Finite Group Databasebeta

← D9×#912 #13149898

Subgroups — D9×#912 #13149898

Order 1728.

860 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D9×#237C212yesmaximal, complemented
2#11721791C211yesmaximal, characteristic, complemented
3C9×#912C211yesmaximal, characteristic, complemented
4C3×D8×D9C211yesmaximal, characteristic, complemented
5#11721186C212yesmaximal, complemented
6D9×#214C212yesmaximal, complemented
7#11721169C212yesmaximal, complemented
8#11720831C211yesmaximal, characteristic, complemented
9D9×Dic12C211yesmaximal, characteristic, complemented
10#11720811C211yesmaximal, characteristic, complemented
11D3×C8×D9C211yesmaximal, characteristic, complemented
12D3×#912—31nomaximal
13D9×#135—31nomaximal
14#88742C2212yescomplemented
15C9×#237C2212yescomplemented
16C3×D4×D9C2212yes—
17D9×#67—22no—
18#88663—22no—
19#88658—22no—
20C2×Dic3×D9—22no—
21C4×D3×D9C2211yescharacteristic
22#88641C2212yescomplemented
23#88634C2212yescomplemented
24D9×Dic6C2212yes—
25C3×C9×D8C2211yescharacteristic, complemented
26#88543C2212yescomplemented
27C8×C3⋊D9C2211yescharacteristic, complemented
28#88521C2211yescharacteristic, complemented
29C9×#214C2212yescomplemented
30C3×#3673C2212yescomplemented
31C9×Dic12C2211yescharacteristic, complemented
32D3×C8×C9C2211yescharacteristic, complemented
33C3×D72C2211yescharacteristic, complemented
34C3×C8×D9C2211yescharacteristic
35#88428C2212yescomplemented
36#88425C2212yescomplemented
37D3×C9⋊C8C2211yescharacteristic, complemented
38#88415C2212yescomplemented
39(C3⋊D9)⋊C8C2211yescharacteristic, complemented
40D9×C3⋊C8C2211yescharacteristic
41D3×#237—32no—
42#64121—31no—
43C3×#912D311yescharacteristic
44C3×D3×D8—31no—
45#63936—32no—
46#63933—32no—
47D3×#214—32no—
48#63810—31no—
49#63807—31no—
50D3×Dic12—31no—
51D32×C8—31no—
52#41×D9—32no—
53C9×#135—31no—
54#63513—32no—
55#63481—31no—
56Q16×D9—31no—
57#63478—32no—
58SD16×D9—32no—
59#63474—31no—
60#63474—61no—
61D8×D9D311yescharacteristic, complemented
62#63467—31no—
63C2×C8×D9—31no—
64C22×C3×D9—22no—
65C2×D3×D9D411yescharacteristic
66C3×D4×C9C2312yes—
67#6233—22no—
68C2×C3⋊Dic9—22no—
69C4×C3⋊D9C2311yescharacteristic
70#6226C2312yes—
71C9×#67—22no—
72C3×#611—22no—
73C2×C9×Dic3—22no—
74C4×D3×C9C2311yescharacteristic
75C3×D36C2312yes—
76C3×C4×D9C2311yescharacteristic
77C9×Dic6C2312yes—
78#6194—22no—
79D3×Dic9D411yescharacteristic
80#6192—22no—
81(C3⋊D9)⋊C4D411yescharacteristic
82Dic3×D9—22no—
83Dic3×D9—22no—
84Dic3×D9—22no—
85Dic3×D9D411yescharacteristic
86#6189—22no—
87C3×C8×C9C2311yescharacteristic
88C3⋊(C9⋊C8)C2311yescharacteristic
89C9×C3⋊C8C2311yescharacteristic
90C3×C9⋊C8C2311yescharacteristic
91D3×#135—91no—
92C2×#912—91no—
93#3830—32no—
94C3×#237C2×D312yes—
95C3×D3×D4—32no—
96D3×#67—62no—
97#3805—62no—
98#3804—62no—
99C2×D3×Dic3—62no—
100C4×D32—31no—
101#3796—32no—
102#3795—32no—
103D3×Dic6—32no—
104C32×D8C2×D311yescharacteristic
105#3754—32no—
106#3746—31no—
107C8×C3⋊D3—31no—
108C3×#214C2×D312yes—
109C3×#213—32no—
110C3×Dic12C2×D311yescharacteristic
111C3×D24—31no—
112C3×D3×C8—31no—
113C3×D3×C8C2×D311yescharacteristic
114#3719—32no—
115#3716—32no—
116#3715—32no—
117#3710—31no—
118D3×C3⋊C8—31no—
119D3×C3⋊C8—31no—
120C9×#41—32no—
121#3701—32no—
122Q8×D9—32no—
123#3699—32no—
124#3699—62no—
125#3699—62no—
126D4×D9—62no—
127D4×D9C2×D312yes—
128#3697—32no—
129#3697—62no—
130C2×C4×D9—31no—
131C2×C4×D9—62no—
132C9×Q16—31no—
133C9×SD16—32no—
134C9×D8—21no—
135C9×D8C2×D311yescharacteristic, complemented
136C2×C8×C9—31no—
137#3675—32no—
138#3674—32no—
139#3673C2×D312yescomplemented
140#3672—32no—
141#3672—62no—
142C2×C9⋊C8—31no—
143D72C2×D311yescharacteristic, complemented
144#3664—32no—
145C8×D9—31no—
146C8×D9—31no—
147C8×D9—31no—
148C8×D9—61no—
149C8×D9C2×D311yescharacteristic
150Dic36—31no—
151Dic36—61no—
152C22×C3×C9—22no—
153C2×C3⋊D9C2×D411yescharacteristic
154C2×D3×C9C2×D411yescharacteristic
155C2×C3×D9—22no—
156C2×C3×D9—44no—
157C2×C3×D9C2×D411yescharacteristic
158D3×D9—22no—
159C3×C4×C9C2411yescharacteristic
160C3⋊Dic9—22no—
161C3⋊Dic9C2×D411yescharacteristic
162C9×Dic3—22no—
163C9×Dic3C2×D411yescharacteristic
164C3×Dic9C2×D411yescharacteristic
165D3×#41—92no—
166C2×#237—92no—
167C3×#135—31no—
168C2×C3×D8—91no—
169#950—92no—
170C2×#214—92no—
171#919—91no—
172D3×Q16—91no—
173#916—92no—
174D3×SD16—92no—
175#912—92no—
176#912—91no—
177#912—91no—
178#912—92no—
179#912—91no—
180#912—91no—
181#912—181no—
182#912D911yescharacteristic, direct product factor
183D3×D8—31no—
184C2×Dic12—91no—
185#904—91no—
186C2×D3×C8—91no—
187C2×D3×C8—91no—
188C22×C3×D3—62no—
189C2×D32—31no—
190C32×D4C22×D312yes—
191#638—62no—
192C2×C3⋊Dic3—62no—
193C4×C3⋊D3—31no—
194#634—32no—
195C3×#67—22no—
196C3×#67—62no—
197C2×C3×Dic3—22no—
198C3×D12—32no—
199C3×C4×D3—31no—
200C3×C4×D3C22×D311yescharacteristic
201C3×Dic6C22×D312yes—
202#627—62no—
203#626—62no—
204#625—62no—
205#624—31no—
206D3×Dic3—31no—
207D3×Dic3—31no—
208D3×Dic3—62no—
209D3×Dic3—62no—
210D3×Dic3—62no—
211C22×D9—22no—
212C22×D9—31no—
213C32×C8C22×D311yescharacteristic
214C3⋊(C3⋊C8)—31no—
215C3×C3⋊C8—31no—
216C3×C3⋊C8C22×D311yescharacteristic
217Q8×C9—32no—
218D4×C9—22no—
219D4×C9—62no—
220D4×C9C22×D312yes—
221C2×C4×C9—31no—
222C2×C4×C9—62no—
223#611—22no—
224#611—62no—
225#611—62no—
226#611—122no—
227C2×Dic9—31no—
228C2×Dic9—62no—
229C2×Dic9—122no—
230D36—62no—
231D36C22×D312yes—
232C4×D9—31no—
233C4×D9—31no—
234C4×D9—31no—
235C4×D9—61no—
236C4×D9—62no—
237C4×D9—62no—
238C4×D9—62no—
239C4×D9C22×D311yescharacteristic
240Dic18—32no—
241Dic18—62no—
242Dic18—62no—
243C8×C9—21no—
244C8×C9—31no—
245C8×C9C22×D311yescharacteristic
246C9⋊C8—31no—
247C9⋊C8—61no—
248C9⋊C8C22×D311yescharacteristic
249C2×#135—271noSylow
250C2×C3×C9—42no—
251C2×C3×C9C22×D411yescharacteristic
252C3⋊D9—21no—
253D3×C9—21no—
254C3×D9—42no—
255C3×D9#13512yescomplemented
256C3×#41—32no—
257C2×C3×D4—92no—
258C2×#67—182no—
259C22×Dic3—182no—
260#239—92no—
261D3×Q8—92no—
262#237—92no—
263#237—92no—
264#237—92no—
265#237—92no—
266#237—182no—
267#237—182no—
268#237—182no—
269#237—182no—
270#237C2×D912yescomplemented
271D3×D4—32no—
272D3×D4—182no—
273#235—92no—
274#235—182no—
275C2×C4×D3—91no—
276C2×C4×D3—91no—
277C2×C4×D3—182no—
278C2×Dic6—92no—
279C3×Q16—31no—
280C3×SD16—32no—
281C3×D8—21no—
282C3×D8—92no—
283C3×D8—91no—
284C3×D8C2×D911yescharacteristic, complemented
285C3×D8D3211yescharacteristic
286C2×C3×C8—31no—
287C2×C3×C8—91no—
288#216—92no—
289#215—92no—
290#214—92no—
291#214—92no—
292#214—92no—
293#214—92no—
294#214—182no—
295#214C2×D912yescomplemented
296#213—32no—
297C2×C3⋊C8—91no—
298C2×C3⋊C8—91no—
299Dic12—92no—
300Dic12—91no—
301Dic12—91no—
302Dic12—181no—
303Dic12C2×D911yescharacteristic, complemented
304D24—31no—
305#204—92no—
306D3×C8—31no—
307D3×C8—91no—
308D3×C8—91no—
309D3×C8—91no—
310D3×C8—91no—
311D3×C8—91no—
312D3×C8—91no—
313D3×C8—181no—
314D3×C8C2×D911yescharacteristic, complemented
315C22×C32—22no—
316C2×C3⋊D3—31no—
317C2×C3×D3—31no—
318C2×C3×D3—62no—
319C2×C3×D3—124no—
320C2×C3×D3D3×D411yescharacteristic
321D32—62no—
322C32×C4C23×D311yescharacteristic
323C3⋊Dic3—31no—
324C3⋊Dic3—62no—
325C3×Dic3—22no—
326C3×Dic3—31no—
327C3×Dic3D3×D411yescharacteristic
328C22×C9—22no—
329C22×C9—31no—
330C22×C9—42no—
331C2×D9—22no—
332C2×D9—31no—
333C2×D9—44no—
334C2×D9—61no—
335C2×D9—62no—
336C2×D9D3×D411yescharacteristic
337C4×C9—21no—
338C4×C9—31no—
339C4×C9—62no—
340C4×C9C23×D311yescharacteristic
341Dic9—31no—
342Dic9—61no—
343Dic9—62no—
344Dic9—122no—
345Dic9D3×D411yescharacteristic
346C2×#41—272no—
347#135—31no—
348#135—271no—
349#135—272no—
350#135—271no—
351#135—272no—
352#135—271no—
353C2×Q16—271no—
354C2×SD16—272no—
355C2×D8—91no—
356C22×C8—271no—
357C3×C9C2×#13511yescharacteristic, complemented, Sylow
358C23×C3—182no—
359C22×D3—62no—
360C22×D3—91no—
361C22×D3—91no—
362C3×Q8—32no—
363C3×D4—22no—
364C3×D4—62no—
365C3×D4—92no—
366C3×D4—182no—
367C3×D4C22×D912yes—
368C3×D4C2×D3212yes—
369C2×C3×C4—31no—
370C2×C3×C4—62no—
371C2×C3×C4—91no—
372#67—22no—
373#67—62no—
374#67—182no—
375#67—182no—
376#67—182no—
377#67—182no—
378#67—182no—
379#67—362no—
380C2×Dic3—22no—
381C2×Dic3—91no—
382C2×Dic3—91no—
383C2×Dic3—182no—
384C2×Dic3—182no—
385C2×Dic3—182no—
386C2×Dic3—182no—
387C2×Dic3—182no—
388C2×Dic3—362no—
389D12—32no—
390D12—182no—
391C4×D3—31no—
392C4×D3—91no—
393C4×D3—91no—
394C4×D3—91no—
395C4×D3—91no—
396C4×D3—91no—
397C4×D3—91no—
398C4×D3—181no—
399C4×D3—182no—
400C4×D3—182no—
401C4×D3—182no—
402C4×D3C22×D911yescharacteristic
403Dic6—92no—
404Dic6—92no—
405Dic6—182no—
406Dic6—182no—
407Dic6—182no—
408Dic6C22×D912yes—
409C3×C8—21no—
410C3×C8—31no—
411C3×C8—91no—
412C3×C8C22×D911yescharacteristic
413C3×C8C2×D3211yescharacteristic
414C3⋊C8—31no—
415C3⋊C8—91no—
416C3⋊C8—91no—
417C3⋊C8—181no—
418C3⋊C8C22×D911yescharacteristic
419C2×C32—42no—
420C2×C32C2×D3×D411yescharacteristic
421C3⋊D3—61no—
422C3×D3—21no—
423C3×D3—32no—
424C3×D3—122no—
425C2×C9—21no—
426C2×C9—42no—
427C2×C9—61no—
428C2×C9—82no—
429C2×C9C2×D3×D411yescharacteristic
430D9—42no—
431D9—61no—
432D9—121no—
433D9#91212yesdirect product factor
434#41—32no—
435#41—272no—
436#41—272no—
437#41—272no—
438#41—542no—
439#41—542no—
440C2×Q8—272no—
441C2×D4—92no—
442C2×D4—542no—
443C22×C4—271no—
444C22×C4—542no—
445Q16—31no—
446Q16—272no—
447Q16—271no—
448SD16—32no—
449SD16—272no—
450SD16—272no—
451SD16—272no—
452D8—92no—
453D8—91no—
454D8D3×D911yescharacteristic, complemented
455C2×C8—31no—
456C2×C8—91no—
457C2×C8—271no—
458C2×C8—271no—
459C2×C8—271no—
460C2×C8—271no—
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—31no—
464C22×C3—42no—
465C22×C3—91no—
466C22×C3—182no—
467C22×C3—364no—
468C2×D3—31no—
469C2×D3—62no—
470C2×D3—91no—
471C2×D3—91no—
472C2×D3—124no—
473C2×D3—181no—
474C2×D3—182no—
475C2×D3—182no—
476C2×D3D4×D911yescharacteristic
477C3×C4—21no—
478C3×C4—31no—
479C3×C4—62no—
480C3×C4—91no—
481C3×C4C23×D911yescharacteristic
482C3×C4C22×D3211yescharacteristic
483Dic3—22no—
484Dic3—31no—
485Dic3—91no—
486Dic3—91no—
487Dic3—181no—
488Dic3—182no—
489Dic3—182no—
490Dic3—362no—
491Dic3D4×D911yescharacteristic
492C32D3×#13511yescharacteristic
493C9—21no—
494C9C2×#91211yescharacteristic, complemented
495C23—182no—
496C23—271no—
497Q8—32no—
498Q8—272no—
499Q8—542no—
500D4—62no—
501D4—92no—
502D4—182no—
503D4—542no—
504D4—542no—
505D4—542no—
506D4C2×D3×D912yes—
507C2×C4—31no—
508C2×C4—62no—
509C2×C4—91no—
510C2×C4—271no—
511C2×C4—271no—
512C2×C4—271no—
513C2×C4—271no—
514C2×C4—542no—
515C2×C4—542no—
516C2×C4—542no—
517C2×C4—542no—
518C8—31no—
519C8—91no—
520C8—271no—
521C8C2×D3×D911yescharacteristic
522C2×C3—21no—
523C2×C3—42no—
524C2×C3—42no—
525C2×C3—61no—
526C2×C3—82no—
527C2×C3—92no—
528C2×C3—362no—
529C2×C3C2×D4×D911yescharacteristic
530C2×C3D32×D411yescharacteristic
531D3—21no—
532D3—32no—
533D3—122no—
534D3—181no—
535D3—181no—
536D3—361no—
537C22—22no—
538C22—31no—
539C22—91no—
540C22—182no—
541C22—271no—
542C22—364no—
543C22—542no—
544C4—31no—
545C4—62no—
546C4—91no—
547C4—271no—
548C4—542no—
549C4C22×D3×D911yescharacteristic
550C3—21no—
551C3D9×#13511yescharacteristic, complemented
552C3D3×#91211yescharacteristic
553C2—42no—
554C2—61no—
555C2—92no—
556C2—362no—
557C2—541no—
558C2D3×D4×D911yescharacteristic
subgroupG/H
centerC2D3×D4×D9
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC3×C9×D8C22
socleC2×C32C2×D3×D4
hypercenterD8D3×D9
hypocenterC3×C9C2×#135
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C4 ◁ D8

Complements

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

NH ≅ G/Ntype
1. D9×#237C2semidirect
2. #11721791C2semidirect
3. C9×#912C2semidirect
4. C3×D8×D9C2semidirect
5. #11721186C2semidirect
6. D9×#214C2semidirect
7. #11721169C2semidirect
8. #11720831C2semidirect
9. D9×Dic12C2semidirect
10. #11720811C2semidirect
11. D3×C8×D9C2semidirect
14. #88742C22semidirect
15. C9×#237C22semidirect
22. #88641C22semidirect
23. #88634C22semidirect
25. C3×C9×D8C22semidirect
26. #88543C22semidirect
27. C8×C3⋊D9C22semidirect
28. #88521C22semidirect
29. C9×#214C22semidirect
30. C3×#3673C22semidirect
31. C9×Dic12C22semidirect
32. D3×C8×C9C22semidirect
33. C3×D72C22semidirect
35. #88428C22semidirect
36. #88425C22semidirect
37. D3×C9⋊C8C22semidirect
38. #88415C22semidirect
39. (C3⋊D9)⋊C8C22semidirect
61. D8×D9D3semidirect
135. C9×D8C2×D3semidirect
139. #3673C2×D3semidirect
143. D72C2×D3semidirect
182. #912D9direct (N × H)
255. C3×D9#135semidirect
270. #237C2×D9semidirect
284. C3×D8C2×D9semidirect
295. #214C2×D9semidirect
303. Dic12C2×D9semidirect
314. D3×C8C2×D9semidirect
357. C3×C9C2×#135semidirect
433. D9#912direct (N × H)
454. D8D3×D9semidirect
494. C9C2×#912semidirect
551. C3D9×#135semidirect

Maximal subgroups

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

subgroupindex
D9×#2372
#117217912
C9×#9122
C3×D8×D92
#117211862
D9×#2142
#117211692
#117208312
D9×Dic122
#117208112
D3×C8×D92
D3×#9123
D9×#1353