C3×C4C4×D3211yescharacteristic
509C3×C4C22×D3211yescharacteristic
510Dic3—31no—
511Dic3—31no—
512Dic3—92no—
513Dic3—92no—
514Dic3—182no—
515Dic3C2×C4×D912yes—
516C32D3×#13111yescharacteristic
517C9—21no—
518C9C2×#90711yescharacteristic, complemented
519C23—91no—
520C23—272no—
521Q8—31no—
522Q8—272no—
523Q8—271no—
524D4—32no—
525D4—31no—
526D4—272no—
527D4—272no—
528D4—272no—
529D4—272no—
530D4—271no—
531C2×C4—32no—
532C2×C4—91no—
533C2×C4—91no—
534C2×C4—91no—
535C2×C4—91no—
536C2×C4—91no—
537C2×C4—272no—
538C2×C4—272no—
539C2×C4—272no—
540C2×C4—272no—
541C2×C4C2×D3×D911yescharacteristic
542C8—31no—
543C8—31no—
544C8—92no—
545C8—271no—
546C8—271no—
547C8C2×D3×D912yes—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—41no—
552C2×C3—62no—
553C2×C3—92no—
554C2×C3—181no—
555C2×C3C22×C4×D911yescharacteristic
556C2×C3C2×C4×D3211yescharacteristic
557D3—22no—
558D3—32no—
559D3—61no—
560D3—182no—
561D3—182no—
562D3—362no—
563C22—32no—
564C22—91no—
565C22—91no—
566C22—182no—
567C22—272no—
568C22—544no—
569C22C4×D3×D911yescharacteristic
570C4—32no—
571C4—91no—
572C4—91no—
573C4—272no—
574C4C4×D3×D911yescharacteristic
575C4C22×D3×D911yescharacteristic
576C3—21no—
577C3D9×#13111yescharacteristic, complemented
578C3D3×#90711yescharacteristic
579C2—21no—
580C2—62no—
581C2—92no—
582C2—181no—
583C2—542no—
584C2C2×C4×D3×D911yescharacteristic
subgroupG/H
centerC4C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC3×C4C22×D32
FittingC3×C9×M16C22
socleC2×C32C22×C4×D3
prime radicalD9×#235C2
cosocleC3×C4×C9C24
hypercenterM16D3×D9
hypocenterC3×C9C2×#131
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. #11721782C2semidirect
3. C9×#907C2semidirect
4. C3×M16×D9C2semidirect
5. #11720878C2semidirect
6. #11720862C2semidirect
7. C2×D9×C3⋊C8C2semidirect
8. #11720809C2semidirect
9. #11720808C2semidirect
10. D9×ΓL(1,𝔽25)C2semidirect
11. D3×C8×D9C2semidirect
25. C3×C9×M16C22semidirect
26. C2×C3⋊(C9⋊C8)C22semidirect
27. #88523C22semidirect
28. C8×C3⋊D9C22semidirect
29. C2×C9×C3⋊C8C22semidirect
30. C3×#3667C22semidirect
31. C9×ΓL(1,𝔽25)C22semidirect
32. D3×C8×C9C22semidirect
33. C3×#3663C22semidirect
35. #88420C22semidirect
36. D3×C9⋊C8C22semidirect
37. #88414C22semidirect
38. (C3⋊D9)⋊C8C22semidirect
41. D9×C3⋊C8C22semidirect
42. D9×C3⋊C8C22semidirect
61. M16×D9D3semidirect
140. C9×M16C2×D3semidirect
144. #3667C2×D3semidirect
151. #3663C2×D3semidirect
187. #907D9direct (N × H)
268. C3×D9#131semidirect
296. C3×M16C2×D9semidirect
311. C2×C3⋊C8C2×D9semidirect
319. ΓL(1,𝔽25)C2×D9semidirect
331. D3×C8C2×D9semidirect
374. C3×C9C2×#131semidirect
455. D9#907direct (N × H)
473. M16D3×D9semidirect
518. C9C2×#907semidirect
577. C3D9×#131semidirect

Maximal subgroups

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

subgroupindex
D9×#2352
#117217822
C9×#9072
C3×M16×D92
#117208782
#117208622
C2×D9×C3⋊C82
#117208092
#117208082
D9×ΓL(1,𝔽25)2
D3×C8×D92
D3×#9073
D9×#1313